CSL 859: Advanced Computer Graphics

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CSL 859 Advanced Computer Graphics

• Dept of Computer Sc. Engg.
• IIT Delhi

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Volumetric Models
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Not a Surface
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Not a Surface
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Volumes

• 3D fields
• Temperature, Pressure
• Generally sampled
• Constructed by Interpolation
• Filter convolution, in general
• Measured or Simulated
• CT-scan, Xray
• CFD Simulation
• Conversion from surface (B-rep)

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Translucence
Note b is absorbed, scatterred and
transmitted.
Color f (a, b)
Color a
Color b
f should depend on Degree of translucence
a Depth of a and b
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Alpha Blending
f s (1-a)d
0 lt a lt 1
Color f (s, d)
f as (1-a)d
Color s
Opacity a
Color d
f ass add
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Over Blend Operator

• Porter Duff 84
• Final fraction
• aout aa ab(1- aa)
• Final effective Color
• cout aaCa ab(1- aa) Cb?
• Recall pixel has aout fraction of the color
• Cout aaCa ab(1- aa) Cb / aout

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Order Dependent

• d' as s (1-as) d
• Examples aA 1, aB 0.4

A over B d' 1CA(1-1)CB
A
B
B over A d'
0.4CB(0.6)CA
A
B
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Order Dependent
Each quad has an opacity of 0.5
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Billboarding etc.
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Volume Representation

• BRDF for every point in space?
• Sometimes only a limited information is
  available, eg, tissue density in CT scan
• Sample Volume
• Computable function at any required point
• Subdivide space into a 3D grid and sample at grid
  points
• Voxel
• Could be a structured or unstructured grid
• With scalar or vector data (sample) per vertex

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Grids
Tetrahedral
Curvilinear
Tetrahedral
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Volume Illumination

• Layers of surfaces
• Particles in space
• Voxel emissions
• Continuous model
• Drebin 98
• Surfaceness at every point
• Based on differense in material densities
• Surface normals from density gradient
• Direct sample value -gt ColorOpacity map
• Transfer function often domain-dependent

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Volume Rendering Techniques

• Iso-surface display
• Ray-casting
• Image-space accumulation
• Back-to-front or Front-to-back
• Sample Ray at many t values
• Composite
• Slice Projection
• Object space accumulation
• Splatting
• Shear-warp factorization
• 3D textures
• Multiple transparent planes
• Or some other shape
• Particles
• Slices

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Slicing Planes
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Particles

• Depth Cues
• Animation
• Shape/Density

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Iso-surfaces

• Create boundary representation (B-rep)
• Usually triangles
• Render as surfaces
• Possibly multiple translucent surfaces
• Surfaces need to be carefully chosen
• Large change across them

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Iso-surfaces

• Multiple Surfaces with varying opacity
• Can impart thickness
• Clipping planes

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Marching Cubes

• Corners of a voxel are inside or outside
• 28 possible configurations
• Only 15 are unique
• Modulo reflection, rotation, inversion
• Create triangles for each case
• Store in a table lte1, e2, e3gt
• Compute vertex location by interpolation

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Ambiguity
2D Example
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Ambiguity Resolution

• Its an aliasing issue
• Increase resolution
• Make consistent with adjacent voxel
• There may be loops
• Separate out into more cases
• Cube 33 Chernyaev 1995
• Sample at the center of the voxel
• Subdivide voxel cube into tetrahedra
• Marching tetrahedra

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Efficiency

• Large number of voxels visited
• Hierarchical structures to cull voxels
• Span-space tree
• Large number of triangles generated
• Polygon simplification

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Min-Max Kd-Trees

• Construction

• Root has median min value left subtree has
  smaller and right larger
• For each sub-tree, root has median max value
• Alternate

Livnat, Shen Johnson 1996
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Search

• Compare threshold (t ) with min of root cell,
• If greater, test cell at this node, then test
  both children
• If less, traverse left

m0
M10
M11
m21
m22
m20
m23
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Lattice Sub-Division

• Shen, Hansen et al. 1996
• A range can be represented as a single point in
  the span space

• Subdivide span space into a lattice and place
  each data cell into a lattice element

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Lattice Sub-Division

• Five cases
• No intersection
• Definite intersection
• Test max only
• Test min only
• Test min and max

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Voxel Splatting

• Traverse voxels front to back
• Normally one slice at a time
• Project voxel into screen-space
• Hexagonal foorprint in general
• Accumulate color and opacity to covered pixels
• Filter Kernel drops of contribution away from
  center of the voxel

