1. Fast, Arbitrary BRDF Shading for Low-Frequency Lighting Using Spherical Harmonics
Jan Kautz, MPI Informatik
Peter-Pike Sloan, Microsoft Research
John Snyder, Microsoft Research

2. Motivation – BRDF vs. Light Complexity
Lighting ?
area lights
point lights
Initially HW accelerated rendering was only able to render objects with point light sources and the Phong model.
In recent years, research tried on the one hand to allow for arbitrary BRDFs with point lights and on the other hand
limited BRDFs (Phong again) but with area lights (often given as an environment map).
Our work tries to fill in the last gap, i.e. arbitrary BRDFs and area lights
BRDF Complexity
Phong + diffuse
arbitrary aniso. BRDFs

3. Motivation – What we want
Illuminate objects with environment maps
Use arbitrary BRDFs
Change lighting on-the-fly
Possibly include self-shadowing and interreflections
At real-time rates
Phong
Anisotropic

4. Related Work – Interactive Techniques
Lighting
DiffSH
RefSpace
high-frequency area lighting
PhoDiff
FreqSpace
ApproxEnv
HomEnv
low-frequency area lighting
PRT
Our Technique
ArbBRDF
point lights
BRDF Complexity
diffuse
Phong
isotropic
anisotropic
Our Technique
Phong/Diffuse Prefiltered Environment Maps [Miller84] [Greene86] [Heidrich99]
Precomputed Radiance Transfer [Sloan02]
Frequency Space Environment Mapping [Ramamoorthi02]
BRDF Approximation for Environment Maps [Kautz99]
Reflection Space Rendering [Cabral99]
Diffuse Environment Maps using Spherical Harmonics [Ramamoorthi01]
Homomorphic Factorization of Environment Maps [Latta02]
Arbitrary BRDFs with Point Lights [Kautz99] [McCool01]

5. Related Work Previous use of Spherical Harmonics
[Cabral87] Bidirectional Reflection Functions from Surface Bump Maps
[Westin92] Predicting Reflectance Functions from Complex Surfaces

6. Background – Spherical Harmonics
Orthonormal basis over the sphere
Analogous to Fourier transform over 1D circle
Important properties:
Rotational invariance  no aliasing artifacts
Projection:
Integration:
Rotation: linear xform on coefficients

7. Background – Spherical Harmonics
Basis functions (examples)
i = 1
i = 2
i = 3
i = 4
i = 8
i = 12
i = 15
i = 19

8. Background – Spherical Harmonics
Example: projection of environment
n=4
n=9
n=25
n=262
original

9. Environment Mapping + Spherical Harmonics
Rendering Equation (no shadows):
Rewrite with
Project Lighting and BRDF
Mention cosine explicitly as important
Rewrite: mention tabulate
light function:
into SH
BRDF:

10. Evaluating the Integral
The integral becomes
But BRDF defined in local frame Rotate lighting (or BRDF) to match:

11. Preprocessing – BRDF Texture
Project BRDF into SH:
Put coefficients in texture map
Use parabolic parameterization for …
Project the BRDF into Spherical Harmonics for every (local) view direction v individually and tabulate the results in a texture map.
One texture map here shows the coefficients for some basis function i, but for all view direction v.
i=1
i=3
i=4
i=5
i=6
i=7

12. Rendering Lookup (local ) Project lighting per object * =…
Lookup (local )
Project lighting
per object * =
Rotate lighting (to local) =
Compute integral
per pixel/vertex

13. Anisotropic brushed in X Anisotropic brushed in Y
Examples
Phong
Anisotropic brushed in X
Anisotropic brushed in Y

14. Rendering – Fixed Light
Lookup (local )
Project lighting …
ONCE
Rotate lighting (local) * =
Compute integral =

15. Rendering – Fixed View Lookup (local ) Project lighting…
Rotate BRDF (to global) * =
ONCE
Compute integral =

16. Example Bird model 48K vert. Measured Vinyl FPS: 6.04 free light/view
28.4 fixed light
128 fixed view

17. Precomputed Radiance Transfer
[SIG02]
Tell cosine again
Without PRT
PRT: Shadows+Interrefl. SIG02: Phong only

18. Precomputed Radiance Transfer – Transfer Matrix
Precompute how global incident lighting  local incident * p1 p1
lighting p2 p2 *
transfer matrices
transferred radiance

