Signal-Specialized Parameterization for Piecewise Linear Reconstruction

Geetika Tewari, Harvard UniversityJohn Snyder, Microsoft Research Pedro V. Sander, ATI Research

Steven J. Gortler, Harvard University Hugues Hoppe, Microsoft Research

Two Scenarios

Authoring: map a texture image onto a surfaceAuthoring: map a texture image onto a surfaceAuthoring: map a texture image onto a surfaceAuthoring: map a texture image onto a surface

Sampling: store an existing surface signalSampling: store an existing surface signalSampling: store an existing surface signalSampling: store an existing surface signal

Signal Source

Procedural texture

High-resolution image texture

High-resolution geometry

Our Goal

ReconstructedReconstructedReconstructedReconstructed

OriginalOriginalOriginalOriginal

Signal-Specialized Parameterization

Signal Independent Parameterization

Related Work

Pinkall and Polthier. 1993. Computing Discrete Minimal Surfaces… Eck et al. 1995. Multiresolution Analysis of Arbitrary Meshes. Hormann and Greiner. 2000. MIPS – global parameterization method. Levy et al. 2002. Least Squares Conformal Maps. Desbrun et al. 2002. Intrinsic Parameterizations of Surface Meshes.

Sloan et al. 1998. Importance Driven Texture Coordinate Optimization.

Hunter & Cohen. 2000. Uniform Frequency Images. Balmelli et al. 2002. Space Optimized Texture Maps. Sander et al. 2002. Signal-Specialized Parameterization.

Signal Specialized Parameterization

Signal Independent Parameterization

Our Contribution

Sander et al. 2002 Metric derived using piecewise constant reconstruction

assumption

New metric for signal-specialized parameterization Assumes piecewise linear reconstruction (more realistic). We empirically evaluate and compare our metric’s results

with Sander et al.

Signal-Specialized Parameterization

ff

h = g . fh = g . f gg

surfacesurface

Signal rangeSignal rangeSignal rangeSignal range

domaindomain originaloriginaloriginaloriginal

reconstructedreconstructedreconstructedreconstructed

signal signal approximation approximation

errorerror

signal signal approximation approximation

errorerror

Piecewise constant

reconstruction

Piecewise constant

reconstruction

Signal-Specialized Parameterization

originaloriginaloriginaloriginal

reconstructedreconstructedreconstructedreconstructed

signal signal approximation approximation

errorerror

signal signal approximation approximation

errorerror

Signal Signal rangerange

Signal Signal rangerange

ff

h = g . fh = g . f gg

domaindomaindomaindomain

Signal height range on domain scanline

surfacesurfacesurfacesurface

Piecewise linear

reconstruction

Piecewise linear

reconstruction

Derivation of Metric

Signal approximation error

= h – (reconstructed h)

Texture domainTexture domain

ij

si

tj

t

s

Single sample

Neighborhood (single red square of area 2)2(

How well is approximated when reconstructed from a discrete sampling over the texture domain D?

fg h

Metric: how to derive

Define error at each point Represent signal as Taylor series Assume reconstruction is linear Error is dominated by 2nd order term

Derive error integrated over ij

Sum ij over whole surface

In the limit, (more and more samples) this sum becomes an integral

... And error vanishes Divide by to obtain convergence

rate Use this as “energy” metric

Partition integral into sum over triangles.

ij

si

tj

t

s

4

Texture domainTexture domain

Error Metric

),(),(),(

),(

),(

),(

iittiistiiss

iitt

iist

iiss

tshtshtsh

tsh

tsh

tsh

Define 3 by 3 matrix at each point (squares of second derivatives).

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Error Metric

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~

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),(ˆ),()(~

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i

i

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ts

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tsAdtsHH

Integrate over each triangle to get “tilded” 3 by 3 matrix, for each triangle

Error Metric

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HSHTi

i

Sum over all all triangles to get “tilded” H for entire surface

Error Metric

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1~)(~

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4)(~

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2)(~

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1)( SSSSSE

Add up 4 of the terms in the matrix (the entire matrix is kept for the upcoming affine transform rule).

Numerical Computation of Metric

We need to compute

Compute Compute Numerical Integration of HNumerical Integration of H

Compute H:Compute H: Function FittingFunction Fitting Isometric flatteningIsometric flattening

its

i

i

i

i

i

i

i

i

i

i tsAdtsHH),(

)(~

)(~)(~

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~)(~

),(ˆ),()(~

)(~

iH

Subdivide faces into subfaces (1-to-4)

Numerical Integration

Compute H at center of each subface

Sum up matrices over all subfaces.H~

Multiply by corresponding surface area to get approximate H~

Function Fitting

Assumption: the signal can be point sampled at parameter domain points (s,t)

least squares solver

FEtDsCstBtAs 22

FEDCBA ,,,,,

BCA

B

C

A

H 22

2

2

Second derivatives:

CBA ,2,2

n

k

kkk

k

k

k

BCA

B

C

A

H1

)2()2(

)2(

)2(

Choosing Points for Local Signal Fitting

During local signal fitting in numerical computation of H: How do we choose (s,t) coordinates for local signal fitting?

