MAKING CHOICES IN MULTI-DIMENSIONAL PARAMETER SPACES CHOICES IN MULTI-DIMENSIONAL PARAMETER SPACES...

MAKING CHOICES IN MULTI-DIMENSIONAL

PARAMETER SPACES

by

Steven Bergner

MSc, Otto-von-Guericke University of Magdeburg, 2003

a Thesis submitted in partial fulfillment

of the requirements for the degree of

Doctor of Philosophy

in the

School of Computing Science

Faculty of Applied Sciences

c© Steven Bergner 2011

SIMON FRASER UNIVERSITY

Fall 2011

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APPROVAL

Name: Steven Bergner

Degree: Doctor of Philosophy

Title of Thesis: Making choices in multi-dimensional parameter spaces

Examining Committee: Dr. Mark Drew

Chair

Dr. Torsten Moller, Senior Supervisor

Professor of Computing Science

Dr. Derek Bingham, Supervisor

Associate Professor — Industrial Statistics

Dr. Steven J. Ruuth, Internal Examiner

Professor of Applied and Computational Mathematics

Dr. Min Chen, External Examiner

Professor of Scientific Visualization

University of Oxford

Date Approved:

ii

thesis

Typewritten Text

16 December 2011

Last revision: Spring 09

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Abstract

Visualization techniques are key to leveraging human experience, knowledge, and intuition

when establishing a connection between computational models and real world systems. At

this interface my dissertation enables effective choices of parameter configurations for dif-

ferent levels of user involvement.

Based on a characterization of several domains of computer experimentation that include

a model of biological aggregations, image segmentation methods, and rendering algorithms,

I derive a set of requirements to propose paraglide — a framework for user-driven analysis of

parameter effects. One outcome of the workflow I suggest is a partitioning of the continuous

space of model configurations into distinct regions of homogenous system behaviour.

To facilitate progressive exploration of a parameter region, I develop a space-filling sam-

pling method by constructing point lattices that contain rotated and scaled versions of

themselves. All levels of resolution share a single type of Voronoi polytope, whose volume

grows independently of the dimensionality by a chosen integer factor as low as 2.

To optimize rendering time while ensuring image quality when viewing data in a 3-

dimensional volume, I perform a Fourier domain analysis of the effect of composing two

functions. Based on this, I relax a previous lower bound for a sufficient sampling frequency

and apply it to adaptively choose the numerical integration step size in raycasting.

By assigning optical properties to data using a spectral light model, it becomes possible

to improve physical realism and to create colour effects that scale the level of distinguishable

detail in a visualization. To help modellers to cope with the freedom in a large design space

of synthetic lights and materials, I devise a method that generates a palette of presets that

globally optimize user-specified criteria and regularization. This is augmented with two

alternative user interfaces to unobtrusively choose a desired mixture.

iii

Acknowledgments

On my way through graduate school I had the luck to cross paths with many remarkable

people. I am grateful to my supervisors Torsten Moller and Derek Bingham for inspiring

discussions, constructive input, and ongoing support. Min Chen and Steve Ruuth deserve

credit for their willingness to examine my thesis on rather short notice and still providing

very good feedback.

The folks at the Graphics, Usability, and Visualization laboratory (GrUVi) at Simon

Fraser University made this an inspiring and fun place to work. While I had many inspiring

discussions throughout the years in the lab, I would in particular like to acknowledge Ramsay

Dyer, Alireza Entezari, Ahmed Saad, Tai Meng, Zahid Hossain, and Tom Torsney-Weir for

their input and collaborations. In particular, I would like to thank Usman Alim, Niklas

Rober, and Nhi Nguyen for proof reading parts of my thesis.

There are also several people beyond the lab that I had the honour to learn from during

different stages of my research: Mark Drew, Dave Muraki, Thierry Blu, Dimitri Van De

Ville, Melanie Tory, Tamara Munzner, Michael Sedlmair, and Stephen Ingram.

Also, I would like to thank past students for their great work that they did under my

(co-)supervision (or despite of it): Vincent Cua, Vladimir Kim, Rishabh Iyer, Matt Crider,

Theresa Sanchez, and Sareh Nabi Abdolyousefi.

Apart from proof reading, I am indebted to Nhi Nguyen for always having an open ear

and for being and amazing friend and partner in all regards beyond grad school.

Last, but not least, none of this journey would have been possible without the encour-

agement and support of my parents and my whole family to whom I would like to extend

my deepest gratitude.

iv

Contents

Approval ii

Abstract iii

Acknowledgments iv

Contents v

List of Tables ix

List of Figures x

1 Connecting formal and real systems 1

1.1 Contributions of this dissertation . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Problem domains that require parameter tuning . . . . . . . . . . . . . . . . 4

1.2.1 Mathematical modelling: Collective behaviour in biological aggregations 5

1.2.2 Bio-medical imaging: Segmentation algorithm . . . . . . . . . . . . . 7

1.2.3 Engineering: Fuel cells . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2.4 Visualization: Scene setup and rendering algorithm configuration . . . 9

1.3 Task structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.4 Data abstraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2 Acquisition and visualization of multi-variate data 19

2.1 Effects of dimensionality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.2 Quadrature error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.3 Discretization of multi-dimensional functions . . . . . . . . . . . . . . . . . . 25

2.3.1 Useful concepts for metric data representation . . . . . . . . . . . . . 25

v

2.3.2 Experimental design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.3.3 Figures of merit: Packing and covering radii, density, and thickness . . 31

2.3.4 Optimal packing and bounds by Minkowski and Zador . . . . . . . . 32

2.4 Visual interfaces for multi-variate computer model data . . . . . . . . . . . . 34

2.4.1 Computational modelling tasks . . . . . . . . . . . . . . . . . . . . . 36

2.4.2 Reconstruction and refinement of spatial data . . . . . . . . . . . . . . 39

2.4.3 Direct volume rendering . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.4.4 Visualization systems for discrete multi-variate data . . . . . . . . . . 41

3 Sampling lattices with low-rate refinement 43

3.1 Change of lattice basis and similarity . . . . . . . . . . . . . . . . . . . . . . . 44

3.2 Construction of sampling lattices with low-rate rotational dilation . . . . . . 47

3.2.1 Constructing rotational dilation matrices . . . . . . . . . . . . . . . . 49

3.2.2 Characteristic polynomial of a scaled rotation matrix in Rn . . . . . . 50

3.2.3 Construction algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.3 Further dimensions and subsampling ratios . . . . . . . . . . . . . . . . . . . 53

4 A sampling bound for composed functions 57

4.1 Frequency domain analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.1.1 Visual inspection of the frequency transfer kernel K(ω,ν) . . . . . . . 60

4.1.2 Determining the boundary of the cone . . . . . . . . . . . . . . . . . 62

4.1.3 Error analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.1.4 Limits of the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.1.5 Relationship to Carson’s rule . . . . . . . . . . . . . . . . . . . . . . . 65

4.2 Application to volume rendering . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.3 Discussion and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5 Designing a palette for spectral lighting 71

5.1 Related work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

5.1.1 Previous approaches to constructing spectra . . . . . . . . . . . . . . . 75

5.1.2 Linear light models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5.1.3 Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

5.2 Designing spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.3 Matrix formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

vi

5.3.1 Combined optimization function . . . . . . . . . . . . . . . . . . . . . 82

5.3.2 Free metamers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.4 Evaluation and visual results . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

5.4.1 Example palette design . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.4.2 Design error with respect to number of constraints . . . . . . . . . . . 85

5.4.3 Spectral surface graphics . . . . . . . . . . . . . . . . . . . . . . . . . 89

5.4.4 Rendering volumes interactively . . . . . . . . . . . . . . . . . . . . . 90

5.5 Discussion and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

5.5.1 Future directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5.5.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

6 Interactive parameter space partitioning 95

6.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

6.1.1 Interactive parameter adjustment in computer experiments . . . . . . 97

6.1.2 Parameter space partitioning . . . . . . . . . . . . . . . . . . . . . . . 98

6.1.3 Unfulfilled design requirements . . . . . . . . . . . . . . . . . . . . . . 99

6.2 Design of the paraglide system . . . . . . . . . . . . . . . . . . . . . . . . . . 99

6.2.1 System components . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

6.2.2 Browsing computed data . . . . . . . . . . . . . . . . . . . . . . . . . 102

6.2.3 Representing a region of interest . . . . . . . . . . . . . . . . . . . . . 105

6.2.4 Non-linear screen mappings . . . . . . . . . . . . . . . . . . . . . . . . 107

6.3 Excursion: Steering a multi-dimensional cursor . . . . . . . . . . . . . . . . . 109

6.3.1 A light dial to control additive mixtures . . . . . . . . . . . . . . . . 109

6.3.2 Enabling simultaneous parameter adjustments using a mixing board . 111

6.4 Validation of paraglide in different use cases . . . . . . . . . . . . . . . . . . . 118

6.4.1 Movement patterns of biological aggregations . . . . . . . . . . . . . . 118

6.4.2 Bio-medical imaging: Tuning image segmentation parameters . . . . 120

6.4.3 Fuel cell stack prototyping . . . . . . . . . . . . . . . . . . . . . . . . 123

6.5 Discussion and future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

7 Discussion and conclusion 128

7.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

vii

List of Tables

1.1 Summary of the requirement analysis. . . . . . . . . . . . . . . . . . . . . . . 12

7.1 Summary of thesis contributions where the level of user involvement decreases

and the inclusion of theoretical analysis increases from top to bottom, with

rows corresponding to different chapters in the order 6, 5, 4, and 3. . . . . . 130

ix

List of Figures

1.1 Classification of a slice of d-PET data using two different parameter config-

urations. The classes are 1: background (BG), 2: skull (SK), 3: grey matter

(GM), 4: white matter (WM), 5: cerebellum (CM), and 6: putamen (PN). . 8

1.2 (a) SPloM view evaluation of the effect of value reconstruction filter (top

row) and sampling distance (bottom row). The columns show different image

comparison metrics by comparing with a best possible ground ’truth’ image

(b) showing the 643 hipiph data set using a transfer function that includes

smooth and sharp opacity transitions. Value filters are sorted by order of

approximation (ef:1,2,3,4) and, within each order, by degree of smoothness

(c:1,2,3) [MMMY97]. The sampling step size along a ray is colour coded

from bottom to top / blue to orange for [1/200, 1/32, 1/16, 1/8, 1/4, 1/2, 1]

grid spacing units. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.3 Abstraction of data, interaction, and computational components. Lines in-

dicate shared data among processing steps and arrows prescribe an order of

execution. On a more detailed level, Red is required input and blue denotes

information that is available after a processing step. . . . . . . . . . . . . . . 16

2.1 Semi-log plot of volumes of n = 1 . . . 30 – dimensional p-norm unit spheres. . 21

2.2 Set of 20 randomly distributed sample points in [0, 1]2 with Voronoi regions

outlined in blue and Voronoi relevant neighbours connected by grey lines.

Notice how the convex hull and the minimum spanning tree are part of the

grey Delaunay graph connecting Voronoi relevant neighbours. For points in

general position the Delaunay graph turns out to be a triangulation or a

simplicial complex in higher-dimensional spaces. . . . . . . . . . . . . . . . . 26

x

2.3 Best lattice packings in dimensions n = 1 . . . 64. The symbol δ refers to the

center density defined below Equation 2.13. Note that the linear interpolation

between the densities given at the named abscissae may be in disagreement

with the actual best packing. The packing radius of the Cartesian lattice is

0.5 in any dimension, which is represented in the figure by the horizontal axis. 33

2.4 Zador’s bounds for the mean squared quantization error of optimal quantizers

in Rn. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.5 Schematic overview of tasks related to studying effects in model parameter

spaces. The blue coordinate axes symbolize the construction of the parame-

terization with one dependent response variable indicated by iso-lines in the

background. The two-sided blue arrow in the center represents the task of

fitting a model to observed field data. The blue chip along this line repre-

sents the possiblity to fit in a digital substitute model that can include model

assumptions to make up for missing data or is simply more efficient to eval-

uate than a more complex model or direct field measurements. The green

itinerary or schedule of parameter adjustments could be provided by com-

putational steering interfaces for a time-evolving simulation model. The red

target indicates the goal of a search for an optimal configuration. The region

labels and outlines in black illustrate a partitioning of the parameter space

into regions of homgenous behaviour of selected responses. An important

part of this picture, but not part of the drawing, is the human observer who

is responsible for interpretation of the analysis in the context of a particular

purpose. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.1 2D lattice with basis vectors and subsampling as given by G and K in the

diagram title. The spiral points correspond to a sequence of fractional sub-

samplings GKs for s = 0..1 with the notable feature that for s = 1 one obtains

a subset Λ(GK) (thick dots) of the original lattice sites Λ(G) (small black

dots). This repeats for any further integer power of K, each time reducing

the sampling density by |det K| = 2. . . . . . . . . . . . . . . . . . . . . . . . 44

3.2 The best 3D lattice obtained for a design with dilation matrices having

|det K| = 2. The letters f and v in the title line indicate faces and vertices,

respectively. The different colours encode the different zones. . . . . . . . . . 54

xi

3.3 Three non-equivalent 2D lattices obtained for a design with dilation matrices

having |det K| = 2. The lattice in the first column is the known quincunx

sampling with a rotation of θ = 45. The other two are new schemes with

different rotation angles. The thick dots show the sample positions that are

retained after subsampling by K. The second row shows the same lattice at

twice the density, with more iteration levels of similarity transformed Voronoi

cells. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.4 Four non-equivalent 2D lattices obtained for a design with dilation matrices

having |det K| = 3. The lattice on the left is the well known hexagonal lattice

with a θ = 30 rotation. The other three are new schemes with different

rotation angles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.5 Comparison of packing radii of best known packings, Cartesian packing, and

the upper bound as in Figure 2.3 on page 33. In addition, some designs of

this chapter are shown that enjoy the low-rate rotational reduction property

of Equation 3.2 for rates β ∈ 2, 3. If proceeding directly from a companion

K as provided by Algorithm 2 unoptimized constructions can be generated

instantly for any n. The optimized designs are obtained by maximization of

the packing radius over choices of S in Equation 3.9. The depicted results

beat Cartesian packing in all cases except for n ∈ 1, 3. . . . . . . . . . . . 56

4.1 Sampling comparison. The data y = f(x) (a) is composed with a transfer

function g(y) (b). Figures (c) and (d) show sinc-interpolated samplings of

g(f(x)). The tighter bounding frequency (d) suggested in this chapter re-

sults in 5 times fewer samples for these particular f and g, still truthfully

representing the composite signal. . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.2 The frequency map K(ω, ν) for a function f(x) determines how much a fre-

quency ν of g contributes to a frequency ω of the spectrum of the composed

function g(f(x)). The examples are (a) single and (c) mixed non-normalized

Gaussians, using ϕ(µ, σ, x) = exp(−(x− µ)2/(2σ2)

)and their correspond-

ing K(ω, ν) in (b) and (d), respectively. The upper and lower slopes of the

low-valued cones (black) are given by the reciprocal of the maximum and the

minimum values of f ′, respectively, as shown in Section 4.1.2. . . . . . . . . . 61

xii

4.3 The graph of the Airy function Ai(t). It decays exponentially toward positive

t with exp(−(2/3)t3/2

). Also notice that its maximum occurs for negative t.

The value for t = 0 in Equation 4.18 is attained at the band edge θ = θe. . . . 64

4.4 Same sampling rates are suggested by both estimates if a single sinusoidal

signal is composed, using the lower frequency of the example in Figure 4.1a,

with the mapping in Figure 4.1b. Both estimates have been 2× over-sampled,

using a sampling frequency that is four times the respective limit frequency. 65

4.5 Examples of the hipiph data set sampled at a fixed rate (0.5) (a) and sampled

with adaptive stepping (b). The adaptive method in (b) uses about 25% fewer

samples than (a) only measuring in areas of non-zero opacity to not account

for effects of empty-space skipping. The similarity of both images indicates

that visual quality is preserved in the adaptive, reduced sampling. . . . . . . 67

4.6 Quality vs. performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.7 Visual comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.1 Spectral design of two material reflectances shown on the left of their repre-

sentative rows. The colors formed under two different illumination spectra

are shown in the squares in the respective columns where D65 (right column)

produces a metameric appearance. . . . . . . . . . . . . . . . . . . . . . . . . 80

5.2 The reflectance spectra on the left of each row are designed to be metameric

under daylight (colours column 1) and to gradually split off into 3 and 5

distinguishable colours under two artificial ‘split light’ sources. The resulting

reflectance spectra are given below the figure. . . . . . . . . . . . . . . . . . . 86

5.3 Each graph shows the average L∗a∗b∗ error in the design process for palettes

of given sizes, constraining all light-reflectance combination colours for several

palettes of different sizes. Changing spectral models and constraints results

in different design error: a) the positivity constrained 31D model, b) the

positivity constrained 100D colour model, c) 31D without positivity constraint. 87

xiii

5.4 Preservation of colour distances for a 10× 10 palette size. Each point in the

graphs represents a certain pair of colour entries in the palette. Its position

along the horizontal axis indicates the L∗a∗b∗ distance between the desired

colours and the vertical position indicates distance of the resulting colour

pair after the design. A position close to the diagonal indicates how well the

distance within a pair was preserved in the design. a) 31-D spectra, uncon-

strained; b) and c) positivity constrained spectra with 31 and 100 dimensions,

respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.5 Car model rendered with PBRT. (a) The spectral materials used in the tex-

ture are metameric under daylight D65, resulting in a monochrome appear-

ance. (b) Changing the illumination spectrum to that of a high pressure

sodium lamp, as used in street lighting, breaks apart the metamerism and

reveals additional visual information. . . . . . . . . . . . . . . . . . . . . . . . 89

5.6 Engine block rendered using metamers and colour constancy. The three im-

ages in the figure are re-illuminated without repeating the raycasting. . . . . 90

6.1 Paraglide GUI running inside a MATLAB session to investigate the animal

movement model of Section 1.2.1. Initially, deliberately chosen parameter

combinations are imported from a switch/case script (a) by sampling the

case selection variable of that script and recording the variables it sets. An

overview (b) of the data is given in form of a scatter plot matrix (SPloM) for

a chosen dimension group (h). Jython commands can be issued inside the

command window (c) demonstrating the plug-in functionality of the system

by manually importing the experiment module, which adds a new item to

the menu bar (d). This allows to create a set of new sample points inside the

region that is selected for parameters qa and qal (e). The configuration dialog

for the MATLAB compute node (f) sets up a show command that produces a

detail view of the spatio-temporal pattern (1D+time) (g). For the configu-

ration point highlighted in yellow in the SPloM, this results in a pattern of

two groups that merge and then progress upwards in a ’zigzag’ movement. . 100

6.2 Dialog to set up a MATLAB compute node . . . . . . . . . . . . . . . . . . . . 102

xiv

6.3 The light dial – interface to control the mixture of lights using normalized

inverse distances of the mixture selector (yellow circle) to the light nodes

(bulb icons) (see Eq. 6.1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

6.4 (a) Graphical user interface (GUI) for the BCF2000 mixing board (b) showing

an experimental trial in progress. . . . . . . . . . . . . . . . . . . . . . . . . 112

6.5 Recorded slider motion paths for item 6: [ . . . . . . 110 110]+/-1 shown in

front of non-manipulation intervals (gray) and mistake intervals (red). The top row

is mixer interaction of a participant performing item ID 6 of forthcoming Figure 6.6

in three trial blocks for each of the two input methods in Figure 6.4. An error moving

irrelevant slider 6 is indicated in the third block. The path correlation matrix in the

right column is discussed in Section 6.3.2.1. . . . . . . . . . . . . . . . . . . . . . 113

6.6 Simultaneous manipulation of sliders as indicated by the maximum (in abso-

lute value) of the normalized slider-slider velocity cross-correlations of Equa-

tion 6.2. Simultaneity varies for different items and participants. . . . . . . . 115

6.7 Illustrating the sample creation in a sub-region of the parameter space it-

erating from coarse to finer sampling. (Un-)filled circles indicate parameter

configurations that lead to an (un-)stable steady state. . . . . . . . . . . . . 119

6.8 Scatter plot matrix view that compares the point embedding (lower left) with

the objective measures that went into computing its underlying similarity

measure. The numbering of the responses corresponds to the class labels of

Figure 1.1 (ID 5: cerebellum and 6: putamen). . . . . . . . . . . . . . . . . . 121

6.9 Scatter plot matrix view of the good cluster (yellow) identified in Figure 6.8

viewed in the subspace of input parameters. In this view sigma and alpha3

indicate clear thresholds beyond which the good configurations are found. . 122

6.10 Two layouts for 204 example experiments. a) input space showing variation

in current and input temperature, b) embedding of the same samples where

spatial proximity reflects plot similarity for cell current density, c) similarity

embedding for membrane electrode assembly (MEA) water content using the

same clusters as assigned in (b). Cluster representatives are shown in Fig-

ure 6.11 and Figure 6.12. These screenshots are from the 2007 C++ version

of paraglide, and are also attainable in the currently discussed Java implemen-

tation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

6.11 Cell current density plots for the clusters in Figure 6.10b . . . . . . . . . . . 125

xv

6.12 MEA water content plots for the clusters labelled in Figure 6.10c . . . . . . 125

A.1 Illustration of the concept of forward cones F of Equation A.1 in a di-

rection u for the vertices of the white quadrangle, similar to Leydold and

Hormann [LH98, Fig. 3]. F (v1) begins in white, F (v2) and F (v3) in light

red and F (v4) in dark red. All cones extend infinitely towards the right, but

are artificially cut of in the picture. Even in practical computations this step

could be required to ensure that the valuations in Equation A.2 stay finite. . 135

B.1 (a) The sample density inside the unit cube jumps when scaling Cartesian

lattices in Rn for n = 2..4 using a scaled identity generating matrix R = αI.

(b) Discrepancy of various 3D point sets including some randomly rotated

regular lattices at differently scaled density. . . . . . . . . . . . . . . . . . . . 141

xvi

Chapter 1

Connecting formal and real systems

To record observations that are either made directly or are acquired with the aid of carefully

crafted devices is the starting point of the scientific method that prescribes how to construct

models that represent real-world systems. Here a system is understood as a collection of

things or concepts that exist by certain mechanisms or rules that in a formal system can

be laid out and verified rigorously. A model denotes a system that is intended to mimic

certain aspects of another system. A computational model is a formal system, where an

algorithm can be run for a given input configuration to compute an output description of the

simulated system. While such an algorithm is assumed to be deterministic, additional input

may introduce randomized behaviour. Having a model whose behaviour fits with available

observations, it becomes possible to diagnose or predict phenomena that occur at different

levels of structural organization, allowing to choose actions that influence development of

the system, or to utilize the observed effects in artificial constructions1 that, for instance,

improve quality of life.

Within this broad picture, the focus of this dissertation is to facillitate development,

analysis, and interpretation of computational models through efficient human interfaces.

This objective is approached at different stages of the modelling process through the lens of

parameter adjustment. Typical tasks and research challenges that arise when working with

computer model parameters are discussed in Section 2.4.1. The different approaches pursued

in the following chapters include a) methods for direct parameter choices, b) specification

of criteria that characterize good choices, and c) theoretical study of model properties that

1Note that the meaning of artificial as in human-made is a special case of natural.

1

CHAPTER 1. CONNECTING FORMAL AND REAL SYSTEMS 2

guide general parameter choice. The work of Chapter 6 enables manipulation of control

parameters in an interactive manner (a). A less direct path is taken in the computer

graphics setting of Chapter 5, where the user specifies criteria (b), such that globally optimal

parameter choices can be made automatically. Pursuing the adjustment of internal model

parameters, Chapter 4 performs a theoretical study (c) of properties of a fundamental data

processing step and uses the findings to determine a volume rendering parameter. On a

similarly theoretical level, Chapter 3 uses general criteria of uniformity in the design of a

space-filling sampling pattern where user input is merely required in form of a constraining

region and a budget for the number of sample points.

The underlying theme of human guidance is somewhat contrary to typical objectives of

algorithm development that seek to eliminate the need for human input as much as possible

in order to obtain consistent results efficiently. When asking someone on the hallway of the

computing science department how this is supposed to be done, the typical answers boil down

to: “Think harder about the problem, incorporate into your algorithm what is known and

improve the theory where it is insufficient.” While it is imaginable to eventually automate

information collection [PBMW99] and inference mechanisms [BWB09], thinking about a

problem in its original domain and improving a model abstraction of it are still very much

human-centric tasks. Also, when a ready-made model is employed that influences choices

that affect people, e.g., regarding the health of a human being, the livability of a city in

planning, or if it informs decisions regarding the economy of a country or our environment

— why would we want to give up the ability a) to convince ourselves that the model does

the right thing (is valid); b) to give hints that make up for missing information from our

experience, knowledge, or intuition; or c) to make human inventiveness available to get a

better2 model more quickly?

The specific examples that I will give in Section 1.2 are a bit smaller scale than what

was just mentioned. However, these computational modelling settings also share a need for

human input. This need provides a basic motivation for visualization3 as a discipline at

the core of scientific study. The particular focus on visual methods to map properties of

data to the screen is justified due to our well developed visual perception. The simultaneity

2Better here means a model with a strong linkage to the real-world system that it is intended to repre-sent. [SRA+08, Ch. 1]

3Beyond the technical scope, the term visualization is also understood as a method of thinking, learn-ing, and communication that works with mental imagery and its transformation, e.g., to improve athleticperformance [GC08]. However, in our research context the meaning is confined to the computational domain.


and spatial resolution of visual processing are unparalleled by other senses, such as touch

or hearing [Gol10, pp. 888]. It enables us to integrate a massive stream of cues for location,

shape, motion, colour, or textural properties that are presented by a still image or an ani-

mated sequence. Visualization research seeks to utilize this natural processing apparatus by

devising effective visual encodings and interaction metaphors that work with data [Mun12].

However, the quality of any data display hinges on the relevance of the given data for

the driving questions. Hence, in extension of the classical scope of visualization research,

my dissertation is looking at user involvement in the acquisition of data. Specifically, this

concerns sampling of continuous phenomena, as well as adjustments in parameter spaces of

computational models.

1.1 Contributions of this dissertation

The remainder of this first chapter will perform a characterization of several domains of

computer experimentation to derive a set of requirements for human-guided study of models

in Section 1.3. Building on this analysis towards the end of this thesis, Chapter 6 then

proposes paraglide — a framework for user-driven analysis of parameter effects. Its main

point is to facilitate a workflow that results in a partitioning of the continuous space of

model configurations into regions of distinct system behaviour.

Establishing a model often involves the adjustment of multiple parameters. The contri-

bution of the theoretical survey4 in Chapter 2 is to show effects that arise when discretizing

multi-dimensional Euclidean space with a minimal budget of points. This also prepares the

background for the remaining chapters.

One feature of paraglide is to elevate user interaction from specifying point locations in

parameter searches to a higher level of outlining constraints of a region that can then be

filled with a given budget of points (see Section 6.2.3). To be able to fill this space efficiently

is the motivation behind Chapter 3, where a class of point sets is designed to directly fulfill

a number of quality criteria. To facilitate progressive regional exploration, point lattices

are constructed that contain rotated and scaled versions of themselves. Their distinguishing

property is that different levels of resolution share a single type of Voronoi polytope, whose

volume grows independently of the dimensionality by a chosen integer factor as low as 2.

4The human interface in this context are systematic methods of thinking provided by mathematics.


Many real world phenomena can be viewed in 3-dimensional Euclidean space together

with a time coordinate. Hence, a common type of data in numerical visualization are

grids that represent some physical quantity for each point in a bounded volume. In order to

reduce sampling costs while maintaining image quality when rendering such data, Chapter 4

presents a Fourier domain analysis of the effect of composing a data signal with a function

that assigns visibility to relevant values. Based on this, a previous lower bound for a sufficient

sampling frequency is relaxed and applied to adaptively choose the step size in raycasting.

Using a spectral light model, it is possible to improve physical realism and to create

colour effects that scale the level of distinguishable detail in a visualization. To help mod-

ellers to make useful decisions in a high-dimensional design space of palettes of lights and

materials, Chapter 5 devises a method that generates a palette of presets that optimally

fulfill a set of design criteria. A brief excursion in Section 6.3 discusses two alternative user

interfaces for steering a multi-dimensional cursor that can also be applied to unobtrusively

search for a suitable mixture.

A word on the narrator’s perspective: While I take care to not claim contributions

in this thesis that are not mine, I recognize that ideas are not born in a vacuum and will

therefore in the following adapt the pronoun we to refer to me as a researcher, the group I

was working with, or to engage the readership, which should be clear from context. I will

explicitly state external contributions where sources are otherwise not clear.

1.2 Problem domains that require parameter tuning

In order to get a more detailed understanding of needs and requirements for user-driven

parameter space navigation, we engaged in a problem characterization phase by conducting

contextual interviews with six experts from three different domains: engineering, mathe-

matical modelling, and segmentation algorithm development. Based on that, we summarize

design requirements in Section 1.3 that are more general yet grounded in real-world appli-

cation areas.

Our methodology corresponds to Munzner’s nested model [Mun09], which casts good

practices of software engineering into the realm of visualization research validation. With

this background, we prepare the discussion of effective visual encodings and interaction

techniques in Chapter 6, where our tool paraglide is presented and validated. This motivates

the remaining chapters that discuss efficient algorithms for related purposes.


1.2.1 Mathematical modelling: Collective behaviour in biological aggre-

gations

Our first target group are two researchers studying properties of a mathematical model that

describes spatial and spatio-temporal patterns of biological aggregations. Furthering the

understanding of such patterns helps to predict animal migration behaviour, e.g., to better

understand how, where, and when fish aggregations form to suggest more efficient fishing

strategies [Par99]. Modelled patterns can also inform measures to contain plagues of locusts

and positively affect quality of life in developing countries [BSC+06].

To study those spatio-temporal patterns, our participants developed a mathematical

model [FE10, LS02] consisting of a system of partial differential equations (PDEs) that

express in one spatial dimension how left and right travelling densities of individuals move

and turn over time, with details provided at the end of this section. The basic idea is to

take three kinds of social forces into account — namely attraction, repulsion, and alignment

— that act globally among the densities of individuals. Attraction is the tendency between

distant individuals to get closer to each other, repulsion is the social force that causes indi-

viduals in close proximity to repel from each other, and alignment represents the tendency

to sync the direction of motion with neighbours. Solving the model for different choices of

coefficients produces many complex spatial and spatio-temporal patterns observed in na-

ture, such as stationary aggregations formed by resting animals, zigzagging flocks of birds,

milling schools of fish, and rippling behaviour observed in Myxobacteria.

Our use case is part of a Master’s thesis on this subject with a focus on comparing two

versions of their model [Abd11]. In the first one the velocity is constant. In the second

one the individuals speed up or slow down as a response to their social interactions with

neighbours. Comparing these models requires to solve them numerically for several different

configurations. Each instance of the model corresponds to one specific choice of the 14 model

parameters that include the coefficients for the three postulated social forces. The output

of the simulation is a spatio-temporal pattern of population densities. The number of basis

functions is an internal parameter that gives the resolution in space and time and influences

the trade-off for accuracy vs. runtime, which for reasonable output lies between 2 minutes

and half an hour. With 5 minutes each, one can perform a full computation of close to 300

sample points in the duration of a single day.

To better understand the space of possible solutions, our participants manually explored


the parameter combinations of their model and demonstrated its capability to reproduce a

variety of complex patterns. While it is difficult to classify all possible patterns, there are

a few standard solutions among them, for which established analysis techniques exist. In

particular, they focus on the solutions of the system that do not change over time and space

— so called spatially homogeneous steady states. A linear stability analysis of these steady

states results in negative or positive growth rates for different perturbation frequencies,

which respectively indicate stable and unstable solutions.

There is a hypothesized relationship between the stability of steady states and the po-

tential for pattern formation. This leads to a derived, more specific goal of the study. In

particular, it enables comparison of constant and non-constant velocity models by inspect-

ing the change in shape of the parameter regions that lead to (un-)stable steady states. A

discussion on how paraglide affects our participant’s workflow involving, for instance, quick

prototypic of new dependent feature variables, is given in Section 6.4.1. An overview of the

parameter space is part of Figure 6.1.

Details about the mathematical model of biological aggregations: Eftimie, de

Vries, Lewis, and Lutscher [FE10, LS02] introduced the one-dimensional model discussed

above to describe animal aggregations as

∂tu+(x, t) + ∂x(Γ+u+(x, t)) = −λ+u+(x, t) + λ−u−(x, t)

∂tu−(x, t)− ∂x(Γ−u−(x, t)) = λ+u+(x, t)− λ−u−(x, t)

u±(x, 0) = u±0 , x ∈ R,

where u+(x, t) and u−(x, t) represent the density of the right and left moving individuals at

position x and time t. Γ represents the velocity. Eftimie et al. considered two different cases

for Γ: constant and density dependent velocities. The functionals λ+ and λ− amount to the

turning rates for initially right or left moving individuals that turn left or right respectively.

Attraction, repulsion, and alignment interactions are modeled in these operators via convo-

lutions with three different kernels that represent each type of response with respect to the

current environment of densities u±. The amount of individuals that turn to left/right, but

were initially moving to the right/left is given by the terms λ+u+ and λ−u−, respectively.


1.2.2 Bio-medical imaging: Segmentation algorithm

Saad et al. [SHMS08] use a kinetic model to devise a multi-class, seed-initialized, iterative

segmentation algorithm for molecular image quantification. Due to the low signal-to-noise

ratio and partial volume effect present in dynamic-positron emission tomography (d-PET)

data, their segmentation method has to incorporate prior knowledge. In this noisy setting,

the segmentation of a basic random walker [Gra06] would just result in Voronoi regions

around the seed points. An extension by Saad et al. makes this method usable for noisy

data by adding energy terms that account for desirable criteria, such as data fidelity, shape

prior, intensity prior, and regularization.

In order to attain the superior segmentation quality of the algorithm, a proper choice

of weights for the energy mixture is crucial. To facilitate this choice of weight parameters,

their code provides numerical performance measures that assess the quality of each segment.

One such measure is the Dice coefficient [Dic45], which gives a ratio of overlap with labelled

training data. A second measure expresses an error of the quality of the kinetic modelling.

Overall, the algorithm is influenced by eight factors (parameters). Ten response variables

provide two quality measures per segment, disregarding background.

The model calibration could proceed by numerical optimization of the performance.

However, for that a choice of importance weights has to be given, which again is a step where

in the general setting human input is required. However, for instance the Dice coefficients

that indicate agreement of the segmented shape with given training data for putamen,

using the two configurations of Figure 1.1(c) and (d), are both above the 90th percentile

of the sampled configurations and less than 0.003 standard deviations apart. Numerically,

this means that both segmentations are of the same, near optimal quality. Yet by visual

inspection, it is possible to tell that the putamen (PN) shape in (d) is favourable over the

one obtained in (c). Hence, guidance of a domain expert is desirable to sort among several

candidate solutions in order to find an improved segmentation, which is hard or impossible

to choose automatically. A workflow for such a procedure is subject of Section 6.4.2.

1.2.3 Engineering: Fuel cells

A fuel cell takes hydrogen and oxygen as gaseous input and converts them into water and

heat, while generating an electric current. Affordable, high-performance fuel cells have the


(a) raw data (b) ground truth

(c) config# 13, dice6 = .8136 (d) config# 44, dice6 = .8128

Figure 1.1: Classification of a slice of d-PET data using two different parameter configurations.The classes are 1: background (BG), 2: skull (SK), 3: grey matter (GM), 4: white matter (WM), 5:cerebellum (CM), and 6: putamen (PN).

potential to enable more environmentally friendly means of transport without any CO2 emis-

sion. To manufacture a prototypical cell stack costs tens of thousands of dollars. Hence, a

reliable synthetic model can greatly bring down the price of finding an optimal configuration

for production.

