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

nD

3D

2D

1D

0D

1D 2D 3D nD

I J

G H

C D E F

A B

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

RIRI 2

3





• Function plot for a 1D scalar field

2D Points

1D manifold: line

}|))(,{( RIssfs

4





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

2D manifold: surface

• Surface representations Wireframe

Hidden lines

Shaded surface

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

5




„The Geometry of Images“

6





7





8




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

RIf :

}),(|),{( cyxfyx

9




Isolines

10




Isolines

11




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 )

12




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

13




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

+

++

+ + +

+ +

+ + +

+

–

–––

14




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

+ +

+ +

+ -

+ +

+ -

+ -

+ -

- +

?

+ -

- +

+ -

- +

?

15




-+

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

+ -

- +

+ -

- +

)(4

11,11,,1,center jijijiji fffff

16




Isolines

• Asymptotic decider Consider the bilinear interpolant within a cell

The true isolines within a cell are hyperbolas

Asymptote

Cell boundaries

Asymptote

17




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)

18




Isolines

• If c we chose the right case, otherwise we

chose the left one

+

+ -

- +

-

+ -

- +

19




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

20




3D Scalar Field Visualization

21




Volumetric Representations

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

• Signed distance function

• Sample on a uniform cartesian grid

• Trilinear interpolation

22





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

23

F000 F100

F110

F111

F011

F001F101





• Indirect Rendering:

Isosurface extraction: Marching Cubes

• Direct Rendering:

Ray Casting

24




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

25




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

26




Surface Sampling

in which voxel ?

27




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

28




Surface Sampling

in which voxel ?

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

with min Fijk c max Fijk

29




Surface Sampling

in which voxel ?



how to find these voxels efficiently ?

30




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

31




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

n

• propagate min/max values upwards

• surface extraction:

traverse only the relevant voxels

O(n3) O(n2 log n)

32




Surface Sampling

• per voxel : ray intersection

• ray ... P + Q

• intersection ... G() = F(P + Q) = c

• solve cubic equation ...

33




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 ...

34




Problems / Bad Approximation

35




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

36




Error Quadrics

• squared distance to plane:

37

• sum distances to vertex' planes:

TTdcbapzyxv ,,,,1,,,

vQvvppvvpvp P

TTTT :,,dist22

2

2

2

2

dcdbdad

cdcbcac

bdbcbab

adacaba

QP

vQvvQvvQvvp v

T

p p

p

T

p

T

p

:,dist planes planes planes

2




Error Quadrics

• optimal point that minimizes the error:

38

• solve using pseudo-inverse (SVD)

(underdetermined vs. overdetermined)

T

p

nn

ppp

p

pp

p

T

pp dcbapndvnn

pp

,,,

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

39




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

40




Visit Each Gray Cell ...

41




Compute Samples on Edges

42

kjdd

di

kjikji

kji,,

,,1,,

,,




kdd

dji

kjikji

kji,,

,1,,,

,,


43




1,,,,

,,,,

kjikji

kji

dd

dkji


44




Lookup Mesh Connectivity

45

. . .




Cell Connectivity Table

46




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

47




Normal cone

for a smooth (flat)

non-feature region

Feature Classification

48





49

Normal cone

for a smooth (flat)

non-feature region





50

Normal cone

for an edge feature

(one large angle)





51





52

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

53




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

54




Lookup Mesh Connectivity

55

. . .




Modified Lookup Table

• Generate triangle fans centered around the

feature sample

56

. . .




Feature Reconstruction

57

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 ...

58




CSG ... CAD / CAM

59

65 x 65 x 65




CSG ... Milling Simulation

60

257 x 257 x 257




Iso-Surface Extraction

61

65 x

65 x

65




Iso-Surface Extraction

62





• Indirect Rendering:

Isosurface extraction: Marching Cubes

• Direct Rendering:

Ray Casting

63




Volume Rendering

64




Volume Rendering

65




Volume Rendering

66




Volume Rendering

67




Ray Casting

• Maximum Intensity Projection

68

rays 0

Scalar value S

Maximum Smax




Maximum Intensity Projection

69




Absorption

70

Ray

s0

I(s0)

s




Absorption

71

Ray

s0

I(s0)

s

Ray

ss~

~q(s)

s0

I(s0)




Ray Casting

• Image order: for each pixel ...

• Shoot rays from eye through each image plane

pixel

• Integrate color c & opacity along the ray r

72

dsescrc

s

dttL

0

)(

0

)()(





73




Opacity

• base color c, opacity

damping factor 1- = e-

emitted color c

• resulting (effective) color

c + (1-) cbackground

74




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)

75




Opacity Accumulation

• Continous compositing

76

• Discrete compositing

decucdttL

i

0

)(

0

)()(DE

DE

1)( ,)(with

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

1

0

kbkgk

K

j

j

m

mjji

xcxc

xxxcuc





77

i

ic

K

Kc

0

0

c





78

C[0..K]

i

ic

K

Kc

0

0

c





79

C[1..K]„back-to-front“

i

ic

K

Kc

0

0

c





80

C[0..K-1]„front-to-back“

i

ic

K

Kc

0

0

c




Back-to-Front Compositing

81

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

)()(

))(1()()(

0

1

1

1

00

0

1

0

xxxxc

xxc

xxxc

K

j

j

m

mjj

K

j

j

m

mjj




Front-to-Back Compositing

• Iterative accumulation j = 0 .. K

with

82

)ˆ1(ˆˆ

)ˆ1(

j

jjcCC

1

0

)1()ˆ1(j

m

m




Transfer Functions

• how to define the

base color c

opacity

• for each voxel ?

• CT, MRI, ...

• diffusion tensor imaging

83




Transfer Functions

84

scalar value s

emission RGB

absorption A

T(s)




Transfer Functions

85

0 1

α




Transfer Functions

86

0 1

α




Transfer Functions

87

0 1

α




Transfer Functions

88

0 1

α

+ Shading




Transfer Functions

89

0 1

α

+ Shading

+ Color coding




Absorption vs. MIP

90




Compositing Schemes

Depth

Inte

nsity Max

Average

Accumulate

First

91




Compositing Schemes

• Compositing: First

• Extracts isosurfaces

Depth

Inte

nsity

First

92




Compositing Schemes

• Compositing: Average

• Produces basically an X-ray picture

Average

Depth

Inte

nsity

93




Compositing Schemes

• Maximum Intensity Projection (MIP)

• Often used for magnetic resonance angiograms

• Good to extract vessel structures

Max

Depth

Inte

nsity

94




Compositing Schemes

• Compositing: Accumulate

• Emission-absorption model

• Make transparent layers visible (- classification)

Accumulate

Depth

Inte

nsity

95




Volume Shading

• Lighting / Shading

Phong model, Gouraud shading

Estimate normal by

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

96

),,(

),,(),,(

zyxF

zyxFzyxN

),,(),,(

),,(),,(

),,(),,(

2

1),,(

11

11

11

iiiiii

iiiiii

iiiiii

zyxFzyxF

zyxFzyxF

zyxFzyxF

zyxF




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

97

otherwise

)()(0)(

)(0)(

,

,

,

)(

)(

0

11

1

)( iii

ii

i

ivi xFvxFxF

vxFxF

xF

vxF

rx





98

1

F





99

r

r F





100

r

r F

r

1




Ray Casting

• Acceleration techniques

Spatial data structures (octree)

Find and skip empty regions

Find homogenous regions, use lower sample

rate

Early ray termination

Opacity threshold terminates ray traversal

Requires front-to-back traversal

Fast cell traversal (Bresenham-like)

101

Documents

Lec 4 Scalar Field Visualization