Lec 4 Scalar Field Visualization

Scalar Field Visualization

Leif Kobbelt

2Visual Computing Institute | Prof. Dr. Leif Kobbelt

Computer Graphics and Multimedia

Data Analysis and Visualization

Types of Data

1D 2D 3D nD

C D E F

dimension of

domain

dimension of

data type

Function Plots and Height Fields

• Visualization of 1D or 2D scalar fields

1D scalar field:

2D scalar field:

RIRI 2

• Function plot for a 1D scalar field

2D Points

1D manifold: line

}|))(,{( RIssfs

• Function plot for a 2D scalar field 3D Points

2D manifold: surface

• Surface representations Wireframe

Hidden lines

Shaded surface

}),(|),(,,{( 2RItstsfts

„The Geometry of Images“

Isolines

• Visualization of 2D scalar fields

• Given a scalar function

and a scalar value 𝑐 ∈ 𝑅• Isoline consists of points

• If f() is differentiable and grad(f) 0, then isolines are

curves

• Contour lines

}),(|),{( cyxfyx

Isolines

• pixel by pixel contouring

• straightforward approach: scanning all pixels for isovalue

• input f : (1,...,xmax) x (1,...,ymax) R

Isovalues I1,..., In and isocolors c1,...,cn

• algorithmfor all (x,y) (1,...,xmax) x (1,...,ymax) do

for all k { 1,...,n } do

if |f(x,y)-Ik| < then

draw(x,y,ck)

• problem: isolinie can be missed if the gradient of f() is too large

(despite range )

Isolines

• Marching squares representation of the scalar function on a rectilinear grid

scalar values are given at each vertex ffij take into account the interpolation within cells

isolines cannot be missed

divide and conquer: consider cells independently of each other

Isolines

• Which cells will be intersected ? Initially mark all vertices by + or – , depending on the

conditions fij c , fij < c

• No isoline passes through cells (=rectangles) which have

the same sign at all four vertices So we only have to determine the edges with different signs

–––

Isolines

• Only 4 different cases (classes) of combinations of signs

• Symmetries: rotation, reflection, change + -

• Compute intersections between isoline and cell edge,

based on linear interpolation along the cell edges

Isolines

• We can distinguish the cases by a decider

• Mid point decider Interpolate the function value in the center

If fcenter < c we chose the right case, otherwise we chose the left case

Not always correct solution

11,11,,1,center jijijiji fffff

Isolines

• Asymptotic decider Consider the bilinear interpolant within a cell

The true isolines within a cell are hyperbolas

Asymptote

Cell boundaries

Asymptote

Isolines

• Interpolate the function bilinearly

• Transform f() to

• is the function value in the

intersection point

of the asymptotes

xyfyxfyxfyxfyxf jijijiji 111,,1, )1()1()1)(1(),(

))((),( 00 yyxxyxf

(x0, y0)

Isolines

• If c we chose the right case, otherwise we

chose the left one

Isolines

• Explicit transformation f() to

can be avoided

• Idea: investigate the order of intersection points either

along x or y axis

• Build pairs of first two and last two intersections

))((),( 00 yyxxyxf

valid not possible

3D Scalar Field Visualization

Volumetric Representations

• Implicit representation F(x,y,z) = 0

• Signed distance function

• Sample on a uniform cartesian grid

• Trilinear interpolation

F(i+u,j+v,k+w) =

F000 (1-u)(1-v)(1-w) +

F100 u (1-v)(1-w) +

F010 (1-u) v (1-w) +

F111 u v w

F000 F100

F001F101

• Indirect Rendering:

Isosurface extraction: Marching Cubes

• Direct Rendering:

Ray Casting

Volume Data

• discrete representation:

Fijk = F(i x, j y, k z)

• piecewise polynomial (linear) interpolation

• tri-linear functions per cell („voxel“)

algebraic iso-surfaces of degree 3

• approximation by a polygonal mesh

Surface Extraction

• Find point samples on the iso-surface

exploit voxel structure

use function values

• Connect neighboring samples

exploit voxel neighborhood structure

relation between voxels and polygons

Surface Sampling

in which voxel ?

