1. Working Group « Pre-Filtering »

2. Why Pre-filtering?
Complex light transports and complex materials extensively studied
Today key issue = management of details
Very large scenes
Complex objects
High quality, antialiased rendering requires costly integrals per pixel
Adaptive multisampling not a solution
Pre-filtering = pre-integrate as much as possible

4. General equation Screen space Integral over pixel Change of variables
Integral over pixel footprint
Introduction of a
Local Illumination Model
Change of variables
Integral over surface maps

5. Uncorrelation hypothesis
Consequence: average of product = product of averages
Application (when hypothesis valid – cf discussion):
Average color
Average BRDF
Average « visibility »
Average shadow

6. Parallax effects

7. Outline Using uncorrelation hypothesis & neglecting parallax:
Color map filtering
Normal map filtering
Horizon map filtering
Shadow map filtering
Procedural map filtering
Summary
Discussion of correlation & parallax effects
Conclusion

8. Color map filtering IF parallax effects neglected
THEN simple linear filtering of color map
→ Can use hardware anisotropic filtering
gives

9. Normal map filtering

10. Normal map filtering IF parallax effects neglected
THEN « simple » linear filtering of BRDF
but BRDF = 4D function, not easy to store
Simplification hypothesis:
BRDF represented with 2D distribution of normals
gives
normal map

11. Normal map filtering Direct methods
Represent fx with linear parameters, mipmap them
Convolution methods
Further assume that
To use ergodicity relation
We get
Represent px with linear parameters, mipmap them

12. Normal map filtering Representing 2D distributions:
Single Gaussian lobe represented with mean and second moments (linear parameters)
3D lobe, 2D lobe in tangent plane, isotropic or anisotropic, etc
Second moments sometimes deduced from the mean (Toksvig)
Multiple lobes
Spherical harmonics

13. Horizon map filtering

14. Horizon map filtering Fraction of S visible from viewer AND light
Visibility represented with horizon map
Reformulate with horizon distribution functions
Or write V in a basis (e.g. Legendre Polynomials)
(parallax effects neglected)
H = Heaviside function, non linear
Horizon map
V now linear in px!

15. Horizon map filtering Representing horizon angle distributions:
Gaussian with mean and second moment
Correlation between view and light directions:
approximations
[Tan et al.]
hotspot effect
[Ashikhmin et al.]

16. Shadow map filtering

17. Shadow map filtering Average shadow (parallax neglected)
Reformulate with depth distributions
Or write S in a basis (e.g. use Fourier series)
H = Heaviside function, non linear
Shadow map (= depth map from light)
S now linear in ps!
gives
Other metod

18. Shadow map filtering Representing depth distributions
Gaussian with mean and second moment

19. Procedural map filtering

20. Procedural map filtering
Global fading
Smooth transition from f(x) to its average
Drawback: attenuate ALL frequencies
Analytic integration
Drawback: almost always impossible to compute analytically
Workaround: apply on each term separately
Frequency clamping
For methods using frequency synthesis (sum of sinusoids, of trochoids, of noise functions, etc)

21. Summary To pre-filter a map M used in a non-linear function f(M,y)
Direct methods:
use a « change of variable » to get f(M,y)=g(K,y) with g linear in the new variables K
Pre-filter the new K variables
Convolution methods:
Introduce the parameter distribution (= histogram) to get f(M,y)=∫f(m,y)px(m)dm, with px(m)=δ(m- Mx)
Pre-filter linear parameters representing the px functions (e.g. mean and second moments, coefficients in a basis, etc). AND, at runtime, evaluate « convolution » f*pA

22. Summary Finding linear parameters characterizing a function:
Use moments
Use a basis of functions Bk
Use a spanning set of functions OR
→ spare representation
ki generally not linearly interpolable (counter-example: k = moments),
requires alignment of « lobes »

23. Discussion Uncorrelation hypothesis False for small footprints
Correlations between color and reflectance
For each color channel, weight normals with reflectancel
Correlations between color and visibility
Correlations between reflectance and visibility
Correlations between view and light visibility
Approximate solutions (cf above)

24. Discussion Parallax effects Parallax offset
Could be handled separately
Parallax Jacobian
Not handled in existing works
Possible solution for normals

25. Discussion Silhouettes and curvature Curvature and normal maps
Curvature of base surface influences normals
Unless normals stored in global frame (instead of tangent frame)
Curvature also influences horizon angles
Silhouettes
Raw mesh resolution becomes visible
Silhouette clipping, shell mapping, relief mapping, etc
Very anisotropic footprints

26. Discussion Mesh filtering and extreme filtering
At large distance, mesh itself must be pre- filtered
Removed mesh details must be incorporated in surface maps (bump maps, normal maps, etc)
Topology changes can occur
e.g. Tree foliage becomes a « volume » at large distance

27. Conclusion
Many hard problems remain to be solved!