The example investigated here is a simulation of a fuel cell stack developed by Chang

et al. [CKPW07]. Their stack model is a system of coupled one-dimensional PDEs de-

scribing the individual cells in the stack. It can be adjusted with about 100 parameters,

where suitable choices of values are known for most of these parameters from fitting to

available measurements. Computing a simulation run outputs 43 different plots that show

how certain physical quantities, such as current density, temperature, or relative humidity

vary across the geometry of the cell stack. The computer model can be rerun for differ-

ent configurations and, thus, allows for much broader exploration of design options than

real prototyping. In particular, engineers are interested in studying failure mechanisms

and to optimize performance under varying conditions for different groups of parameters


that represent the geometry of the assembly (size and number of cells in a stack), material

properties (permeability), or running conditions (temperature, pressure, concentration for

cathode and anode). Experiments demonstrating the use of the suggested interactions are

given in Section 6.4.3.

1.2.4 Visualization: Scene setup and rendering algorithm configuration

The following two use cases are located in a research laboratory, which is a natural envi-

ronment for model development. A topic of particular interest at the Graphics, Usability,

and Visualization (GrUVi)-Lab at Simon Fraser University is to improve techniques for the

visualization of volumetric data. This kind of data can capture physical quantities, such

as density, flow, or tensor information in a bounded spatial domain and is used to answer

diagnostic and predictive questions in fields as diverse as health care, environmental studies,

or engineering. In both cases, I was part of the group, but the objectives and developed

tool chain apply to different follow-up projects, beyond the initial one described here.

1.2.4.1 Rendering algorithm parameters

The goal of this project in 2005 was to make headway into the open problem of providing

guidelines to choose optimal configuration parameters of a volume rendering algorithm.

Ideally, this should lead to informative views of the studied data that are of sufficient

quality, do not miss any important features, and that can be computed at minimal cost.

While the previous cases where suggesting interactive experimentation, such a proce-

dure amounted to current practices in this particular use case that is described in the

following. To facillitate a pilot study that would allow first statements about the rendering

problem, I developed an object oriented implementation of a raycaster based on C++ tem-

plate mechanisms and integrated it into the lab software vuVolume, which is published on

sourceforge.net.

The customizable raycaster could be interacted with at runtime by writing and read-

ing a hierarchical XML description of its state. This way, computation for many different

configurations could be done offline with bash scripted for-loops that would combine pre-

set XML snippets to set data, transfer function, lighting model, camera, and reconstruction

parameters [MMMY97]. Different configurations influence the resulting image quality and

processing costs for time and memory. For each setup we generated one image with a

http://sourceforge.net/projects/vuvolume/


best possible and expensive choice of reconstruction parameters, which in a slight abuse of

terminology was referred to as ground truth.

(a) scatter plot matrix (SPloM)

(b) ground truth image using best parameter settings

Figure 1.2: (a) SPloM view evaluation of the effect of value reconstruction filter (top row) andsampling distance (bottom row). The columns show different image comparison metrics by compar-ing with a best possible ground ’truth’ image (b) showing the 643 hipiph data set using a transferfunction that includes smooth and sharp opacity transitions. Value filters are sorted by order ofapproximation (ef:1,2,3,4) and, within each order, by degree of smoothness (c:1,2,3) [MMMY97].The sampling step size along a ray is colour coded from bottom to top / blue to orange for[1/200, 1/32, 1/16, 1/8, 1/4, 1/2, 1] grid spacing units.

It was then possible to experimentally validate the quality of the resulting renditions

with a number of numerical and perceptually based image distance metrics (e.g., [Dal93]),

whose implementation is due to Alireza Entezari. The resulting table of factor choices vs.

truth distance responses was then visualized using a scatter plot matrix (SPloM) view pro-

vided by the weka toolkit. An example for one particular volumetric scene setup is shown

in Figure 1.2a. This is part of a SPloM that focusses on the effects of two parameters,

namely the type of basis function that is used for value reconstruction, and the sampling

http://www.cs.waikato.ac.nz/ml/weka


distance that is used when numerically accumulating radiance along rays through the re-

constructed volume. When combined for all wavelengths this gives the colour information

that is displayed in each pixel of the rendered image as shown in Figure 1.2b.

While these graphs do not lend themselves to derive concise, generalizable statements

about the rendering problem, they do show significant effects of sampling distance, and of

the order of approximation of the reconstruction basis. This pilot study also showed that a

satisfactory exploration of all rendering parameter effects using nested for-loops is infeasible.

1.2.4.2 Scene parameters

In order to render a scene it has to be constructed first. As will be explained in Section 5.1,

modelling techniques for geometry have already received much attention in graphics re-

search, whereas the construction of synthetic materials still has important open questions

to offer.

An aspect of particular interest in our lab, was the support of more complete light models

considering full spectral power distributions (SPDs) instead of tri-chromatic (red,green,blue)-

tuples In our previous work [BMDF02], we outlined the usefulness of a more complete light-

ing model, by pointing out that it enables two complementary directions: improved physical

realism, as well as increased design freedom for artificial effects.

Materials and light sources: In order to find SPDs to configure light sources and mate-

rials in a scene, one can either acquire them using a measurement device, obtain data bases

of spectra from the web, or construct new ones by mixing combinations of a smaller set.

Note that the inclusion of lights in this construction is already a step beyond classical

computer graphics, where one simply sets up (red, green, blue)-triples for each surface

point and illuminates with a white light. Designers of a scene that care about correct

colour reproduction when illuminating with something else than just white light need to

find spectra that produce specific colours for particular combinations of illuminant and

surface. Alternatively, one could also ask for perceived colour differences between certain

materials to be zero. The task of a designer who constructs a scene is to come up with these

criteria and to find materials that fulfill them.

Considering the huge size of the design space in relation to the small size of desirable

solutions, this is a very difficult, if not infeasible adjustment problem. This poses a research

problem that is addressed in Chapter 5, where also the lack of current practices for this


purpose and directions for improvement are elaborated further.

Light mixtures: In the scenes we consider the superposition principle of light applies. So,

given a designed palette with light sources, a remaining task is to define a linear mixture

of these lights or a transition of mixtures that produce a convincing visualization. For

a complete setup one would adjust the geometry of the lights and move them around to

achieve the desired effects. More abstractly, this amounts to choosing a vector of weights

that results in a set of desired colours for all surfaces in view. Due to the central role

of human judgement in this setting, this poses an interactive adjustment task for which

Section 6.3 investigates two different practical solutions.

1.3 Task structure

All previous use cases are motivated by questions about a real-world system. In each

setting, domain knowledge about the problem has been expressed in form of a computational

model and the parameter space of the model is explored in order to relate observations

about its properties to the corresponding real setting. Please refer to the previous

subsections for details about the parameter spaces in the respective settings. The studies

share a set of requirements as summarized in Table 1.1.

Table 1.1: Summary of the requirement analysis.

R1: integrate with existing practices and code

R2: specify parameter region of interest (ROI)

R2a: sample ROI and compute data set

R3: browse data providing overview (R3a) and detail (R3b)

R4: construct feature variables (manually labelled or computed)

R4a: combine features to derive a distance metric

R5: identify region(s) of similar outcome in parameter space

R6: find optimal point for a particular dependent variable or user notion

R7: analyse sensitivity of feature values to change in input

R8: save state of the project for later reproduction

R9: perform theoretical study that derives knowledge to improve the model


Biological aggregation patterns: During our interviews, it became clear that it is impor-

tant for these target users to inspect the behaviour of an existing MATLAB implementation

for their PDE system (R1). This allows to invoke the simulation for different combinations

of parameter values. Since multiple different parameter combinations have to be explored,

it is necessary to narrow down the computations to suitable regions in parameter space (R2)

and to assist the choice of sample points in these regions (R2a). Visual judgement of the

computed solutions (R3) is one method to enable a qualitative distinction among different

patterns of movement (R3b) and to determine, which different sets of parameter choices

produce a given behaviour (R3a and R5). The growth rate of a linear perturbation can be

included as a feature variable (R4). Due to the size of the space of possible solutions, com-

putational help to generate an overview of all possible behaviours is desirable. To ensure

findings are reproducible, it should be possible to save the state of the project (R8).

Bio-medical imaging: In this setting, assistance in choosing a suitable parameter region

to sample is again an important task, where a visual approach is required (R2). At this

early stage of model development the chosen law-driven approach [SRA+08, p. 5] is most

effective. In order to determine plausibility of a solution the ability to use visual judgement

is crucial (R3). While initially mathematical modellers perform this task, input from biology

experts is imaginable at a later stage. This could also yield ideas for primary sources of

field data that could feasibly be included. Also, dependent feature variables (objective

measures) are already constructed for segmentation quality (Dice coefficients) and kinetic

modelling error. Requirements R1-4 apply here, except for the sample creation R2a. The

complexity of the segmentation problem can only be captured by multiple performance

measures. The main goal is to find a robust parameter configuration that leads to good

performance and is robust under a number of varying factors. In particular, performance

should be invariant for different noise levels or patient scans. Automatic optimization (R6)

is challenging with multiple competing quality measures. One step towards that goal is to

produce a weighted sum of performance measures (R4). For the algorithm to work robustly

under different factors, it is important that the segmentation quality does not decay too

quickly for slight changes to the chosen input parameter configuration (R7). To enable the

user to assess robustness of a performance optimum, it is helpful to identify the region of

parameter configurations that lead to ’good’ segmentation results (R3a+5), in order to make

a robust choice within it.


Fuel cell stack design: This case of simulation model inspection invokes basic require-

ments R1-3. The goal of constructing a high-performance cell stack is akin to R4 and R6.

Also, fitting the model can be seen as an inverse problem of finding parameter settings that

match measured output behaviour. Instead of introducing a new requirement, we recognize

that this can be achieved through an optimization (R6). Related to that is the need to have

a reliable and trusted model requires to identify parameter region boundaries that indicate

transitions in stack behaviour. Such a decomposition (R5) can greatly support reasoning

about plausibility of the model.

Renderer configuration: The use case in Section 1.2.4.2 is different from the previous

ones in that interactive experimentation has already been done using a tool chain involving

vuVolume, bash, MATLAB, and weka. The outcome of a pilot study in Figure 1.2a gave first

insight into the complexity of the rendering problem, indicating the limits of an exhausive

exploration via a factorial design that combines all possible design options.

However, the overview in Figure 1.2 is not the solution, but rather the initial problem

setting in this case. Open questions arising from this study are to find theoretical approaches

that address suitable choices of reconstruction kernel and sampling distance. The PhD thesis

of Entezari [Ent07] made significant contributions to the former aspect that were also applied

in the volume renderer of the tool chain developed in this pilot study. The latter aspect

regarding the choice of sampling distance gave inspiration for the theoretical analysis (R9)

provided in Chapter 4.

Setting up spectral illumination for a scene: In order to see how the problem of find-

ing spectral power distributions for lights and materials fits in the frame of multi-parameter

optimization, consider a palette for a linear light model (see Section 5.1.2) to be a point in

a Euclidean space with a dimensionality of [(#materials + #lights) × #lightmodel com-

ponents]. Some cases that we encoutered involved as many as 248 dimensions, such as the

example of Figure 5.2 on page 86 with 5 materials under 3 lights using a 31-dimensional

spectral model.

The design criteria in this setting are specified by combination colours for each light

vs. each material. For human tri-chromatic perception it is sufficient to specify a perceived

colour with 3 components. In the 5 × 3 example this amounts to 45 dependent colour

components for that both, a function to compute them and a desired target (R,G,B)-tuple

are given (R4).


The task of finding suitable spectra in this setting is an inverse problem where one has to

choose factors (light and material SPDs) that produce given combination colour responses.

Rather than defining a new requirement item, this can be framed as an optimization prob-

lem (R6), for which manual search through this space is virtually impossible and, hence,

automatic assistance would be desirable. Rather than analysing sensitivity (R7) in this

case it actually needs to be controlled by introducing regularization terms that are decribed

further in Section 5.2.

Some of the spectra may be required to match real-world measurements and all of them

should be positive in order to be physically plausible, restricting the search to the positive

orthant. Constraining the solutions in this way can be seen as non-interactive construction

of a feasibility region (R2).

1.4 Data abstraction

A common theme of our use cases is that, unlike classical data visualization settings, they

do not start from given data, but work with a computational model, whose parameter space

is sampled in order to construct a set of data points for interactive analysis. Here, the

notion of a data source, which was recently introduced as abstraction of a file loader by

Ingram et al. [IMI+10], could be extended to include basic information about the available

variables, their types, and valid ranges. Enhanced with a capability to query new points

that are not yet stored in a data table, this could be used as an interface to static data

as well as dynamic computation, similar to a function call in a procedural programming

language. This results in a synthetic data source, which we refer to as compute node. The

design of a basic user interface for this abstraction will be discussed in Section 6.2.1. The

conceptual organization of the required tasks along with their inputs and outputs is shown

in Figure 1.3. It separately considers user interaction and computational pipeline, where all

modules operate on the same data and share one flow of control.

A numerical model in our cases consists of relations among a set of variables or dimen-

sions. Some background on relevant mathematical concepts is summarized in Appendix A.

Variables may be inherent to the problem domain or internal to the model. Further possible

distinction can be made considering the distribution of their values, which can be contin-

uous or discrete, (un-)known, (un-)observable, (in-)expensive to sample from, determined

empirically from data, or structurally inferred.


Set up compute node

run

default point

show

derived variables

file IO

Group variables/dims

Specify ROI

View data (sub-)space

overviewbi-variate viewhistogramdetail view

Assign variables

assign manually

trigger computation for points and resolution

Sample inputs

Derive variables

Compute outputs

User Interaction Computation

Distance metric

featuresobjectivesembedding coordinatescluster membership

#dims

region of interest

restrict to ROI

#dims

210

for resolution

Figure 1.3: Abstraction of data, interaction, and computational components. Lines indicate shareddata among processing steps and arrows prescribe an order of execution. On a more detailed level,Red is required input and blue denotes information that is available after a processing step.

The overall model is represented by a function f : Rn → Rr. It is parametrized

over a multi-variate Euclidean domain, in which a point is denoted in vector notation as

x = (x1, x2, . . . , xn). A point in the multi-field range of f , can be computed as f(x) =

y = (y1, y2, . . . , yr). The combination of domain × range of f gives its data space. Occa-

sionally [WB94], the xi are referred to as independent and the yi as dependent variables.


However, this notion does not apply in general, since the presence of constraints may intro-

duce dependencies among the xi. Alternative terminology for inputs and outputs of f are

factors and responses, or parameters and derived variables, respectively5.

We assume that code to compute f is given as a black box and can be invoked for a set

of points X = xk ⊂ Rn of finite size m = |X|. This set X is referred to as a design or a

sample [SWN03, p. 15]. With a prescribed ordering this amounts to the construction of a

design matrix X ∈ Rm×n containing the points in its m rows (R2a). The mapping f gives a

set of responses Y = f(xk). By concatenating these values as [X Y] ∈ Rm×(n+r) the data

table is obtained, giving the main input to further processing or visualization. Applying the

concept of a function f , we impose that the output of the code is deterministic. Uncertainties

of the system can still be modelled by specifying probability distributions for additional

environmental variables xi [SWN03, p. 121]. Even for non-deterministic code, such an

additional variable could simply index the order or the time of a particular invocation.

Using the conceptual ingredients of Figure 1.3, f is formed by composing a potentially

costly image computation h (R1) and variable derivation g (R4) to give f = g h. The

image of h is meant in a mathematical sense, but can represent an actual picture or a disk

image that captures the result of the computation for a particular configuration x. This

indirection should emphasize the possibility to cache output images y ∈ D, but in simple

cases the derived response variables yi are computed directly and g is just the identity

D = Rr → Rr.

Depending on what derived variable yi = gi(y) is specified (R4), its information may be

interpreted as a feature, embedding coordinate, cluster membership label, likelihood, dis-

tance from a template point, or objective measure (R6) — to give a few practical examples.

In each case it may be possible to compute gi or to assign values manually, depending on

whether a function definition or a user’s concept is available.

Some processing steps (R5+7) require a notion of distance or similarity among points

(R4a). Technical background on distances and norms is provided in Appendix A.1.1.

One method considered here is the Euclidean distance dr between feature vectors in Rr.

Beyond that, distance dc combines all information about each configuration point, including

its parameter coordinates x ∈ Rn or a domain specific function operating on the disk images.

In order to accommodate simulations with a large number of variables n + r, an early

5This is a summary of terminology found in the referenced data analysis literature.


step of the interaction allows the user to divide variables into groups of smaller sizes nl.

This way one can separate input and output, indicate other semantic information inherent

to the simulation model, and produce more focussed multi-variable views (R3).

An important aspect of Figure 1.3 is that the sample creation is split into a specification

of a region of interest M ⊂ Rn+r, where areas in the input space are outlined (R2), see also

Section 6.2.3. This is input to an automatic method that generates a point distribution of

good numerical quality in that region that also fulfills a given budget m. What good quality

means in the context of multi-dimensional point distribution is subject of the following

chapter.

Chapter 2

Acquisition and visualization of

multi-variate data

Enabling people to inspect and understand complex data sets is a core objective of computa-

tional visualization. While algorithms may apply to general data types, staying aware of the

original problem domain is crucial to allow for meaningful interpretation of a visualization.

Aside from this cognitive motivation there are also computational reasons to maintain the

connection to the data generating source. In particular, if a computational model is used

to generate the data set, it may be invoked to obtain further data to refine or extend the

region of interest to sample from. While data in its original Latin meaning is “something

given” its acquisition can be influenced by deliberate choices, turning it into a response or

“something asked for”. This more active perspective on data explains two related threads in

this chapter, namely to discuss criteria and methods for sampling to “ask good questions”

in Section 2.3 that leads into a discussion of “making sense of the answers”, which on a

numerical level begins with the topic of integration in Section 2.2 and reconstruction in the

subsequent sections. This structure repeats in Appendix B at a different level of depth. A

closing discussion of interactive visual interfaces in Section 2.4 that facilitate comprehension

of the so-acquired numerical data gives background for Chapters 4 and 6.

The following historical excursion will show that without a notion of continuity of the

underlying space, a special treatment of the multi-dimensional setting is not required — all

variables could be folded into one without loosing any of the non-existent structure. Because

a fundamental notion of continuity is readily implied in our multi-dimensional setting, the

title of this thesis does not mention it explicitly.

19

CHAPTER 2. ACQUISITION AND VISUALIZATION OF MULTI-VARIATE DATA 20

2.1 Effects of dimensionality

An intuitive notion of dimension goes back to Euclid’s “Elements” (300 BC), in which he

begins: “1. A point is what has no part. 2. A line is what has lengths but not width.

(...) 5. A surface is what has length and width only.”1 The dimension of these objects is

determined by the number of parameters required to refer to each of their elements: line

1, surface 2, solid 3. This notion of dimension, while intuitive, has a remarkable counter

example.

In 1887 Jordan proposed a rigorous definition of a curve to be a continuous function

of a single parameter, whose domain is the unit interval [0, 1]. Soon after, Peano and also

Hilbert [Hil91] devised a continuous mapping of the unit interval onto the full unit square

creating a space-filling curve that one can follow and pass through all points of the two-

dimensional square. Extensions of these mappings cover the entire unit cube [0, 1]n with a

Jordan curve, still depending on a single parameter only.

All of these curves are densely self-intersecting one-to-many mappings. In particular,

Hilbert points out that with a slight modification of his square filling curve the number

of self-intersections at a point can be reduced to three. The Lebesgue covering theorem

mentioned in Appendix A.1 asserts that this number may not be reduced further.

To restore the intuition about the dimensional number as the number of parameters

needed to represent each element of a set, one has to add the property of uniqueness to

the continuous mapping that parameterizes the set. Such a homeomorphism maps one

set into another leaving all its topological invariants intact, such as dimensional number,

number of connected components, or genus (number of holes). To return the focus to

data analytic aspects, the definition of a topology, continuous functions, and manifolds are

deferred to Appendix A. After briefly leading into topological topics involving basic notions

of neighbourhood, the following discussion is again of geometric nature.

Volume of the hyper-sphere in Minkowski p-norm: A basic mathematical object is

the n-dimensional p-norm sphere

Spn = x ∈ Rn : xp1 + xp2 + · · ·+ xpn ≤ rp (2.1)

of radius r with p ∈ [1,∞], defined here to contain its boundary. It is of relevance in

1Some of these observations follow a historical survey by Duda [Dud79].


numerous theoretical and practical geometric settings and arises often in the context of

metrics and norms discussed in Appendix A.1.1. In Euclidean space Rn with Lebesgue

measure voln the sphere is the set with the smallest surface area [Mat02, pp. 222]. A closed

form expression for its volume is derived by Newman [New72, p. 101] as

voln(Spn) = 2nrnΓ(1 + 1/p)n

Γ(1 + n/p), (2.2)

using the Gamma function Γ as a continuous extension of the factorial, giving Γ(n) = (n−1)!

for n ∈ Z+. The graph in Figure 2.1 shows its behaviour for increasing n and p. A curious

0 5 10 15 20 25 3010−25

10−20

10−15

10−10

10−5

100

105

1010volume of n−sphere for p−norm

n

volu

me

p = 1

p = 2

p = 3p = 4p = 5p = 10

Figure 2.1: Semi-log plot of volumes of n = 1 . . . 30 – dimensional p-norm unit spheres.

observation is that the 2-norm sphere hyper-volume reaches its peak at n = 5 and volumes of

all p-norm spheres ultimately converge to 0 for growing n except in the case of the hyper-cube

for p = ∞ with a volume of 2n. It is somewhat misleading to perform this interpretation

along the n-axis, because a 3-dimensional sphere, for instance, contains infinitely many non-

intersecting 2-dimensional disks of non-zero area. However, vertical comparison for different

choices of p is fine and makes a striking case for non-box shaped regions, when using p-spheres

to mark out regions of interest for exploration. The discussion in Section 6.2.3 on page 105


will come back to this aspect. Rearranging Equation 2.2 one obtains the radius

rn =Γ(1 + n/2)1/n

2Γ(3/2)(2.3)

for an n-dimensional 2-norm sphere of unit volume. This radius will be relevant in Sec-

tion 2.3.4 on page 32 to provide an upper bound for the density of periodic sphere packings.

The implications of this discussion are: the more variables or dimensions in a metric

space are to be inspected, the larger is the volume to cover. Volume is directly proportional

to the number of configuration points that need to be computed in order to maintain a

certain density. In most settings, this directly corresponds to computational cost, which we

would like to minimize. One way to do so, is to determine first, how many variables actually

matter. While algorithmic development on this topic is current research not covered in this

thesis, the following brief technical discussion of the issue can provide an entry point in the

future.

Estimating dimensionality: Lebesgue’s covering theorem points out a connection be-

tween the dimensional number n of a region M and the minimum number of n+ 1 simulta-

neous intersections when covering M with small open neighbourhoods. In a metric space one

can use interiors of spheres of radius R for this purpose. Let mR(M) denote the minimum

number of such neighbourhoods of diameter < R needed to cover a set M and note that if

M has an n-dimensional volume, this count fulfills mR(M) ∼ R−n. This is the idea behind

the so-called capacity dimension, also called Hausdorff- or fractal-dimension that estimates

the largest polynomial degree n of the above count for arbitrarily small neighbourhoods as:2

dimcap(M) = − limR→0+

lnmR(M)

lnR. (2.4)

A more easily computed lower bound to this number is given by the correlation dimension.3

It is also considered as a measure of intrinsic dimension by Levina and Bickel [LB05]. To

allow for a statistical analysis, they construct a Poisson process for the point set that is uni-

formly distributed inside M counting the number of points that lie within a growing radius

2Note that this works for any non-zero volume of M , as it comes out of the logarithm in the enumeratoras an additive constant and then vanishes against the magnitude of the denominator.

3The correlation dimension [Wei09] is counting pairs of points of a set M within a radius R of each other asa measure of connectedness. This count rises in the order of Rn, with the monomial degree n correspondingto the correlation dimension dimcor(M).


of each other. The growth rate of this process depends on the local density, dimensionality,

and implicitly the sphere volume Vn = voln(S2n) derived in Equation 2.2. Based on this, a

closed form solution is derived for the maximum likelihood estimate of the dimensionality

that is averaged over all points of M . Their discussion points out that dimension may vary

with position x and with a choice of scale, which is specified in their method by either a

maximum radius or an index of the k-furthest neighbour to consider in the estimate.

Beyond these geometric perspectives on dimensionality, the approximation of functions

in linear spaces also requires choices of sample points for which a discussion is provided in

Appendix A.2. This setting can be interpreted as looking at a family of integrals — one

member for each point to reconstruct. The following discussion more specifically considers

the computation of a single integral.

2.2 Quadrature error

An important class of integration methods are the so-called quadrature (or cubature) rules

that approximate an integral of a function f over a finite domain M ⊂ Rn as a weighted

sum of point samples in a set X ⊂M of size m = |X|

If =

∫Mf(x)dnx ≈ Qf =

∑xk∈X

wkf(xk). (2.5)

Different choices for X and w distinguish the respective integration methods. For instance,

consider a factorial design, e.g., implemented via nested for-loops that iterate each com-

ponent variable xi of the vectors x ∈ X over a set of values Xi. This leads to an overall

number of m =∏ni=1 |Xi| points. Even if only two distinct values per variable are desired,

this design has m = 2n points, growing exponentially in the number of variables. We will

mainly focus on the placement of the points and assume uniform weights w = 1m . However,

prepending a brief discussion of non-uniform weighting will help to illustrate a basic effect.

Using a linear function space model, the integrand f is approximated by a sum of basis

functions (see Appendix B.2, also Appendix B.2.2 or Phillips [Phi03, pp. 119])). This linear

model determines the weighted contribution wk that each point sample has to the integral

estimate. More specifically, the composite Newton-Cotes method uses Lagrange polynomials

of degree s < m to properly represent a certain level of smoothness of the integrand.

The resulting error term is proportional to hs+2f (s+1)(ξ) for some point ξ contained in


the integration domain with sampling distance4 h = m−1/n. This leads to an integration

error ε of order O(m−(s+2)/n

), where the bounded (s+ 1)st-derivative of f disappears as a

constant factor. One can turn this around to estimate the cost m for a desired accuracy as

O(ε−n/(s+2)

), which is pointed out by Bungartz and Griebel [BG04] as an illustration of two

important effects. Firstly, there is an exponential growth in the need for sample points with

increase in dimensionality n as it also occurred for the factorial design mentioned above. It is

a quite ubiquitous effect and, hence, acquired its own name curse of dimensionality [Bel61].

Secondly, additional assumptions on the smoothness of the integrand, such as bounded

derivatives f (s+1), may reduce this need and can thus be considered a blessing of regularity.

With uniform weights w in Equation 2.5, a popular choice for X is a set of random

points uniformly distributed over the domain of the integrand resulting in the Monte Carlo

method [ES00]. For integrands f ∈ L2 the standard deviation σ of this estimate decays5 as

O(m−12 ) independent of the dimensionality n. This is a very desirable property, especially

compared against the O(m−1n ) error given above for a composite Newton-Cotes rule of

order s = 1 on regularly spaced points (a.k.a. trapezoidal rule using linear interpolation).

While this seems to be a perfect antidote against the dimensional ‘curse’ it does have its

downsides [Nie92, p. 7]: (i) The error bound is only probabilistic and does not offer the

same guarantee as a maximum error bound. (ii) Possible known regularity of f does not

improve the efficiency 1εm [PH04a, p. 663]. (iii) The computational construction of random

point sets typically involves pseudo-random number generation, which may bring its own

difficulties to bear. These observations give rise to search for constructions of deterministic

point sets with guaranteed error bounds that have a superior efficiency for certain classes

of functions. Further pointers in this direction are provided in Appendix B.1.1.

This concludes our discussion of general numerical effects that one should be aware of

when working in the context of multi-variate data acquisition and processing. The following

part of this chapter provides background that is more specific to possible applications of the

chapters to follow.

4One way to obtain the distance h from the numberm of sample points is as edge length of the fundamentalparallelepiped of a Cartesian lattice Zn scaled to have m points in a unit volume.

5Calling the k-th independent random sample Fk = f(xk), the variance of the Monte Carlo estimator is

derived as V [Qmf ] = V[

1m

∑mk=1 Fk

]= 1

m2

∑mk=1 V [Fk] =

σ2f

m.


2.3 Discretization of multi-dimensional functions

As pointed out in the data abstraction Section 1.4, we view a simulation model as a function

f with deterministic code to compute it. In order to apply the visualization techniques that

will be pointed out in Section 2.4, we first need to turn this functional representation into

a finite, discrete data set.

The setting of numerical integration has already been introduced in Section 2.2. Another

operation required for digital numerical representation is quantization. Often also referred to

as clustering, it obtains a discrete representation of a probability distribution over a metric

space, which involves criteria discussed in Section 2.3.1.2. While quantization of the range

of f typically incurs some loss of information, discretization of its domain can in certain

cases be perfectly undone. The theory related to this topic of reconstruction and prediction

of unobserved points is reviewed from the angle of signal processing and approximation

theory in Appendix B.2 with a statistical perspective from the field of experimental design

provided in Section 2.3.2.

2.3.1 Useful concepts for metric data representation

First, some basic constructs are defined that will find application in later chapters.

2.3.1.1 Voronoi cells and the Delaunay graph

A basic construct when studying point set geometry is the nearest neighbour cell of a sample

point, also denoted as Voronoi cell. It contains the portion of space around a point that is

closer to this point than to any other one. An illustration of what this means geometrically

is given in Figure 2.2. Algebraically, such a cell around a point xk ∈ X is defined as

Ω(xk) = x ∈ Rn : ‖x− xk‖ ≤ ‖x− u‖ for all u ∈ X. (2.6)

The boundary of this region is formed by bisecting hyperplanes between xk and the Voronoi

relevant neighbours

N(xk) = 2Ω(xk)− xk ∩X, (2.7)

namely all points of X that are covered when uniformly scaling Ω(xk) around xk by a factor

of 2.


Figure 2.2: Set of 20 randomly distributed sample points in [0, 1]2 with Voronoi regions outlined inblue and Voronoi relevant neighbours connected by grey lines. Notice how the convex hull and theminimum spanning tree are part of the grey Delaunay graph connecting Voronoi relevant neighbours.For points in general position the Delaunay graph turns out to be a triangulation or a simplicialcomplex in higher-dimensional spaces.

2.3.1.2 Measuring quantization error via the second order moment

A partition of the continuous domain into a finite number of cells can be used to analyze

the action of a quantizer that rounds each point in Rn to the nearest quantization site

in X. This application motivates a first quality measure for the set X. The normalized,

dimensionless second order moment of inertia measures the mean squared error (MSE) per

quantized symbol6 [CS82, Eq. 8],[CS99, Ch. 21]

Gn(Ω(xk)) =1

n

∫Ω(xk)‖x− xk‖2dnx

voln(Ω (xk))(n+2)/n

, (2.8)

for signals that are uniformly distributed over the range containing Ω(xk). This figure is

lowest for a sphere and may also serve as a measure for sphere dissimilarity of the Voronoi

region. The functional that computes the MSE Gn(X) for the entire domain of the quantizer

X, is formed as a sum of the second order moments of each of the Voronoi cells of the points

in X. It can be computed analytically with the method of Appendix A.1.3 and is applied

in the context of Figure 3.2 on page 54.

6The root of this error (RMSE) would be in the unit of the original distance measure and is numericallyeasier to interpret. We keep the squared form, since it is mathematically easier derive from.


Numerical relaxation: The idea underlying Lloyd’s algorithm [Llo82] is to iteratively

improve a quantizer X by moving the quantization sites xk to the centers of their Voronoi

regions. Convergence of this algorithm has been analyzed for one-dimensional [SG86] and

two-dimensional settings [DEJ06], but is still open for quantization in higher-dimensional

spaces. Lloyd’s algorithm can also be studied as a dynamical system [CB09], e.g., of agents

maximizing their mutual distances from each other, with interesting patterns for the tra-

jectories of the point locations that arise from using different approximations to the true

Voronoi cell centroid as local attractor. The algorithm has numerous applications, including

half-toning [BSD09, Han05], photon-tracing [SJ09], and the optimization of experimental

designs [HA02]. The latter is due to the fact that the mean squared quantization error when

representing a region with a point as in Equation 2.8 is minimized when this representative

point is the centroid of the region. An alternative to numerical optimization is to directly

construct a quantizer that is a local optimum of the second moment error functional, i.e.

that is a fixed point of Lloyd’s algorithm. An important example of that is given in the

following.

2.3.1.3 Sampling lattices

For any periodic structure that can be seen as n-dimensional, additive subgroup of Rn, it is

possible to construct a set of basis vectors [Mat02, p. 22] whose integer multiples generate

the set of periods or self-similar displacements of the structure. This set is referred to as

point lattice7 Λ. Such a set Λ is defined as subset of Rn that is closed under addition and

inversion

Λ(G) = Gk : k ∈ Zn ⊂ Rn. (2.9)

Its non-singular generating matrix G ∈ Rn×n is omitted as an argument if it is clear from

context. To retain the uniqueness of k, we exclude non-square G of rank n from consider-

ation. Scaling and angles between the basis column vectors of G are implicitly represented

by its Gram matrix

A = GTG. (2.10)

7This has nothing to do with the algebraic structure called lattice that refers to a partially ordered setwith a unique infimum and supremum. A collision could occur when talking about the face lattice of apolytope that represents containment of vertices, edges, or facets (see Appendix A.1). The attributes point-or sampling lattice both refer to the above definition.


The dual Λ⊥ of a lattice is defined as

Λ⊥ = x ∈ Rn : x · u ∈ Z for all u ∈ Λ (2.11)

= (G−1)Tk : k ∈ Zn.

The second, equivalent definition gives a generating matrix G−T for the dual lattice [Nie92,

p. 132], which consequently has A−1 as Gram matrix.

Due to the translational group structure of Λ, the neighbourhoods around any lattice

point are identical, resulting in only one type of Voronoi cell, as defined in Equation 2.6. We

use Ω to denote the cell centered around the origin and occasionally provide the generating

G as argument or a point x to denote the cell around it.

Voronoi cells are not the only space-filling tiles that repeat at lattice points. Other

periodic, space-filling tiles must have the same volume as the Voronoi cell of the lattice

underlying the tiling, but may differ in shape. The fundamental parallelepiped with edges

along the basis vectors in the columns of G is another example. This gives an easy way to

compute the volume of the Voronoi cell as

voln(Ω(G)) = |det G| . (2.12)

An efficient method to compute the Voronoi relevant set N(0) of Equation 2.7 for a lattice

point set is described by Agrell et al. [AEVZ02]. It used in Chapter 3 to determine properties

of the Voronoi cell of a lattice.

It is also possible to tile a rational lattice having a G ∈ Qn×n with axis aligned “bricks”

as obtained by Gilbert [Gil93] from the Hermite normal form of its integer scaled G, which

provides displacements and sizes of the bricks.

Two generating matrices G1 and G2, are similar or equivalent if they produce the same

lattice up to rotation and scaling [AEVZ02], i.e. αQG1 = G2 for a non-zero α and some

QTQ = In. Because of that it is always possible to equivalently represent a lattice with a

triangular generating matrix that can be obtained via QR-decomposition [TI97, p. 48]. If

|α| = 1 then Λ(G1) and Λ(G2) are congruent. Allowable changes of basis that lead to similar

lattices are subject of Section 3.1 on page 44 and key to the construction idea of that chapter.

Despite their simplicity, lattices enjoy remarkably deep theoretical connections as well as

numerous practical applications, excellently accounted for in a reference by Conway and


Sloane [CS99]. The relevance of lattice sampling for multi-variate function reconstruction

becomes apparent considering the special structure these regular sample point distributions

impose on the resulting interpolation matrix, which is discussed in Appendix B.2.2.

2.3.2 Experimental design

The task of generating a data set from a function f has been abstracted in Section 1.4

as designing a sequence of points xk = X ⊂ Rn that discretize the domain of f to

then run an algorithm to compute the output responses for the set of input configurations.

Before giving an overview of related methods for multi-dimensional sampling, let us consider

different purposes of sampling:

1. Exploratory designs facilitate a comprehensive analysis of the function f , a model of

its full joint probability distribution PX ≤ x, Y ≤ f(x) could be constructed [Bis06,

p.13], addressing R2a. To obtain such a complete model, a possible approach is to use

space-filling designs, as discussed in Appendix B.1.

2. Prediction-based designs: In order to compute a single aggregate statistic, such as

the mean value of f , one can numerically perform integration, which amounts to the

application of a linear functional. Further settings might involve the application of

a family of such functionals, e.g., to approximate function values at new positions,

which is also studied under the name of reconstruction or interpolation. In all of these

settings it is possible to estimate an error or infer a confidence interval, which may or

may not take newly acquired data into account.