Surface Sampling

in which voxel ?

F((i+u) x, (j+v) y, (k+w) z) =

(1-u)(1-v)(1-w)Fijk + (1-u)(1-v) w Fijk+1 +

(1-u) v (1-w)Fij+1k + (1-u) v w Fij+1k+1 +

u (1-v)(1-w)Fi+1jk + u (1-v) w Fi+1jk+1 +

u v (1-w)Fi+1j+1k+ u v w Fi+1j+1k+1 +

(u,v,w) [0,1]3 min Fijk F(u,v,w) max Fijk

Surface Sampling

in which voxel ?

Sc[F] passes through all (and only) voxels

with min Fijk c max Fijk

Surface Sampling

in which voxel ?

how to find these voxels efficiently ?

Surface Sampling

in which voxel ?

how to find these voxels efficiently ?

- store min/max {Fijk,Fijk+1,Fij+1k,Fij+1k+1,

Fi+1jk,Fi+1jk+1,Fi+1j+1k,Fi+1j+1k+1}

- build octree

Min/Max Tree

• store min/max for every voxel Vijkn

• octree Vijkn-1 has children

V2i 2j 2kn ... V 2i+1 2j+1 2k+1

• propagate min/max values upwards

• surface extraction:

traverse only the relevant voxels

O(n3) O(n2 log n)

Surface Sampling

• per voxel : ray intersection

• ray ... P + Q

• intersection ... G() = F(P + Q) = c

• solve cubic equation ...

Surface Sampling

• per voxel : ray intersection

• ray ... P + Q

• intersection ... G() = F(P + Q) = 0

• solve cubic equation ...

• ray parallel to coordinate axis

solve linear equation ...

Problems / Bad Approximation

Surface Sampling

• use additional information to improve

iso-surface approximation

• normal vector to iso-surface Sc[F]:

F(x,y,z) = [ F/x, F/y, F/z ]

• find surface samples by (approximately)

intersecting tangent planes

Error Quadrics

• squared distance to plane:

• sum distances to vertex' planes:

TTdcbapzyxv ,,,,1,,,

vQvvppvvpvp P

TTTT :,,dist22

dcdbdad

cdcbcac

bdbcbab

adacaba

vQvvQvvQvvp v

:,dist planes planes planes

Error Quadrics

• optimal point that minimizes the error:

• solve using pseudo-inverse (SVD)

(underdetermined vs. overdetermined)

pp dcbapndvnn

1 with :

Surface Data

• polygon meshes / triangle meshes

M = ({pi},{Tj})

• compute samples on grid edges within voxels

• compute faces (vertex connectivity) for each voxel for each grid edge

Marching Cubes [Lorensen/Cline ´87]

• extract surface patch for every grid cell

classification by the signs at the corners

(black, white, gray)

• approximate samples

linear interpolation of scalar distance values

along cell edges

• look-up table with pre-computed triangulations

28 entries, many symmetries

Visit Each Gray Cell ...

Compute Samples on Edges

kjikji

Lookup Mesh Connectivity

Cell Connectivity Table

Extended Marching Cubes

• same algorithmic principle

• for each cell ...