19. Arbitrary BRDF with PRT
Lookup (local )
Project lighting …
per object
Transfer & rotate light
per pixel/vertex * * =
Compute integral =

20. Example Stanford buddha 50K vert. Ashikhmin- BRDF FPS: 4.05 no xfer
15.6 fixed light
127 fixed view

21. Example 2: PRT with different BRDFs
Phong [SIG02]
Measured Vinyl

22. Results – Different BRDFs

23. Results – Brushed Metal-Patch
Anisotropic AS brushed radially
Anisotropic AS brushed tangentally

24. Results – Spatially Varying BRDF
Varying Exponent
Varying Anisotropy

25. Comparison of SH order vs. Glossiness

26. Conclusions Pros: Fast, arbitrary dynamic lighting
Works for arbitrary BRDFs
Combined with PRT: includes shadows and interreflections
Cons:
Works only for low-frequency lighting
Not real-time (yet)

27. Thank you!
Questions?
Please visit us at

28. Glossy Transfer – Rendering
transfer matrix
lighting coefficients
transferred radiance
exiting radiance *
BRDF kernel
evaluate
at R
convolution
transfer computation
lookup

29. Precomputation – Transfer Matrix
Glossy Transfer
More difficult, but works similarly
Have to compute matrix instead of vector
Update matrices for interreflections
Neighborhood Transfer
Same as for glossy, just for points not on surface

30. Precomputation – Diffuse Transfer
Visually: .
Basis 16
Basis 17
Basis 18
illuminate
result

31. Rendering Project lighting into SH Per-vertex:
Project into local tangent frame
Lookup :
Rotate lighting:
Compute dot-product: … * = =

32. Results Glossy object, 50K mesh
No Shadows/Inter Shadows Shadows+Inter
Glossy object, 50K mesh
Runs at 3.6/16/125fps on 2.2Ghz P4, ATI Radeon 8500

33. Dynamic Lighting Sample incident lighting on-the-fly Results
Precompute textures for SH basis functions in cube map parameterization
Render 6 cube map faces around p
Read them back
Projection: simple dot-product between cube maps
Results
Low-resolution cube maps sufficient: 6x16x16 Average error: 0.2%, worst-case: 0.5%
Takes 1.16 ms on P3-933Mhz, ATI 8500

34. Introduction – Light Integration
Integrate over all incoming light
Emitter 1
Emitter 2
Diffuse: * cos<n, s>
Glossy: * f(v, s) * cos<n, s>
Receiver

35. Background – Spherical Harmonics
Projection:
Reconstruction:
Integration:
Convolution/Rotation:
Simple and efficient formulas

36. Overview Previous Work Background Fast Environment Mapping with SH
Spherical Harmonics
Fast Environment Mapping with SH
Theory
Rendering
Combine with Precomputed Radiance Transfer
Results
Conclusions

37. Comparison – Size Light vs. SH Order0°
20°
40°
n= n= n= n= n= n= n= RT linear quadratic cubic quartic quintic windowed

38. Introduction – Filtered Environment Maps
Environment map over sphere
Source
Target
BRDF maps to:
Shift-variant & radially symmetric kernel  2D filtered environment map
But:
General anisotropic BRDF  5D filtered environment map
apply filter
Filter kernel

39. Precomputed Radiance Transfer
Precompute how global incident radiance is transferred to local incident radiance at points :
For self-shadowing and interreflections
Transfer is represented as a
Transfer Matrix
global
local
transfer

40. Introduction – Environment Maps
Approximate incident light field with a single sample at object’s center
Assumptions:
Environment at infinity
No self-shadowing or interreflections (concave object)

41. Motivation – BRDF Complexity
HW rendering: Increased BRDF complexity
But only for point light sources!
[Heidrich98]
[Kautz00]

42. Results – Head in Various Environments
Max Planck head
50K vertices
Ashikhmin- BRDF
FPS:
5.24 no xfer
4.18 xfer
25.3 fixed light
130 fixed view

43. Results – Different BRDFs

44. Motivation – Environment Maps
Store light incident at single position
Prefilter to get glossy reflections:
Only limited Phong-like BRDFs
Complex BRDFs require up to 5D table
Dynamic lighting difficult, no self-shadowing
[Miller & Hoffmann84]
filter