3. Reconstruction model adjustment: To further adapt the reconstruction or regression

model to field data, sample points are used to determine an appropriate model family

(linear/non-linear), a suitably reduced dimensionality, or other regularization param-

eters. For instance, dimensional reduction and the choice of a correlation function for

Gaussian Processes fall into this category [JMY90, PM62].

4. Analyzing variability: Uncertainty analysis determines variability of a response based

on the distributions given for the environmental variables. This provides confidence in-

tervals for the responses that can be used to guide selection of relevant features [Fra07].

Sensitivity analysis extends this analysis to determine how output variability is affected

by each of the input variables [SRA+08] (R7). Similar measures appear in the analysis

of (backward) stability of numerical algorithms [TI97, p. 104] and a recent study on


the stability of iso-surfaces [PH10].

5. Optimization-based designs: In this setting, only those parts of the domain of f are

relevant that are likely to contain an optimum. Starting from an initial design, it is

possible to steer concentration of the sample density in subsequent updates [MAB+97,

BFG08, JSW98] (R6).

These tasks are somewhat ordered by decreasing degree of comprehensiveness. For instance,

the general case (1) will require an exhaustive sampling where (5) – after suitable initializa-

tion – may focus further points on small, promising regions in parameter space.

Space-filling criteria are optimized by minimax/maximin designs of Johnson et al. [JMY90].

They show equivalence of some function analytic and geometric optimality criteria, i.e. the

different optimal designs either minimize maximum variance (G) of the predictions, volume

(D) of the m-dimensional confidence ellipsoid of the predictions, or average variance (A)

of the estimator. Respectively, for the Voronoi cells around each sample point these are

equivalent to minimizing the distance of the furthest vertex (synonymous: optimal covering

radius, minimax distance design, G-optimal for near-independent correlation), maximizing

distance of closest facet (best packing radius, maximin design, D-optimal), and minimizing

their second order moment (best uniform quantizer, A-optimal).

Somewhat complementary to these properties are Latin hyper-rectangle designs [MB06].

These designs care about uniformity of the projections of the point set onto each axis. It is

possible to construct hybrid designs that maximize both, axis projection- and space-filling

criteria.

The book by Lemieux [Lem09] gives an accessible overview on mostly non-adaptive

sampling methods that are for instance relevant to provide space-filling initializations for

purposes (1), (2), or (4) in Section 2.3.2. Another exposition by Santner et al. [SWN03]

provides more background on model-adaptive sequential sampling, including an introduction

to Gaussian process models (for purposes (2), (3), or (5)).

Of the above list, it will initially be aspect (1) that will matter in solving the problems

laid out above. As more insight on the model behaviour is gained and included in the

analysis, the adaptive techniques of categories (2) and (5) will address sampling requirements

more effectively.

Bias: Another important aspect of the quality of a sample when obtaining dependent

measures from it is the absence of sampling bias [CK95, p. 89]. Using different sampling


methods and, in particular, different regions to sample the input variables, the shape of the

distribution of output variables is likely going to be affected. The experimenter should be

aware of this effect when interpreting the requested observations. This problem will not be

discussed much further in this thesis, but should at least be pointed out.

2.3.3 Figures of merit: Packing and covering radii, density, and thickness

In the context of quantization distributions of distances from the domain M of interest

to the set of sample points X gives a quality measure involving the second order moment

G of the Voronoi cells defined in Equation 2.8. Due to its translational invariance, the

choice of a lattice X = Λ has only one type of cell Ω. While G(Ω) is minimized by cells of

spherical shape it is not possible to tile space of dimension greater than one with spheres.

The objective of the packing problem is to do as good as possible of a job by minimizing the

amount of empty space when packing a large number of equally sized spheres into a sub-

region of Rn. The practical relevance of good solutions for this problem extends to efficient

storage methods for real-world objects such as oranges and ball bearings [CS99, EDM09].

Related to this objective is the covering problem asking to arrange a large set of over-

lapping spheres to cover a large region of Rn with the minimum amount of overlap among

the spheres8. In the following, we will discuss a number of quality measures arising from

covering and packing problems.

The packing radius ρ is the maximum radius of non-overlapping spheres centered at the

sample points, which is also the largest sphere inscribed to Ω. The packing density ∆ of

a lattice is defined to be the ratio of the volume of a sphere of packing radius ρ over the

volume of the Voronoi cell Ω of the lattice [CS99, Ch. 1, Eq. 20].

∆ =Vnρ

n

voln(Ω), (2.13)

where Vn = voln(S2n) is the volume of an n-dimensional 2-norm sphere of unit radius given

in Equation 2.2. The alternative measure of center density δ = ∆Vn

can be interpreted as

a ratio of volumes as well. Instead of a sphere of radius ρ, a cube or parallelotope of edge

length ρ is measured in proportion to voln(Ω). The compensation for the sphere volume is

the main distinguishing factor between the center density δ and packing density ∆, which

8In both previous problem statements, the attribute “large” is used to avoid infinity and boundary issues.


can also be seen from this equality (see also Equation 2.12)

voln(Ω) =Vnρ

n

∆=ρn

δ= |det G| . (2.14)

The smallest radius for spheres centered at the points of Λ that entirely cover the domain

is referred to as the covering radius

R = maxx∈Rn

minu∈Λ‖x− u‖. (2.15)

The argument x ∈ Rn that maximizes the distance to the nearest lattice sites is also referred

to as a deep hole. It is a furthest vertex of Ω, which provides one method to compute R from

the circumsphere of Ω. A point set that minimizes R for general point sets is referred to as

minimax design [JMY90], also discussed in Section 2.3.2, where the number of design sites

equidistant to the deep hole are referred to as index I∗∗ of the design. A quality measure

arising from the covering problem is the thickness Θ = VnRn

|det G| representing the average

number of spheres that contain a point in space [CS99, pp. 31]. Similar to packing density

one can define a normalized thickness θ = Rn

|det G| .

A problem related to packing is that of maximizing the kissing number τ , which denotes

the number of adjacent spheres that just touch a center sphere at equal distance. It is also

the number of distance minima ρ on the facets of Ω. The hexagonal lattice has kissing

number 6, the 2D Cartesian lattice 4, and a randomly chosen generating matrix G of any

dimension almost surely produces a lattice with kissing number 2, counting the nearest

neighbour and its inverse. Maximizing τ can be interpreted as a packing problem on the

surface of a sphere with an optimum angle of 60 between adjacent centers spanning an

equilateral triangle [CS99, p. 24].

2.3.4 Optimal packing and bounds by Minkowski and Zador

Solutions to the packing problem that optimize the quantitative measures just considered

are given by Conway and Sloane [CS99, p. 15] with a list of generating matrices provided

by Hamprecht and Agrell [HA02]. An overview of the densest lattices in dimensions up to

n = 64 is shown in Figure 2.3. The upper bound in this plot is based on Minkowski’s theorem

([Mat02, pp. 17], [New72, pp. 92]) asserting that a convex body K, centrally symmetric

around the origin of the Cartesian lattice Zn with voln(K) > 2n, contains at least one lattice


0.5

1

1.5

2Packing radii

!="1/

n

n

Z A2 A3 D4 D5 E6 E7 E8 K12

L16

L24

L25

L26

B27

B28

B29

Q30

Q31

Q32

Q33

Q34 B35

KS36 D37

D38

P48_

39P4

8_40

P48_

41P4

8_42

P48_

43P4

8_44

P48_

45P4

8_46

P48_

47 P48

MW

54 L56

NE64

densest known latticesarithmetic minimum upper bound rn

Figure 2.3: Best lattice packings in dimensions n = 1 . . . 64. The symbol δ refers to the center densitydefined below Equation 2.13. Note that the linear interpolation between the densities given at thenamed abscissae may be in disagreement with the actual best packing. The packing radius of theCartesian lattice is 0.5 in any dimension, which is represented in the figure by the horizontal axis.

point other than the origin. By inclusion of a linear transform, the same volume bound can

be shown to hold for general lattices scaled to unit density |det G| = 1. This can be used

to prove the packing bound shown in Figure 2.3 as follows.

The unit volume sphere radius rn of Equation 2.3 is half the radius of a sphere of volume

2nVn. According to Minkowski’s theorem, spheres centered at lattice points having radius

larger than rn have to intersect. This provides the upper bound for ρ shown in Figure 2.3.

The bound is directly related to Hermite’s constant γn limiting the arithmetic minimum,

which is the maximum squared distance ρ2 of a point closest to the origin over all possible

ΛG with |det G| = 1 in n-dimensions [New72, pp. 202]. In the computation of Voronoi

relevant neighbours (Equation 2.7) this bound can be used to check whether an obtained

packing radius is too good to be true. This is applied for verification of results in Chapter 3.

An interest in dense packings is further motivated by the open conjecture of Conway

and Sloan [CS82] stating that an optimal lattice quantizer, i.e. a lattice with lowest pos-

sible Gn(Ω), is the dual lattice of an optimal packing. Considering the task of quantizing

a sequence of scalars, an observation due to Zador [CS85] is that it is more efficient to

simultaneously quantize several values as vectors rather than each value at a time. The

reason lies in the difference in second order moment (as defined in Section 2.3.1.2) between

Gn = 0.08333 for the n-dimensional cube corresponding to a separate encoding of each

dimension, and Gn of the Voronoi cells of good coverings in higher dimensions, e.g., 0.0801


0 5 10 15 20 25

123456789

10

Zador’s upper bound for MSEof n−dimensional optimal quantizer

n

100*

Gn

upperlower (sphere)CC (cube)

Figure 2.4: Zador’s bounds for the mean squared quantization error of optimal quantizers in Rn.

for the hexagon, 0.0785 for the truncated octahedron of BCC, or 0.0657 for the Voronoi

cell of the 24-dimensional Leech lattice. Each of these figures directly corresponds to the

mean squared error per symbol when encoding chunks of 2, 3, or 24 numbers by the nearest

points on the respective lattice. A plot of Zador’s bound as summarized by Conway [CS85]

is shown in Figure 2.4. The derivation of the upper bound, as described by Zador [Zad82,

Eq. 16-23 and 43], involve the second moment of the distribution of round-off errors of a

uniform random quantizer with a large number m of quantization sites. For m → ∞ it

converges to the second moment of an n-dimensional sphere, which serves as lower bound

for any optimal quantizer and is shown as green line in Figure 2.4.

2.4 Visual interfaces for multi-variate computer model data

The discussion in Section 1 has remarked that computational models in domains of human

interest significantly benefit from good human interfaces. Regardless of their domain, the

development of any computer model has to go through certain stages including to convince

developers and users of its validity. At a basic level, this concerns development methodology


and the choice of programming language9, but this topic is explored further here.

In Section 1 a point was made for using the visual channel in order to create efficient

interfaces. The visualization process [dSB04, HM90, CJ10] classically starts with a given

data set, possibly performing further refinement or filtering to then map data points to

graphical primitives that are subsequently rendered to the screen. Recent work by Chen and

Janicke [CJ10] considers an information theoretic perspective on the visualization pipeline.

Their survey of applicable theory from that domain proposes a unified view on the amount

of information that a particular data set provides relative to a set of possible ones, and, in

the same light, how much information different visual mappings (e.g., overview and detail)

of a data set provide in relationship to each other. This promising abstraction opens up

several directions for further pursuit by the visualization community: a) A large size for

an encoding is bad by typical information theoretic notions. However, as pointed out by

the authors, a large image may be significantly easier to read than a small one. In general,

what matters for the comprehension of a visualization is the computational or temporal

complexity of the decoding procedure, probably more so than the spatial complexity of the

code. b) Information measures based on entropy typically are sensitive to noise, since it is

rated as something unexpected and in that is similar to valuable information. A possible

extension of this work could attempt to make distinctions between relevant and irrelevant

information, as it could for instance be derived from a driving question or task. The task

overview that will be given in Section 2.4.1 can be seen as a first step into a task-driven

organization of research. However, it is focussed on interacting with computer models,

rather than visualization algorithms.

With data input at one end of the visualization process and the human observer at

the other it is natural to start by looking at the quality of the input. The first part of

this chapter did this from a numerical perspective, where two main factors where discussed

that impact sample complexity: the volume of the region to cover (Section 2.1) and the

uniformity of the point distribution that is generated inside a given region (Section 2.3.2).

Beyond numerical point data, the following distinction of data types provides a possible

structure to the visualization research landscape.

9Choices of high-level scripting languages, such as Python or MATLAB (see also Section 6.2.1)can already be considered among the human-friendly ones. See http://www.webmonkey.com/2011/02/

cussing-in-commits-which-programming-language-inspires-the-most-swearing/ for a humorous sta-tistical argument in this direction. However, having noted the issue of bias earlier, in this sample differentlanguages imply different types of problems programmers are working on.

http://www.webmonkey.com/2011/02/cussing-in-commits-which-programming-language-inspires-the-most-swearing/

http://www.webmonkey.com/2011/02/cussing-in-commits-which-programming-language-inspires-the-most-swearing/


Data taxonomies: While the data is represented on a computer in discrete form, the

nature of the system it is describing can be either continuous or discrete. This distinction

forms a basis for the visualization research taxonomy by Tory and Moller [TM04], where

continuous data refers to a domain with Euclidean topology of Rn. The attribute spatial

typically means n ∈ 1, 2, 3 and temporal data includes a time variable. For discrete

data the topology can be more arbitrary10 and quite frequently is the main subject of

the visual representation as for instance in graph drawing. Other criteria to distinguishing

different types of visualization settings could consider the continuity of the range of measured

attributes (categorical, ordinal, numeric) [CM97] rather than its domain, or the availability

of a computational model (simulation) or streaming data source (providing feature space

trajectories). Further perspectives could consider the type of task [PVF05] addressed by

the method and the role of the user (understand, annotate, steer, etc.). The latter is not

further considered here, but will appear as a topic again in Chapter 7.

Classifying data by degrees of indirection: In the context of model fitting, human

provided input could also be considered real data. To still enable a distinction to measured

field data, one could adapt terminology from historical studies, where primary sources refer

to original material or evidence, secondary sources are reports about this material, and

tertiary sources are summaries of common knowledge about the original events that could

be gathered in encyclopedias. With these distinctions, human input that adjusts model

behaviour to match with experience can be considered secondary data. Synthetic data that

is generated by a model or reconstruction method also falls into this secondary category.

Finding theory and incorporating it to improve a model would in this setting amount to

adding rules and variable relations, which in this abstract view is tertiary data.

2.4.1 Computational modelling tasks

An overview of tasks that arise when working with computational models is given in Fig-

ure 2.5. The topic of model construction is not directly dealt with in this thesis, since this

task is considered to be solved by given code. However, enhancement or adjustment of a

model are required as per R4 of Table 1.1 and a possible interface to facilitate this will be

subject of Section 6.2.1.

10A related concept defined in Appendix A is the discrete topology that consists of the set of all subsetsand thus imposes no particular neighbourhood structure or all possible ones at the same time.


Figure 2.5: Schematic overview of tasks related to studying effects in model parameter spaces.The blue coordinate axes symbolize the construction of the parameterization with one dependentresponse variable indicated by iso-lines in the background. The two-sided blue arrow in the centerrepresents the task of fitting a model to observed field data. The blue chip along this line representsthe possiblity to fit in a digital substitute model that can include model assumptions to make upfor missing data or is simply more efficient to evaluate than a more complex model or direct fieldmeasurements. The green itinerary or schedule of parameter adjustments could be provided bycomputational steering interfaces for a time-evolving simulation model. The red target indicates thegoal of a search for an optimal configuration. The region labels and outlines in black illustrate apartitioning of the parameter space into regions of homgenous behaviour of selected responses. Animportant part of this picture, but not part of the drawing, is the human observer who is responsiblefor interpretation of the analysis in the context of a particular purpose.

Human interfaces with computational models: The interplay between user and com-

putational model that is facilitated by a visual interface can, in one direction, allow us to

steer the computation to focus on the most interesting aspects first. The other way around,

the human observer can be used by the algorithm as a feature extractor, e.g., to supply

annotations or tags to sets of data points that can serve as further input. The inclusion

of the user in the generation and analysis of data sets allows i) to learn about the user’s

objective, e.g., in form of a query, that can be refined as new information becomes available,

ii) to determine a region of interest in parameter space, where further adaption of sample

density per spatial region or per dimension is possible, or iii) to guide sample refinement

using error measures that are based on perceptual impact.

Fitting Models to Data: A comprehensive review of multi-dimensional data visualization

is provided by Holbrey [Hol06], who indicates a trend towards indirect visualization that

includes additional levels of processing, mostly involving fitting of alternative representa-

tions, that are then visualized instead of the original data. This goes along with a survey by

De Oliveira and Levkowitz [FdOL03] who relate multi-dimensional visual exploration and


data mining, observing a need for methods that more tightly integrate visualization and

analysis. This recent trend to use visualization methods to make data analysis algorithms

more understandable is pursued by Wickham [Wic08], discussing the usage of statistical

models on top of the given data. To sharpen the terminology, a distinction is made between

the family of a model (linear regression, clustering, etc.), the form (which variables are

factors or responses), and the fitted model finally refers to a particular instance adapted to

the underlying data. Some characteristic tasks that are encountered in visual data mining

are recognized by Himberg [Him04], including predictive modelling, descriptive modelling,

discovering rules and patterns, exploratory data analysis, and retrieval by content.

Data-centric visualization research: Since multi-dimensional numeric data is the result

of discretizing the parameter space of the types of computational models considered in Sec-

tion 1.2 any applicable visualization technique should support it. To determine the relative

interest of the visualization community in multi-dimensional data, a survey of publications

of the three conferences of IEEE VisWeek 2008 has been assembled. A grouping by data

set type11 gives the following distribution of paper counts:

• Vis: scalar volumes (12), flow and tensor (8), general multi-field12 (6), segmented

data (2), surface models (15)

• InfoVis: Graphs/Trees (14), (multi-)relational tables (8), continuous/spatio-temporal (3)

• VAST: (Geo-)spatial (2), time stamped feature space trajectories (3), relational ta-

bles (5), integrating mixed types (6)

The first three categories of the visualization conference alone amount to over 26 publi-

cations that deal with data originating from continuous models. Most of the models are

computational and likely would allow for user adjustment of configuration parameters.

To provide quick pointers to further background discussions, related work specific to

certain design components of the paraglide system are embedded in Section 6.2. Work related

to the spectral palette design use case described in Section 1.2.4.2 is discussed in Section 5.1.

11A raw text form of this informal supplementary survey is available at http://www.cs.sfu.ca/~sbergner/personal/proj/thesis/dataset-survey.html

12See Section 1.4 for a definition.

http://www.cs.sfu.ca/~sbergner/personal/proj/thesis/dataset-survey.html

http://www.cs.sfu.ca/~sbergner/personal/proj/thesis/dataset-survey.html


2.4.2 Reconstruction and refinement of spatial data

Reconstruction takes a discrete data set as input and fits to it a continuous model that

can be queried at any position of its domain. Intermediate continuous representations are

an integral part of any volume rendering algorithm [HHS94]. For reasons mentioned at

end of Section 2.3.1.3 regular sampling lattices have been the method of choice for this

application [EDM09]. Due to the uniform neighbourhood structure, regular sampling is

also amenable to parallel processing, enabling fast reconstruction and rendering on the

GPU [EKE01a, KPH+03]. For efficient projection, also randomized arrangements of data

points have merit as can be seen in Monte-Carlo volume rendering [CSK03].

Reconstructing continuous density functions: Instead of reconstructing a continuous

function that fits the given data points, it is also possible to compute a continuous descrip-

tion of the density distribution of its range (histogram) or the combined domain×range

space. The projection to one- or two-dimensional subspaces for presentation on the screen

amounts to computing so-called marginal or dependent probabilities. The classical ap-

proach of kernel density estimation [Bis06, Ch. 2] is a distant relative to point based splat-

ting [Wes90, MMC99, HE03] with additive compositing, where spherical reconstruction

kernels are projected to image space prior to interpolation. Novotny and Hauser [NH06]

pre-process and bin a data set into 2D histograms for each pair of dimensions. This data

structure is useful for speeding up rendering of parallel coordinates [BBPD08], but could

also be used to render discretized scatterplots.

If input/output relationships are known : An alternative to value-based binning is

to use sample point adjacency information to project kernels for each cell rather than each

response value, as published by Bachtaler and Weiskopf [BW08] to reconstruct a continuous

density for display in a scatter plot13. Similar to Scheidegger et al. [SSD+08] they use

the co-area formula from level-set theory to obtain an asymptotically accurate histogram

computation for continuous domains.

Also exploiting an input-output relationship, Adams et al. [ABD10] provide a com-

putational representation for multi-dimensional continuous phenomena based on a regular

simplicial decomposition of the domain, where the vertices of each simplex lie on the permu-

tohedral lattice (dual of d-dimensional root lattice Ad [CS99]). Barycentric weights for any

13A related observation that I communicated to the second author in March 2007 is described in Sec-tion 6.2.2.3.


given point in space are used as multi-dimensional analysis and reconstruction filter. Since

the permutohedral lattice is a projection of the separable Zd+1 lattice, an efficient implemen-

tation of Gaussian filtering is possible that is used to perform bi-lateral filtering [TM98] and

non-local means de-noising [BCM05] on the combined domain×range space of a function

(e.g., a colour image). While the lattice spans the entire space Rd, most given functions will

only cover a fraction of it. The indices of this portion are stored in a hash table ignoring all

empty space. The approach requires a fixed choice of resolution that can not be adapted as

more samples are added to the representation.

2.4.3 Direct volume rendering

The following work is related to the use case of Section 1.2.4.1 and provides background to

Chapter 4.

The problem of how to properly evaluate the function under the rendering integral has

been debated since the beginnings of volume graphics. Wittenbrink et al. [WMG98] made

the observation that it is important to interpolate f in order to properly super-sample (gf),

while Younesy et al. [YMC06] pointed out that it is important to low-pass filter g in order

to sub-sample (g f). Recently, an experimental exploration of the rendering parameter

space has been conducted by Kronander et al. [KUMY10] that is more complete than what

is shown in Figure 1.2 on page 10, giving further directions for theoretical study. The large

number of factors to consider indicates that it would be useful to decompose them into

groups that can be studied independently.

To our knowledge, the work by Kraus [Kra03] and Schulze and Kraus [SKLE03] is the

only previous work that investigates the sampling of the volume rendering integral by means

of Fourier analysis and the sampling theorem. For the function models they use in their

derivations, the essential Nyquist rate of (g f) is πνgνf , where νg and νf are the maximum

frequencies in g and f , respectively. This statement is in accordance with a similar conjecture

by Engel et al. [EKE01b].

Related work in the field of signal processing considers properties of time-warping, which

can be interpreted as a composition of a warping function with a signal. However, works by

Clark et al. [CPL85], Azizi et al. [ACM02], and others in that field [BJ95] focus on invertible

(monotonous) warping functions. For that reason, their work can be used to gain insight

on the subject, yet their results are based on too restrictive assumptions in order to still be

applicable in our setting.


Adaptive sampling: The main benefit of understanding the required sampling rate for the

volume rendering integral is that the sampling rate can be adapted to the lowest possible

value in order to reduce the computational load. Various approaches to adaptive sampling

are known in the literature (see Glassner [Gla95, Ch. 9] for theoretical background). A sim-

ple example is empty space skipping, which identifies regions of vanishing contribution to the

integral and skips those regions [Kni00, KW03, LMK03, SK00, YS93]. Other methods flexi-

bly adapt the sampling rate to the requirements of volume rendering. For example, adaptive

sampling can be employed for hierarchical splatting [LH91], GPU ray casting [RGWE03],

or texture-based volume rendering [LHJ99]. Adaptive sampling is related to purpose (5) of

the experimental design discussion in Section 2.3.2.

2.4.4 Visualization systems for discrete multi-variate data

There is a variety of software systems with capabilities suitable for multi-dimensional data vi-

sualization, including MATLAB, Octave, Weka integrated into MATLAB via Spider, GGobi, Xmdv,

SciLab, Extrema, Fityk, and KNIME. Most related to our design is probably Chekanov’s [Che10]

jHepWork, which uses the versatility of Jython scripting to quickly prototype visualization

systems. However, all of the above systems are for general data analysis and only partially

geared to address the requirements of Table 1.1.

Workflows and operators: RapidMiner (formerly Yale) by Mierswa et al. [MWK+06] sup-

ports a flexible choice of methods for pre-processing and data analysis using visual program-

ming. Kietz et al. [KSBF10] point out that due to the large number of available operators in

RapidMiner and the size of a typical training and classification workflow, its construction and

validation can be an overwhelming task for a human user. Automatic planning techniques

need to elicit too much background information before making useful decisions for work-

flow construction, which leaves their proposed cooperative-interactive approach as a viable

alternative. Being limited to a pre-defined set of parameterized operators is too restrictive

for the dynamic development settings of the use cases of Section 1.2.

The use of generic building blocks to impose more structure to the data analysis process

is also pursued by the DimStiller system of Ingram et al. [IMI+10]. Their user population

analysis distinguishes experts and novices in either data analysis or the problem domain and

concludes that these different skills often do not sufficiently coincide. To address this gap

http://www.mathworks.com

http://www.gnu.org/software/octave/

http://www.cs.waikato.ac.nz/ml/weka/

http://www.kyb.tuebingen.mpg.de/bs/people/spider/

http://www.ggobi.org/

http://davis.wpi.edu/xmdv/

http://www.scilab.org/

http://exsitewebware.com/extrema/

http://fityk.nieto.pl/

http://www.knime.org/

http://rapid-i.com/content/view/10/32/lang,en/


for domain experts, pre-configured operators can be chained together to form new or pre-

configured workflows. The multi-field data viewing capacities of this pipeline architecture

are suitable for the purposes indicated in Section 1.2. However, the lack of a simulation

operator that can be dynamically enhanced at runtime (R1), the need for a user-interaction

to specify a multi-dimensional region in a potentially empty data subspace (R2), and missing

methods to produce a sampling pattern in this region (R2a), as well as missing support

for (R6+7), reveal that the main focus of the system is different from that of user-driven

experimentation, and justifies further research into these open directions.

The consequences of these framing requirements are discussed in Chapter 6. The in-

termediate chapters provide alternative approaches to sample parameter spaces with the

connecting theme laid out in Section 1.1.

Chapter 3

Sampling lattices with low-rate refinement

The goal of this chapter is the construction of point lattices that permit sub-sampling by

a dilation matrix that produces a similar lattice, rotated and scaled by a dimensionally

independent, low integer rate [BVBM09]. Before going through the relevant definitions,

Figure 3.1 gives an illustration of the rotationally nested scales of resolution of a design in

n = 2 dimensions.

Motivation to study lattices: The simple construction of a point lattice in Equation 2.9

as a linear transform of the Cartesian lattice Zn = Λ(In) provides an integer multi-index

k for each point. This means that a computational representation does not have to store

positions. The fundamental role of lattices in the context of function approximation becomes

apparent considering that the family of functions ei2πuTω : u ∈ Λ(G) completely spans

the space of periodic functions in L2(Rn) with periods in Λ⊥ = Λ(G−T ) (see [PM62, KAH05]

or Appendix A.2). This means that any function in this space has a unique expansion in

terms of a discrete sequence of coefficients for these basis functions1.

As drawn out in Figure 2.3 on page 33, lattices also give optimal or very good packing

radii in many different dimensions n. This means that those lattice point sets (or designs)

fulfill the maximin criterion of Johnson et al. [JMY90], who have shown that this maximizes

the volume of the posterior confidence ellipsoid of estimates of a Gaussian process with a

diagonal covariance matrix. This has practical applications for sparse initial sampling, where

measurements can be assumed to be near independent.

1Considering the statement at the beginning of Section 2.3.1.3, this means that the Fourier transform ofany periodic function is discrete.

43

CHAPTER 3. SAMPLING LATTICES WITH LOW-RATE REFINEMENT 44

G =

[0 −0.33071 −0.375

], K =

[2 −14 −1

], θ = arccos 1

2√

2≈ 69.3

−1.5 −1 −0.5 0 0.5 1 1.5−1.5

−1

−0.5

0

0.5

1

1.5

Figure 3.1: 2D lattice with basis vectors and subsampling as given by G and K in the diagram title.The spiral points correspond to a sequence of fractional subsamplings GKs for s = 0..1 with thenotable feature that for s = 1 one obtains a subset Λ(GK) (thick dots) of the original lattice sitesΛ(G) (small black dots). This repeats for any further integer power of K, each time reducing thesampling density by |det K| = 2.

3.1 Change of lattice basis and similarity

The definitions and properties of lattices that are useful in the discussion of this chapter are

given in Section 2.3.1.3 on page 27. In the following, a few useful Lemmas are proven that

are required to justify some aspects of our construction in Section 3.2. In particular, we

will need to construct different congruent bases and determine for two given bases whether

they generate congruent lattices. The key property enabling such a test is given by Propo-

sition 3.1. This property is stated without proof by Conway and Sloan [CS99, p. 10] and

papers on the subject of lattice construction [AEVZ02]. Hence, the following derivations

are included to contribute to a better understanding of lattice properties that will be used

later. The informed reader can skip right ahead to the construction section.


Lattice congruence and identity: A unimodular matrix is defined to have integer ele-

ments and unit determinant. This gives the following property.

Lemma 3.1. The inverse of a unimodular matrix is also unimodular.

Proof. The inverse of a matrix can be constructed from its adjugate (transposed co-factor

matrix) as T−1 = adj(T)/ det T. Since all co-factors2 of an integer matrix must be integer,

unimodularity of T carries over to T−1.

Possible generators for the group of unimodular matrices are discussed by Newman [New72,

pp. 23]. Our implementation, referred to as genUnimodular(n), uses a construction of

T = LU from several random integer lower and upper triangular matrices having ones on

their diagonal. Diagonal elements of −1 are also potentially interesting [EMV04], but this

increase of the construction space by a factor of 2n is omitted from the discussion in this

chapter.

Lemma 3.2. T is unimodular ⇔ Λ(T) = Tk : k ∈ Zn = Zn.

Proof. (⇒) Given that T is unimodular, it is obvious that Tk ∈ Zn for any k ∈ Zn. Further,

for any x ∈ Zn there is a k = T−1x ∈ Zn, because by Lemma 3.1 T−1 is an integer matrix.

(⇐) Since Λ(T) has the same geometry as Zn, we can conclude from Equation 2.12 that

|det T| = 1. To establish that T is also an integer matrix, note that Tk = x ∈ Zn must

hold for any k ∈ Zn. Picking a row vector of T and denoting it by tT , it is required that

tTk ∈ Z. Since k could be any unit vector ei, all elements ti of t have to be integer. This

must hold for all rows of T. Hence, T has to be unimodular.

Two lattices with generating matrices G1 and G2 that produce the same point sets Λ(G1) =

Λ(G2) are called identical. About the generating matrices of two identical lattices we can

now say the following.

Lemma 3.3. Two lattices Λ(G1) and Λ(G2) are identical, if and only if their generating

matrices are related by G1 = G2T for a unimodular T.

Proof. By construction in Equation 2.9, any lattice is a linear transform of Zn. Applying

G−12 to both lattices, the first sentence above follows from Lemma 3.2 with a unimodular

T = G−12 G1 = (G−1

1 G2)−1.

2A co-factor is (−1)i+j times the determinant of a matrix, whose ith row and jth column are omitted.


An immediate consequence of this is the following statement by Conway and Sloane [CS99,

p. 10, Eq. 23].

Proposition 3.1. The Gram matrices of two congruent lattices are related via A1 =

TTA2T for some unimodular T.

Proof. By Equation 2.10 and Lemma 3.3 we have A1 = GT1 G1 = TTGT

2 G2T = TTA2T.

The matrix T manipulates each column vector of G by adding integer multiples of other

column vectors. While not changing the position of the points in the lattice, this transfor-

mation does affect their integer coordinates (multi-indices).

Reduction: A good reason to search for a particular unimodular transformation T is to

obtain a reduced basis for a given lattice. Here, the objective is to choose equivalent basis

vectors such that the indices k of the Voronoi relevant neighbourhood (Equation 2.7 on

page 25) around the origin are contained within a small radius. This is a rough outline of the

different formal reduction criteria by Hermite, Minkowski, and others [AEVZ02, ZAM08].

Equivalence check: Even for triangular generating matrices scaled to unit determinant, a

check for equivalence of two generating matrices is an NP-complete problem [vEB81, GJ79].

An equivalent test can be done by extracting and comparing the Voronoi polytopes of two

lattices, which fall into the same complexity class.

Our method employs a simpler necessary test for equivalence by looking at the first few

elements of the set q(A) = kTAk : k ∈ Zn using the Gram matrix A of a reduced lattice

basis scaled to unit determinant. If the sorted lists q(A1) and q(A2) of two lattices disagree

in any element, G1 and G2 are not equivalent (also suggested by Newman [New72, p. 60]).

Equivalently, it is possible to restrict the list for this test to the Voronoi relevant indices

K = G−1N(0) of Equation 2.7 instead of all k ∈ Zn.

Dilation: Regular subsampling of a lattice can be expressed via a dilation matrix K ∈ Zn×n

to form a new lattice with generating matrix GK. The rate of reduction in sampling density

corresponds to

|det K| = αn = β ∈ Z+, (3.1)

which is the factor that scales the volume of the fundamental parallelepiped (Equation 2.12).

In the context of wavelet and multi-channel filter bank design Kovacevic and Vetterli [KV92]

define conditions for admissible sub-sampling matrices K:


• Λ(GK) ⊂ Λ(G),

• the eigenvalues λi of K must satisfy |λi| > 1, and

• ideally, all |λi| = α to preserve the directional geometry of the signal.

The last property ensures that that the subsampled signal is treated equally in all basis

directions. Only few iterations of sub-sampling should produce the same type of lattice that

the construction starts from, i.e. for a small number c of iterations one would like Kc = αcI

for αc ∈ Z.

Dyadic subsampling with K = 2I discards every second sample along each of the n

dimensions resulting in a reduction rate of β = 2n that grows exponentially in n. Entezari

et al. [EMV04] construct an admissible dilation with a smaller reduction rate for the body

centered cubic (BCC) lattice giving β = 4 and c = 3 in n = 3.

Van De Ville et al. [VDVBU05b] consider the 2D quincunx subsampling, which is an

interesting case permitting a two-channel dilation (β = 2) in n = 2. With the implicit

assumption of starting from the Cartesian lattice they show that dilations with β = 2 do

not exist for n > 2.

The main idea of the following construction is to relax the starting condition to include

general generators G ∈ Rn. It is then shown that a dimensionally independent low-rate

dilation becomes possible for any n. To allow for fine-grained scale progression we are

particularly interested in low subsampling rates, such as β = 2 or 3 that are not affected by

the dimensionality n. This means that point distances change by a factor as low as α = 21/n.

3.2 Construction of sampling lattices with low-rate rotational

dilation

We are looking for a non-singular lattice generating matrix G that, when sub-sampled by a

dilation matrix K with reduction rate β as in Equation 3.1, produces a similar lattice, that

is, it can be scaled and rotated by a matrix Q with QTQ = α2I. These are formal properties

behind the example given earlier in Figure 3.1. Algebraically, such an equivalence between

Λ(G) and the dilated Λ(GK) can be expressed as

QG = GK, (3.2)


leading to the observation that dilation K and scaled rotation Q are related by a similarity

transform

G−1QG = K. (3.3)

Suitable K for equivalent dilation: Using a matrix J2 =

[1 j

1 −j

]it is possible to

diagonalize a rotation matrix in R2×2 by the following similarity transform

1

αQ2 =

[cos θ − sin θ

sin θ cos θ

]= J−1

2

[ejθ 0

0 e−jθ

]J2 = J−1

2 ∆J2, (3.4)

where ∆ contains a pair of complex conjugate eigenvalues. Substituting this decomposition

for Q in Equation 3.2 leads to

K = αG−1J−1n ∆JnG, (3.5)

K = αP∆P−1, (3.6)

where the first equation gives the structure of K and the second one its eigendecomposition,

which can be computed. Putting this together allows to obtain lattice basis and rotation as

G = J−1n P−1

Q = αJ−1n ∆Jn.

