1. © 2003 University of Wisconsin
Last Time
Sub-structuring
Adaptive Meshing
Hierarchical Radiosity
02/03/03
© 2003 University of Wisconsin

2. © 2003 University of Wisconsin
Today
Importance
Better Form Factors
Different Basis Functions
Meshing
02/03/03
© 2003 University of Wisconsin

3. © 2003 University of Wisconsin
Importance
For a single image, only the visible surfaces are directly important
Don’t care what’s “down the hall”
Surfaces illuminating visible surfaces are also important, and other surfaces illuminating those surfaces…
Importance propagates backwards from the “primary” sources of importance
Visible surfaces are typically prime sources of importance
02/03/03
© 2003 University of Wisconsin

4. Solving for Importance
Ri is the inherent importance of a given element
When radiosity is gathered, importance is shot
Note the form factor in the above equation, and the reflectance term
Use BIF as error estimate in determining subdivision
Combine with multi-grid method
Note: must use different push-pull operations for importance
Push down area-weighted, pull up non-weighted
02/03/03
© 2003 University of Wisconsin

5. © 2003 University of Wisconsin
Form Factor Errors
Hemicube replaces the form factor double integral with a point to area form factor
Assumes small changes over the source patch (Fij is Fsource-receive)
Violated if patches are close – cosine terms and distances vary greatly
Violated if visibility varies across source patch
Limited hemicube resolution results in aliasing
Small patches may not be seen, or may “beat”
We talked about this last time
Reciprocity relation used to compute progressive radiosity form factor is applied incorrectly
02/03/03
© 2003 University of Wisconsin

6. © 2003 University of Wisconsin
Hemicube Beating
Raw patch intensities
Interpolated
02/03/03
© 2003 University of Wisconsin

7. Decoupling Hemicube Duties
Hemicube computes two distinct things:
Visibility
Sub-area form factors – uses pixel form factor to estimate point to element form factor
Replace one of both with alternate methods
Use hemicube visibility but compute patch to element form factors accurately
Visibility from ray-casting
Sub-area form factors from disc approximation
02/03/03
© 2003 University of Wisconsin

8. © 2003 University of Wisconsin
Shooting to Vertices
“Standard” progressive radiosity shoots from points to areas (due to hemicubes)
Point is middle of shooting patch
Area is the area of a receiving element
Better to shoot from area to points:
Area is source patch, point is receiving vertex
Progressive radiosity really needs Fji
Less aliasing under certain circumstances
Radiosity is needed at vertices for display
Greater control over receivers
02/03/03
© 2003 University of Wisconsin

9. Patch-to-Element Form Factors
Use hemi-cube to determine visible elements
For each pixel, compute form factor from visible point back to the source patch, ie compute Fji directly
Each pixel is a small piece of receiving element, and source patch may be large, so point-patch form factor is more accurate
For each receiving element, area-weight sum the point-patch radiosities
Which violated assumption of hemicube does this address?
02/03/03
© 2003 University of Wisconsin

10. © 2003 University of Wisconsin
Ray-casting
Aim: compute
Cast rays from y to sub-patches, Pik, of Pi:
Ray determines (constant) visibility for each sub-patch
Then:
02/03/03
© 2003 University of Wisconsin

11. © 2003 University of Wisconsin
Delta Form Factors
We need to compute Fk
Several possibilities
Assume constant, evaluated using angles and radius for a point at the center of the patch (can be very bad)
Use analytic methods, point-polygon (costly)
Use disc approximation
Assume sub-patch is a small disc
Form factor for point-disc is known
02/03/03
© 2003 University of Wisconsin

12. © 2003 University of Wisconsin
Enhancements
Sample points on source can be varied
Keep subdividing source patch until the estimate for each sub-patch is similar to that for its children
Place receiving vertices on all surfaces - no chance to miss surfaces entirely
Stratified sampling
Arbitrary precision with more rays
But eventually the “constant radiosity over patch/element” assumption bites
02/03/03
© 2003 University of Wisconsin

13. © 2003 University of Wisconsin
More ways to improve
Ray casting is expensive
Visibility and all the standard ray-tracing speedups apply
Note that test for visibility is like shadow-ray test, so use caching
Spatial subdivisions (eg octree) help too
We are still assuming constant radiosity across patches, and computing meshes on the fly
Allow different basis functions (non constant)
Better meshing
02/03/03
© 2003 University of Wisconsin