Canonical parameterization

(0,0) (1,0)

(0,1)

Isometric Flattening

It might be necessary to include samples over neighboring faces

Isometrically map face to (s,t). Isometrically flatten three neighbors. For 3 subdivisions we use 15 points.

Parametrization algorithm

Start with uniform parametrization. Iterate: for each vertex, try random line searches Minimize:

But: This is too time consuming on large meshes - Need multigrid

method. We modify the parameterization algorithm by Sander et al

[2002]

)(5

1~)(~

9

4)(~

9

2)(~

5

1)( SSSSSE

Neighborhood Optimization

OptimizatioOptimizationn

OptimizatioOptimizationn

H4’H4

1H

2H

5H

3H

5H

2H

3H

1H

4H

4H

Affine Transformation Rule

needs to be evaluated every time we change the parameterization.

Useful trick: Precompute with respect to some chosen s,t coordinates.

When triangle is warped, can be updated in closed form without resampling the signal (affine transform rule).

New New parameterizationparameterization

Initial (fixed) Initial (fixed) parameterizationparameterization

Affine Affine transformtransform

H~

H~

H~

H~

H~

Affine Transformation Rule

J matrix: Jacobian of the mapping from new triangle parameterization to old.

Linear system of untransformed second derivatives:

Thus H can be transformed via:

Summary:

New parameterization

yielding

tt

st

ss

tt

st

ss

tt

st

ss

h

h

h

Q

h

h

h

J

JJ

J

JJ

JJJJ

JJ

J

JJ

J

h

h

h

222

2221

221

2212

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212

1211

211

2

)(

2

TTttstss

tt

st

ss

QQQhhh

h

h

h

Q

TQHQH

TQSHQH )(~~

Initial (fixed) Initial (fixed) parameterizationparameterization

Relationship to Approximation Theory

Goal (Approximation Theory): Approximate some bivariate scalar function g(x,y) using linear interpolation with a given number of triangles over the (x,y) plane.

Result Can minimize the error of piecewise linear approximation (L2 sense) by [Nadler, 1986]:

As number of triangles An optimal orientation of a triangle is given by the

eigenvectors of the Hessian of g An optimal aspect ratio of a triangle is given by:

2

1

min

max

Correspondence with Nadler’s Result

Hessian of g:

Nadler’s result: optimal triangles are axis aligned and with aspect ratio

Nadler’s solution minimizes our energy functional!

If this bivariate scalar function is quadratic: FByAxyxg 22),(

B

A

20

02

2

1

B

A

Experiments

Signal Approximation Error (SAE): RMS difference between the original signal and its bilinear reconstruction.

We compare our results with the signal-stretch metric of Sander et al. [2002]

reconstructionreconstruction

reconstructedreconstructedreconstructedreconstructed

signasignall

surfacsurfacee

Parameterization Parameterization algorithmalgorithm

Hardware: Bilinear

Assume: linear

1

10

100

1.E+02 1.E+03 1.E+04 1.E+05 1.E+06

Texture size (texels)

Sig

na

l ap

pro

x. e

rro

r

Signal Error - Piecewise Constant Reconstruction

Signal Error - Piecewise Linear Reconstruction

Parasaur's head: 3,870 vertex parameterization

Parasaur’s head

Sander et al.

Ours

Factor of 4 savings in texture area

256x256128x128

SAE: 5.5 SAE: 5.2

Ours

Signal consists of surface normal. Signal is obtained by normal-shooting from geometry to a

high resolution mesh.

Sander et al.

Fandisk

128x128

SAE: 5.3

Signal consists of surface normal.

128x128

SAE: 2.9

Ours

Sander

et al.

Fandisk

OursSander et al.

Comparison with Previous Work

ComparisonMeasure

Terzopoulos & Vasilescue

1991

Sloan et al. 1998

Hunter &

Cohen 2000

Balmelli et al.2002

Sander et al.2002

Ours

Applications

Adaptive sampling

and reconstructio

n

Compact Texture Maps

MetricAdjustable

spring energy

Wavelets &

User- specified

scalar importanc

e

Fourier Wavelets

SAE(1st

order Taylor)

SAE(2nd order

Taylor)

Remarks

Requires optimizati

on procedure

Allows user

control

Fast greedy algorith

m

Texture deviation error

Constant re-

construction

Captures signal

anisotropy Faster

Linear re-

constriction

Captures signal

anisotropy

Slower by 1-2

minutes

Conclusions and Future Work

Future Directions Perceptual measures More optimal treatment of signal discontinuity Non-asymptotic analysis Optimization of texture samples

Significant savings in texture space for the same level of signal approximation error compared to metric by Sander et al.

Thank you…