(3.7)

Thus, given a matrix K that has an eigendecomposition corresponding to that of a uni-

formly scaled rotation matrix Q, we can compute the lattice generating matrix G as in

Equation 3.7. To allow for later optimization, it would be nice to be able to generate more

than one or, ideally, all possible solutions to this construction. The non-unicity of the eigen-

decomposition allows to insert an additional diagonal matrix S that scales the otherwise

unit eigenvectors in the columns of P

K = αG−1J−1n S∆S−1JnG (3.8)

again using Equation 3.6 gives a different G for the same rotation Q as

G = J−1n SP−1. (3.9)


The resulting G may be complex valued. Turning it into a real matrix by taking its real or

imaginary components does, however, not destroy the similarity relationship of Equation 3.3.

With G normalized to unit determinant this basis vector scaling introduces n−1 additional

degrees of freedom into the design. Below, we will refer to this construction as function

formGQ(K,S) using S = I by default.

3.2.1 Constructing rotational dilation matrices

The following proposition is key to the construction of integer K that facillitate the rota-

tional dilation of Equation 3.2.

Proposition 3.2. The similarity of K with a rotation matrix Q (Equation 3.3) and its

diagonalization α∆ (Equation 3.4) imposes restrictions on the coefficients ck of their shared

characteristic polynomial

dK(λ) = det(K− λI) =n∑k=0

ckλk, (3.10)

where dK(λ) = dQ(λ) = d∆(λ/α) with α as given in Equation 3.1 and β being a simple

product of primes.

For even n: the only non-zero integer coefficients are c0 = β, c2n/2 < 4β, cn = 1. This

leaves a finite number of options for cn/2.

For odd n: a single polynomial d(λ) is possible with non-zero coefficients c0 = −β, cn = 1.

The proof is given in Section 3.2.2.

Finding all possible integer K that fulfill Proposition 3.2 is a hard problem to which we

have not yet found a complete solution. Our initial approach used an exhaustive search,

which is possible for all integer elements within certain bounds. For n = 2 and β = 2 this

gave all three cases shown later in the results.

However, the following observation enables a direct construction for any n. The con-

straints in Proposition 3.2 prescribe monic polynomials with cn = 1. This makes it possible


to directly construct a suitable K via the companion matrix [TI97, p. 192]

K =

0 −c0

1 0 −c1

1 0...

. . .. . . −cn−2

1 −cn−1

. (3.11)

that fulfills Equation 3.3. We refer to the above procedure as function compoly(n, α, cn/2)

that returns a companion matrix (Equation 3.11) with a characteristic polynomial as in

Equation 3.15 or 3.18.

Obtaining alternative dilation matrices: With this starting point it is possible to

construct additional suitable dilation matrices via a similarity transform with a unimodular

matrix T

KT = TKT−1 = PT∆P−1T . (3.12)

By Lemma 3.1 KT remains an integer matrix. By Equation 3.5 and Equation 3.7 a call

to formGQ(KT ) leads to a new basis GT = GT that by Lemma 3.3 generates an identical

lattice. So, this method can not be used to search for new lattices, but allows to perform a

reduction of G as explained in Section 3.1.

Optimizing further non-equivalent solutions: The introduction of diagonal eigenvector

scaling S in Equation 3.7 as a second parameter to formGQ allows to construct non-equivalent

lattice bases G and introduces n−1 degrees of freedom into the construction. In this search

space it is possible to optimize further criteria, such as the ones discussed in Section 2.3.1.2

and Section 2.3.3 to select the “best” lattice.

3.2.2 Characteristic polynomial of a scaled rotation matrix in Rn

The similarity relationship between K and Q in Equation 3.2 implies that they share the

same characteristic polynomial d(λ) of Equation 3.10. First, note that the zero crossings of

d(λ) determine the eigenvalues d(λk) = 0 and determinant β = d(0) = c0 fixes the constant

of the polynomial [TI97, p. 184]. Further, since K is an integer matrix the polynomial

d(λ) ∈ Z[λ] has integer coefficients ck.

In order to find integer matrices K with the eigenvalues of a scaled rotation matrix,


it will be important to distinguish the two different forms of the diagonal matrix ∆ in

Equation 3.4, 3.5, and 3.8 for the case of even n

∆ = diag[ejθ1 e−jθ1 . . . ejθn/2 e−jθn/2 ]

and the case of odd n

∆ = diag[1 ejθ1 e−jθ1 . . . ejθ(n−1)/2 e−jθ(n−1)/2 ]

with analogue block-wise constructions to form Jn from J2 of Equation 3.4.

The following proof of Proposition 3.2 is due to a collaboration with Thierry Blu.

Proof. For even dimensionality n the characteristic polynomial of K and Q fulfills

d(λ) =

n/2∏k=1

(αejθk − λ)(αe−jθk − λ)

=

n/2∏k=1

(α2 − 2λα cos θk + λ2)

=

n/2∏k=1

[(α4

λ2− 2

α3

λcos θk + α2

)λ2

α2

]= d

(α2

λ

)(λ

α

)n.

(3.13)

Thus, if

d(λ) =n∑k=0

ckλk

=

n∑k=0

ck

(α2

λ

)k (λ

α

)n=

n∑k=0

cn−kαn−2kλk

⇔ ck = αn−2kcn−k = β1− 2kn cn−k.

(3.14)

If ck 6= 0 and ck, β ∈ Z then β1− 2kn ∈ Q. This is impossible for 0 < 2k < n, assuming small

values of β, such as 2, 3 or any simple product of primes. This implies that ck = cn−k = 0


for k = 1, 2, . . . n2 − 1. For k = n2 the ck can be non-zero leading to

d(λ) = λn + Cλn2 + αn (3.15)

with the requirement that C2 < 4αn so that the complex eigenvalues d(λk) = 0 are evenly

distributed on the complex circle of radius |λk| = α.

For odd dimensionality n the polynomial fulfills

d(λ) = (α− λ)

(n−1)/2∏k=1

(αejθk − λ)(αe−jθk − λ)

⇒ d(λ) = −(λ

α

)nd

(α2

λ

).

(3.16)

Thus, if

d(λ) =n∑k=0

ckλk

= −n∑k=0

ck

(α2

λ

)k (λ

α

)n= −

n∑k=0

cn−kαn−2kλk

⇔ ck = −αn−2kcn−k = −β1− 2kn cn−k.

(3.17)

By the same reasoning as for the even case, ck = 0 for all k = 1, 2, . . . n−12 results in only

one possible characteristic polynomial

d(λ) = λn − αn. (3.18)

3.2.3 Construction algorithm

The steps for constructing lattices with the desired subsampling matrices are summarized

in Algorithm 1. The function compoly(n, α,C) is defined in Section 3.2.1. A possible imple-

mentation for the function genUnimodular(n) is described in Section 3.2.1 and formGQ(K,S)

is defined below Equation 3.7.


Algorithm 1 genLattices(n, β,S)

1: LatticeList ← 2: Ks ← genKompan(n, β)3: for all K ∈ Ks do4: (G,Q)← formGQ(K,S)5: LatticeList← LatticeList∪(K,G,Q)6: end for7: return LatticeList

Algorithm 2 genKompan(n, β)

1: Ks = 2: if n is even then3: for all C ∈ Z : C2 < 4β do4: Ks ← Ks ∪ compoly(n, β

1n , C)

5: end for6: else n is odd7: Ks ← compoly(n, β

1n )

8: end if9: return Ks

It should be noted that the lists of lattices returned by genLattices for different S

may contain several equivalent copies of the same lattice. Here, we perform a reduction to

non-equivalent cases using the check developed in Section 3.1.

3.3 Further dimensions and subsampling ratios

For the case n = 2 we have created lattices permitting a reduction rate 2 in Figure 3.3 and

rate 3 in Figure 3.4, shown at the end of the chapter. In both cases, familiar examples arise

in the quincunx and the hex lattice for the ratios β = 2 and 3, respectively.

We performed the continuous optimization mentioned at the end of Section 3.2.1 for

n = 3 by changing S using an optimization based on numerical gradient estimation (MAT-

LAB’s fminunc). The goodness of a lattice was evaluated by constructing its Voronoi cell

using Equation 2.7 on page 25 and computing the dimensionless second order moment

(Equation 2.8) by analytically integrating a quadratic polynomial over the polyhedral do-

main using a method similar to the one described in Section A.1.3. For n = 3 and β = 2

all searches with this criterion converged to the same optimum lattice shown in Figure 3.2.

The dimensionless second order moment for the Voronoi Cell of this lattice is G = 0.081904.


−0.50

0.5−0.5

0

0.5−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

kissing # = 2, # f = 14, # v = 24, G(P) = 0.081904, # zones = 6

Figure 3.2: The best 3D lattice obtained for a design with dilation matrices having |det K| = 2. Theletters f and v in the title line indicate faces and vertices, respectively. The different colours encodethe different zones.

For comparison, the Cartesian cube has Gcc = 0.0833 and the truncated octahedron of the

BCC lattice has Gbcc = 0.0785.

The direct construction method that was developed in this chapter enables the extension

of the lattice design to larger n. A summary of results for n = 2 . . . 9 and β = 2, 3 is provided

in Figure 3.5.

Further directions: Our subsampling schemes have admissible properties mentioned in

Section 3.1 and allow for multidimensional wavelet construction [VDVBU05a]. Another di-

rection for investigation is the construction of non-Cartesian sparse grids that are applicable

in the context of high-dimensional integration and approximation adapting to smoothness

conditions of a given function space [Gri06].

Before going into these directions, however, a quality criterion that can directly be built

into the construction without the need for iterative optimization would be desirable. Also, a

complete enumeration of all non-equivalent K that obey Proposition 3.2 is an open problem.


quincunx, θ = 45 θ = arccos√

22 ≈ 69.3 θ = 135

G =

[ √2 0

0√2

], K =

[−1 −11 −1

]G =

[0 0.61

−0.93 −0.23

], K =

[−1 −12 0

]G =

[0 0.58

−1.22 0.41

], K =

[1 −13 −1

]

2 1 0 1 22

1.5

1

0.5

0

0.5

1

1.5

2

2 1 0 1 22

1.5

1

0.5

0

0.5

1

1.5

2

2 1 0 1 22

1.5

1

0.5

0

0.5

1

1.5

2

4 2 0 2 44

3

2

1

0

1

2

3

4

4 2 0 2 44

3

2

1

0

1

2

3

4

4 2 0 2 44

3

2

1

0

1

2

3

4

Figure 3.3: Three non-equivalent 2D lattices obtained for a design with dilation matrices having|det K| = 2. The lattice in the first column is the known quincunx sampling with a rotation ofθ = 45. The other two are new schemes with different rotation angles. The thick dots show thesample positions that are retained after subsampling by K. The second row shows the same latticeat twice the density, with more iteration levels of similarity transformed Voronoi cells.

0 50

1

2

3

4

5

1: R = [−0 0.93;1.1 −0.54] K = [−1 2;−2 1] θ=90

0 50

1

2

3

4

5

2: R = [0 0.84;−1.2 0] K = [1 −1;2 1] θ=54.74

0 50

1

2

3

4

5

3: R = [0 0.74;−1.3 0.22] K = [1 −1;3 0] θ=73.22

0 50

1

2

3

4

5

4: R = [0.66 0;1.1 −1.5] K = [−3 4;−3 3] θ=90

Figure 3.4: Four non-equivalent 2D lattices obtained for a design with dilation matrices having|det K| = 3. The lattice on the left is the well known hexagonal lattice with a θ = 30 rotation. Theother three are new schemes with different rotation angles.


1 2 3 4 5 6 7 8 9

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

Packing radii

ρ=δ1/n

Cartesiandensest known latticesupper bound r

n

β=2−optimizedβ=3−optimizedβ=2−companionβ=3−companion

n

Figure 3.5: Comparison of packing radii of best known packings, Cartesian packing, and the upperbound as in Figure 2.3 on page 33. In addition, some designs of this chapter are shown that enjoy thelow-rate rotational reduction property of Equation 3.2 for rates β ∈ 2, 3. If proceeding directly froma companion K as provided by Algorithm 2 unoptimized constructions can be generated instantlyfor any n. The optimized designs are obtained by maximization of the packing radius over choicesof S in Equation 3.9. The depicted results beat Cartesian packing in all cases except for n ∈ 1, 3.

Chapter 4

A sampling bound for composed functions

A fundamental problem in image synthesis is the evaluation of the rendering integral [PH04c].

In particular, volume rendering is based on a version that requires a mapping of given data

values f(x) to optical properties. By slightly abstracting the integrand within the volume

rendering integral, this mapping can be viewed as a composite function g(f(x)) = h(x),

where g is the mapping that assigns optical properties (i.e. opacities) to values of the data

f . In the context of volume graphics g is referred to as a transfer function. The actual input

to the rendering algorithm is the signal g(f(x)) = (g f)(x). Besides the chosen quadrature

formula for evaluating the integral, a crucial parameter determining the accuracy of the

numerical solution to the integral is the sampling distance. Since it is common to use linear

interpolation, it is important to use at least twice the Nyquist rate in order to guarantee an

accurate evaluation of the integral.

Despite the common use of this approach of sampling a composed data function it has

not yet undergone a satisfactory mathematical analysis. In particular, there were no clear

statements on how the mapped function is to be sampled appropriately. The only known

exception was the previous work [EKE01b, Kra03, SKLE03], which suggests that the proper

essential Nyquist rate of (gf) is proportional to the product of the respective Nyquist rates.

However, this estimate is too restricted for many data models and is mostly over-estimating,

as demonstrated in Figure 4.1. The knowledge of the proper sampling rate of the function

(g f)(x) will enable us to not only predict a proper error behaviour, but allows us to

accelerate rendering algorithms by skipping over regions that need less sampling in order

to guarantee a particular error behavior. While fast, high-quality solutions for quantized 8-

bit data exist in form of pre-integrated transfer functions, adequate sampling rates for high

57

CHAPTER 4. A SAMPLING BOUND FOR COMPOSED FUNCTIONS 58

−3 −2 −1 0 1 2 30

0.5

1

x

(a) Example data signal:f(x) = 0.5 + 9

20sin( 2π

4x) + 1

20sin(4πx)

0 0.2 0.4 0.6 0.8 1

0

0.5

1

y

(b) A transfer function: g(y) = 12(1− cos(2πy))

−3 −2 −1 0 1 2 3

0

0.5

1

x

(c) Sampling g(f(x)) at 4 times the boundingfrequency π

2νgωh

−3 −2 −1 0 1 2 3

0

0.5

1

x

(d) Sampling g(f(x)) at 4 times the boundingfrequency νg max |f ′|

Figure 4.1: Sampling comparison. The data y = f(x) (a) is composed with a transfer functiong(y) (b). Figures (c) and (d) show sinc-interpolated samplings of g(f(x)). The tighter boundingfrequency (d) suggested in this chapter results in 5 times fewer samples for these particular f andg, still truthfully representing the composite signal.

dynamic range volumes and multi-modal or multi-dimensional data, such as (f, |f ′|), are yet

unknown. Typical estimates are based on a proper sampling of f alone, which neglects the

effect of the transfer function. In the following we present an estimate for suitable sampling

that takes the effect of the transfer function into account.

For that purpose, we investigate [BMWM06] the effects of function composition in the

form g(f(x)) = h(x) by means of a spectral analysis of h.1 Building on related work

discussed in Section 2.4.3, in Section 4.1 we provide a rigorous mathematical treatment that

decomposes the spectral description of h into a scalar product of the spectral description of

g and a term that solely depends on f and that is independent of g. We then use the method

of stationary phase to derive the essential maximum frequency of g(f(x)) bounding the main

portion of the energy of its spectrum. This limit is the product of the maximum frequency of

g and the maximum derivative of f . This leads to a proper sampling of the composition h of

the two functions g and f . We apply our theoretical results to a fundamental open problem

in volume rendering—the proper sampling of the rendering integral after the application of

a transfer function. In particular, Section 4.2 demonstrates how the sampling criterion can

1Note that the order of f and h is exchanged from the abstraction in Section 1.4. Also, interactiveparameter space exploration does not apply in this chapter.


be incorporated in adaptive ray integration for direct volume rendering. We summarize our

contributions in Section 4.3 and give some directions for future exploration.

4.1 Frequency domain analysis

In the subsequent analysis, the data is represented by f(x), which typically maps from R3 to

Rr, with r being the number of modalities. Our transfer function g maps Rr to a scalar value

in R, which could be one channel of the optical properties, such as opacity. The composite

function is

h(x) = g(f(x)). (4.1)

Considering g(ν) to be the Fourier transform of g(y), h(x) results from the inverse transform

of g as

h(x) = g(f(x)) = (2π)−r/2∫Rr

g(ν)eiν ·f (x)drν. (4.2)

This is the inverse Fourier transform giving g(y) for y = f(x). The Fourier transform of

h(x) can be written as

h(ω) = (2π)−(r+3)/2

∫R3

∫Rr

g(ν)eiν ·f (x)drν e−iω·xd3x (4.3)

= (2π)−(r+3)/2

∫R3

∫Rr

g(ν)ei(ν ·f (x)−ω·x)drνd3x. (4.4)

Next, we want to switch the order of integration of the product measure drνd3x. This

is permitted [Bre02, p. 255], if we can ensure that the integrand is non-negative (Tonelli’s

Thm.) or is in L1 (Fubini’s Thm.), as defined in Equation A.3 on page 137. The latter holds,

due to the constant magnitude of the complex exponential, if g ∈ L1. This is not restricted

to, but includes g with bounded support B = ν ∈ Rr : ‖ν‖2 < νg having band-limit νg,

i.e. g(ν) = 0 for ν /∈ B and otherwise g(ν) < ∞. It is also possible to add a finite sum of

unbounded bkδ(ν − ak), where δ is the Dirac-delta function, for ak ∈ B and bk ∈ C without

affecting the membership of g in L1.

Switching the order of integration now yields

h(ω) = (2π)−(r+3)/2

∫Rr

g(ν)

∫R3

ei(ν ·f (x)−ω·x)d3xdrν. (4.5)


Noticing that the inner integral is independent of g, we use

K(ω,ν) =

∫R3

(2π)−(r+3)/2ei(ν ·f (x)−ω·x)d3x to write more concisely (4.6)

h(ω) =

∫Rr

g(ν)K(ω,ν)drν (4.7)

= 〈g∗(·), K(ω, ·)〉ν . (4.8)

This shows that forming the frequency spectrum h(ω) of the composition g f can be

interpreted as application of a linear operator with kernel K to the spectrum g(ν), which

can be implemented by means of a dot product with the complex conjugate of g for each

target frequency ω, as defined in Equation A.4. In the following, we will take a closer look

at the properties of the frequency transfer kernel K(ω,ν).

4.1.1 Visual inspection of the frequency transfer kernel K(ω,ν)

The kernel K(ω,ν) of Equation 4.6 solely depends on properties of f and is independent of

the mapping g. Further, in its role as a kernel of the linear operator used in Equation 4.8 it

can be interpreted as a map that determines how much of the energy of a certain frequency

component ν of g is mapped to a frequency ω in the target spectrum h.

To get an intuition for the properties of this function we will first inspect it visually.

Figure 4.2 shows K(ω, ν) for different one-dimensional scalar functions f(x), which is why

the bold notation is dropped at this point. In particular we have chosen a single Gaussian

function and a combination of two Gaussian functions. K(ω, ν) is computed as the discrete

Fourier transform of eiνf(x), which is a possible interpretation of a distretized Equation 4.6.

The picture has to be imagined periodically continued in ω (horizontally), which is due to

the discrete computation. In fact, the bright lines cutting off the corners in Figure 4.2d

are due to that effect. This does not apply to the continuous case that we deal with in the

subsequent analysis. Further, the function is point symmetric, or more precisely Hermitian,

since from Equation 4.6 it follows that K(ω, ν) = K∗(−ω,−ν).

A significant property apparent from Figure 4.2 is the low-valued cone in the middle,

starting narrow at ν = ω = 0 and increasing in size towards larger ω. According to

Equation 4.8 the spectrum of the composite function h(ω) is formed by the dot product

with the conjugate spectrum g∗(ν). In order to determine at which maximum wavenumber

ω = ωh the function h(ω) ceases to have a significant contribution, we have to figure out


0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1f 1(x)

x

(a) f1(x) = ϕ(1/2, 1/12, x)

!/"

#/"

−1 −0.5 0 0.5 1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

−7

−6

−5

−4

−3

−2

−1

0

(b) Frequency map K(ω, ν) of f1(thesholding magnitudes at e−7)

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

f 2(x)

x(c) f2(x) =ϕ(1/2, 1/12, x)+ϕ(1/4, 1/30, x)

!/"

#/"

−1 −0.5 0 0.5 1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

−7

−6

−5

−4

−3

−2

−1

0

(d) Frequency map K(ω, ν) of f2

Figure 4.2: The frequency map K(ω, ν) for a function f(x) determines how much a frequency ν ofg contributes to a frequency ω of the spectrum of the composed function g(f(x)). The examplesare (a) single and (c) mixed non-normalized Gaussians, using ϕ(µ, σ, x) = exp

(−(x− µ)2/(2σ2)

)and their corresponding K(ω, ν) in (b) and (d), respectively. The upper and lower slopes of thelow-valued cones (black) are given by the reciprocal of the maximum and the minimum values of f ′,respectively, as shown in Section 4.1.2.

at which ω the main spectrum of g(ν) is mostly contained inside the low-valued cone of

K(ω, ν), indicated in Figures 4.2(b+d), to produce a negligible contribution to the integral

of Equation 4.7. From this, a sampling rate for the reconstruction of h(ω) will follow.


4.1.2 Determining the boundary of the cone

In Equation 4.6 it is apparent that K(ω,ν) is an integral over an oscillating function eiu(x)

with unit magnitude and phase u(x) = ν · f(x) − ω · x. For the following analysis we will

restrict ourselves to the one-dimensional case. This is appropriate when performing the

analysis along a single ray. Further, we assume f(x) to be a scalar-valued function.

As an introductory example, consider a linear function f(x) = ax. This yields the

integral I = K(ω, ν) =∫∞−∞ e

i(aν−ω)xdx = δ(aν − ω). If the phase of the integrand is zero,

i.e. aν−ω = 0, the integral is infinite. However, if the phase changes at a constant non-zero

rate a, the integral is zero. This behavior is well-known as Dirac’s delta function δ.

For general functions f(x) it can be said that the integral in Equation 4.6 has significant

cancellations in intervals where the phase u(x) = νf(x)−ωx is changing rapidly. The largest

contributions occur where the phase of the integrand varies slowest, in particular where its

derivative u′(xs) = 0. An approximate solution for the integral can be obtained by only

considering the neighborhood around xs, which are the so-called points of stationary phase.

The previous statement only applies if the term u(x) for the phase can be split up into

the product of a large scalar and a function in the order O(1). To facilitate this split, we

change the parameterization of the integrand from K(ω, ν) to polar coordinates K(κ, θ).

Hence, the phase becomes u(x) = κ(f(x) sin θ − x cos θ). The points xs of stationary phase

are then given by

du

dx=

d

dxκ(f(x) sin θ − x cos θ) = 0 (4.9)

f ′(xs) sin θ − cos θ = 0 (4.10)

1

f ′(xs)= tan θ. (4.11)

In the following approximation we replace the integrand of Equation 4.6 by a second-order

Taylor expansion around each xs resulting in2

Ixs ∼∫ ∞−∞

eiκ(f(xs) sin θ−xs cos θ+ 12f ′′(xs)x2 sin θ)dx (4.12)

Ixs ∼ eiκ(f(xs) sin θ−xs cos θ)

(2π

κ|f ′′(xs) sin θ|

)1/2

eiπ4sgnf ′′(xs) sin θ. (4.13)

2We do not need to consider (x− xs), because we can substitute x with x = x′ + xs (and then rename x′

back to x).


For f ′′(xs) considerably different from zero the integrand vanishes quickly as (x − xs)2

increases. The full integral is approximately obtained by summing all Ixs for all xs fulfilling

Equation 4.11. This case is relevant for points min(f ′) < 1tan θ < max(f ′). Outside this range

we do not have any points of stationary phase and the overall integral forming K(κ, θ) is

close to zero.

This observation establishes the main insight of our analysis: the extremal slopes of f

form the boundary of the cone observed in Figure 4.2. Therefore, the primary result of this

chapter is that the composite function has an essential maximum frequency of

ωh = νg maxx|f ′(x)| , (4.14)

where νg is the maximum frequency of g. The corresponding sampling rate should be chosen

just above the essential Nyquist rate of 2ωh.

Rapid decay at the boundary edge:3 An interesting case arises if one considers the

boundaries of the essentially band-limiting interval. They form the boundaries of the cone.

To inspect the range around this band edge we define a critical angle θe fulfilling tan θe =

1/f ′(xe) and f ′′(xe) = 0 with xe being a maximum point of f ′(x). Here, the second derivative

vanishes, which requires a third-order Taylor approximation of u(x). In the vicinity of the

band edge for θ h θe the resulting integral is

K(κ, θ) ∼∫ ∞−∞

exp[iκ(f(xe) sin θ − xe cos θ + (f ′(xe) sin θ − cos θ)x+1

6f ′′′(xe)x

3 sin θ)]dx

(4.15)

substituting x = αx using α =(

2κf ′′′(xe) sin θ

)1/3

K(κ, θ) ∼= eiκ(f(xe) sin θ−xe cos θ)α ·∫ ∞−∞

exp

[i

(κα(f ′(xe) sin θ − cos θ)x+

x3

3

)]dx (4.16)

3The discussion in this section is due to a collaboration with David J Muraki.


considering eis + e−is = 2 cos(s)

K(κ, θ) ∼= 2πeiκ(f(xe) sin θ−xe cos θ)α · 1

π

∫ ∞0

cos

[ακ(f ′(xe) sin θ − cos θ)x+

x3

3

]dx (4.17)

= 2πei(f(xe)κ sin θ−xeκ cos θ)

(2

f ′′′(xe)κ sin θ

)1/3

·Ai

[κ(f ′(xe) sin θ − cos θ

)( 2

f ′′′(xe)κ sin θ

)1/3]

(4.18)

which has a solution involving the Airy function4 Ai, whose graph is shown in Figure 4.3.

-5

0.4

0.2

0

-0.2

-0.4

1050-10

t

Ai(t)

Figure 4.3: The graph of the Airy function Ai(t). It decays exponentially toward positive t withexp

(−(2/3)t3/2

). Also notice that its maximum occurs for negative t. The value for t = 0 in

Equation 4.18 is attained at the band edge θ = θe.

4.1.3 Error analysis

The result in Equation 4.18 gives us an idea of how K(ω, ν) behaves near the band edge5.

The first factor is a complex exponential that changes in phase with ω and ν and is fixed

in magnitude 2π. The second factor is decaying in O(ν−1/3). The rapid decay of the third

factor is indicated in Figure 4.3 toward increasing κ. Important to note is that the main

contributions from Ai(t), including its maximum, occur for t < 0. That means by choosing

our cutoff to be at tan θ = 1/max |f ′| we obtain an estimate for the band-limitedness of h.

Since the resulting spectrum h is in the general case not band-limited at all, but will still

4Ai is defined as Ai(t) = 1π

∫∞0

cos(tx+ x3

3)dx with Ai(0) = 0.355028 . . .

5For the interpretation recall that ν = κ sin θ and ω = κ cos θ.


have most of its energy concentrated below the cutoff, we refer to it as essentially limiting

frequency or to twice the limiting frequency as the essential Nyquist rate.

4.1.4 Limits of the model

Since the above derivation is based on approximations, it is important to be aware of the

limitations arising from the assumptions made to facilitate the analysis. The most important

one is that the method of stationary phase is only applicable if the phase is amplified by

a large constant. In our case this means that the derivation does not necessarily hold for

small κ. This is a reasonable assumption as long as we consider (ω, ν) not too close to (0, 0),

which is the case for ν near the band-limit νg of practical transfer functions g and max |f ′|considerably different from 0 for typical data.

4.1.5 Relationship to Carson’s rule

A previous analysis [Kra03, SKLE03] suggests that the maximum frequency to be expected

in transfer function composition is given by π2 ωf νg, multiplying the two band-limiting fre-

quencies of the data f and the transfer function g, respectively. Using Carson’s rule and

making the assumption that f is normalized to have values in the range [0, 1] (not required

in our derivation), the statement was derived that over 98% of the energy is preserved within

the cutoff frequency.

−3 −2 −1 0 1 2 3

0

0.5

1

x

(a) Sampling with 2πωf νg

−3 −2 −1 0 1 2 3

0

0.5

1

x

(b) Sampling with 4 max |f ′| νg

Figure 4.4: Same sampling rates are suggested by both estimates if a single sinusoidal signal is com-posed, using the lower frequency of the example in Figure 4.1a, with the mapping in Figure 4.1b.Both estimates have been 2× over-sampled, using a sampling frequency that is four times the re-spective limit frequency.

However, this previous discussion replaces the original function f by a single sinusoid of

the maximum frequency of f . At this point we believe that this leads to an over-estimate of

the required sampling rate. Our estimate leads to the maximum derivative of f as the key

factor determining the proper sampling frequency. Considering the example in Figure 4.1, for


a mixture of sinusoids where the higher frequencies contribute less to the overall amplitude,

our estimate produces a tighter sampling of the signal. In the case of a single sinusoid our

estimate suggests the same sampling distance as the conservative one as shown in Figure 4.4.

Here, both examples have been 2× over-sampled and interpolated using sinc interpolation.

4.2 Application to volume rendering

In the following we are investigating the implications of the above theory when applied in

volume visualization.

Adaptive sampling: A direct application is to use the maximum frequency of (g f) in

order to determine the sampling rate for the volume rendering integral. Here, the maximum

value of |f ′| is computed in the whole volume to calculate a fixed, overall sampling rate.

Unfortunately, a (possibly small) region of the data set containing the maximum of |f ′|would solely determine the sampling, even if the data set were slowly changing in other

parts. A better solution is adaptive sampling: the rate is chosen spatially varying to reflect

the local behavior of the data set.

The space-varying step size can be determined by identifying the maximum value of |f ′|in a small neighborhood around the current sampling point. In other words, the discussion

from Section 4.1 is applied only to a window of the full domain of f . The step size in this

window region is equal or greater than the step size for a global treatment. Therefore, we

typically obtain fewer sample points, without degrading the sampling quality. By working

within a given window our implementation actually is a space-frequency technique using

the results from a frequency analysis within a local neighborhood around a sample.

There are numerous previous papers on adaptive volume rendering, a few of which

are mentioned in Section 2.4.3. Most of the adaptive approaches need some kind of data

structure that controls the space-varying steps size. Our approach also follows this strategy.

The distinctive feature of our approach is not the fact that an adaptive step size is used,

but that we provide a mathematically based criterion for choosing the step size. In fact,

most of the existing adaptive rendering methods could be enriched by this criterion.

Our implementation consists of the following parts. First, a volume of gradient magni-

tudes is computed for the scalar data set. Second, the gradient-magnitude volume is filtered

using a rank-order filter that picks out the maximum in a given neighborhood around a

grid point. The size of the neighborhood is user-defined; its shape is a cube. The size of


the neighborhood is a 3D version of the ray-oriented window size that is used to derive

the step size criterion. By using the maximum gradient magnitude in a 3D neighborhood,

the isotropic step size is chosen conservatively in this neighborhood. The third step is the

actual volume rendering. We currently use a CPU ray caster that selects the sampling

distance at a point based on the filtered gradient-magnitude volume. The maximum step

size is clamped to the size of the neighborhood to avoid sampling artifacts that may arrive

through the construction of the gradient-magnitude volume. If the sampling rate were to

exceed a certain user-defined threshold (e.g., a hundred times the frequency of the data

grid), it will be artificially clamped to that threshold value to avoid excessive sampling.

Note that, for a fixed transfer function, steps one and two of the above pipeline are

pre-processing steps that do not have to be re-computed during rendering. To speed up the

change of transfer function, additional acceleration data structures should be considered.

For example, ideas for the efficient computation of space-leaping (see [KW03]) could be

explored.

(a) (b)

Figure 4.5: Examples of the hipiph data set sampled at a fixed rate (0.5) (a) and sampled withadaptive stepping (b). The adaptive method in (b) uses about 25% fewer samples than (a) onlymeasuring in areas of non-zero opacity to not account for effects of empty-space skipping. Thesimilarity of both images indicates that visual quality is preserved in the adaptive, reduced sampling.

Results of our technique are shown in Figure 4.5. Using adaptive sampling visually

similar results are obtained with 25% fewer samples than in the uniform stepping. Note

that adaptive sampling based on gradient magnitude automatically performs empty space

skipping if these regions are homogeneous in value with low gradient magnitude. To allow


for a fair comparison to uniform sampling stepping we only count samples taken in ranges

of the volume having non-zero opacity. That way the empty space in the images does not

skew the statistics.

For a more quantitative evaluation we have repeated several experiments (using different

sampling distances) for fixed and adaptive sampling and have compared the resulting image

against a ground-truth image, computed at a fixed sampling distance of 0.06125 relative

to a unit grid point spacing of 1. The result of the evaluation is shown in Figure 4.6.

The error plot is based on the signal to noise ratio (SNR), computed as SNR(x,y) =

10 log10

(‖x‖2‖x−y‖2

). We have conducted the image comparison with several different error

metrics, including numeric and perception-based ones. For all metrics the adaptive sampling

clearly outperforms uniform sampling in terms of the number of samples needed to achieve

a given image quality, just like in the SNR-based example given in Figure 4.6.

0 1 2 3 4x 106

0

10

20

30

40

50

60

70

80

90

# samples

Sign

al to

Noise

Rati

o

Quality vs. Performance

adaptiveuniform

Figure 4.6: Quality vs. performance, where quality is measured using signal to noise ratio (SNR)and performance is indicated by the number of samples taken along all rays cast into the volume.Adaptive sampling clearly outperforms the uniform (fixed) sampling. Only samples in areas of non-zero opacity are taken into account, i.e., both sampling schemes equally make use of empty spaceskipping.

For the illustration in Figure 4.7 the adaptive sampling is manually adjusted by multi-

plying a constant (amount of oversampling) onto the suggested sampling distances to match

the number of samples of the uniform sampling Figure 4.7a. This time the numerical error

of both images just matches (both at SNR of about 63). Still, the adaptive sampling shown

in Figure 4.7b has much less prominent artifacts near the surface transition. In case of noisy


(a) (b)

Figure 4.7: Visual comparison of two renditions of the tooth data set (CT scan) both using about thesame number of samples: (a) uniform sampling distance 1, (b) using adaptive sampling (2% fewersamples than (a)). The artifacts near the surface transitions are considerably less prominent in (b),which is also due to the fact that non-uniform sampling replaces structured aliasing with noise.

data, as it can be the case with acquired data, the gradient magnitude based adjustment of

the sampling distance might be less efficient than it would be for smooth data sets.

4.3 Discussion and outlook

This paper closes a gap in understanding and accurately estimating of the volume rendering

integral. Namely, it closes perhaps the most important theoretical gap still existing – the

proper sampling rate to be used during the rendering step. Hence, the main contribution

of this paper is an analysis of the frequency behavior after a transfer function has been

applied to spatial data. The resulting rule is that the essentially band-limiting frequency of

the composite function h(x) = g(f(x)) is given by

ωh = νg maxx

∣∣f ′(x)∣∣ . (4.19)

This is not a strict band-limit, but frequency components decay exponentially beyond ωh.

Because of the tight estimate of this limit, we suggest a slight oversampling. For most prac-

tically used interpolation methods twice the critical sampling rate (four times the limiting

frequency) should suffice.


The treatment in this paper is independent of the application and hence can be applied

in other fields of signal processing and applied mathematics. However, the focus of our

application has been rendering. In addition to the theoretical findings we have applied the

result to a method for adaptive sampling based on the maximum gradient magnitude. We

have been able to apply our theoretical results for an adaptive rendering algorithm that

achieves the same quality in the rendered images by reducing the number of samples needed

significantly.

Conclusion: The effects of composition can be interpreted in different ways. The above

analysis and discussion has considered the mapping of optical properties g(f(x)) on a multi-

field, multi-variate data function f . One can also consider f as a change in parameterization

of g. Both views amount to fundamental data processing operations and an interpretation

of our results beyond the realm of volume graphics seems worthwhile.