14. © 2003 University of Wisconsin
Return to Assumptions
Recall the two fundamental assumptions of the standard radiosity approach:
Constant radiosity over each element
Perfectly diffuse surfaces
We wish to more accurately represent the radiosity at every point:
Regardless of the quality of the solution (form factors, subdivision, etc.), if the correct radiosity cannot be represented, it won’t be computed
02/03/03
© 2003 University of Wisconsin

15. Better Radiosity Representations
Two approaches:
Work around the constant radiosity assumption by constructing meshes where elements really should have constant (or bilinear) radiosity
Use more expressive representations for radiosity (work under different assumptions)
02/03/03
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16. © 2003 University of Wisconsin
Meshing Artifacts
Where does poor meshing really show?
Find the (many) artifacts in the image
02/03/03
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17. © 2003 University of Wisconsin
Typical Meshing
Lays down an orthogonal grid in some orientation
May subdivide, but the principle directions remain unchanged
Require subdivision down to pixel resolution to remove artifacts
02/03/03
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18. © 2003 University of Wisconsin
Staircase Effect
Most evident at shadow boundaries:
Elements are dark if they have more than half shadow, light otherwise
Solution:
Subdivide the mesh to sub-pixel resolution (costly)
Blur the boundaries (introduces other problems)
Align element boundaries with the shadow boundaries (discontinuity meshing, later)
02/03/03
© 2003 University of Wisconsin

19. © 2003 University of Wisconsin
Staircase Effect
02/03/03
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20. © 2003 University of Wisconsin
Light and Shadow Leaks
Occur when the mesh does not respect geometric boundaries
Interior walls meeting the floor
The wall appears to float
Shadows leaking out from under objects
No reasonable perceptual interpretation – these look worse
Solutions:
Ensure that there are element boundaries at geometric intersections (hard to do without user help)
Discontinuity meshing
02/03/03
© 2003 University of Wisconsin

21. © 2003 University of Wisconsin
Shadow Leaks
02/03/03
© 2003 University of Wisconsin

22. Interpolation Artifacts
Have to “fill in” radiosity between vertices, or blur element boundaries
Interpolation schemes are frequently poorly implemented, particularly those in hardware
Solutions:
Meshes that make interpolation easier
More expensive interpolation
02/03/03
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23. © 2003 University of Wisconsin
Poor Interpolation
02/03/03
© 2003 University of Wisconsin

24. Preprocessing the Mesh
Seek to remove inconsistencies before starting the solution
Ensure invariants are met:
Objects are solid
Objects don’t intersect
This is very hard to do automatically - it should be part of the modeling process
But, commercial modelers are generally poor in this regard
02/03/03
© 2003 University of Wisconsin

25. © 2003 University of Wisconsin
Fixing Interpolation
Pre-process to remove concave polygons
Replace them with triangles - not too hard
Enforce subdivision constraints
Neighboring patches cannot differ by more than one subdivision level
Post-process the remove T-vertices
Subdivide mismatched neighbors and interpolate new vertices correctly
02/03/03
© 2003 University of Wisconsin

26. Smoother Interpolants
Bi-linear (hardware accelerated) interpolation uses only vertices of given polygon
Build higher order interpolants
Must make sure not to use a high-order interpolant across a low order discontinuity
Gather to a texture or cloud of points
Gather to pixels in the image (slow, but good results)
02/03/03
© 2003 University of Wisconsin

27. Discontinuity Meshing
Identify expected discontinuities and mesh around them
Sharp boundaries due to point light sources or object contact
Derivative discontinuities due to area sources and multi-object shadows
Related to aspect graphs in computer vision
Places where the set of visible things changes
02/03/03
© 2003 University of Wisconsin

28. Two Types of Discontinuities
Assume polygonal environment
Vertex-Edge events
Discontinuities where the plane defined by a vertex and an edge intersects other objects
Edge-Edge-Edge
Higher order discontinuities at places where three edges appear to meet at a point
Produce quadric curves as shadow boundaries, which are hard to mesh
Generally ignored
02/03/03
© 2003 University of Wisconsin

29. Meshing With Discontinuities
Construct VE planes
Intersect them with surfaces
Mesh the resulting edges
Constrained triangulation is a difficult problem
Mesh must be able to store different radiosity values at one point, because radiosity is different on each side of the edge
02/03/03
© 2003 University of Wisconsin

30. Using Discontinuity Meshes
Very high number of possible discontinuities: O(n6) for n vertices
Only find discontinuities due to bright light sources
Try to only find visible discontinuities
Research topic: Integrate into hierarchical scheme
Use discontinuities as splitting planes in hierarchy
02/03/03
© 2003 University of Wisconsin