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Clustered Principal Components for Precomputed Radiance Transfer

Clustered Principal Components for Precomputed Radiance Transfer. Peter-Pike Sloan Microsoft Corporation Jesse Hall, John Hart UIUC John Snyder Microsoft Research. Demo. PRT Terminology. PRT Terminology. PRT Terminology. PRT Terminology. PRT as a Linear Operator.

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Clustered Principal Components for Precomputed Radiance Transfer

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  1. Clustered Principal Components for Precomputed Radiance Transfer Peter-Pike Sloan Microsoft Corporation Jesse Hall, John Hart UIUC John Snyder Microsoft Research

  2. Demo

  3. PRT Terminology

  4. PRT Terminology

  5. PRT Terminology

  6. PRT Terminology

  7. PRT as a Linear Operator • l : light vector (in source basis) • Mp : source-to-exit transfer matrix • ep: exit radiance vector (in exit basis) • y(vp) : exit basis evaluated in direction vp • ep(vp) : exit radiance in direction vp

  8. PRT Special Case: Diffuse Objects transfer vector rather than matrix [PRT02] SH [Xi03] Directional [Ng03] Haar [Ashikhmin02] Steerable • independent of view (constant exit basis) • matrix is row vector • previous work uses different light bases • image relighting

  9. PRT Special Case: Surface Light Fields transfer vector rather than matrix [Miller98] [Nishino99] [Wood00] [Chen02] [Matusik02] • frozen lighting environment • matrix is column vector

  10. Factoring PRT (BRDFs) • Tp: source → transferred incident radiance • Rp : rotate to local frame • B: integrate against BRDF [Westin92] • y(vp) •ep: evaluate exit radiance at vp

  11. Hemispherical Projection • exit radiance is defined over hemisphere, not sphere • spherical harmonics not orthogonal over hemisphere • how to project hemispherical functions using SH? • naïve projection assumes “underside” is zero • least squares projection minimizes approximation error • see appendix

  12. Factoring PRT (BRDFs)

  13. Extending PRT to BSSRDFs • already handled by original equation • use [Jensen02], only multiple scattering (matrix with only 1 row) • mix with “conventional” BRDF

  14. Problems With PRT • Big matrices at each surface point • 25-vectors for diffuse, x3 for spectral • 25x25-matrices for glossy • at ~50,000 vertices • Slows glossy rendering (4hz) • Frozen View/Light can increase performance • Not as GPU friendly • Limits diffuse lighting order • Only very soft shadows

  15. Compression Goals • Decode efficiently • As much on the GPU as possible • Render compressed representation directly • Increase rendering performance • Make non-diffuse case practical • Reduce memory consumption • Not just on disk

  16. Compression Example Surface is curve, signal is normal

  17. Compression Example Signal Space

  18. VQ Cluster normals

  19. VQ Replace samples with cluster mean

  20. PCA Replace samples with mean + linear combination

  21. CPCA Compute a linear subspace in each cluster

  22. CPCA • Clusters with low dimensional affine models • How should clustering be done? • Static PCA • VQ, followed by one-time per-cluster PCA • optimizes for piecewise-constant reconstruction • Iterative PCA • PCA in the inner loop, slower to compute • optimizes for piecewise-affine reconstruction

  23. Static vs. Iterative

  24. Related Work • VQ+PCA [Kambhatla94] (static) • VQPCA [Khambhatla97] (iterative) • Mixture PC [Dony95] (iterative) • More sophisticated models exist • [Brand03], [Roweis02] • Mapping to current GPUs is challenging • Variable storage per vertex • Partitioning is more difficult (or requires more passes)

  25. Equal Rendering Cost VQ PCA CPCA

  26. Rendering with CPCA

  27. Rendering with CPCA Constant per cluster – precompute on the CPU Rendering is a dot product Compute linear combination of vectors Only depends on # rows of M

  28. Non-Local Viewer • Assume: • vp constant across object (distant viewer) Rendering independent of view & light orders - linear combination of colors

  29. + = + Rendering

  30. 2 2 1 3 2 2 2 2 2 2 2 2 1 3 2 2 2 1 Overdraw • faces belong to 1-3 clusters • OD = 1  face drawn once • OD = 2  face drawn 2x • OD = 3  face drawn 3x • coherence optimization: • reclassification • superclustering

  31. Texture Constants Exit Rad. GPU Dataflow Vertices Vertex Shader PixelShader

  32. Demo

  33. Results All examples have 25x25 matrices, 256 clusters, 8 PCA vectors

  34. Conclusions CPCA • works in “signal space”, not “surface space” • uses affine subspace per-cluster • compresses PRT well • is used directly without “blowing out” signal • requires small, uniform state storage • provides • faster rendering • higher-frequency lighting

  35. Future Work • time-dependent and parameterized geometry • higher-frequency lighting • combination with bi-scale rendering • better signal continuity

  36. Questions? • DirectX SDK for PRT available soon. • Jason Mitchell, Hugues Hoppe, Jason Sandlin, David Kirk • Stanford, MPI for models

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