Chapter 5

Designing a palette for spectral lighting

In this chapter [BDM09], we employ a more complete spectral representation of

light in a computer graphics setting. This approach allows for more physically

correct renditions of a scene under different lighting conditions. Beyond that,

our prior work [BMDF02] has demonstrated that full spectra allow for additional

visual color effects, that conventional (red,green,blue)-based tri-chromatic rep-

resentations would fail to reproduce.

Giving designers the power to render spectral scenes, also faces them with an

increased challenge to construct them, since classical production pipelines are

typically based on the RGB model. To alleviate this problem this chapter de-

vises a tool to augment a palette of given lights and material reflectances with

additional constructed spectra that yield specified colors. This enables to specif-

ically construct spectral effects such as metamerism or objective color constancy.

We utilize this to emphasize or hide parts of a scene by matching or differen-

tiating colors under different illuminations. These color criteria are expressed

as a quadratic programming problem that can be solved for a globally optimal

solution and can include additional positivity constraints.

The proposed method has the potential to create expressive visualizations. A

key application of our technique is to use specific spectral lighting to scale the

visual complexity of a scene by controlling visibility of texture details in surface

graphics or material details in volume rendering.

71

CHAPTER 5. DESIGNING A PALETTE FOR SPECTRAL LIGHTING 72

Introduction: Light interacting with matter and its processing by the human visual sys-

tem is the basis of imaging. In graphics it is common to simply use RGB values for all

interactions, although it is well known that such a coarse approximation of full spectra can

lead to disastrous errors [JF99]. Hence, for both surface graphics, including transmissions,

as well as volume graphics, we should be using a good approximation of spectra.

Spectra in the context of this chapter will refer to both light as a spectral power distri-

bution (SPD) over different energy levels (wavelengths) of the electromagnetic field, as well

as wavelength dependent reflectance of the material. The latter arises from the fact that

a material reflects or re-emits different wavelengths of incoming light to different degrees.

Both light SPD and reflectance are modeled as scalar functions of wavelength and their basic

interaction can be described as a product of the two, resulting in a reflected spectrum (color

filtering). Additional effects involve shifts in emission of energy towards lower wavelengths

(fluorescence) or at later points in time (phosphorescence). However, in the following we will

consider color filtering alone, since this is the dominant effect if typical current RGB based

photo-realistic scenes are converted and enhanced within a spectral rendering framework.

The field known as physically-based rendering aims at producing photo-realistic im-

agery [PH04b]. Employing an appropriate algorithm that renders a convincingly realistic

impression in practice is only a part of the solution. First, we need a synthetic scene that

contains sufficient detail and has a close correspondence to a physically plausible setting. In

a static setting one can distinguish three aspects: geometric modeling of shapes or density

distribution of volumes, determining material appearance by setting reflection properties,

and configuring the lighting of the scene.

Making a scene: Discrete surface representations can either be acquired from direct sen-

sor measurements [LPC+00] or be extracted indirectly, for instance by depth reconstruc-

tion from multiple images [PKG99, BZS+07] or by matching shape models to individual

pictures [BAZT04, ERT05]. Alternatively, a scene can be modelled manually (e.g., using

Blender or 3D Studio max) or be created procedurally [DHL+98, KM07]. Since the creation

of synthetic geometry has already received significant attention in research, we will focus,

in the following, on the creation of synthetic materials.

Consequences of changing the light model: When switching the light model from RGB

(red, green, blue) components to a spectral representation, the geometric modeling remains

unaffected. Also, directional dependence of material shading properties as it is expressed by

http://www.blender.org/

http://usa.autodesk.com/3ds-max/


a bi-directional reflection distribution function (BRDF) can still be used. However, modeling

the wavelength dependence of the BRDF as well as the light requires a new tool replacing

the classical color picker.

While in the classical approach the reflectance or the light spectra are chosen separately,

what we often need to model is the result of their interaction. In other words, we would like

to input the resulting light-reflectance interaction as a constraint into our modeling system

and have the system choose proper materials and lights.

The design tool devised in this chapter is not limited to picking specific colors for cer-

tain light-material combinations. As well as that, it is aimed at taking advantage of certain

effects that are inherent to the use of spectra. They are based on the notion of metamers

— materials that look the same under one light, but may have clearly distinguishable color

under another. In typical color matching problems metamerism is increased to make mate-

rials match under different lights. Our goal is to ensure metamerism only for one light, but

to exploit the visual effect when the reflectances do not match if illuminated with a different

light source. This can be employed to control visibility of additional detail in a scene as will

be shown for a surface graphics example in § 5.4.3.

Another effect is that of a metameric black [WS82] a surface spectral reflectance function

that produces a zero RGB triple under a particular light. Under another light the surface

appears colored, not black. Highly (but not completely) transparent materials tend to

virtually disappear from a scene when they turn black. Another light that is reflected

brightly may bring their appearance back to the user’s attention. The question of how to

incorporate such behavior into the design of spectra is the subject of § 5.2.

The obverse of the situation is one in which a user controls the appearance of a scene

by changing the lighting. In a design involving metameric blacks as the difference between

reflectances those materials retain their color as the light changes — we refer to this as

objective color constancy. Then clearly, if one material is designed to change appearance, as

the lights change, while other materials stay the same, we have a means to guide the user’s

attention, which can be used for the exploration of data sets.

Sampling the visible spectrum from 400 nm to 700 nm in 10 nm steps results in a

31-component spectral representation. Instead of our usual 3-vector component-wise multi-

plication, we have an order of magnitude more computational work. In a raytracing scenario

where we may have billions of light-material interactions this will be very computationally


expensive. Similar computational costs arise for a raycasting setting with many volume sam-

ples taken along each ray. Hence, we require a much more compact representation, which

preserves the benefits of 31-component spectra under certain conditions. Fortunately, a rep-

resentation is available that accurately represents full spectra using about twice the number

of components of tri-stimulus representations [DF03] (5 to 7 basis coefficients here). More-

over, the new representation has the virtue that interactions can again be carried out using

a component-wise multiplication, which we call a spectral factor model: instead of sim-

ple component-wise multiplication of RGB, we substitute a novel simple component-wise

multiplication of basis coefficients.

The following section examines related work and properties of linear color models, con-

cluding in an efficient representation for spectra. Our design method is proposed in § 5.2.

There, different criteria are introduced and expressed in § 5.3 as a least-squares problem.

In § 5.4 an example scenario is described and the effects of the different criteria are ex-

plained. In addition, an application of the design framework to spectral volume rendering

is discussed. The contributions of the approach are summarized in § 5.5.2 by providing a

discussion of limitations and possible future directions. Supplementary material to this work

contains Matlab code to perform all design steps as well as a Java implementation. The

material may be obtained at http://www.cs.sfu.ca/gruvi/Projects/Spectral_Engine.

5.1 Related work

A number of spectral rendering frameworks have been compared by Devlin et al. [DCWP02],

indicating a lack of open source solutions which has recently been changed by PBRT [PH04b]

only requiring a minor source modification to allow for rendering with a spectral color

model. Spectral light models have found application in volume graphics before [NvdVS00,

BMDF02, ARC05, SKB+06]. In particular, Abdul-Rahman and Chen [ARC05] and Streng-

ert et al. [SKB+06] improved the accuracy of selective absorption by employing Kubelka-

Munk theory that has previously been used to improve realism of renditions of oil paint-

ings [GMN94]. For that, spectral BRDFs for layered paint are acquired by analyzing sim-

ulated micro-structure as a pre-processing step. This is a promising extension to analytic

models for interference colors or diffraction [Sta99, SFD00]. A generative model to produce

realistic impressions of human skin [DJ06] also considers cross-effects between directional

and energy dependence of reflectance. However, in the majority of appearance models the

http://www.cs.sfu.ca/gruvi/Projects/Spectral_Engine


terms for directional and energy dependence of the reflectivity can be considered indepen-

dently. Hence, in our design method we will concentrate on the wavelength dependence of

the appearance only.

5.1.1 Previous approaches to constructing spectra

In this work we consider a method to obtain spectra to equip a synthetic scene according

to certain appearance criteria. Spectral representations of light have their origin in more

accurate models of physical reality. Hence, if the required materials or lights are available,

a first choice would be to determine the reflectance or emission spectra by measuring them,

e.g., via a spectrometer [MPBM03]. Such measurements and imagery can be used to learn

about chemical constituents of a light emitting or reflecting object of inspection. For in-

stance, one can learn about the gas mixtures of stars via their emitting spectra, or one can

find out about different vegetation on the surface of our planet using knowledge of distinct

reflectances of plants and soils. When looking for sampled spectra online one typically finds

graphs, but not the underlying data.

There is also a history of work on estimating spectral information from color filter array

data, as used inside digital cameras or simply based on RGB images [DF92]. To resolve the

problem of increased dimensionality of the spectral space over the measured components

(usually three), assumptions are included about the illumination as well as the color filters.

Both of these may be controlled as described by Farrell et al. [FCSW99] who obtain spectral

approximations from digital photos of artwork.

For spectra already measured, one may also pick spectra from a database. Wang et

al. [WXS04] enhance this selection, by automatically choosing a neighborhood of 8 spectra

surrounding a point of user specified chromaticity. These are linearly interpolated to produce

an artificial spectrum close to physical spectra. Also, they use Bouguer’s law to create

additional spectra.

But a sufficiently large database may not be available to contain solutions satisfying all

design constraints. Also, only using real physical spectra might not be a requirement in a

computer graphics setting, which could instead also benefit from the creation of completely

artificial spectra. Since linear basis functions are a successful model to represent measured

spectra, they are the common choice to form a basis for modeling new spectra. Previous

choices include delta functions at distinct wavelengths [Gla89], boxes covering certain bands,

and exponential functions, such as Fourier or Gaussian [SFCD99]. These approaches produce


mixtures of three functions for red, green and blue (RGB). To obtain flatter spectra [Smi99]

also includes complementary colors (CMY).

All of the above methods to create artificial spectra have one common problem: they

consider the design of a reflectance for a given color without considering the illumination

spectrum. Such a design essentially only applies in controlling the appearance of self emitting

objects. But for typical materials it is only the reflected spectra that we can see. These are

related to the light spectrum, via a product with the wavelength dependent reflectance, but

they are not the same. Thus, the color of a surface should indeed be chosen with respect

to the reflected spectrum, but what really needs to be assigned in the scene description is a

reflectance and a light.

This observation is the main motivation for our design method. The second main dif-

ference over previous methods is that we consider a design of an entire palette of several

reflectances and lights instead of just single combination colors. This allows us to take ef-

fects of combined material appearance or lighting transitions into account. In the following,

we will provide some background on linear color models, leading to a choice of basis for

efficient component-wise illumination computations, called the spectral factor model. The

design method and its description in § 5.2, however, are independent of a particular choice

of linear model.

5.1.2 Linear light models

Linear representations for spectra of lights and reflectances are attractive for rendering com-

plex scenes for several reasons. Firstly, all illumination calculations can be performed in a

linear subspace of reduced dimensionality; and the basis can be specialized for a set of rep-

resentative spectra, thus improving accuracy. In general, each illumination computation in

the linear subspace implies a matrix multiplication. The following discussion will construct

this illumination matrix R.

The basic idea of using a linear model is to describe a spectrum C(λ) (e.g., a color

signal [WS82] formed from the product of light and surface spectral reflectance functions)

by a linear combination of a set of basis functions Bi weighted by coefficients ci:

C(λ) =

m∑i=1

ciBi(λ). (5.1)


The choice of basis functions can be guided by different criteria. Marimont and Wan-

dell [MW92] discuss different approaches to finding a basis that minimizes perceivable errors

in the sensor responses. Peercy [Pee93] devised a framework for using linear color models in

illumination calculations. The quality of a particular basis can be summarized as the trade-

off between accuracy and computational complexity. There are two different approaches

to this issue. We can form specialized bases tailored to the particular set of spectra in a

scene. Then these spectra have only minimal error when projected to the subspace spanned

by the linear model. Spectra differing from the prototype spectra may have larger error

from projection. Alternatively, general bases are suitable for a wider set of spectra. For

instance, using exponential functions, such as a Fourier basis, only assumes some degree of

smoothness of the modeled spectra. Certainly, all spectra that are smooth enough will be

well represented; however, a Fourier representation may have negative coefficients for valid

physical (i.e., nonnegative) spectra, making the model problematic to use in a hardware

implementation.

In order to computationally model interactions between spectral power distributions

(SPDs) we represent the power distribution C(λ) as a discrete vector. A full spectrum then

consists of equidistant point samples taken over the visible range from 400 nm to 700 nm

at 10 nm intervals forming a vector ~C ∈ R31. The basis in Eq. 5.1 then becomes a 31 ×mmatrix B comprised of the set of m basis vectors ~Bi and the coefficients ci become a vector

~c approximating Eq. 5.1 as

~C =m∑i=1

ci ~Bi = B~c. (5.2)

Modeling illumination we will restrict ourselves to non-fluorescent interactions — no

energy is shifted along different wavelengths of the spectrum. Hence, the observed reflected

color spectrum equals a component-wise product of the two SPDs. We will use diag(~S) as

a diagonal matrix composed of the elements of ~S to define a component-wise multiplication

operator ∗ between ~E and ~S:

~C = diag( ~E) ~S = diag(~S) ~E = ~E ∗ ~S = ~S ∗ ~E. (5.3)

The coefficients ~c for a spectrum ~C can be obtained via the pseudo-inverse B+ of B defined

in Equation B.13:

~c = B+ ~C. (5.4)


The spectra forming ~C can also be expressed in the linear subspace as ~S =m∑k=1

sk ~Bk and

similarly ~E. We combine the previous equations:

ci = ~B+

i (

m∑j=1

ej ~Bj ∗m∑k=1

sk ~Bk). (5.5)

The two operands inside the parentheses are ~E and ~S from Eq. 5.3. The result of the spectral

multiplication (in the full space R31) is then projected back onto the basis functions, as in

Eq. 5.4. The vector ~B+

i denotes the ith row of the inverted basis B+. By reordering, we

obtain

ci = ~B+

i

m∑j=1

m∑k=1

( ~Bj ∗ ~Bk) ejsk. (5.6)

This equation can be rewritten as a matrix multiplication, but one of the two coefficient

vectors has to be integrated into it. To do so, we define a new matrix R written in terms

of either ~s or ~e:

~c = R(~s)~e = R(~e)~s,

with Rij(~v) = ~B+

i

m∑k=1

( ~Bj ∗ ~Bk) vk (5.7)

The m × m matrix R carries out any reflectance computation inside the linear subspace.

Equation 5.7 also shows that an arbitrary choice of some basis does not necessarily lead to

a diagonalization of R. However, it is at least possible to use specialized hardware to apply

this matrix at every reflection event [PZB95]. Current commodity graphics hardware do

also allow for an implementation using the GPU.

In the following, we discuss how to modify the basis functions such that componentwise

multiplications alone, with diagonalization of R, will suffice for such computations.

5.1.3 Accuracy

One problem with using linear models is that only the spectra from which the basis functions

are derived are likely to be represented with good accuracy. For this we make use of principal

component analysis (PCA) for a set of given spectra, and the first m significant vectors are

taken to span an orthonormal linear subspace for the spectra. Other spectra, which have

not been considered during the construction of this basis may be very different from their


projections into that space.

Particularly in the case of fluorescent (spiky) lights or sharp cutoff bands, we should

make use of a dedicated, or ‘specialized’, basis. Each illumination step as described in

Eq. 5.7 makes use of a result projected back into the linear subspace and hence at every

interaction the linear representation may move farther from an accurate representation of

the product spectrum. This problem is especially relevant if we use multiple scattering or

spectral volume absorption. The highest accuracy is achieved when only very few illumina-

tion calculations are performed. In case of a local illumination model in combination with

‘flat’ absorption (alpha blending), only one scattering event is considered with no further

transmission events. Another technique especially appropriate for linear color models is

sub-surface scattering [HK93]. This method uses only very few reflections beneath a sur-

face. Yet the spectral absorption (participating medium) is important for realistic results, so

using spectra can greatly improve correctness; since there are only relatively few absorption

events the accuracy is still acceptable.

5.2 Designing spectra

The technique described in this section seeks to extend a scene containing real world re-

flectance and light spectra by creating additional artificial materials and lights that fulfill

certain criteria:

1. a constructed light/reflectance should produce user-chosen colors in combination with

given reflectances/lights;

2. spectra may also be represented in a lower dimensional linear subspace model for

which the approximation error should be minimal;

3. to regularize the solution we minimize the second order difference of the discrete

spectrum; this provides smoothness of the solution and improves convergence;

4. physically plausible spectra should be positive, which enters the optimization as a

lower bound constraint.

The first three of these points are expressed as linear least squares problems. This allows

us to weight and combine different criteria and to employ standard solution methods.


400 500 600 7000

0.2

0.40.6

0.8

400 500 600 7000

0.5

1

400 500 600 7000

0.2

0.4

400 500 600 7000

0.1

0.2

refl 1

refl 2

sodium hp D65 daylight

Figure 5.1: Spectral design of two material reflectances shown on the left of their representativerows. The colors formed under two different illumination spectra are shown in the squares in therespective columns where D65 (right column) produces a metameric appearance.

All settings involved in the design process are represented as a palette of spectra and

combination colors as shown in Fig. 5.1. The display uses columns for lights and rows

for reflectances. In the example the lights act as input to the design algorithm while the

reflectances were open for redesign. For any light-reflectance combination, the user may

define a desired color that should result from the illumination. It is displayed in the framed

sub-square of the color patch. Its surrounding area shows the color that the actual resulting

spectra produce in combination with each other. The design was successful if the desired

and the actual colors are similar enough. The appearance of ‘refl 2’ under the high-pressure

sodium lamp is dark brown instead of the desired gray, which is acceptable in this example.

5.3 Matrix formulation

It is possible to approach the design problem by solving a linear equation system for a

spectrum ~x

Qrgb,31diag( ~E)~x = ~c, (5.8)

where Qrgb,31 is the spectrum to RGB conversion matrix1. The solution ~x will be a re-

flectance producing the user-specified color tri-stimulus ~c under the given illumination

spectrum ~E. Further, one might ask for multiple lights ~Ek to produce colors ~ck with

1The matrix may be formed as Qrgb,31 = Qrgb,xyz ·Qxyz,31, where the rows of Qxyz,31 are the 3× 31 setof color matching functions in the CIE XYZ model [WS82] and Qrgb,xyz is a hardware (monitor) dependent3× 3 matrix to transform XYZ to RGB.


reflectance ~x. One can solve for this by vertically concatenating the illumination matrices

Qrgb,31diag( ~Ek) = Q( ~Ek)rgb,31 into a matrix M and their respective color forcing vectors ~ck into

a vector ~y. As there might not be a spectrum that fulfills all conditions exactly we switch

from solving an equation system to a quadratic minimization problem:

min~x||M~x− ~y|| = min~x

[~xTMTM~x− 2~yTM~x

], (5.9)

An unconstrained solution would be available via the pseudo-inverse M+ = (MTM)−1M

as ~x = M+~y. Alternatively, we use quadratic programming (QP), because it allows the

inclusion of lower and upper bound constraints for the components of ~x. Note that the

entire design could be carried out, such that ~x gives a light ~E instead of a reflectance ~S, by

replacing ~E with ~S in Eq. 5.8. The solution ~x would then contain a light ~E producing color

~c when illuminating the given reflectance ~S. This outlines the main idea behind the design

method. We will refine it in the following by adding more optional criteria, such as linear

subspace model error minimization and smoothness via minimal second order differences.

Finally, all criteria are weighted and combined by concatenating them to form M and ~y in

a single QP problem.

As shown in § 5.1.2, color computation can also be performed in the linear subspace. The

3×m matrix that takes an m-vector representation in basis B of Eq. 5.2 directly to RGB

is Qrgb,m = Qrgb,31 B. The least squares problem that minimizes the error when computing

illumination in the subspace is expressed as:

min~s‖Qrgb,mR(~e)~s− ~c‖, corresponds to

min~S‖Q( ~E)

rgb,m,31~S − ~c‖, with

Q( ~E)rgb,m,31 = Qrgb,m R(Qm,31

~E) Qm,31, (5.10)

where R(~e) is the matrix from Eq. 5.7 that performs the illumination calculation of light ~E

and surface ~S using their Rm subspace representations ~e = Qm,31~E or ~s in analog form.

From the previous discussion we can express color objectives for illumination in the

point sampled 31-dimensional spectral model and in the m-dimensional subspace model.

The following criterion seeks to minimize the difference between the resulting colors from


these two illumination methods to be close to a zero 3-vector ~03:

min~S‖F( ~E) ~S −~03‖, (5.11)

F( ~E) = Q( ~E)rgb,31 −Q

( ~E)rgb,m,31. (5.12)

The third criterion is smoothness. While the previous two criteria are aimed at accurate

color reproduction, this one is introduced to allow control over the general shape of the

spectrum and to provide regularization reducing the search space of the optimization. An

optimal solution for given design colors with minimum error can lead to spiky spectra with

large extrema. A commonly used indicator for roughness of a curve is the integral over

the squared second derivative or second order differences in our discretized model. Other

indicators are possible, but this one can easily be expressed in the following form:

min~S‖D ~S −~031‖, with

D = Toeplitz([ −1 2 −1 0 · · · 0 ]) , ~031 = zero 31-vector. (5.13)

D is a tri-diagonal matrix having 3-vector [−1, 2, −1] on the three middle diagonals and zero

everywhere else, which is also called a Toeplitz matrix. The whole matrix D is normalized

by 1/(√

31 ‖ − 1 2 − 1‖): the√

31 takes care of the number of rows of the matrix so as not

to make smoothness more important than the design color matrices. Those we normalize

by 1/√

3 in order to have comparable importance. This is relevant when the residues of all

above criteria are combined in a sum of squares, as we will discuss next.

5.3.1 Combined optimization function

Each of the design criteria is expressed as one of the matrices Q( ~X)rgb,31,Q

( ~X)rgb,m,31,F,D with

accompanying objective vectors (target colors or zero vectors). The design matrix M is

formed by vertically concatenating these criteria matrices. Similarly, the associated forcing

vectors are stacked to form ~y. The different criteria are weighted by ωij|F |D for design

colors ~cij , error matrix F, and smoothness D respectively. These weights provide control

over the convergence of the minimization and may all be set to 1. We compute a minimum

error solution for an overdetermined system Mi ~x = ~yi for a surface ~Si corresponding to the


set of stacked equations: ωi1Q( ~E1)rgb,31

ωFωi1F( ~E1)

ωDD

· ~x = ~y =

ωi1~ci1

~03

~031

, (5.14)

We solve this system for a minimum error solution using the form of Eq. 5.9. The solution

~x will contain the desired reflectance ~Si producing color ~ci1 with light ~E1. If there are

several colors that should be produced in combination with different lights ~Ej , the upper

two blocks are vertically repeated for each ~Ej , since the smoothness criterion D only needs

to be included once. In the following we will consider the simultaneous creation of several

spectra.

5.3.2 Free metamers

In the above formulation the design of one spectrum ~Si is independent of the other spectra

that are to be designed. However, it is possible to solve for all needed spectra simultane-

ously, by combining their individual design matrices Mi in a direct sum. This means to

concatenate the matrices diagonally and to fill the remaining elements with zeros ∅ in the

form M = [M1 ∅; ∅M2], where the semicolon denotes vertical concatenation. The respec-

tive forcing vectors ~yi are stacked as well and the solution vector ~x will contain several

spectra concatenated.

We will use this to include ’free’ colors into the design: we create two spectra ~Si, ~Sj

and instead of defining their desired color as part of ~y, we will leave the actual color open

and retrieve it as part of the solution in ~x. This is useful if we want those two reflectances

to look the same under a light ~Ea, but do not care what specific color they will actually

form as long as they are metameric. This can then be combined with further design colors

for a different light ~Eb. More formally, we solve for a weighted solution ~cia, ~Si, ~Sj of the

system

Q( ~Ea)rgb,31

~Si = Q( ~Ea)rgb,31

~Sj ,

Q( ~Eb)rgb,31

~Si = ~cib

(5.15)

using an illumination matrix as defined after Eq. 5.8 and with color ~cib given for a surface

~Si under light ~Eb: i.e., we ask this surface ~Si under another light ~Ea to have the same color


as surface ~Sj under that light. As a stacked matrix Eq. 5.15 becomes

−ωia 0 0 ~031

0 −ωia 0 ωiaQ( ~Ea)rgb,31

~031

0 0 −ωia ~031

−ωja 0 0 ~031

0 −ωja 0 ~031 ωjaQ( ~Ea)rgb,31

0 0 −ωja ~031

0 0 0 ~031

0 0 0 ωibQ( ~Eb)rgb,31

~031

0 0 0 ~031

· ~x = ~y =

~03

~03

ωib~cib

. (5.16)

The involved weights ω can be changed from the default value 1 to steer the importance

of this color criterion over others. The resulting ~x contains the ‘free’ color ~cia = ~cja in the

first three components, and after that two 31-component vectors for reflectances ~Si and ~Sj .

This setup becomes interesting when used with upper and lower bounds on ~x, because then

the ‘free’ color can be forced into a given interval without being specified precisely. The

blue metameric color under light D65 in Fig. 5.1 was obtained using this method.

5.4 Evaluation and visual results

In the following, we will demonstrate the use of our spectral design method in several

contexts. We will start with a palette design, followed by an error evaluation for different

design conditions. Beyond considering palettes by themselves we will also show their use

in practical spectral rendering examples in 3D surface graphics and volume rendering. The

set of Matlab scripts for the design method and example setups of this paper along with a

Java version of the spectrum generator are available as supplementary material at the URL

given at the end of the introduction.


5.4.1 Example palette design

An example palette design is shown in Fig. 5.2. The target colours (shown in the framed

sub-squares) under the light in the third column were taken from a palette of the Color

Brewer tool [HB03]. A light fulfilling these colours is generated with our method and is

denoted SplitLight 2. The center column colours are chosen to visually merge each of the

two red tones in column 3 and separate the two blue tones, replacing them by their average.

These colours result when switching to designed illuminant SplitLight 1. In contrast, the

first column, illuminated by measured standard daylight D65, is set to a single metameric

average colour for all five reflectances. The spectra of the two artificial ‘split light’ sources are

initially chosen as shifted positive sine waves of different period lengths (60nm and 85nm).

For the setup the smoothness weight ωD was set to 1 and subspace error minimization was

omitted. All light sources are initially normalized to a luminance Y = 6 and the reflectances

are allowed to have magnitudes in [0, 1]. This scaling factor was determined experimentally

through a preliminary unbounded design.

In a first design phase we create reflectances fulfilling the given colours. Here, we choose

importance weights ωi,1 = 4 for the first column, leaving the remaining weights at 1. This

gives correct metamers under daylight D65 and gets the remaining palette colours approxi-

mately right. In a second phase we use the newly created reflectances to re-create the two

‘split lights’ to produce the given colours more exactly. The resulting spectra are shown in

the graphs in Fig. 5.2.

5.4.2 Design error with respect to number of constraints

Expressing the design criteria as soft constraints allows us to always obtain a solution, but

possibly with errors depending on how well the criteria agree with each other. In order to

obtain a better understanding of these errors we performed a number of automated tests

on palettes of varying sizes. For each of the tests (with errors displayed in Fig. 5.3), we are

requesting fixed random combination colours between each reflectance and light (uniformly

distributed in RGB space). Lights and reflectance are formed by first creating reflectances

for fixed random lights and then re-computing lights for the new reflectances. The bottom

left and right axes of each graph in Fig. 5.3 indicate the numbers of reflectances and lights,

respectively. The vertical axis denotes the average L∗a∗b∗ distance between designed and

actual light-reflectance combination colours over the entire palette of a given size. While


400 500 600 7000

0.5

1

400 500 600 7000

0.1

400 500 600 7000

0.5

400 500 600 7000

0.2

400 500 600 7000

0.5

400 500 600 7000

0.5

Daylight D65

400 500 600 7000

0.5

1

SplitLight1

400 500 600 7000

0.5

1

SplitLight2

d65 = [0.4702 0.5197 0.5304 0.4924 0.5958 0.6645 0.669 0.6525 0.6582 0.6179 0.6213 0.6122 0.5952 0.6117 0.5929

0.5906 0.5679 0.5469 0.5441 0.5038 0.5111 0.5089 0.4981 0.4731 0.4754 0.4543 0.4555 0.4674 0.4447 0.3958 0.4066];

split1 = [0.6678 1.102 1.107 0.6561 0.1297 0.043 0.4404 0.8899 0.9883 0.7145 0.273 0 0.2365 1.07 1.065

0.7561 0.292 0.0444 0.1839 0.858 1.136 0.8329 0.288 0.05081 0.2354 0.5776 0.8495 0.959 0.8936 0.6813 0.3666];

split2 = [0.5569 0.9856 1.172 1.064 0.7426 0.3406 0.03923 0 0.2016 0.6381 1.012 1.013 0.8544 0.8999 0.3158

0 0.01159 0.3446 0.6981 1.119 1.049 0.8205 0.5588 0.2338 0 0.07294 0.4685 0.8727 1.039 0.9004 0.5156];

refl1 = [0 0 0 0 0 0.1908 0.337 0.08154 0 0 0 0 0 0 0

0.1881 0.5014 0.1729 0 0 0.005627 0.1097 0 0 0 0.09737 0.524 0.8975 1 0.8346 0.4686];

refl2 = [0 0.006213 0.03533 0.0938 0.155 0.1646 0.1112 0.04728 0.0334 0.0771 0.1367 0.1579 0.1164 0.04379 0.005049

0.03718 0.1136 0.1759 0.1892 0.1624 0.1227 0.08678 0.05953 0.04195 0.03305 0.02917 0.02617 0.02203 0.01681

0.01118 0.005552];

refl3 = [0.5278 0.7863 0.6334 0.2164 0.02129 0.01284 0.07832 0.09049 0 0 0 0 0.1197 0.5766 0

0 0 0 0 0.4177 0.1081 0 0 0 0.3824 0.3816 0.03062 0 0 0 0];

refl4 = [0.158 0.2753 0.3143 0.2531 0.1204 0.01347 0.01566 0.09516 0.1335 0.08491 0.02991 0.09995 0.2566 0.2573 0.07249

0 0 0 0.141 0.1585 0.000607 0.001365 0.2275 0.3689 0.2555 0.0623 0 0 0.02455 0.04152 0.03065];

refl5 = [0.1036 0.1798 0.2026 0.1579 0.07112 0.01515 0.04265 0.134 0.2162 0.2319 0.1768 0.1033 0.07003 0.08917 0.1286

0.1434 0.1117 0.04218 0 0 0 0 0.07356 0.3759 0.5742 0.5092 0.2921 0.09508 0.003234 0 0];

Figure 5.2: The reflectance spectra on the left of each row are designed to be metameric underdaylight (colours column 1) and to gradually split off into 3 and 5 distinguishable colours under twoartificial ‘split light’ sources. The resulting reflectance spectra are given below the figure.


a)0

510

15

0

5

10

150

10

20

30

# ligths

Avg. Lab error for random color palettes of different sizeusing 31−component positive spectral model

# refls

ΔLa

b

b)0

510

15

0

5

10

150

10

20

30

# ligths

Avg. Lab error for random color palettes of different sizeusing 100−component positive spectral model

# refls

ΔLa

b

c)0

510

15

0

5

10

150

10

20

30

# ligths

Avg. Lab error for random color palettes of different sizeusing 31−component unbounded spectral model

# refls

ΔLa

b

Figure 5.3: Each graph shows the average L∗a∗b∗ error in the design process for palettes of given sizes,constraining all light-reflectance combination colours for several palettes of different sizes. Changingspectral models and constraints results in different design error: a) the positivity constrained 31Dmodel, b) the positivity constrained 100D colour model, c) 31D without positivity constraint.

an error ∆Lab < 3 is visually indistinguishable, we found that errors up to 10 are still

acceptable. The 31-dimensional positive spectral model of Fig. 5.3a lies in this acceptable

error range for palette sizes of up to 7×7. Each RGB combination colour adds 3 constraints,

which for 10 colours are matched by the degrees of freedom of a 31-dimensional spectrum

to be designed. Thus, without positivity constraint an error is expected to occur after a

palette size of 10× 10 as observable in Fig. 5.3c.

Methods to reduce the error are to increase the dimensionality of the colour model

(Fig. 5.3b) or to drop the positivity constraint (Fig. 5.3c). The drastic reduction in error

shows that positivity imposes a major restriction on the design process. Reducing the

weight of the smoothness term ωD has a similar error decreasing effect as increasing the


dimensionality of the colour model since both are different ways to regularize the solution.2

For our experiments we have kept a fixed ωD = 0.001.

a)

0 50 100 150 200 2500

50

100

150

200

250

desired colour distances

actu

al c

olou

r dis

tanc

es (a

fter d

esig

n)

Lab distance preservation, 10x10 palette, 31D spectra, unconstrained

b)0 50 100 150 200 250

0

50

100

150

200

250

desired colour distancesac

tual

col

our d

ista

nces

(afte

r des

ign)

Lab distance preservation, 10x10 palette, 31D spectra, positive

c)

0 50 100 150 200 2500

50

100

150

200

250

desired colour distances

actu

al c

olou

r dis

tanc

es (a

fter d

esig

n)

Lab distance preservation, 10x10 palette, 100D spectra, positive

Figure 5.4: Preservation of colour distances for a 10 × 10 palette size. Each point in the graphsrepresents a certain pair of colour entries in the palette. Its position along the horizontal axisindicates the L∗a∗b∗ distance between the desired colours and the vertical position indicates distanceof the resulting colour pair after the design. A position close to the diagonal indicates how well thedistance within a pair was preserved in the design. a) 31-D spectra, unconstrained; b) and c)positivity constrained spectra with 31 and 100 dimensions, respectively.

2See the end of Appendix A.2 for a possible connection.


Preserving distance between colours: When setting up several colours a designer some-

times need not closely specify just what colour is actually produced, but rather that the

colours of two objects should be very similar or notably different. This idea is the motivation

for our second type of evaluation. Here, we do not look at the preservation of the actual

colour in the design, but rather the distances between them. In an N ×M palette setup,

we consider each of the 12(N ×M)2 colour pairs, excluding duplicate pairings.

In particular, we want to see how well the (perceptual) distance between the desired

colours matches the (perceptual) distance between the actual colours produced by the de-

signed spectra. Similar to the previous analysis, the evaluation in Fig. 5.4 shows that posi-

tivity is a rather strong constraint, but that increasing the dimensionality of the underlying

spectral model can be used to compensate for it.

5.4.3 Spectral surface graphics

a) b)

Figure 5.5: Car model rendered with PBRT. (a) The spectral materials used in the texture aremetameric under daylight D65, resulting in a monochrome appearance. (b) Changing the illumina-tion spectrum to that of a high pressure sodium lamp, as used in street lighting, breaks apart themetamerism and reveals additional visual information.

To implement spectral raytracing we have used the physically based rendering toolkit

(PBRT) [PH04b]. Its modular design allows us to easily extend the colour model to a 31

component linear model with the appropriate colour space transformation matrices. Also,

we have added support to load spectral textures, and a Python script to replace RGB values

by linear combinations of red, green, and blue spectra to facilitate rendering of conventional

RGB scenes with the modified spectral renderer. Fig. 5.5 shows two impressions of the same

scene under different illumination. Lighting is provided by an area light source emitting one

of the two illumination spectra of Fig. 5.1. The spectral texture for the flame job on the

side of the car uses the reflectances from the same palette. Thus, the daylight metamerism


makes the texture disappear, while the nighttime sodium street lamp illumination breaks

apart the metamerism and makes texture details appear. Since here we needed the metamers

to match exactly, we set their importance weight to 10, while leaving the weighting for the

colours under the sodium lamp at 1.

5.4.4 Rendering volumes interactively

To further illustrate the efficacy of the design method and the palette tool, we consider a

volumetric data set that contains an engine block. The data is represented by a volumetric

grid of scalar density values. Optical properties, such as opacity and spectral reflectance are

assigned to different densities via a spectral transfer function. For each data set a specific

palette is produced that contains light spectra and reflectances that are assigned to distinct

parts of the data (ranges of density values) via the transfer function. The volume is rendered

via a post-illumination step: the images are first rendered with a flat white light, or rather,

lighting having all-unity basis coefficients. Then lighting is changed. The actual raycasting

is performed once for a given viewpoint and all subsequent images for changing light can

be computed in real time by simply multiplying the reflected coefficients for spectra from

flat white illumination recorded in an image pixel by the new light’s basis coefficients. See

Section 6.3.1 for further details about how to influence a light mixture, which provides an

application case for an alternative multi-dimensional cursor control widget.