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Raycasting
Sample ray at least once per voxel intersected
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Ray Integration
x(t) ray, parameterized by t
s(t) scalar value c(t) color emitted
light a(t) absorption coefficient
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Discrete Ray Integration
n
i-1
C S C (1- a )
i i
0
0
D
0
C
C C (1-a ) C
i i i1
Requires back to front blending step from n-1
to 0
Alternatively Use Over operator
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Illumination

• Phong/Blinn Model
• For each sample, evaluate
• C ambient diffuse specular
• constant I Kd (N.L) I Ks
  (N.H)n
• I emission color at the sample
• N normal at the sample

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Phongs Model Applicable
C(p) Cl Ka Cl / (K1 K2 d(x))
Kd (N(p) . L Ks (N(p) . H )n
Cl color of light ka,kd,ks ambient,
diffusive, specular coefficient K1,K2
constant (used for depth attenuation) N(x)
normal at x
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Normal Estimation

• Compute the gradient at each corner
• Interpolate the normal using central difference

y1
N(x,y,z) (f(x1) - f(x-1))/2,
(f(y1) - f(y-1))/2,
(f(z1) - f(z-1))/2
z1
x-1,
y-1,
z-1
x1
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Transfer Function

• Maps from scalar value to opacities
• Region of interest high opaicity (more opaque)
• Rest translucent or transparent
• Opacity is typically specified by the user
• Examples for computing opacity Levoy 88
• Isosurface
• Region boundary (e.g. between bone and fresh)

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Transfer Function
Red
v
Green
(R, G, B, A)
v
Blue
v
Alpha
value
v
Fragment Shader
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Draw Region of Interest

• Goal Visualize voxels that are near a
• selected threshold value t
• No iso-surface extraction
• Alpha test
• void AlphaFunc (enum func, clampf ref )
• Assign maximum opacity to voxels with value t
• Construct a transition area with reducing opacity
• Normally of constant thickness

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Opacity Construction
Maintain a constant isosurface thickness, R
Opacity k (v-t) ?
Distance to value t is a non-local operation
involves a search
opacity a
opacity 0
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Opacity Construction
Assuming a constant gradient, dv/dx Dv/Dx
(value change per voxel)
thickness R
opacity a value t
opacity a a (t - v(x)) / (R ?v/?x)
opacity 0 v t ?v/ ?x R
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Texture Mapping

Single slice 2D image
Textured-mapped polygon
2D polygon
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Volume Texture Mapping
2D Textures
Side view
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Volume Textures
Proxy Geometry

• Each xz slice is a texture on a proxy polygon
• Blend polygons (traditionally back to front)

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Changing Viewing Direction
Simply change the view matrix and re-render
Until
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Changing View Direction
Until the view direction is near parallel to the
slice planes Ray through the pixel intersects no
polygon
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Re-slice Volume

• Solution
• Re-orient the polygons
• In general, pick orientation most perpendicular
  to the viewing direction

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Three Textures

• Must reorganize the input textures for different
  views
• Pre-process and store three separate texture sets

xz slices
yz slices
xy slices
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Texture Volume Rendering
Algorithm Disable depth test Enable
blending For (each slice from back to front)
- bind appropriate texture - Create a
quad corresponding to the slice - Assign
texture coordinates to vertices - Render
(using OpenGL alpha blending)
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Varying Sampling Rate
d
d
d
d gt d gt d
Artifacts due to different sampling at
neighboring pixels
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Solution

• Insert intermediate slides to maintain the
  sampling rate

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Slice switching
There is a sudden change of slicing direction
when the view vector transits from one major
direction to another. The change (popping
artifact) in the image intensity can be quite
visible
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Solution
Use Image-space axis-aligned slicing plane
Modify slicing plane every frame, resulting in
small change
Must clip polygons to volume!
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3D Texture Mapping
Input texture space is 3D Texture coordinates
(r,s) -gt (r,s,t)
(0,1,1)
(1,1,1)
(0,1,0)
(1,1,0)
(1,0,0)
(0,0,0)
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3D Texture Mapping

• Every point inside the volume can now be sampled
• Interpolated from neighboring points
• No need to ensure polygons aligned with slices
• Slicing independent
• No need to re-organize into separate 2D textures

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Unstructured Grid

• Tetrahedral Models
• Re-Sample into a uniform grid (texture)
• Render B-rep of each polyhedron
• Transparent polygons
• Ray-cast
• Need to compute the next tetrahedron
• Splatting
• Ordering is expensive
• Order per scan-line in the plane passing that
  scanline
• Reduces to 2D sorting

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Curvilinear Grid

• Ordering a little simpler than unstructured grids
• Two step warp
• Grid points are effectively arranged into a
  3D-array
• Inverse-map from Euclidean to grid space
• Texture look-up