Find surface samples on the grid cell edges

Evaluate surface normals at the sample points

Feature detection

Feature sampling

Feature reconstruction

Normal cone

for a smooth (flat)

non-feature region

Feature Classification

Normal cone

for a smooth (flat)

non-feature region

Normal cone

for an edge feature

(one large angle)

Normal cone

for a corner feature

(two large angles)

Feature Detection

• opening angles of the normal cone = maxi,j <ni, nj> = mink <nk, ni x nj>

• edge features > 0, > 0 sharp edge

• corner features > 0, <= 0 sharp corner

Feature Sampling

• Each pair ( pi , ni) defines a tangent element

• Intersect tangent planes to approximate

a piecewise smooth surface

• Find least squares solution by SVD

Overdetermined cases

Underdetermined cases

Suppress smallest singular value at edge

features

Lookup Mesh Connectivity

Modified Lookup Table

• Generate triangle fans centered around the

feature sample

Feature Reconstruction

feature

detection

feature

samplingedge-flipping

Results

• Mesh complexity Two additional triangles per feature sample Typically 10% overhead

• Computation time 20% to 40% overhead

• Improved approximation

• Parameters 0 and 0

False positive feature detection !?

• Applications ...

CSG ... CAD / CAM

65 x 65 x 65

CSG ... Milling Simulation

257 x 257 x 257

Iso-Surface Extraction

• Indirect Rendering:

Isosurface extraction: Marching Cubes

• Direct Rendering:

Ray Casting

Volume Rendering

Ray Casting

• Maximum Intensity Projection

rays 0

Scalar value S

Maximum Smax

Maximum Intensity Projection

Absorption

Ray Casting

• Image order: for each pixel ...

• Shoot rays from eye through each image plane

• Integrate color c & opacity along the ray r

dsescrc

Opacity

• base color c, opacity

damping factor 1- = e-

emitted color c

• resulting (effective) color

c + (1-) cbackground

Ray Casting

set color c(.) and opacity (.) for all voxels

for all pixels ui = (ui, vi)

find ray from eye through pixel ui : eye + d

init cacc(ui) = acc(ui) = 0

for all samples xj = (xj, yj, zj) = eye + j d

get c(xj) and (xj) by tri-linear interpolation

composite with cacc(ui) and acc(ui)

Opacity Accumulation

• Continous compositing

• Discrete compositing

decucdttL

)()(DE

1)( ,)(with

))(1()()()(0

xxxcuc

C[0..K]

C[1..K]„back-to-front“

C[0..K-1]„front-to-back“

Back-to-Front Compositing

))(1())(1()()(

))(1()()(

Front-to-Back Compositing

• Iterative accumulation j = 0 .. K

)ˆ1(ˆˆ

)1()ˆ1(j

Transfer Functions

• how to define the

base color c

opacity

• for each voxel ?

• CT, MRI, ...

• diffusion tensor imaging

Transfer Functions

scalar value s

emission RGB

absorption A

Transfer Functions

+ Shading

Transfer Functions

+ Shading

+ Color coding

Absorption vs. MIP

Compositing Schemes

nsity Max

Average

Accumulate

Compositing Schemes

• Compositing: First

• Extracts isosurfaces

Compositing Schemes

• Compositing: Average

• Produces basically an X-ray picture

Average

Compositing Schemes

• Maximum Intensity Projection (MIP)

• Often used for magnetic resonance angiograms

• Good to extract vessel structures

Compositing Schemes

• Compositing: Accumulate

• Emission-absorption model

• Make transparent layers visible (- classification)

Accumulate

Volume Shading

• Lighting / Shading

Phong model, Gouraud shading

Estimate normal by

Approximate F(x,y,z) by central difference

),,(),,(

zyxFzyxN

),,(),,(

iiiiii

zyxFzyxF

Volume Rendering Modes

• Classification

assign spatial opacity values (xi, yi, zi)

assign for isovalues F(x,y,z) = (transfer

function)

isosurface ... = (=0) ... blur with radius r

otherwise

)()(0)(

)( iii

ivi xFvxFxF

Ray Casting

• Acceleration techniques

Spatial data structures (octree)

Find and skip empty regions

Find homogenous regions, use lower sample

Early ray termination

Opacity threshold terminates ray traversal

Requires front-to-back traversal

Fast cell traversal (Bresenham-like)

Lec 4 Scalar Field Visualization

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