(a) (b) (c)

Figure 5.6: Engine block rendered using metamers and colour constancy. The three images in thefigure are re-illuminated without repeating the raycasting.

By assigning distinct reflectances we are able to separate the appearance of materials.

However, we also have metameric materials that under a particular light source result in


colours that are identical for our visual sensors. Hence, we have the ability to merge certain

materials and therefore guide the user’s attention towards or away from these materials.

Fig. 5.6 illustrates this effect. Under light 1 (Fig. 5.6a) both materials of the block look

the same. The entire engine is perceived as a homogeneous structure. Blending to light

2 (Fig. 5.6b), the metamer effect breaks down and the materials become distinguishable.

These details emerge gradually as part of a user interaction that will be described later in

Section 6.3.1 on page 109.

Another principle of our spectral material design is to make use of the effect of colour

constancy. In Fig. 5.6 we have created light sources, in connection with materials, that leave

the colour of one material the same and change only the colour of the materials we intend

to influence with that light — the engine block keeps it’s green colour under both lights.

Since traditional colour theory usually describes only phenomena that occur in the real

world, this restricts us to non-artificial metameric materials. However, it would be desirable

to design materials that disappear entirely from the image, if so desired by the user. In

colour science, these materials are called metameric blacks [WS82], i.e., materials with zero

RGB under some specific set of lights. Such a material is assigned to the surrounding

reconstruction noise in Fig. 5.6. This ‘smoke’ becomes visible as we blend over to light 3 in

Fig. 5.6c. In the other images it is rendered as well, but under the first two lights it remains

black. In rendering, alpha compositing still takes place, so occlusion of other materials is in

fact not undone. Nevertheless, it is possible to introduce materials that only scatter light

but do not actually have an alpha value to diminish light that comes from the back. If those

X-ray materials turn black under a certain light they entirely disappear from the image sum

and free the view to things that they obscure.

5.5 Discussion and conclusions

In computer graphics the step from RGB to full spectral colour models for illumination

calculations is typically made to increase realism of the renditions, since the increased di-

mensionality of a spectral light and reflectance model allows for more accurate illumination

computations. At the same time it leaves the creator of a spectral scene with the tasks of

choosing the additional degrees of freedom to improve appearance to the best effect, where

the highest level of realism would of course be attained using real world measurements.

Alternativley, it becomes possible to make adjustments in order to attain new effects that


are specific to spectra. Since the colour appearance of a material is dependent on the illu-

mination and typically differs under varying illumination, the key idea pursued here was to

conduct a combined design for pairs of reflectances and lights.

The example palette in Fig. 5.1 showed a simple setup, using metamerism under one light,

that breaks apart into distinguishable colours under a different light. This palette is used

in a spectral texture of Fig. 5.5 to illustrate how non-/metamerism under specific lighting

can be used to reveal texture details selectively via lighting change. A possible extension to

such a setup would be a dynamic scene in which the user introduces the metamer-breaking

light via a flashlight. In that case, the flashlight would become a metaphor for revealing

additional detail in the scene.

The same idea of merging and splitting metamers may be applied hierarchically to a

palette, in Fig. 5.2. The idea here is that under one illumination (daylight) the entire

scene has a homogeneous appearance. Under additional ‘split’ lights there are 3 and then

5 classes distinguishable leading to different levels of discernability, e.g., in a rendition of a

map. Using this in a spectral texture similar to Fig. 5.5 forms a new way to scale visual

complexity by selectively introducing additional texture detail.

Also, controlling the lighting in the spectrum design addresses a problem pointed out by

Smits [Smi99] that some colours can only be modeled with physically implausible reflectances

that have magnitudes larger than 1. However, when considering colour as perceived from

an image one does not deal with reflectances, but rather reflected spectra, which involve

the multiplication of a light source spectrum. By considering colours for light-reflectance

combinations, the [0,1] bounds for reflectances can be maintained by scaling up the intensity

of the light sources. One way to obtain a suitable scaling is to initially produce reflectances

without upper bound. If their maximum magnitude is above 1, we may scale up all lights

by that magnitude value and repeat the design with a forced upper bound of 1 for the

reflectances.

The design errors discussed in § 5.4.2 were shown to depend on the number of constraints

and also on enforcement of positivity. In response to this situation, the user has to make

a decision. Firstly, the user could adjust the importance weights to drive the compromise

of conflicting criteria, e.g., favoring the colour combinations that have the highest visible

errors or that matter most to the final result. This in particular applies to the metamers,

that are supposed to look exactly the same. To achieve this we set the importance of the

metameric colour patches 4 or 10 times higher than that of the remaining ones. Another


solution are ‘free’ colours, since these allow us to choose a ‘natural’ metameric colour that

is easiest to fulfill in concert with the other given colour conditions.

As an additional option, in case precise colour fulfillment is very important, the positivity

constraint could be dropped to allow spectra with negative components. As a result, the

optimization may more likely fulfill the given criteria without any error (see Fig. 5.3c). The

same figure also indicates that choosing an even higher dimensional spectral model is yet

another possibility for satisfying more constraints.

This kind of iterative design process of adjusting criteria to what is possible has not

been performed in our automatic evaluation, but is feasible in practice. An interactive

‘live’ update feature of our Java implementation was found to be helpful in this regard. It

performs the re-design (constructing and solving the criteria matrix) in a background thread

whenever changes are made. This immediate feedback allows the user to adjust conflicting

colours or constraints the moment they are introduced.

5.5.1 Future directions

The design has so far only considered colours of single illuminants combined with single

reflectances. In rendering practice, one may also be interested in mixtures of different

illuminants as well as materials of combined reflectances. While the light source mixture

would be linear due to additive superposition of the electromagnetic field, the same may

not hold for reflectances of mixed materials.

More advanced illumination models could be taken into account. For instance a bi-

reflectivity matrix relating illuminating wavelengths to re-emitted wavelengths [DCWP02]

could be used to model fluorescence. In fact, one could readily apply such a matrix, replacing

diag( ~E) in Eq. 5.8 and design a light ~E to produce a given colour on a fluorescent surface.

A feature similar to the free colour approach that creates two or more colours that are

as distinct as possible rather than equal could be useful. In that case, it might be helpful

to switch to a more perceptually uniform colour space. A linear approximation to CIELAB

[SHT97] could be of use here. Work in such a direction should also consider perceptual

issues of palette creation [HB03].

Just like physical simulation can replace manual geometrical modeling, e.g., for cloth or

water surfaces employing spectral rendering could achieve the same for lighting and colour

appearance. That is, with sets of materials designed to produce a well balanced and distinct

appearance under various lights. Instead of changing the palette to create a different mood


the change would be implicit under a different light.

With a proper paint mixture model one could look for feasible spectra that could be

mixed from paints or in combination with other base materials, such as metals or textiles

or even skin as in the case of face masks and make-up.

All our examples are given in the context of image synthesis. But a method to construct

spectra might also be used for reconstruction of spectra in a computer vision setting, e.g.,

using multiple images of the same scene taken under different colour filters or different

controlled illumination. Here, each colour filter would give rise to a distinct spectrum to

camera RGB transformation matrix that takes the effect of the filter into account via a

pre-multiplied diagonal matrix containing the filter absorbance spectrum. The optimization

of the design method could then reconstruct a spectrum that simultaneously fulfills the

different recorded RGB values of a given image pixel under the respective colour filtered

transformation matrices.

5.5.2 Conclusions

Even though spectral rendering is fairly easy to implement, it is still not widely used.

Performance issues are only partly a reason. In fact, due to cache coherency and with

cheap component-wise illumination calculations, the drop in performance is not significant.

Rather, we felt that it is the lack of spectral materials and the difficulty of reusing existing

RGB setups and textures within a spectral setup that have posed an obstacle to users.

To close this crucial gap in the design pipeline, we have devised a spectral palette tool

that allows the user to create a set of lights and reflectances such that different parts of a

rendering can be enhanced, or be made to disappear, in real time and interactively. We

formulated the design process as a least squares problem, yielding global minimization of

the criteria. The design scheme and optimization is novel in both graphics as well as in

colour science. The resulting set of spectra and colours have utility in the visualization

of volume data, but their usefulness is not restricted to this arena or to surface graphics.

In fact, with the liberty to inject actual physical spectra for any of the components, one

may design appropriate lights to attain specific perceived colours when viewing real physical

subjects.

Chapter 6

Interactive parameter space partitioning

Computational power today enables researchers to build and study algorithmic models that

may involve large numbers of variables and complex relations among them. The gist of

Chapter 1 was that in order to draw any practically relevant conclusions from a simulation,

it remains crucial to ensure a close correspondence between formal model and the real-world

system under scrutiny.

A possible step to achieve this correspondence is to match model output with measured

field data. However, in early modelling stages such data may not be available and a law-

driven approach has to be chosen. Beyond that, even after fitting given observations there

may still be free model parameters that can be controlled to adjust the behaviour of the

computer simulation. This can happen, if the expressive power of the model exceeds the

number of available measurements, or if the measurements are so noisy that several different

model instances are equally acceptable. To formally address this case, it is possible to

introduce additional regularizing criteria that a solution has to fulfill. This was, for instance,

done in the context of our spectral palette design in Equation 5.13 on page 82 by minimizing

the curvature of the generated distribution functions. Instead of using numerical criteria,

the user could be given a method to interactively tune free parameters of the model to favour

more plausible solutions that match prior experience. In such a case, a domain expert could

be involved to interactively tune parameters of the model or to prescribe ranges that favour

solutions that match prior experience, theoretical insight, or intuition.

Towards that goal, we recognize that the optimization of parameters for some notion

of performance is distinct from the objective to discover regions in parameter space that

exhibit qualitatively different system behaviour, such as fluid vs. gaseous state, or formation

95

CHAPTER 6. INTERACTIVE PARAMETER SPACE PARTITIONING 96

of various movement patterns in a swarm simulation. Optimization is one focus of statistical

methods in experimental design and has great potential for integration with visual tools, as

for instance demonstrated recently by Torsney-Weir et al. [TWSM+11].

The focus of this chapter is on the latter aspect of qualitative discovery. This can

support the understanding of the studied system, strengthen confidence in the suitability of

the modelling mechanisms and, thus, become a substantial aid in the research process.

In the context of modelling this is a novel viewpoint, since typical approaches calibrate

one best version of the model and then study how it behaves. To put regional parameter

space exploration into practice, a number of challenges have to be overcome. To identify and

address those, we (a) performed a field analysis of three application domains and derived

a list of requirements in Section 1.2, (b) present paraglide, a system that addresses these

requirements with a set of interaction and visualization techniques novel for this kind of

application area, (c) conducted a longitudinal field evaluation of paraglide showing practical

benefits. In summary, paraglide sets out to make the following contributions to computational

modelling:

• Parameter region construction is promoted as a separate user interaction step dur-

ing experimental design. This allows to address different efficiency issues of multi-

dimensional sampling.

• A common step in explorative hypothesis formation is the construction of additional

dependent feature variables and goal functions. Paraglide facilitates this with inter-

preter based back-ends. Also, this seamlessly integrates model code from sources such

as MATLAB, R, or Python.

• Qualitatively distinct solutions are identified and the parameter space of the model is

partitioned into the corresponding regions. This allows to visually derive global state-

ments about the sensitivity of the model to parameter changes, which traditionally is

studied locally.

6.1 Background

Research that is related to visual analysis of multi-dimensional data has been discussed in

Section 2.4 and specifically Section 2.4.4. Hence, the following review can focus specifically


on user interfaces that adress some aspects of interactive parameter adjustment of compu-

tational models. Methods that are more specific to the design of particular components of

paraglide will be discussed in the respective sections of Section 6.2.

6.1.1 Interactive parameter adjustment in computer experiments

Computational steering systems focus on adjustment of parameters during execution of

time-dependent simulations [MvWvL99]. Since our users do not modify parameters during

the run of a simulation, our problem setting is different from classical steering in that we do

not need to handle live updates of variables that are shared between simultaneously running

modules. However, there is enough similarity to benefit from a comparison.

Wijk et al. [vWVLM97] performed an evaluation of their computational steering en-

vironment (CSE) and recognized major uses for debugging, presentation, and assistance

in technical discussions that progress faster when “What if?” questions can be answered

immediately. A follow-up survey by Mulder et al. [MvWvL99] identified further uses for

model exploration, algorithm experimentation, and performance optimization. While these

systems inspire numerous design decisions, the specific requirements for efficient regional

sampling and an easy integration of end-user codes for simulation and derived variable

computation are either not fulfilled or could be improved.

Berger et al. [BPFG11] discuss a system to visualize engineering and design options

in the neighbourhood around an optimal configuration of a computer simulation. Based

on a continuous function abstraction they provide a local analysis method that benefits

domain experts. In order to not get stuck in local maxima, optimization methods usually

benefit from an additional global perspective on the problem domain. The qualitative

decomposition pointed out in the cases of Section 1.2 and pursued in the following provides

such a complementary view.

The challenge of devising a user interaction for sample construction has recently been

taken on in the Paranorama system of Pretorius et al. [PBCR11]. Their users can specify

different ranges of interest for individual variables along with the number of requested

distinct values per range. The sample points are then constructed via a Cartesian product

of the value sets. Integrating this method into an image segmentation system, received

positive feedback from users. However, combining many value ranges with this method may

result in large sampling costs. Beyond numerical arguments, also screen space real estate

is used up more quickly when viewing data sets with large numbers of variables, and a


significantly increased cognitive cost arises when investigating and interpreting the effects

of many factors on possibly multiple responses. Due to their significant impact on sampling

and processing costs, Section 6.2.3 will give careful consideration to the number of involved

variables and the volume of the region of interest.

6.1.2 Parameter space partitioning

The computational model analysis cases of Section 1.2 all benefit from an overview of regions

of distinct system behaviour marked out in their input parameter space (R5 of Table 1.1

on page 12). So far, there is no prior research in the visualization community that provides

such a representation. After considering different names for the method, such as parameter

space segmentation, clustering, or partitioning, it is the latter term that relates us to two

prior contributions from an old and a recent member of the sciences, namely physics and

psychology.

Bhatt and Koechling [BK95] study the behaviour during impact of two solid bodies with

finite friction and restitution, which results in a tangential sliding velocity that continuously

changes direction after impact. The problem is characterized by 9 parameters, three for

the impulse direction of the impacting body and six for its rotational moment of inertia

tensor. The first step of their analysis determines a reduced set of three dimensionless

parameters that completely define the tangential flow of sliding velocities. An important

observation is that the qualitative behaviour of the flow is characterized by 2 or 4 solution

curves of invariant direction, as well as the critical points and sign changes of the velocity

along these straight lines. This results in 4 main cases with up to 3 sub-cases each. An

implicit expression of the boundary between the cases is derived that is quartic in terms

of the 3 dimensionless parameters. By fixing one parameter and showing slices through

this boundary, the enclosed regions can be visually distinguished and are labelled with the

different cases they represent. This provides a comprehensive overview of all possible sliding

behaviour. While providing a sophisticated algebraic analysis of a specific phenomenon, their

discussion does not deal with numeric aspects involved of general computational models.

Pitt et al. [PKNM06] also promote parameter space partitioning with an example anal-

ysis of a model to predict, whether visual stimuli are recognized as words or non-words. In

their overview of analysis techniques, they distinguish two axes that separate quantitative

from qualitative and local from global techniques. In this view, partitioning is a global,

qualitative method, and sensitivity analysis a case of more local, quantitative inspection.


Their method proceeds from a notion of equivalence among model configurations and a

set of valid seed configuration points. The regions around these points are sampled using a

Metropolis-Hastings algorithm with uniform target distribution. Rejected points have fallen

into non-equivalent, adjacent regions and are explored subsequently.

An important point made by Pitt et al. is to also address the need to improve the user’s

confidence in the plausibility of a given model. The studies they present are supported

by showing the variety of qualitatively distinct model behaviour. However, a discussion of

suitable analysis system design and considerations of required user interactions are not the

focus of their exposition.

6.1.3 Unfulfilled design requirements

Methods to visually inspect multi-variate point distributions (to address R3a of Table 1.1)

are available in several of the frameworks listed earlier in Sections 2.4.4 and 6.1.1. However,

the required capability to also generate data points (R2a) or to add derived dimensions (R4a)

is missing from most systems that are mainly geared towards visualization of a static data

set. Systems for experimental design, on the other hand, take care of the sampling require-

ments (R2a), but often lack interactive, visual methods to solicit required user input (R2).

Computational steering systems combine sampling and visualization, but specifically focus

on live-adjustments to parameters of a simulation that evolves over time. Their focus is

often on some sort of interactive investigation, which could benefit from further support for

broader state discovery (R5).

6.2 Design of the paraglide system

We will now discuss aspects of the design of paraglide, giving individual consideration to the

graphical user interface (GUI), the software system, and choices of algorithms or methods

for particular tasks. Paraglide was developed in a user-centered design process with five users,

one or two from each domain. We met with our participants in person covering longitudinal

time ranges of four years (fuel cell engineering), two years (mathematical modelling), with

monthly meetings, and five months (image segmentation) with weekly meetings. In these

meetings we discussed design mockups and prototypes, observed our users working with

paraglide, and gathered formative feedback in terms of usability and feature requests that

we used to improve paraglide’s design. In addition, these meetings contributed to refining


our understanding of user practices and design requirements (see Section 1.2), as well as

gathering summative feedback and anecdotal evidence (see Section 6.4).

a

c

b

d

e

f

g

e

h

Figure 6.1: Paraglide GUI running inside a MATLAB session to investigate the animal movementmodel of Section 1.2.1. Initially, deliberately chosen parameter combinations are imported froma switch/case script (a) by sampling the case selection variable of that script and recording thevariables it sets. An overview (b) of the data is given in form of a scatter plot matrix (SPloM)for a chosen dimension group (h). Jython commands can be issued inside the command window(c) demonstrating the plug-in functionality of the system by manually importing the experimentmodule, which adds a new item to the menu bar (d). This allows to create a set of new samplepoints inside the region that is selected for parameters qa and qal (e). The configuration dialog forthe MATLAB compute node (f) sets up a show command that produces a detail view of the spatio-temporal pattern (1D+time) (g). For the configuration point highlighted in yellow in the SPloM,this results in a pattern of two groups that merge and then progress upwards in a ’zigzag’ movement.

6.2.1 System components

The snapshot of Figure 6.1 shows the paraglide GUI and provides a brief overview of the

main steps of the interaction. In the left of the main window (Figure 6.1d) dimension group

tabs are shown that can be used to switch between selected subsets of variables. Right next

to it appears the view for an individual group of dimensions (h), which shows histograms

indicating the distribution of values for the respective variables. If a group has more than 8

dimensions, compact range selectors are shown instead of histograms. This frees up screen

space and eliminates computational costs for keeping their information updated, e.g., when


the data set or the filtered selection changes. In Figure 6.1 the larger area in the right of

the main window (d) provides a display of the data points (R2/a + R3a/b). In the example,

a scatter plot matrix (SPloM) is shown (b) that arranges scatter plots on a grid, where

each row or column is associated with one variable for the vertical or horizontal plot axes,

respectively. Alternatively, it is possible to configure individual, enlarged scatter plots to

inspect pairs of variables and show them in this area. In the console in (c) one can enter

commands for MATLAB, Java, or the Jython interpeter that paraglide is running.

Workflow integration via scripting: Using the Jython import command to load mod-

ules readily takes care of managing dependencies among plug-ins. It is possible to script

workflows at runtime and add them to the menu. Dependent scripts can be stored and re-

covered along with an XML description of the state of the current project. This also creates

a separate folder that contains all cached disk images and other meta data.

Data management, view, control, and state: The core system is structured along the

model–view–controller development pattern, which is partly inherited by using components

of the prefuse system [HA06]. Particular use is made of the ability to select points of a

centrally maintained prefuse table by evaluating a boolean expression on its row tuples with

further details given in Section 6.2.3.

Compute node interface: To integrate domain specific computation code we use a com-

pute node abstraction. It can return a list of parameter names with optional description text

and set/get accessors. A node may have more specialized features that it can announce in-

ternally by returning a list of capability descriptors. The main ones are compute solution,

display plot, file IO, and compute feature. One possible interface for the ComputeNode

binding for a MATLAB backend is provided by the configuration dialog shown in Figure 6.2,

where the edit boxes correspond to the capabilities. Respectively, this means the node may

be able to provide different detail plots for a solution, it may compute solutions to a given

configuration or derive named scalar or vector features, which are output quantities simi-

lar to plots. A node with file IO capability can store and retrieve cached solutions, such

as MATLAB data files. In the example of Figure 6.2, the run command creates a sine wave

vt = a sin(ft2π+φ) for 101 values of t = 0 . . . 1, parameterized by phase shift φ, frequency f ,

and amplitude a, with default values 0, 1, and 1, respectively. The show command displays

the graph with axes of fixed height ±5. Due to instant computation, file IO is disabled.

The ‘add dependent variable’ button allows to enter a line of code, whose return value


Figure 6.2: Dialog to set up a MATLAB compute node

is assigned to a variable of chosen name. This implements the derived variables yi = g(y)

described in Section 1.4 and serves requirements R4/a. If a scalar is returned, it can be

shown in the SPloM view along with the input variables. It is also possible to return vector

features that may not be shown directly, but can be used to compute similarity or distance

matrices, or to determine adjacency information. Methods to derive further embedding

coordinates from this information are discussed in Section 6.2.4. The example in Figure 6.2

picks out the deflection of the oscillation at time t = 0 (v0) and half way into the interval

(v 12). When inspecting these two local features, it becomes apparent that v0 depends on

a and φ only, where v 12

is influenced by all three parameters. In this simple setting, it is

possible to make this observation by thinking about the equation given for v, as well as by

studying the scatter plots of input/output variable combinations. For the latter method,

however, we would first need to create a data set of tuples (a, f, φ, v0, v 12).

With a readily configured connection to the simulation back-end, paraglide has a notion

of the input parameter space Rn as well as the output space Rr of the simulation code.

Initially, however, there may only be a single point of default values present, around which

it is possible to expand the data set. To generate additional points it is possible to use some

of the methods discussed in Section 2.3.2. While paraglide does not require to start with a

given data set, the following discussion of system components will assume that an initial set

of points already exists.

6.2.2 Browsing computed data

This stage provides the user with an overview of the data points as they distribute over

input, output, and derived dimensions.


6.2.2.1 Viewing multi-dimensional data spaces

To address requirements R3a/b, we provide multiple simultaneous plots that give different

views of the same data table and are linked to display a common focus point, highlight,

or selection region. The subspaces that can be visualized that way range from multi- to

0-dimensional, to allow the user to relate overview of the whole distribution and detail plots

that represent a single point. We also provide techniques to view 2D subspaces, such as

scatter plots that allow for pair-wise inspection of relationships among variables, and 1D

projections of the marginal densities that can be shown in form of histograms.

There are many more possible techniques for multi-variate multi-field data visualization,

such as parallel coordinates, star plots, biplots, glyph-based visual encodings, or scatter plots

arranged in a matrix or table layout, with discussions of pros and cons provided in various

surveys [Hol06, FdOL03]. Implementations of these techniques are either contained directly

in paraglide or are available via export to MATLAB, R, or protovis.

6.2.2.2 Grouping dimensions

As mentioned at the outset of this chapter, the increased number of variables involved with

modern simulation codes poses a challenge for their cognitive and numerical analysis. The

visual complexity rises and makes data plots more difficult to interpret. The strategy of

combining multiple views has its limits in screen space, as well as perceptual and cognitive

handling. A possible remedy to this problem could be a) indirect visualization of fitted

reduced dimensional models [Hol06], or b) to divide the overall number of dimensions into

groups for more focussed inspection.

Grouping simplifies complexity and can be based on statistical or structural information.

Research on grouping of variables may consider dimension reduction and feature selection.

To allow the user to express semantic information, we provide an interface to construct or

modify a dimension group that simply consists of check boxes that indicate group member-

ship for each variable. During browsing, only dimensions of the currently selected group

are shown, which reduces the required screen space. While automatic assistance in forming

these groups is imaginable, our current approach of manual selection proved sufficient in all

use cases.


6.2.2.3 Scatterplots for points in continuous data spaces

The possible use of scatterplots in the context of sensitivity analysis (R7) is pointed out by

Saltelli et al. [SRA+08, Sec. 1.2.3].

Conventional scatterplots show points directly. However, for the computer simulations

under consideration, there is an underlying continuous phenomenon. This means that a)

interpolated density distributions could be reconstructed and displayed, b) input and output

dimensions (or variables) with clear dependency relationships can be distinguished, and c)

descriptions of continuous region in data space can be visualized.

A classical technique to realize (a) is kernel density estimation, where instead of drawing

a dot for each point, a kernel that smoothly decays with increasing distance from the point

is additively rendered into the screen plane. In the language of direct volume visualization

this amounts to splatting with additive, X-ray-type compositing. A typical issue with this

approach is the proper choice of kernel diameter for proper reconstruction of multi-modal

density distributions.

Being able to separately consider input and output dimensions (b), enables an elegant

way to address this issue in scatterplots of the output. For a generated sampling pattern,

point adjacencies in the input space are known or can be obtained by constructing a Delau-

nay mesh that connects Voronoi relevant neighbours. The faces of this mesh are simplices

in the typical case or bounded cells in general. To reconstruct the projected continuous

density for a scatterplot of output dimensions, it is possible to project the cells between

the points, instead of kernels representing the points themselves [NH06, ST90]. For further

background see Section 2.4.2.

The accuracy of a reconstructed density distribution varies based on distance from the

given points. This introduces an additional challenge to continuous visualizations that

direct point drawings do not have, since a notion of uncertainty is readily perceivable as

gaps between points. This is one reason, why continuous density plots are not used in

paraglide, and we instead opt for a more easily interpretable direct point display.

Apart from a density distribution, it is possible to visualize other continuous objects

that live in the same space along with the data points, as hinted earlier in point (c). In

general, this could ask for a direct visualization of scalar functions over a continuous domain.

More specifically, the construct required to address the presented use cases are 0, 1-valued

indicator functions that describe a continuous selection region.


6.2.3 Representing a region of interest

Choosing a region of interest (ROI) and the construction of a set of sample points inside it, as

described in 2.3.2, are tightly related. The shape of the region and its locally varying level of

detail amount to the support and density of a probability distribution, respectively. Seeing

a finite set of sample points as discrete and a region description as continuous distribution,

one can use distributional distance measures to assess how well one approximates the other.

While conceptually a probability distribution sufficiently describes what is needed to

capture about a region of interest, it may be difficult for the user to grasp or specify,

especially in multi-dimensional domains. A possible simplification is to omit the varying

level of detail and to just consider uniform density. In this view the region M of Figure 1.3

on page 16 is given by the support of the distribution and its volume corresponds to the

inverse density. Defining more complex regions than hyper-boxes in Euclidean spaces of

possibly more than three dimensions, however, is a complex task for a human user. An

algebraic way to express what is wanted, as pursued with the feature definition language

(FDL) of Doleisch et al. [DGH03], can provide complementary input that goes beyond the

expressive power of current interaction widgets. The XML encoding of paraglide’s system

state includes a region description, which can be separately stored and imported.

Beyond the box — filtering derived variables: While this prior abstraction prepares

much of what is needed in our application settings, the principal region template of a hyper-

box might prove impractical in a higher-dimensional setting involving many variables. The

reason for this lies in the drastic way the volume of a hyper-box rises relative to an inscribed

2-norm sphere as their dimensionality increases. This may not be much of a concern when

selecting points from a given set. When generating uniformly distributed point sets, however,

the costs are usually proportional to the volume of the requested region. So, ideally one

would like to keep it as small as possible.

The degenerate form of a region shrunk to a single point constitutes a cursor. It can be

used as a focus point for detail inspection, as well as a method to choose a default anchor for

the generation of new samples. Starting from this, one way to obtain smaller regions simply

amounts to a change in perspective from defining a region that covers certain value ranges

to constructing a meaningful neighbourhood around the focus point. A common construct

for such a point neighbourhood is a sphere bounded by some radius in a given metric.

While the choice of distance metric leaves a lot of possibilities. We opt to use p-norms


that come with just a single parameter and contain the popular hyper-box with p =∞ and

the Euclidean sphere for p = 2. Section 2.1 showed that with decreasing p these regions

strictly decrease in volume, ending up to just select the coordinate axes for the somewhat

degenerate choice of p = 0.

This points to an alternative method of constructing custom regions by forming a hyper-

box of ranges for chosen derived variables: For polytopes these would be linear combinations

of other variables that make up the plane equations for the bounding facets. To obtain

spheres in any metric, one could create a variable that measures the distance from a focus

point. Bounding the maximum of this distance variable implicitly constructs a sphere on

the dimensions that were involved in the distance computation. Boxes in this view are

represented by the∞-norm and a simple switch to the isotropic Euclidean 2-norm sphere can

significantly reduce the volume of interest. A data-adaptive example of dependent variables

that facilitate, interactive multi-dimensional point selection is discussed in Section 6.2.4 and

applied in Section 6.4.2.

Specifying a region of interest: The input method that we use to specify a region

of interest is similar to prior approaches for constructing hyper-boxes. One of the first

approaches is the HyperSlice system by van Wijk and van Liere [vWvL93]. They steer

a multi-dimensional focus point by constraining its coordinates using multiple clicks that

locate its position in different 2D projections that are presented in form of a scatter plot

matrix. Martin and Ward [MW95] extend the possible user interactions to form a hyper-

box shaped brushing region beyond scatterplot matrices to include parallel co-ordinates, or

glyph views. The prosection matrix by Tweedie and Spence [TS98] also provides a similar

form of control where the cursor point can be expanded into a hyper-box. The data inside

the box is projected into different scatterplots. Collapsing an interval to a point changes

the corresponding view from slab projection to slicing.

The previous interfaces map the data space to screen space using multiple axis aligned

projections to 1 or 2-dimensional subspaces that map to range sliders or scatterplots. This

requires the user to make a sequence of adjustments in order to specify a single cursor or

region position. While this allows for a precise placement, the time required to make an

adjustment grows at least linearly in the number of dimensions, while requiring the user to

attend to multiple controls. Two possible solutions to this issue are discussed in Section 6.3.


6.2.4 Non-linear screen mappings

One way to reduce the complexity of multi-dimensional cursor or region control is to reduce

the number of involved linked projections. This could be achieved by providing views that

give more comprehensive information about the data distribution than 2D projections. In

particular, there is a family of techniques for non-linear screen space embeddings that are

designed to reveal most characteristics of the data distribution. Enabling the user to make

selections in such a view may obviate the need to consider views from other angles.

In these methods, each experimental run is again represented as a point, where spatial

proximity among points corresponds to similarity of two runs. Point placement with respect

to the coordinate axes is typically hard to interpret.

Prior work into this direction constructs slider widgets for smooth n-D parameter space

navigation, as developed by Bergner et al. [BMTD05] and by Smith et al. [SPK+07]. Both

present a 2D embedding of sample nodes to obtain coordinates based on the screen distance

between a movable slider and a set of nodes. Any curve the user describes by dragging the

slider results in smoothly progressing weights that interpolate data at the nodes, which could

result in different mixtures of light spectra or shape designs in the respective application

setting.

Dimension reduction: Instead of arranging points in a circle or another prescribed shape,

it is also possible to place them in a data adaptive way using dimension reduction techniques.

Kilian et al. [KMP07] use a distance preserving embedding of points to represent shape

descriptors to control different designs of shapes. Janicke et al. [JBS08] lay out the minimum

spanning tree of the data points for a multi-attribute point cloud, allowing the user to specify

a region of interest in this embedding. Extending this to larger sets of points, the glimmer

algorithm of Ingram et al. [IMO09] is able to produce distance preserving embeddings for

thousands of nodes via stochastic force computation on the GPU. Each of these methods

require some notion of distance or adjacency among points, which is derived from dependent

feature vectors that can be constructed through the interface described in Section 6.2.1.

Spectral embedding: To embed m data points from an n-dimensional space to the 2-D

screen [Lux07], we start from a data matrix X ∈ Rm×n. It can be turned into an affinity

matrix A = XXT ∈ Rm×m, whose elements are normalized as (C)i,j = (A)i,j/√

(A)i,i(A)j,j

to yield a correlation matrix C. This implicitly scales each row-tuple in X to lie on the

surface of the n-dimensional 2-norm unit sphere. Hence, the operation is referred to as


sphering and the resulting elements of C may be interpreted as cosine similarity or Pearson’s

correlation coefficient. The orthogonal eigen-decomposition of this positive semi-definite

matrix C = V DV T gives unit eigenvectors vi = (V ):,i with decreasing eigenvalues λi =

(D)i,i. The m components of λ2v2 and λ3v3 are providing the x- and y-coordinates for the

spectral embedding of the m points, respectively.

Many variations of this technique are possible. Firstly, instead of constructing affinity

A via dot products, it is possible to employ any other monotonously decreasing kernel

(A)i,j = ϕ(xi,xj), such as the Gaussian similarity kernel ϕ(u,v) = exp(−‖u− v‖2/(2σ2)).

Often, sphering is combined with a previous centering, where the mean µ = X · 1/n is

subtracted from the data X. Alternatively, if the rows of X contain frequencies or counts,

one can rescale their sums to 1, which projects the data points onto the surface of the

positive orthant of the 1-norm sphere.

The described method is dual to its popular ancestor — principle component analysis

(PCA), where one instead begins with affinity A = XTX of centred data X. The above

embedding algorithm then gives the n-dimensional principal components vi. When the data

is projected onto an axis of direction v the variance of the resulting coefficients is

σ2v = EX [((x− µ)Tv)2] = vTEX [(x− µ)(x− µ)T ]v = vTCv.

This shows that the correlation matrix can be used to compute variance of the data for

arbitrary axes and explains the special role of its dominant eigenvectors σ2vi = vTi Cvi = λi

to give directions of maximum variance. The different initializations of the data matrix lead

to a family of techniques that include biplots, correspondence analysis, and (kernel-)PCA

— all of them may be efficiently computed via singular value decomposition (SVD) [GH87].

Grouping points: Clusters of similar sample points can have arbitrary shapes. To not

impose too strict assumptions on their distribution we opt for a manual method to assign

cluster labels. For that, the user determines a plot where the clusters of interest are suffi-

ciently separated. After enclosing the points by drawing rectangular regions in the plot, it

is possible to manually assign cluster labels or to directly work with the multi-dimensional

region description. Automatic clustering algorithms could perform poorly in this setting, if

assumptions about cluster shape, such as convexity, are not met.


6.3 Excursion: Steering a multi-dimensional cursor

6.3.1 A light dial to control additive mixtures

The material of this section is published in Bergner et al. [BMTD05, §IV,§VII].

We can design materials and lights such that each light influences exactly one material.

Hence a linear combination of the lights will create a mixed rendition of the scene. Assuming

that we have m materials and n lights, we have an n-dimensional space to navigate to

produce different visualizations. This is a difficult task for the user.

Figure 6.3: The light dial – interface to control the mixture of lights using normalized inversedistances of the mixture selector (yellow circle) to the light nodes (bulb icons) (see Eq. 6.1).

We developed an interaction metaphor that includes all light sources within one interac-

tion widget, which we call the “light dial”. This is a two-dimensional n–gon slider, mapping

a 2D space into the n-dimensional parameter space (see Fig. 6.3). Our n light sources are

the vertices of the n–gon. Any position on the plane characterizes a weighted sum of the

light sources that make up the n-gon. This way, the higher-dimensional parameter space

is mapped to a 2D topology of nodes. This is the user interface that has been applied to

produce the light mixtures used in Figure 5.6 on page 90.

Each control node has a position on the screen and represents a light source, and the

mouse is used to freely move the mixture selector over and between the nodes (e.g., the

yellow dot position in Figure 6.3). Position is used to determine a scalar weight for each

node, with a value that grows as the selector comes closer to a node. We can compute a


weight Li(x) applied to the light for a selector position x in the following manner:

Li(x) =n∏

k=1, k 6=i

‖x− xk‖‖xi − xk‖

. (6.1)

To make the weights usable for combining light sources, we normalize the sum of all weights

Li to one and scale it with a separate intensity slider. When the position of the dot coincides

with the position of a light bulb in the widget, the influence of all other light sources is

eliminated. The influence of a source can be entirely removed by switching it off. Hence,

the light dial allows one to focus on the influence of a single light to the materials used in

the scene. It has not been designed to navigate the full n-D space at once. Instead, the

intention is to conveniently slide from one pure light to the next pure light.1 Additionally,

it is possible to correlate dimensions (making lights have similar intensity) by moving two

light bulbs closer to each other. This modifies the shape of the 2D mapping of the parameter

space and allows the user to reach new parameter constellations.

To summarize, our light dial implements three different interaction metaphors:

• move yellow dot (change weights by distance) — the object is weighted towards the

lights we want to see it under,

• move light nodes — group lights, or drag lights away to lower their influence (math-

ematically speaking: change shape of 2D mapping of higher-dimensional parameter

space, correlate dimensions by decreasing spatial distance in widget plane), and

• switch lights on/off (reducing dimensionality, comparable to taking a light node and

moving it far away; deactivation of the light allows it to keep its position in the

mapping plane.)

This makes the 2D dial a convenient interface to navigate through the colour schemes. The

light dial metaphor may be useful for other tasks, such as navigating between different points

of view, controlling alpha values of different segments, etc.

1Nevertheless, it is still possible to navigate the entire space of weight combinations. Imagine a concentricarrangement of the light bulbs around the light dot in the middle: all lights have the same weights. Now,a light bulb may be dragged instead of the dot, moved closer to or farther away from the center (where thedot is). Since only one distance is changing, it is possible to change only one weight. To actually have justone weight changing, the separate overall intensity slider would have to be moved correspondingly.


6.3.2 Enabling simultaneous parameter adjustments using a mixing board

From the perspective of interface style, a mixing board occupies an interesting point at the

intersection of tangible user interfaces, augmented reality, and haptic user interfaces. The

sliders embody the benefits of a tangible interface: their handles engage the whole hand

and provide feedback on contact, they are physically constrained to a single direction of

motion and have physical backstops of motion at their limits of travel. Augmenting the

mixing board by front-projecting graphical output on it [CBS+07], integrates the space of

the user’s movements with the display space, a form of augmented reality. Finally, some

mixing boards have motorized sliders, allowing software controlled haptic effects such as

detents.

Although a mixing board offers the benefits of these three interface styles, it is also

limited compared to more typical members. Movement and display of a slider are restricted

to a single degree of freedom, with all sliders in the same plane and direction, whereas most

tangible and augmented reality interfaces emphasize movement in three or six degrees of

freedom. Yet restricted as it is, the interaction task supported by the mixing board, entering

several bounded numeric values, recurs in applications ranging from scientific and medical

data analysis, computer-aided design, to setting document margins.

In a prior qualitative study, participants reported a greater sense of engagement and

productivity when using the mixing board for such tasks [CBS+07]. In the study discussed

in the following [BCKM11], we focus on quantifying the performance benefits from the

tangible properties of the mixing board. We compare its acquisition time, movement time,

and workload demands to those of a similar array of graphical sliders. We also compare

subjective, qualitative evaluations of the two interfaces by our 12 participants.

Task: In order to cover different possible application scenarios, we chose to study an abstract

multi-value adjustment task that is representative of different practical settings. In each

trial, participants were presented a tuple of values and asked to set controlling sliders to

those values within a specified precision, requiring one, two, four, or eight sliders to be set.

The screen displayed an interface (Figure 6.4) that resembles the BFC2000 mixing board

used in the study. Adjustments to the graphical or the motorized physical sliders caused

the corresponding movements on the alternate interface.


(a) (b)

Figure 6.4: (a) Graphical user interface (GUI) for the BCF2000 mixing board (b) showing an exper-imental trial in progress.

6.3.2.1 Activity interval analysis and outlier detection

The slider motion path data is collected as a sequence of events of time stamped value

updates. It can be graphed as in Figure 6.5, where the result of the decomposition of the

overall trial time is also displayed. The six trials stem from a two-slider item that was

presented and recorded in three blocks using the mixing board (top row) and three blocks

using mouse/GUI (bottom row). The gray intervals mark idle times when no valid slider

was manipulated. The red sectors mark error times spent manipulating irrelevant sliders.

Types of activity intervals: To clarify the detailed mechanics underlying the decompo-

sition shown in Figure 6.5, movement time was broken down into several distinct subtimes,

which were determined algorithmically using custom MATLAB scripts. Acquisition time

was computed as the time from the start of the presentation of a trial (end of the previous

trial) to the movement of the first slider. For the mouse, it comprised the activities of pos-

sibly moving the hand from the space bar to the mouse (if the user ended the previous trial

with their mouse hand), moving the mouse to the first slider, and clicking. For the mixer,

acquisition time consisted of moving the hand from the trial end button on the right panel

of the mixing board to the first slider. Manipulation time was computed as the time any of

the the sliders involved in the task were actually moving. Between time was computed as

the time spent between slider movements. Total movement time was computed as the time

from the start of movement of the first slider to the end of movement of the last slider. For


Figure 6.5: Recorded slider motion paths for item 6: [ . . . . . . 110 110]+/-1 shown in frontof non-manipulation intervals (gray) and mistake intervals (red). The top row is mixer interactionof a participant performing item ID 6 of forthcoming Figure 6.6 in three trial blocks for each of thetwo input methods in Figure 6.4. An error moving irrelevant slider 6 is indicated in the third block.The path correlation matrix in the right column is discussed in Section 6.3.2.1.

a given trial, the sum of acquisition, manipulation, and between time equals the total time,

not including the end time between the last slider movement and the pressing of the trial

end button (space bar for the mouse, or a dedicated button on the mixer).

We have also determined error time, which is the amount of time between task relevant

slider movements when erroneous sliders have been moved. An example of such an event

can be observed in the top row of Figure 6.5 for the third block of the mixer interaction, in

which slider 6 was moved in error. Error time is contained in the between time and is also

reported separately.

Motion path velocity correlation: A key feature of the mixer interaction is the possi-

bility of simultaneous movement of sliders as the user may use the same hand or both hands

to affect multiple sliders at any given time of the interaction. As a measure of simultaneous

movement we have used path velocity correlation. Slider velocities can be determined as

the time derivatives of positions sa(t) and sb(t), e.g., s′a(t) = dsa(t)dt . The time correlation

of the movement of the sliders is given via an integral over the products of slider velocities


over the duration t = 0..T of an entire trial

ca,b =

∫ T

0s′a(t) · s′b(t)dt. (6.2)

We omit to compute correlation over different displacements in time as we are only interested

in a measure of simultaneity. To ease later interpretation of the correlation measure, we

normalized all non-zero elements ca,b =ca,b√ca,acb,b

, which produces an un-centered correlation

matrix C. The off-diagonal elements are valued in [−1, 1] depending on whether two sliders

a and b have been moved into opposite or the same direction. A value of 1 corresponds to

completely identical movement, and is also found on the diagonal for elements ca,a of sliders

that have been moved.

Experimental design: The number of values to be set and the precision required were

varied systematically using a six-factor, within-subjects design. The primary factor was

technique [mixer or GUI]. Order of technique was counterbalanced, with participants ran-

domly assigned to one of the two possible orders. The other five factors consisted of number

of sliders [1, 2, 4, or 8], precision [loose ±7 or tight ±1], distance of target value [near (±6)

the initial value, far (±46), or at a backstop (0 or 127)], slider layout [adjacent or sepa-

rated], and post-hoc value layout [aligned or opposing] could be distinguished among the

items. To keep the length of the experimental session manageable, these factors were only

varied in a subset of the 192 possible combinations to create the sample targets—i.e. they

were not fully-crossed. The resulting pool of 24 targets, listed in Figure 6.6, was presented

in a randomized order for every block for every participant.

6.3.2.2 Results

Simultaneity: Our analysis of simultaneous adjustments is summarized in Fig. 6.6. We

show the maximum magnitude correlation between different sliders obtained from the off-

diagonal elements of matrix C defined after Equation 6.2. Each cell in the grid represents a

set of mixer trials of a participant for a given item. One can observe the effects of different

strategies. Three participants never made use of any simultaneous adjustments. Three

others even moved sliders simultaneously into opposite directions, as indicated by the blue

cells. Items with target values that required adjustment into the same direction on adjacent

sliders caused more simultaneity than items requiring movement of non-adjacent sliders.


Timing comparison between mixer and mouse: The recorded timing data were dis-

tributed log-normally. Consequently, all analysis of variance was performed on the log of

the times and effect sizes are reported as percentage changes in geometric mean. The mixer

was 24% faster than the mouse for total time (p < .001), 10% faster for acquisition time

(p < .042), not significantly faster for manipulation time, and 81% faster for between time

(p < .002). The study showed the following differences: For total time, acquisition time, and

between time the times for the mouse were larger than those for the mixer. The between

times for the mixer are virtually all 50% or less than between times for the mouse. Over-

all, 7.5% of the non-manipulation time (acquisition time + between time) has been spent

adjusting erroneous mixer sliders. Using mouse and GUI sliders this rate is as low as 1.1%.

Workload analysis: In the opinion questionnaires (NASA TLX [HS88]), all of the par-

ticipants stated that they preferred the mixing board to the screen controls, with three

participants explicitly stating that the mixer was more enjoyable or the mouse more boring.

Most of the participants also used both hands when adjusting physical sliders on some trials

and/or reported bimanual input as an advantage particular to the mixing board. Three

participants reported that the mixing board was more precise than the graphical controls,

and just as many reported that the physical slider knobs were easy to grab.

Summary: The main results of the previous section, the overall time to complete a trial

Figure 6.6: Simultaneous manipulation of sliders as indicated by the maximum (in absolute value)of the normalized slider-slider velocity cross-correlations of Equation 6.2. Simultaneity varies fordifferent items and participants.


with mixer interaction is 78% of mouse/GUI pointer based interaction. For items including

extremal values (backstop) the rate can go as low as 72%, corresponding to a gain in speed

of close to 40% when using the mixer.

Determining activity intervals, as described in Section 6.3.2.1, allowed to distinguish the

different periods of non-interaction into initial acquisition time and in-between acquisition

time. The latter is significantly (p < .001) dependent on the input device. Remarkably, the

manipulation time showed no significant dependence.

A trial task was displayed on the screen while the manipulation devices - mixer and

mouse to its right - were placed on the table in front of the screen. Mouse interaction

works via an intermediate pointer or cursor representation on the screen, and thus in the

same visual space the task is presented. The mixing board has its manipulation elements in

physical space. These need to be found using the hands and potentially moving the eye gaze

over to the board. To deal with this situation several participants adopted the strategy of

keeping their gaze on the screen while blindly operating the sliders and occasionally touching

a wrong one. Most of these erroneous manipulations were short term mini-manipulations in

search of the right one to adjust. As reported in the analysis, the overall amount of erroneous

adjustments in the mixer manipulation made only a small difference to the between time

and were higher than the mouse/GUI. Overall, the between time (containing the error time)

was much shorter for the mixer than for the mouse/GUI based interaction.

Applications: The shortened time of interaction and the ability to make simultaneous

adjustments make the mixing board a suitable device for user-guided search over multi-

parameter spaces as it occurs in a variety of application settings. For example, the sliders

could be used to apply ratings of relevance, influencing an algorithm’s notion of good search

results [BSNA+11, TWSM+11, BM10].

Alternatively, parameters may be influenced directly as it would be useful to explore

a design space or to configure visualizations. The user may determine a most promising

direction of exploration by learning about cross-dependencies among different parameters.

With separate adjustments, such as done via the mouse, the direction of the steps in

which one can move through parameter space will always be along parameter axes. With the

simultaneous adjustment of the mixer sliders it is also possible to progress along diagonal

directions in the planes spanned by any pair of parameter axes. This may be helpful for

steering more directly towards optimal configurations.

Another advantage for the mixer in such exploration settings is the fact that it frees up


screen real estate. With the separate interaction device, the entire display can be used to

show the data in more detail.

In our experiments we have used the motor control of the sliders only to ensure a clearly

defined initial state for each task. However, depending on the application setting there is

more to be gained from this feature. In particular, parameter dependencies and constraints

could be expressed to limit the user’s search of feasible regions, or enable the user to push its

boundaries. In a dynamic query range selection [CBS+07] we have enforced the constraint

that the upper bound slider may always be at a value greater or equal to the lower bound

slider.

Further directions: One could also investigate the integration of a separate slider adjust-

ment as provided by the mixing board with more advanced mouse/GUI interactions [vWvL93,

TS98].

Another comparison could consider touch screen interfaces, such as TouchOSC2, in re-

lation to tactile or haptic ones, like a physical mixing board. The study presented in this

section has inspired follow-up work [STD09] that showed a significantly improved visual

focus on the task display, when physical sliders are used instead of GUI controls, which did

not occur when using touch screen sliders placed in the same prior location of the physical

sliders on a table top display. While they did not obtain the timing improvements docu-

mented in this report, the improved eye gaze focus provides further indication and a possible

explanation of the reduced cognitive load when using the mixer.

The items used in our current study are all fairly structured and contain several identical

values. Further studies could consider more different value combinations, e.g., giving target

points that lie in certain specifically shaped neighbourhoods around the origin. Different

directions of the required steering could be considered, as well as measures of how exhaus-

tively a user explored the feasible options. A readily available binding in the paraglide system

of this chapter could provide the technical setup for such a study.

In summary, the shown characteristics portrayed mixing boards as a good fit for the

purpose of a detailed multi-parameter control. Easy and relatively cheap integration of such

a device into an application is possible. Its strengths should particularly hold in settings

where screen space is precious, and undivided visual focus is crucial.

2http://hexler.net/software/touchosc

http://hexler.net/software/touchosc


6.4 Validation of paraglide in different use cases

In this section, we discuss qualitative feedback and anecdotal evidence from user interviews

and usage sessions during the later stages of our user-centered design process (see Sec-

tion 6.2). The first stable and iteratively improved versions of paraglide were installed in the

work environment of our three lead users. Overall, paraglide has been in use at several occa-

sions over a period of 4 years, accompanied by weekly meetings of our research focus group

for a period of 2 years, and problem-adaptive meeting frequencies in the domain settings.

We present the summative findings in form of usage examples in which our users (a)

were able to do something that they weren’t able to do without paraglide, (b) could gather

new insights into their model by using paraglide, or (c) felt to be able to conduct some task

more efficient with paraglide than with traditional tools.

6.4.1 Movement patterns of biological aggregations

Semi-automatic screening for interesting solutions: In Section 1.2.1 on page 5 a

conjecture was pointed out that interesting spatial patterns only form with parameter con-

figurations for which the PDE has unstable spatially homogenous steady states. This means

that linear stability analysis can be used to detect the potential for pattern formation.

In the course of this research project, the model developer implemented a function to

compute the type of (in-)stability for a given steady state. She had no problems to make

the feature available in paraglide within few minutes using the compute node interface of

Figure 6.2. Colouring the data points by stability type then helped to focus the pattern

search, because computing the feature based on just the input x takes about 5 seconds,

where a full population density would take 5 minutes per configuration point. With this

computationally cheap screening, it became possible to cover larger areas of the parameter

domain before zoning in on sub-areas to compute more comprehensive output that includes

spatio-temporal patterns.

During one meeting a simple positivity test was implemented this way to answer, within

five minutes, whether any solutions with negative densities were present — providing a very

efficient debugging aid.

Discovering structure: To generate a set of sample points simply by specifying the

containing region, the uniform sampling method, and the requested number of points, was

considered a very convenient way to generate data: “You don’t need to worry about the


coding, e.g., for loops, to set up region bounds, or a choice of sampling strategy.” Aside

from saving time, the interaction also puts the user’s focus on core questions of choosing

and combining value ranges. Within the selected range, coarse sampling to provide overview,

followed by more focussed, finer sampling to acquire details, proved to be a good strategy

— the structure in Figure 6.7 was found this way and inspired our user to further analytic

investigation. In particular, that an increase in repulsion leads to increased stability and

less pattern potential corresponds well with biological experience.

Figure 6.7: Illustrating the sample creation in a sub-region of the parameter space iterating fromcoarse to finer sampling. (Un-)filled circles indicate parameter configurations that lead to an(un-)stable steady state.

Investigating pattern formation hypothesis: To investigate the hypothesized relation-

ship between pattern formation and linear stability analysis the customizable detail view

feature proved helpful. The main output shows a full spatio-temporal pattern as given in

Figure 6.1g. Another view created for this application is a bifurcation diagram that shows

how the multiplicity and stability type of all steady states (spatially homogenous solutions of

the PDE system) are changing, as one parameter (qal in this case) is changing its value. This

enables further study of possible relationships between steady states and pattern formation,

as shown in the supplementary video material.3

Comparison of different model versions: The comparison of model versions using non-

or constant velocities was enabled by creating two different feature variables computing the

stability type using either of the two conditions. Switching the colour coding between these

3It is located at http://www.cs.sfu.ca/~sbergner/personal/proj/highd/paraglideswarms.html.

http://www.cs.sfu.ca/~sbergner/personal/proj/highd/paraglideswarms.html


two variables allowed to visually compare the stability regions of the two model versions.

This facilitated a main insight of the Master’s thesis, showing that the instability of most

steady states tends to increase in the presence of non-constant velocities.

Overall, the users in this case found paraglide to be: “a user-friendly tool that makes

creating the sample points and comparing the computations much easier. This tool is capable

of giving the user a better intuition about different solutions that correspond to various

parameter configurations.”

6.4.2 Bio-medical imaging: Tuning image segmentation parameters

In the following, we evaluate the use of paraglide in the context of Section 1.2.2. During

three recorded meetings of overall 6 hours a workflow was developed, implemented, and the

required interaction steps performed. The goal there was to find a robust setting for eight

parameters of a segmentation algorithm that produces good results for different data sets

and noise levels, assessed by ten numeric objective measures. The term good in the sense

of this discussion, refers to all points on a plateau of the optimization landscape that have

target values close to the global optimum. When chosen from an initial, explorative sample

they are also referred to as candidates, or representatives of the good cluster. Since the

optimum is an ambiguous term in the context multi-objective optimization, our method

proceeds by first grouping all points that are similar to each other. This leaves the task of

finding out which cluster of points is a good one. The shape of the plateau of good points

viewed in the space of input parameters informed the developer about which parameters to

keep and which ones to drop. It also leads to a choice of configuration for the algorithm.

To enable faster computation the volumetric patient data was reduced to a single slice that

contains representatives from each class.

Find good candidate points by visual inspection: For easier inspection, the full

set of variables is first broken down into groups. A SPloM view of the input parameters

verified that the sampling pattern indeed uniformly covers the 2D scatterplot projections.

To focus on the problem, the user isolated the group of performance measures described

in Section 1.2.2. Manually chosen configuration points improved one or two performance

criteria, and allowed to verify basic data sanity in a linked data table view. A combined

manual optimization of the 2 performance measures for each of the 5 classes, however, would

require to pay attention to simultaneous changes in 5 scatterplots. The developer considered


Figure 6.8: Scatter plot matrix view that compares the point embedding (lower left) with the ob-jective measures that went into computing its underlying similarity measure. The numbering of theresponses corresponds to the class labels of Figure 1.1 (ID 5: cerebellum and 6: putamen).

this a very difficult to infeasible task that needed to be simplified.

Construct the good neighbourhood: For most points in parameter space, a continuous

change in the input parameters leads to a continuous change in the segmentation algorithm’s

behaviour and the derived performance measures. This means that for each good point, it

is worthwhile to explore the neighbourhood around it to find additional good and better

settings [MH07]. With the distinction of input and output dimensions it is possible to

construct and combine different notions of neighbourhood around a point. The dialog of

Figure 6.2 is used to combine performance measures using weights that equalize the dynamic

ranges. This feature vector space is then viewed using the spectral embeddings described

in Section 6.2.4. Figure 6.8 shows the similarity embedding in the lower left view, where

the good cluster is highlighted in yellow. Judging from the strong diagonal distribution

in two plots in the matrix, the horizontal embedding dimension is dominated by dice6

and the vertical one by dice5. Since both should be maximized by good results, it is not

surprising that manual inspection quickly identified the good cluster in the upper right of

the embedding. The user found it convenient to make the cluster selection in the embedding,

which underlines the point of Section 6.2.3. Apart from making interval selection easier,

the embeddings also proved as an aid in a number of tasks: a) find good candidates, b)

group adjacent good points into cluster(s), and c) check the embedding by inspecting it in

a SPloM view together with the feature variables as in Figure 6.8.


Multi-factor assessment: To determine the relevance of each parameter for the overall

performance of the segmentation algorithm, the developer viewed the distribution of the

good cluster in input parameter space. This also gives a notion of sensitivity, where a large

enough size of the good region indicates stability w.r.t. parameter changes [SRA+08, Sec.

1.2.3].

Figure 6.9: Scatter plot matrix view of the good cluster (yellow) identified in Figure 6.8 viewed inthe subspace of input parameters. In this view sigma and alpha3 indicate clear thresholds beyondwhich the good configurations are found.

When projecting the cluster onto each variable individually, its shape can be either

spread out or localized in one or multiple density concentrations. If the good points in the

example of Figure 6.9 are spread out along a dimension, the corresponding parameter is

unusable for steering between good and bad performance, as in this case for “don’t care”

parameters alpha1,2,7. Observations like that inform the developer of energy terms

to drop and, hence, directly influenced algorithm development. Parameters showing more

localized good points or a clear transition are kept as part of the segmentation model and

are set to some robust value that is further from the boundary, inside the good region.

Usage of the ROI representation: The region abstraction of Section 6.2.3 simplified

several tasks: a) inspect its content by transferring it between sessions, considering different

segmentation noise level or patient data; b) to adjust the region description under these

different experimental conditions; c) refine the sampling of configurations of the current

model; d) communicate data requests via email. The main steps of the user driven opti-

mization perform (a) and (b) iteratively for runs with different noise levels. This results in

a region description for good and robust parameter choices. Refinement (c) was performed

implicitly by applying (a) to a pre-computed denser data sample of 10, 000 points, which

yielded 23 good configurations with a segmentation quality similar to Figure 1.1d. The best

chosen configuration of dice6 = 0.8282 and error6 = 0.0621, was also verified to be visually

convincing.


Verify generalization: While the optimum has been made robust by constructing it

over different experimental conditions, its performance has to generalize well beyond the

condition used during adjustment. Hence, a final verification is run using data from 10

previously unseen patients. Compared to the best configuration, the 10 validation data sets

showed very good Dice coefficient (µ = 0.781540, σ = 0.06) and excellent kinetic modelling

parameter error (µ = 0.062, σ = 0.0001) throughout. This indicates that the configuration

overall delivers high shape accuracy as well as low kinetic error. Two of the 10 data sets yield

just above average, yet acceptable, Dice coefficients, which inspired a separate investigation.

This shows that the interaction steps suggested by paraglide can accelerate and benefit daily

practice in state-of-the art research.

6.4.3 Fuel cell stack prototyping

The following case was introduced in Section 1.2.3 and concerns the simulation of a fuel cell

stack. The model by Chang et al. [CKPW07] depends on about 100 input parameters and

produces 43 different plots of various physical quantities characterizing the behavior of the

cell stack. The parameters are structured into semantic groups describing different parts of

the assembly. Further, the developer of the code has provided short description texts for

the variables and their physical units. These parameter groups and descriptions are passed

on through the ComputeNode interface of Section 6.2.1 and appear in paraglide as prepared

variable groups and tool tips.

A parameter region of interest is chosen by the user, giving value ranges for the selected

dimensions. All other parameters are kept at constant default values. Paraglide interfaces

with the simulation code via a network connection, allowing multiple instances of the sim-

ulator to compute output for the generated sample configurations distributed over several

computers. When all experiments are computed, one can choose an output plot of interest.

The corresponding plots of all experiments are collected and compared using the correlation

measure and layout method described at the end of Section 6.2.4. Since spatial proximity

in these embeddings represents similarity in experimental outcome with respect to the cho-

sen plot, detail inspection and manual labelling of multi-dimensional clusters using simple

rectangle selection become feasible and improve confidence in the resulting decomposition.

Experiments with current and inflow temperature: To keep the initial experiment

simple, we have chosen a region of interest over two input parameters: stack current


(a) x-stack current, y-in temp.(b) cell current density similar-ity

(c) MEA water content similar-ity

Figure 6.10: Two layouts for 204 example experiments. a) input space showing variation in currentand input temperature, b) embedding of the same samples where spatial proximity reflects plotsimilarity for cell current density, c) similarity embedding for membrane electrode assembly (MEA)water content using the same clusters as assigned in (b). Cluster representatives are shown inFigure 6.11 and Figure 6.12. These screenshots are from the 2007 C++ version of paraglide, and arealso attainable in the currently discussed Java implementation.

(10A..400A) and stack inflow temperature (333K..343K). In this region 204 samples are

created with a uniform random distribution shown in Figure 6.10a. The colour coding is

added at a later stage and has no relevance for the initial step.

In Figure 6.10b the sample configurations are arranged according to their similarity in cell

current density. In this embedding simulation outcomes can be inspected and configuration

sample points can be manually labeled using a screen rectangle selection. For comparison,

another similarity based embedding is shown in Figure 6.10c for the water content of the

membrane-electrode assembly (MEA) using the same cluster labels as Figure 6.10b. When

going back to the input space Figure 6.10a, the colour coding reflects the parameter ranges

of distinct behavior. Cluster representatives are shown in Figure 6.11 and Figure 6.12.

Our users found the parameter space partitioning of Figure 6.10 intriguing, giving them

a new method to study their model. For instance, they pointed out that while cluster

representative Figure 6.11e may be physically unreasonable, the (e) region in Figure 6.10a

can be interpreted as a “bad” region, where adverse reactions occur.

6.5 Discussion and future work

The development of computer simulations needs careful setup of the involved parameters.

The discussed use cases indicated that the required process of understanding can be time


(a) (b) (c)

(d) (e) (f)

Figure 6.11: Cell current density plots for the clusters in Figure 6.10b

(a) (b) (c)

(d) (e) (f)

Figure 6.12: MEA water content plots for the clusters labelled in Figure 6.10c


and resource intensive, and that systematic assistance is in order. Our validation of these

investigations showed how the proposed decomposition of the continuous input parameter

space benefits different questions. One finding suggests that even with potentially large

numbers of variables, such a clustering of responses can result in a small number of re-

gions. This can lead to a significant conceptual simplification and narrow down questions

to particular sub-regions. Also, it provides starting points for local sensitivity analyses.

Feasible or interesting regions for large numbers of variables can be relatively small in

volume when compared to their bounding box. This is an example setting, where domain

specific hints can help to guide the choice of sample points. For that, one could seek to

obtain an implicit function representation of the region boundary, that could be used by the

sampling module of Figure 1.3.

ROI usage: In the requirement analysis of Section 1.2 it became apparent that the specifi-

cation of a region of interest (ROI) as discussed in Section 6.2.3 has multiple uses in different

sub-tasks:

• specify a domain or sub-region for sampling to create or refine the data set,

• choose a viewport to focus the overview,

• steer a cursor to set default values or invoke a detail view,

• make a selection of points for subset processing, for instance to manually assign labels,

• filter points to crop the viewed data range in order to deal with occlusion,

• enable mouse manipulation of the region description,

• export/import region descriptions to compare among different data sets.

Basic user interfaces for this purpose have been discussed in Section 6.3.1 and Section 6.3.2.

Exploring these directions further could lead to interesting combinations of user interaction

and dimension reduction. So far, Section 6.2.2.2 presents a very simple, completely manual

approach to this.

Our current implementation (e.g., Section 6.2.2.3) facilitates different point grouping

methods: manual labelling, application of a classifier function, and clustering via a user

determined similarity measure. Since clustering is key to parameter space partitioning,

further research on suitable interactive and (semi-)automatic techniques for this purpose

would be useful.


Conclusions: The paraglide framework can be seen as a general user interface to work

with simulation code. While software engineering aspects for such a system can not be

disregarded, they should not be seen as the focus of this discussion. The main point of

relevance to visualization research is that all use cases studied here exhibit a common need

for region decomposition of the input parameter space based on a clustering or quantization

of output responses. While fitting and optimization are the usual focus in current literature

(see also the task overview in Section 2.4.1), parameter space partitioning has not yet

received much attention in visualization. By showing its applicability and practical benefits

in settings where a comprehensive understanding of the underlying system is required, we

hope to have stimulated further work in this direction.

Such a decomposition method can use feature extractors, as shown in Section 6.4.1,

employ distance measures in Section 6.4.2 and Section 6.4.3. Both approaches required user

control or at least careful setup to yield informative results. The output is in all cases a

set of continuous, multi-dimensional regions. Envisioning this, our initial research focussed

on high-quality projection techniques for scatter plots of this kind of data. After adjusting

research focus to sampling aspects and the use case evaluation presented here, suitable

options to visually represent a region decomposition of a multi-dimensional continuous space

still deserve further study.

Chapter 7

Discussion and conclusion

In the following I will briefly put the individual discussions of the previous chapters into

perspective. The combining theme of Chapter 1 was emphasizing the importance of involv-

ing human judgement in order to establish correspondence between computational models

and the real-world systems they represent. The conclusion, however, does not follow in all

cases and counter arguments can be made easily. For instance, live sensor measurements

can feed directly into a computational model that adjusts to new conditions as streaming

data arrives. This could happen at a much higher speed and accuracy than a human experi-

menter could copy the sensor readings into a spreadsheet, visually inspect the data for basic

sanity, before crunching the numbers with further algorithmic processing. Nevertheless, at

some early stage of model development and validation it might have been done this way.

In the context of machine learning algorithm development Cohen [CK95, p. 12] states

that the confirmation of a model does not start out as crisp as the final publication typi-

cally presents it. He outlines that the model construction process goes through a number

of phases [CK95, p. 7]: Initial, data-intensive exploratory studies give casual hypotheses,

for which assessment studies provide parameter baselines and ranges. With this informa-

tion, observation and manipulation experiments can then establish statistical evidence for

relevance of certain factors and the effects of different value levels.

Coming from the related field of image analysis, where the objective is also to automate

recognition tasks as much as possible, my entry into data visualization research arose from

the observation that the model development process required good interfaces for scrutiny

and control by the developer. For instance, I had to develop ad-hoc tools to assist the gen-

eration of informative training data [Ber03, §4.2] and the adjustment of higher-level model

128

CHAPTER 7. DISCUSSION AND CONCLUSION 129

parameters (e.g., to find weights for combined ratings or energy terms [Ber03, Eq. 4.13]).

My main point is that many situations are conceivable, where it can be very important

that methods and tools are in place that enable a human to take control over computer mod-

els — development in research, perceptually guided design, and monitoring and response in

unexpected conditions are a few such examples. At the same time, there is a risk that hu-

man interfaces are abused to compensate for insufficient algorithm development, burdening

people with repetitive, demotivating tasks that really should better be done by a computer.

After viewing the argument from both sides, it should have become clear that there is

a line to be drawn that separates algorithmic problems that people should be able to get

involved with, from settings where this involvement should be minimized. At this point, I am

not attempting such a distinction. Rather, do the chapters of this dissertation demonstrate

different roles that a human investigator can take. As outlined in Chapter 1, approaches

range from interactive trial and error adjustments in Chapter 6, to criteria specification that

feeds into an automatic optimization of Chapter 5, to theoretical study of a problem that

enables a systematic parameter choice in Chapter 4, to a uniformly space-filling sampling

method for which the user merely outlines a region of interest and prescribes a budget of

points in Chapter 3.

At the end of Section 2.4, a categorization of different data sources was suggested,

that distinguishes field data (primary), human input (secondary), and theoretically derived

relations or laws (tertiary). The different involvement of secondary and tertiary sources

prescribes the order of the rows in the overview Table 7.1, where, in retrospect, each chapter

is motivated by a general purpose of study. This leads to a decomposition into tasks, which

imply more specific sub-goals or objectives. The role of the user is indicated in each case as

well as the amount of theoretical study that was performed to adress the problem. These

two scales turned out to be somewhat inverse to each other.


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7.1 Conclusion

Beginning the exposition of my dissertation by characterizing practical domains of study

that employ computational models in settings that required involvement of human judge-

ment, which included a model of biological aggregations, image segmentation methods, and

rendering algorithms — I derived a set of requirements to propose a framework for user-

driven analysis of parameter effects. A novel outcome of the workflow I suggested was a

partitioning of the continuous space of model configurations into regions of distinct system

behaviour.

To facilitate progressive regional exploration, I devised a space-filling sampling method

by constructing point lattices that contain rotated and scaled versions of themselves. All

resolution levels of this construction share a single type of Voronoi polytope, whose volume

grows independently of the dimensionality by a chosen integer factor as low as 2.

With the goal of reducing sampling costs while maintaining image quality when rendering

volumetric data, I performed a Fourier domain analysis of the effect of composing two

functions. Based on this, I relaxed a previous lower bound for a sufficient sampling frequency

and applied it to adaptively choose the step size in raycasting.

After suggesting that spectral light models in computer graphics can not only be used

to improve physical realism, but also to create colour effects that scale the level of distin-

guishable detail in a visualization, I developed a method that helps designers to choose a

setup from a high-dimensional parameter space of possible palettes of spectral power dis-

tributions, to optimally fulfill a number of user-specified criteria. Given such a palette of

materials and lights, a remaining task is to choose a desired mixture of presets, for which I

studied two alternative interaction methods and showed how each of them either improves

timing performance or reduces cognitive load, both aspects bear the potential to improve

the quality of the outcome of the task.

Despite the apparent diversity, the selection of research that is combined in this disser-

tation partitions a common space of approaches to human-driven improvement of computa-

tional models through the lens of parameter adjustments. Hopefully, this contributes to a

more comprehensive understanding. Related topics, beyond the scope of this thesis, could

study direct interfaces between computational models or — on the other side — seek to

improve interaction among people.

Appendix A

Mathematical concepts related to data

analysis

A relation is a set of I-tuples xii∈I with index set I, where n-ary relations consist of n-

tuples with index set I = 1 . . . n. A function f : X → Y corresponds to a binary relation

R of domain X and range Y such that for x ∈ X and y, z ∈ Y xRy and xRz implies y = z,

i.e. every element in the domain is mapped to only one element of the range. To define a

notion of continuity for such a mapping, we will first consider some topological concepts.

A.1 Topology and geometry

A topology τ on a set X is a collection of subsets of X satisfying: 1. ∅, X ∈ τ , 2. τ is closed

under finite intersections, 3. τ is closed under arbitrary unions. A non-empty set equipped

with topology τ is referred to as topological space (X, τ). An element of τ is also called

neighbourhood or open set in X. The complement of an open set is a closed set. The trivial

topology consists of only ∅ and X. The discrete topology consists of all subsets of X.

A function f : X → Y between topological spaces is continuous if f−1(U) is open in

X for each open set U in Y . One says that f is continuous at the point x, if f−1(V ) is a

neighbourhood of x whenever V is a neighbourhood of f(x) [AB06, p. 36]. An equivalent

definition in metric spaces is given in the next section.

A homeomorphism is a one-to-one correspondence that is continuous in both directions.

Properties of point sets that are not affected by such transformations are called topological

invariants with examples given in Section 2.1.

132

APPENDIX A. MATHEMATICAL CONCEPTS RELATED TO DATA ANALYSIS 133

A simplex is the smallest convex closed set containing r + 1 points (its vertices) that

do not lie in an r dimensional hyperplane. An r-dimensional simplicial complex [Ale65, p.

15] is a set of points in Rn that can be divided into r-dimensional simplices with r ≤ n,

such that each pair of simplices have either no points in common or share a face (of any

dimension). A face F of a convex set P is defined as F = x ∈ Rn : cx = c0 where cx ≤ c0

is valid for all x ∈ P [Zie95, Ch. 2]. Faces of dimension 0, 1, or n − 1 are called vertices,

edges, or facets, respectively.

A topological space is said to be a closed n-dimensional manifold if it is homeomorphic

with a connected polyhedron and if, moreover, its points have neighbourhoods that are

homeomorphic with the n-dimensional interior of a sphere. Note that this common definition

excludes sets that have a locally varying dimensionality. Regarding manifolds essentially as

simplicial complexes is a viewpoint of algebraic topology, which is different from an analytic

definition involving tangent spaces.

These are the ingredients needed to state Lebesgue’s covering theorem: If one covers an

n-dimensional set with a finite number of sufficiently small1, but otherwise arbitrary closed

sets, there must necessarily be points belonging to at least (n + 1) of these sets. On the

other hand, there exist arbitrarily fine covers where this number of (n + 1) intersections is

not exceeded [Ale65, p. 9]. The value of n ∈ Z+ is referred to as covering dimension and

finds application in Section 2.1.

A.1.1 Metrics and norms

In a geometric setting, a topology can be generated by means of a distance metric d :

X ×X → R+ that is symmetric and only zero for two identical points. Further, it satisfies

the triangle inequality d(x, z) ≤ d(x,y) + d(y, z) for all x,y, z ∈ X. The discrete metric is

d(x, y) = 1 if x 6= y and d(x, y) = 0 if x = y. It generates the discrete topology mentioned

earlier.

A metric determines the set of open balls B(x, r) = y ∈ X : d(x,y) < r of radius

r > 0, whose countable unions induce a topology and dimensionality. If a metric on a vector

space is translation invariant d(x + a,y + a) = d(x,y) for all a ∈ X and homogeneous

d(αx, αy) = |α| d(x,y), it is equivalent to a norm ‖x‖ = d(0,x).

With metric domain and range one can refine the earlier definition of a continuous

1in terms of diameter


function, by replacing the neighbourhoods U and V with open balls. This is equivalent to

defining continuity of a function f : X → Y to mean that for every ε > 0 one can choose a

δ > 0, such that for all x, c ∈ X it holds that ‖x− c‖ < δ ⇒ ‖f(x)− f(c)‖ < ε.

Of particular importance is the space `p of infinite sequences or tuples with index set

N, containing Rn as a finite-dimensional subspace. The `p-norm defined on this space gives

the number ‖x‖p = (∑

k∈N |xk|p)1/p for 1 ≤ p < ∞, returning the maximum element of

x if p = ∞. The unit ball in Rn with the 2-norm is a sphere, for the 1-norm in R3 it

is an octahedron (with corners (±1, 0, 0); (0,±1, 0); (0, 0,±1)), and the ∞-norm in any

dimension corresponds to an axis aligned cube with facets at distance 1 from the origin.

Further, the triangle inequality ensures that all ε-balls are open sets [AB06, p. 24]. Volumes

of d-dimensional unit balls for the class of p-norms given in Section 2.1 find further use for

packing bounds in Section 2.3.4.

A.1.2 Properties of centrally symmetric polytopes

A centrally symmetric (c.s.) polytope P ⊂ Rd has a center c ∈ P such that c + x ∈ P ⇔c − x ∈ P. McMullen [McM70] showed that if for some k with 2 ≤ k ≤ d − 2 all k-faces

of a polytope P are c.s., then all its faces of any dimensionality are c.s. In this case Pcan equivalently be viewed as i) a projection of a D-dimensional hypercube (D ≥ d), ii) a

Minkowski sum of D line segments in Rd, or iii) a zonotope [Zie95, pp. 199].

There is a one-to-one correspondence between norms and convex, bounded sets that are

c.s. around the origin [Mat02, p. 344]. Given such a set K, its associated norm can be

defined as ‖x‖K = mint > 0 : xt ∈ K. The convexity of K ensures the triangle inequality

and vice versa [AB06, p. 227].

A.1.3 Measuring polytopes

It can be useful to be able to integrate a function over the domain of a polytope P, for in-

stance to compute its volume, second order moment, or Fourier transform. Lawrence [Law91]

describes a method to measure P that is based on summing vertex terms. The theorem

derived in his work states that

χ(P) =∑

v∈vertices(P)

(−1)e(v)χ(F (v)), (A.1)


V3

V1

V2

V4

forward cone at v3

u

Figure A.1: Illustration of the concept of forward cones F of Equation A.1 in a direction u for thevertices of the white quadrangle, similar to Leydold and Hormann [LH98, Fig. 3]. F (v1) begins inwhite, F (v2) and F (v3) in light red and F (v4) in dark red. All cones extend infinitely towards theright, but are artificially cut of in the picture. Even in practical computations this step could berequired to ensure that the valuations in Equation A.2 stay finite.

where χ(P) denotes the characteristic function, which is 1 inside P and otherwise 0. The

exponent e(v) counts the number of facets S with normal nS for which uTnS > 0. At vertex

v with position xv a forward cone F (v) towards some fixed direction u is constructed as

illustrated in Figure A.1. It is defined as

F (v) = x ∈ Rn : sgn((x− xv)TnS) = sgn(uTnS) for all facets S of P at vertex v.

The theorem can be used to compute integrals over a collection F of polyhedral sets

in Rn that is closed under finite intersections and unions. A valuation on F is a function

V : F → R, such that V (∅) = 0 and for each pair of sets A,B ∈ F the identity

V (A) + V (B) = V (A ∩B) + V (A ∪B) holds. Then

V (P) =∑

v∈vertices(P)

(−1)e(v)V (F (v)) (A.2)

follows as a corollary of the previous theorem stating that an integral over the polytope Pcan be computed by summing the integrals of the forward cones of its vertices. The choice

of u should not align with any of the edges of P and further numerical issues are discussed in

[LH98]. For different valuations V it is possible to analytically compute volume, centroid,

second order moment, or Fourier transform of a polytope. While analytic integration is

desirable for its exact results, there is a large number of practical settings where a direct

solution is not available as discussed in the following.


A.1.4 How the Euclidean norm induces Euclidean topology

The open ε-balls B(p, ε) for all x ∈ X are a basis forming the topology induced by the

underlying norm. Two norms are equivalent if there are two positive constants c1, c2 such

that c1‖x‖a ≤ ‖x‖b ≤ c2‖x‖a for all x ∈ X. Equivalent norms induce the same topology.

A Cartesian product of sets X1 × X2 × ... × Xn is the set of n-tuples (x1, x2, ..., xn),

with xi ∈ Xi. Euclidean space Rn is the Cartesian product of n sets of real numbers.

The topology of the real numbers can be generated from unions and finite intersections of

open intervals (a, b). The multi-dimensional Euclidean topology can be generated from open

boxes (without their boundary) made of open intervals (ai, bi) along each of the n coordinate

axes. To construct a box as intersection of n open sets Ri−1 × (ai, bi) × Rn−i shows how

generators of the topology of the real line extend to generators of Euclidean topology of Rn.

To see the equivalence to the topology induced by a norm, note that these open boxes

coincide with the ε-balls of the ‖·‖∞ norm. For finite dimensional spaces the ‖·‖p norms

are equivalent for p = 1, ...,∞. This shows how the Euclidean 2-norm and, equivalently, all

p-norms on Rn induce Euclidean topology.2

A.2 Function spaces

Function reconstruction with linear models, as will be discussed in Section B.2 on page 145,

is carried out in a vector space F(Rn) of continuous functions over the multi-dimensional

real domain. This space is equipped with an inner product

〈·, ·〉 : F × F → C with the following properties:

1. 〈f, f〉 is real and positive for all f 6= 0 and 〈0, 0〉 = 0,

2. 〈f, g〉 = 〈g, f〉∗ for all f, g ∈ F , where ∗ denotes complex conjugation,

3. 〈αf, g〉 = α〈f, g〉 for any scalar α and f, g ∈ F ,

4. 〈f, g + h〉 = 〈f, g〉+ 〈f, h〉 for all f, g, h ∈ F .

Two vectors or functions are considered to be orthogonal, if their inner product is zero. The

product induces a norm ‖f‖2 =√〈f, f〉, under which all Cauchy sequences converge to an

element in F , which makes the space complete [Gri02, p. 107].

2The description in this section follows a discussion with Ramsay Dyer.


A space F that has an inner product and a countable basis is referred to as Hilbert

space. An application of this theory in the context of function reconstruction is discussed

in Appendix B.2.3.

In F one can define an orthonormal set ϕl ⊂ F by the condition 〈ϕl, ϕk〉 = δ(k− l) for

any l, k ∈ Z and δ(k) = 0 for k 6= 0 and δ(0) = 1. It fulfills Bessel’s inequality∑∞

l=0 |cl|2 ≤

‖f‖2 with cl = 〈ϕl, f〉 for any f ∈ F . This inequality turns into Parseval’s equality, if and

only if ϕl is a basis, giving a unique vector (cl) ∈ `2 (defined in Section A.1.1 for any

f ∈ F [Gri02, p. 191].

Lebesgue spaces, commonly denoted as Lp(Rn) spaces, are defined to contain all functions

of finite norm

f ∈ Lp ⇔ ‖f‖p =

(∫Rn

|f(x)|p dnx)1/p

<∞. (A.3)

The space L2 of square integrable functions has the inner product

〈f, g〉 =

∫Rn

f∗(x)g(x)dnx (A.4)

and contains all f , for which the induced norm ‖f‖2 is finite. One should note that under this

norm each function is equivalent to a number of functions that agree to it almost everywhere.

In particular, two functions of zero distance ‖g−f‖2 = 0 can disagree g(x) 6= f(x) at points

x ∈ X ⊂ R as long as volnX = 0, e.g., X is any countable set of points.

The dimensionality of a function space is the cardinality of a basis for the space or the

number of independent parameters needed to identify an element in the space. To give an

example, each element of∏n−1, the space of polynomials of degree up to n−1, is determined

by n coefficients.

In machine learning settings, one encounters resolution (time steps, pixels) as a mea-

sure of dimension. For instance, it is common to consider a 1000 × 1000 pixel gray-scale

image as a 106-dimensional vector, where each pixel is one basis function. This number

changes depending on the resolution of the image. A method to determine the true intrinsic

dimensionality as discussed in Section 2.1 can help to guide the choice in this case. An-

other approach is given by the band-limit or number of non-zero coefficients in the Fourier

expansion of a function given in Equation B.6, which also implies a dimensionality for the

function space.


Mulero-Martınez [MM07] discusses trade-offs between smoothness and dimensionality

of a function space for an approximation via Gaussian RBFs using neural networks. He

builds on an analysis by Sanner and Slotine [SS92], who consider a representation of multi-

dimensional band-limited functions using Gaussian basis functions centred on a regular

lattice.Through slight oversampling, the class of interpolating functions is extended, allowing

for non-canonical reconstruction kernels when accepting vanishingly small approximation

errors. See Section B.2.3 for further analysis of the order of this approximation scheme.

Appendix B

Background on multi-dimensional

sampling

The background provided in this section is relevant to the topic of discretizing computational

function descriptions and supplementary to Chapter 2.

B.1 Uniformity criteria for integration

In continuation of the discussion of quadrature of Section 2.2 on page 23 this section deals

with sampling criteria as they arise in the context of numerical integration with wide ranging

applications that include solutions to equations governing the transport of light [PH04a, pp.

721] and money [SW98, HSW04].

B.1.1 A measure of uniformity

The goal for Quasi-Monte-Carlo methods is to construct deterministic point sets that lead

to favourable error bounds when used in quadrature rules. A common measure of uniformity

is discrepancy ([PH04a, p. 317]; [Nie92, p. 14]), which is defined as

D(X,A) =

∣∣∣∣∣voln(A)− 1

m

m∑i=1

χA(xk)

∣∣∣∣∣ (B.1)

D(X,A) = supA∈A

D(X,A). (B.2)

139

APPENDIX B. BACKGROUND ON MULTI-DIMENSIONAL SAMPLING 140

The supremum picks out the worst case error among all sets A ∈ A, when approximating

a volume, given by the n-dimensional Lebesgue measure voln, by counting points of X that

fall into the support of an indicator function χA. Notice the technical detail that some

authors omit the normalization by the number of sample points in D(X,A) resulting in a

definition that is increased by the factor m.

The standard choice for the set family are corners C or, more precisely, axis aligned boxes

within the unit cube [0, 1]n with one corner anchored at the origin. For this case, one speaks

of the star-discrepancy D∗. Indicator functions for non-anchored boxes can be assembled

via a weighted sum of 2n anchored boxes and so can a bound of their discrepancy [Nie92,

p. 15].

Instead of using the supremum in Equation B.2 it is also possible to compute an average

of D(A,X)p over all sets A ∈ A giving the so-called Lp discrepancy. However, especially

in a higher-dimensional setting the average volume of a box is very small and, thus, the L2

discrepancy is not a good measure of uniformity, as pointed out by Matousek [Mat99, pp.

13], who also provides a discussion of discrepancy for other set systems, such as spheres and

half-planes.

A variation on this theme are digital nets that yield zero discrepancy for the class of

axis aligned boxes that have edge lengths of b−k for some low b ∈ 2, 3, 4, . . . and k =

0 . . . dlnbme that at each scale level k are translated to tile the unit cube at scale k = 0.

Efficient construction methods for such point sets have been devised by Sobol, Niederreiter,

and Keller to name a few ([Nie92, Ch. 4], [Mat99, pp. 51], [KK02]).

Numerical experiments: To give an intuition for why the Cartesian lattice Zn is a

particularly bad choice for quadrature rules, the illustration in Figure B.1a shows how the

number of points in the unit cube changes when scaling the lattice Zn, which is equivalent to

plotting the discrepancy of Equation B.1 for differently scaled cubes anchored at the origin

as required for D(X, C) of Equation B.2. Worst-case discrepancy occurs at the discontinuous

deviations from the exact volume estimate given by the cyan diagonal line. These jumps

arise from sheets of points that simultaneously enter the axis aligned cube. As their count

grows exponentially with increasing dimensionality n, so does the discrepancy of this lattice.

A simple fix to this situation is possible by applying a random rotation to the Cartesian

lattice, since it disturbs this disadvantageous alignment with the coordinate axes. In 3D,

Figure B.1b shows the resulting improvement in discrepancy between the black and the cyan

curve, representing discrepancy for an aligned and rotated sampling lattice, respectively.


0 1000 2000 3000 4000 50000

1000

2000

3000

4000

5000

6000

7000

1/|det(R)|

#sam

ples

in u

nit c

ube

number of samples obtained by scaling nD Cartesian lattices

2D3D4Dexact

(a)

(b)

Figure B.1: (a) The sample density inside the unit cube jumps when scaling Cartesian lattices in Rn

for n = 2..4 using a scaled identity generating matrix R = αI. (b) Discrepancy of various 3D pointsets including some randomly rotated regular lattices at differently scaled density.

Relationship to integration error: Knowing the star-discrepancy D∗(X, C) of Equa-

tion B.2, it is possible to bound the integration error for a function f of bounded variation

stated by the Koksma-Hlawka inequality [Mat99, p. 23]. Note that the integrand f is as-

sumed to be transformed, such that the domain of integration M of Equation 2.5 coincides

with the n-dimensional hypercube [0, 1)n.∣∣∣∣∣∫

[0,1]nf(x)dnx− 1

m

∑x∈X

f(x)

∣∣∣∣∣ ≤ D∗(X, C)V (f), (B.3)

where V (f) is a measure of regularity of f called variation in the sense of Hardy and Krause,

which is defined following Niederreiter [Nie92, p. 19]: The variation in the sense of Vitali is

defined as

V (n)(f) =

∫ 1

0· · ·∫ 1

0

∣∣∣∣ ∂nf

∂x1 · · · ∂xn

∣∣∣∣ dx1· · · dxn. (B.4)

Further, a restricted form of this measure V (k)(f, i1 . . . ik) performs the integration only over

a facet of the k-dimensional unit cube, which is the set x ∈ Rn : xl = 1 for all l 6= i1 . . . ik.


The variation in the sense of Hardy and Krause is then defined as

V (f) =n∑k=1

∑1≤i1<...<ik≤k

V (k)(f, i1 . . . ik). (B.5)

Star-discrepancy has also been studied as Kolmogorov-Smirnov statistic for testing fit to

the uniform distribution [SWN03, p. 143]. A result is the iterated log law providing

a high-probability estimate of D∗ of a set of |X| = m random points as D∗(X, C) =√log logm

m ([Nie92, p. 166], [Mat99, pp. 37]). This bound for D∗ is surpassed by the

deterministic constructions pointed out earlier for digital nets.

B.1.2 Point lattices for integration

In Section 2.2 sampling on a scaled Cartesian lattice has been given as an example to illus-

trate the curse of dimensionality with a further demonstration of its bad discrepancy bounds

given in the context of Figure B.1. Consequently, random sampling and deterministic, ir-

regular patterns were pointed out as superior alternatives in the previous section. However,

this should not suggest that using regular lattices to discretize an integrand for summation

is always a bad idea. In fact, deviating from the standard Cartesian choice is rewarded with

tremendous increase in efficiency that may exceed the previously discussed state of the art.

In particular, additional regularity of the integrand, going beyond the previously mentioned

bounded variation, may be taken into account to improve convergence. Basic results on the

subject of lattice rules are summarized by Niederreiter [Nie92, Ch. 5], for which the book

by Sloan and Joe [SJ94] can serve as gentle introduction with a recent follow-up by Kuo

and Sloan [KS05].

Most of the discussion on lattice sets is specific to smooth integrands f with period

of a unit cube In

= [0, 1]n. A function f over a bounded, connected domain M can be

transformed into In

via a smooth non-linear transformation [SJ94, pp. 32]. It is then

possible to make this transformed f periodic by summing translates over Zn.

Such a Zn periodic f aligns with an integration lattice that is defined to have the property

Λ ⊇ Zn. Since such a lattice contains the unit vectors, the integer dot products with its

dual lattice defined in Equation 2.11 imply that all components of any point x ∈ Λ⊥ have

to be integers and thus Λ⊥ ⊆ Zn.


It is now possible to analyse the error of the approximation by quadrature Q of Equa-

tion 2.5 in the frequency domain. The smooth, periodic function f can be represented by

an absolutely convergent Fourier series (using imaginary unit j =√−1) as

f(x) =∑

h∈Znf(h)e2πjh·x, (B.6)

where f(h) =

∫[0,1)n

f(x)e−2πjh·xdnx. (B.7)

The approximation error can be obtained by applying Q to each term of the Fourier

expansion of f ([SJ94, p. 27, Thm. 2.8], [Mat99, p. 78])

Qf − If =∑

h∈Znf(h)Qe2πjh·x − If =

∑h∈Λ⊥\0

f(h). (B.8)

The transition from Zn to summing over the dual lattice Λ⊥ as defined in Equation 2.11

arises, because Qe2πjh·x = 1, whenever h ∈ Λ⊥ and is otherwise 0. The origin is removed

from the sum because f(0) = If .

Regularity or smoothness of f results in a lack of high frequency terms and amounts to

a rapid decay of the Fourier coefficients reducing the error in Equation B.8. This suggests

to define a class of functions Enα [Nie92, Def. 5.1] that bounds the magnitude of the Fourier

coefficients f(h) to be of polynomial decay with degree −α. Using r(h) =max(1, |h|) and

r(h) =∏ni=1 r(hi) we define

f ∈ Enα ⇔∣∣∣f(h)

∣∣∣ < Cr(h)−α for some fixed C > 0.

It is interesting to note that a sufficient condition for a function f ∈ Enα is that all partial

derivatives∂k1+...+kn

∂k1x1 · · · ∂knxnf(x) with 0 ≤ ki ≤ α for 1 ≤ i ≤ n

exist and are continuous on Rn [Nie92, p. 103]. A worst-case representant of Enα can be

constructed as fα(x) = C∑

h∈Zn r(h)−αe2πjh·x for x ∈ Rn. Its quadrature error

Pα(Λ) = Qfα − 1 = C∑

h∈Λ⊥\0

r(h)−α (B.9)

provides a quality measure for the lattice Λ, since the error of the approximation in Eq. B.8


for any f ∈ Enα is bounded by CPα(g,m) [Nie92, Thm. 5.3].

In order to make high-dimensional integrals tractable1, it is essential to either weight

the dimensions [SW98, KS05] or to weight each sample in the integration rule with a den-

sity [HSW04], in both cases the absolute sums of the weights have to be bounded.

Measure preserving randomization and scrambling: The low-discrepancy point set

constructions mentioned so far are all deterministic. While this was one of the goals, there

are also reasons to think about additional randomization, i.e. one that does not destroy

the digital net property. One reason is that for some subspaces the Halton-Hammersley

sequence exhibit undesirable regularities [VC06]. Another reason is that randomization via

scrambling allows for error estimates via repeated experiments [Mat99, p. 61].

Including a random shift c ∈ Rn to offset the integration lattice turns integration rule

Qf into a random variable, whose mean is an unbiased estimate of the integral. Similar

to the original Monte Carlo method, the variance of these shifted rules allows for an error

estimate as discussed by Kuo and Sloan [KS05] and in [SJ94, p. 90].

Latin hypercube sampling is a technique in experimental design that is applicable when

only a low number of points can be queried [SWN03, §5.2]. A LH sample of size m in the

n-dimensional unit cube [0, 1)n can be obtained from the rows of an m×n matrix Π, where

each column is a permutation of A = 0m ,

1m , . . . ,

m−1m . The points in the m rows can be

further displaced within each cell by uniform random shifts. When projecting such a LH

design onto any of the n co-ordinate axes, a perturbed regular 1-D grid is obtained. This

desirable property of the point set projection is complementary to the previously mentioned

criteria of packing, covering, or low-discrepancy uniformity. Hence, hybrid design methods

are a promising research direction pointed out by Santner et al. [SWN03, §5.5].

It is further possible in an LH design to accommodate a non-uniform, separable density

function by adjusting the cell boundaries Ak for each dimension k = 1 . . . n. As cell sizes

grow in regions, where lower density is requested, the variability of the integrand is likely

to increase in these larger cells. For a reliable estimate of the integral both effects have to

be taken into account. This is described in the work of Mease and Bingham [MB06], who

devise an algorithm for choosing the boundaries Ak using a stationary random field with

a given measure of dispersion as model for possible integrands. While this provides one

1A problem is tractable if the error can be bounded by a polynomial in n and m−1 with n being thenumber of dimensions and m the number of sample points.


method for non-uniform sampling, some further examples are discussed in the following.

B.1.3 Non-uniformly distributed point sets

The previous sections focussed on methods for producing sets of uniformly distributed points

in the unit cube. A point set X following a non-uniform, positive density p ∈ L1(Rn) can be

obtained from uniformly distributed ones via the inversion method [PH04a, p. 638]. This

method is also used by Hickernell et al. [HSW04] to extend previous discrepancy bounds to

integrals over weighted unbounded domains.

Adaptive sampling: These techniques make use of information from past sample points

before requesting a batch of new ones. Adaptive refinement of a grid (e.g., Octree) counts

into this category, as well as any form of importance sampling, where the importance distri-

bution evolves as information from an acquired sample becomes available. The optimization-

based designs mentioned in Section 2.3.2 choose a sample to optimize measures of utility

and cost [SWN03]. Utility is attributed to an increase in accuracy and confidence in the

resulting estimate, whereas costs for acquiring a sample may arise from computation time,

storage size, financial costs, or health risks, to name a few possibilities.

If the desired target distribution (proportional to utility minus cost) is not normalized

(i.e. its density does not integrate to one over entire domain) one can use Markov-chain

Monte Carlo (MCMC) sampling to draw a sample from a proportional distribution [ES00,

Ch. 7]. In computer graphics this finds application in light transport [VG97] using MCMC

to focus local exploration in the space of light paths around those of high throughput to

improve approximation where the integrand is large. The related technique of particle

filtering [ADFDJ03] can be employed to sample the modes (maxima) of a dynamically

changing distribution, which, for instance, is used for feature tracking [Rek04].

For adaptive sampling, uniform or space-filling designs discussed above [SWN03, Ch. 5]

can provide initialization sites to ensure sufficient exploration of the space.

B.2 Reconstruction and approximation in vector spaces

The reconstruction of a function from observed sample points is a general problem that

appears in a wide variety of applications. Early examples range back as far as 300 BC, when

Greek and Babylonian astronomers filled gaps in tables of recorded positions of celestial

objects using linear interpolation [Mei02]. This facilitated the creation of calendars to


address down to earth questions, such as when to best till the fields or to bring in the

harvest. Making the leap into a more recent computational setting, function approximation,

as a way of fitting a model to given observations, is a core topic of machine learning and

statistics, where it is also referred to as regression or prediction ([Bis06, pp. 137],[SWN03,

pp. 49]). The signal processing perspective on sampling and reconstruction, used in the

following discussion, is laid out by Unser [Uns00].

Formally, one would like to reconstruct a function f : Rn → Rr from given observations

yk = f(xk) made at points of a sample X = xk ⊂ Rn with index k = 1 . . .m. In the

following, we will focus on the simple case n = r = 1 that is then extended to multi-

variate domains with n > 1. The case of multi-field functions with range dimension r > 1

is not explicitly considered here. A simple approach would be to treat every response

separately, while a combined treatment would allow to improve the representation using

vector quantization ideas of Section 2.3.1.1.

In order to compute function values f(x) at any previously unobserved position x 6∈ Xone has to generalize the information given for X to the entire domain of f . To do so,

some additional knowledge is required that can formally be expressed as membership of f

in a vector space F(Rn) of functions over the domain Rn. Modern signal processing ([BU99,

Uns00],[Gri02, Ch. 7],[CL09]) commonly takes F to be a Hilbert space equipped with an

inner product 〈·, ·〉 and properties as discussed in Appendix A.2. Such a space is complete

under the induced norm ‖f‖ =√〈f, f〉. It contains a countable set of basis functions ϕl

that can be scaled and summed to uniquely represent any function f ∈ F(Rn) as

f =∑l∈Z

clϕl. (B.10)

In computational practice one can only work with a finite number of parameters cl that

for l = 1 . . .m give a vector c ∈ Rm. The ϕl associated with these non-zero coefficients span

the reconstruction subspace V ⊆ F .

B.2.1 Projection, generalized interpolation, and best approximation

The mapping from F to V is performed by a linear approximation operator Q. A function f

in its range V can be represented faithfully and remains unchanged by another application

of Q, the implied property Q2f = Qf is defining for a projection. The best approximation


to f ∈ F in a subspace V ⊂ F is given by an orthogonal projection [Gri02, p. 185, 195]2.

The coefficients for the linear expansion in Equation B.10 can be obtained via func-

tionals λl that inherit the linearity from Q. Applied to continuous functions f they can

be implemented by an inner product with an analysis function ϕl as stated by the Riesz

representation theorem [CL09, p. 132] giving [CL09, p. 40]

cl = λl(f) = 〈ϕl, f〉. (B.11)

For any linear functional λ operating on subspace V one has the generalized interpolation

property λ(Qf) = λ(f) [CL09, p. 41]. Considering only one such functional is similar to the

integration problem of Section B.1. The classical notion of interpolation arises when using

point evaluation functionals λl(f) =∫f(x)δ(x− xl)dx = 〈δ(· − xl), f〉 = f(xl).

For perfect reconstruction the analysis functionals λl are bi-orthogonal to the basis func-

tions3 satisfying 〈ϕl, ϕk〉 = δ(k−l). If ϕl and ϕk do not fulfill the bi-orthogonality condition,

they can be made so by pre-filtering the measurements λl(f) [Uns00, BTU99]. A basis that

is bi-orthogonal to point evaluation functionals of the xk ∈ X is said to be a cardinal basis

for X. In this case the coefficients in Equation B.10 are simply given by the function values

cl = f(xl), corresponding to the standard notion of interpolation.

B.2.2 Determining the parameters for an interpolating function

The method to obtain coefficients via Equation B.11 is useful for theoretical understanding,

but in practise a complete description of f to compute the inner product with the analysis

filter is not available. If point samples of f are given, note that the parameters of an

interpolating approximant fc have to obey the equation system

f = Ac, (B.12)

2To see that an orthogonal projector Q maps a point x ∈ F to the closest point v in its range V, use thePythagorean Law‖x− v‖2 = ‖(x−Qx) + (Qx− v)‖2 = ‖x−Qx‖2 + ‖Qx− v‖2 ≥ ‖x−Qx‖2 [CL09, p. 210]. The Pythagoreanlaw holds, since 〈x + y, x + y〉 = 〈x, x〉 + 〈x, y〉 + 〈y, x〉 + 〈y, y〉 = 〈x, x〉 + 〈y, y〉, if x and y are orthogonal[Gri02, p. 179].

3With ϕk being a basis for V one can write∑l∈Z δ(k − l)ϕl = ϕk = Qϕk =

∑l∈Z λl(ϕk)ϕl. The

coefficients on the left and the right of this unique expansion can be identified, giving the bi-orthogonalityλl(ϕk) = δ(k − l).


where f contains the function values f(xk) for k = 1 . . .m and each row of the interpolation

matrix A ∈ Rm×n ([CL09, p. 4]; a.k.a. design matrix [SWN03, p. 54]) picks up values of

a basis function ϕl at the sample points Ak,l = ϕl(xk). The vector c of n coefficients can

be solved for uniquely only if matrix A is non-singular. If X has more points than there

are basis functions (m > n), Equation B.12 is an over-determined system. In this case, one

can still ask for c to minimize the 2-norm error argminc‖f − fc‖. A solution for this least

squares problem can be obtained as c = A+f using the pseudo-inverse [Bis06, p. 142]

A+ = (ATA)−1AT (B.13)

that performs an orthogonal projection onto the range of A [TI97, p. 81].

The form of matrix A is determined by the choice of basis ϕl and sample point

positions xk. The choices discussed in the following impose structure on A, such as

identity, triangular, Toeplitz, or symmetric positive definiteness and make it easier to solve

for the coefficients in Equation B.12.

Cardinal basis: Such a basis obeys ϕl(xk) = δ(l − k) for all xk in a sample X. This

turns A into an identity matrix and the coefficients in the expansion of Equation B.10

become cl = f(xl). On the real line the Lagrange polynomials ϕl(x) =∏nk=1k 6=l

x−xlxk−xl fulfill

this property and span the space∏n−1(R) of polynomials of degree n − 1 [CL09, p. 11].

One can extend these polynomials for interpolation on Rn by replacing the measure of

distance φ(x, y) = x− y in enumerator and denominator by other ones, such as the 2-norm

φ(x,y) = ‖x− y‖ or any metric φ(x,y) = d(x,y) [CL09, p. 62].

Triangular: Another approach is given by Newton’s polynomials, where a polynomial ϕl

associated with xl has its zeros at all xk for k < l. Hence, A turns into a triangular matrix

that can be solved via back substitution [CL09, p. 57].

Toeplitz: If the basis functions are shift-invariant, i.e. translates of a fixed function ϕ(x−xl) = ϕl(x) with a set of shift vectors that have translational group structure (i.e. X = xlis a lattice as defined in Section 2.3.1.3) [CL09, p. 71], then the columns of A can be ordered

to give constant diagonals, which is also referred to as Toeplitz matrix [Sta02, pp. 50]. Such

a matrix A performs a discrete convolution on c. Hence, for non-cardinal ϕl one can

obtain coefficients for an interpolating fc by inverse filtering of vector f [BTU99]. The

computational efficiency of being able to perform multiplication of a matrix of this type via

a convolution provides a strong application case for lattices for function reconstruction.


Symmetric, positive definite: The Schoenberg interpolation theorem [CL09, pp. 101]

states that a positive definite (and thus invertible) matrix A for radially symmetric basis

functions (RBF) ϕl(x) = h(‖x−xl‖2) can be formed with any completely monotone4 func-

tion h. Its orthogonal set of eigenvectors can be obtained via the Cholesky decomposition.

B.2.3 Perfect reconstruction and approximation

Perfect reconstruction is possible, if f lies in the finite dimensional reconstruction subspace

V. Shannon’s sampling theorem states that a function f with band-limit ω, i.e. f(ω) = 0

whenever |ω| > ω, can be perfectly reconstructed from sample points taken at regular

distance T = xk − xk−1 = π/ω. It involves a classical shift-invariant choice of basis on the

real line with generator ϕ(x) = T · sinc(x/T ), where sinc(x) = sin(πx)/πx, which is cardinal

for regularly spaced point samples and spans the space of band-limited functions [Sha49,

Uns00]. Interpolation in general shift-invariant spaces (i.e. φ 6= sinc) amounts to an oblique

projection, because the δ point evaluation functionals and the reconstruction filter do not

span the same space [Uns00, Sec. III-C].

Approximation error kernel: The approximation error in the general framework of

Hilbert space projections is discussed by Blu and Unser [BU99], who devise a quantitative

method of estimating the L2 approximation error ‖f − Qf‖2 of the operator defined in

Section B.2.1 as

εf (T ) =1

2π

∫ ∣∣∣f(ω)∣∣∣2Eϕ,ϕ(Tω)dω, (B.14)

where the error kernel E depends on analysis and synthesis filters ϕ and ϕ only

Eϕ,ϕ(Tω) =∣∣∣1− ˆϕ(ω)∗ϕ

∣∣∣2 +∣∣∣ ˆϕ(ω)

∣∣∣2∑n6=0

|ϕ(ω + 2nπ)|2 . (B.15)

The extension of Shannon’s theorem, mentioned at the beginning of this section, to

multi-dimensional domains has analog properties to the one dimensional case, where the

sampling distance T is reciprocal to the maximum frequency ω. In the multi-dimensional

version, provided by Petersen [PM62], a signal sampled on a lattice Λ has a frequency

domain representation on the dual lattice Λ⊥ (defined in Equation 2.11).

Using Hilbert space abstraction to model second order stationary random processes

4Complete monotonicity of f means 1. f ∈ C[0,∞), 2. f ∈ C∞(0,∞), 3. (−1)kf (k)(t) ≥ 0 for t > 0, k =0, 1, . . . .


Kunsch et al. [KAH05] derive an expression for the reconstruction error. Their results show

that for high-rate sampling of random fields, which corresponds to a wide, monotonously

decreasing correlation function, the lattice that maximizes the packing radius of the dual

lattice is favourable. Hence, for these high-rate sampling conditions in 3D the BCC lattice

provides an optimal choice, since it has the optimally packed FCC lattice as dual. For low

sampling rate cases they obtain a result that is similar to Johnson et al. [JMY90]. Corre-

sponding to a low correlation among adjacent sample points, the lowest average estimation

error in this setting is achieved when sampling with an optimal packing lattice. In R3 an

optimal choice is given by the FCC lattice.

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http://mathworld.wolfram.com/CorrelationDimension.html

Index

adjugate, 45admissible, 46arithmetic minimum, 33

band-limit, 149

capacity dimension, 22Cartesian lattice, 43center density, 31centrally symmetric, 134characteristic polynomial, 49closed, 132co-factor, 45companion matrix, 50computational model, 1compute node, 15, 101continuity, 134continuous, 132continuous at the point, 132continuous function, 20covering dimension, 133covering problem, 31covering radius, 32covering theorem, 20, 133

data source, 15data source, secondary, 36data source, synthetic, 15data space, 16data table, 17deep hole, 32design matrix, 17, 148digital nets, 140dilation matrix, 46dimensions, 15discrepancy, 139

discrete topology, 132Dyadic subsampling, 47

face, 133face lattice, 27factorial design, 23formal system, 1Fourier series, 143function, 132fundamental parallelepiped, 28, 46

Gamma function, 21generating matrix, 27, 46good, 120Gram matrix, 27

homeomorphism, 20, 132hyper-cube, 21

identical, 45image, 17integration lattice, 142interpolation matrix, 148

Jordan curve, 20

Lagrange polynomials, 23, 148lattice, congruent, 28lattice, dual, 28lattice, similar, 28, 47

manifold, 133maximin criterion, 43metric, 133minimax design, 32Minkowski’s theorem, 32model, 1

166

INDEX 167

multi-field, 16multi-variate, 16

neighbourhood, 132norm, 133

open set, 132

packing density, 31packing problem, 31packing radius, 31parameters, 17point lattice, 27projection, 146pseudo-inverse, 77, 148

reconstruction subspace, 146relation, 132

sample, 17sampling, 19shift-invariant, 148similar lattice, 43simplex, 133space-filling curve, 20sphere, p-norm, 20star-discrepancy, 140system, 1

thickness, 32Toeplitz matrix, 82, 148topological invariants, 20, 132topology, 132tractable, 144transfer function, 57

unimodular matrix, 45

variables, 15visualization, 2Voronoi cell, 25, 28

zonotope, 134

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