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Frequency Domain Normal Map Filtering

Frequency Domain Normal Map Filtering. Charles Han Bo Sun Ravi Ramamoorthi Eitan Grinspun Columbia University. Normal Mapping. (Blinn 78). Normal Mapping. (Blinn 78) Specify surface normals. Normal Mapping. ?. A Problem…. Multiple normals per pixel Undersampling Filtering needed.

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Frequency Domain Normal Map Filtering

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  1. Frequency DomainNormal Map Filtering Charles Han Bo Sun Ravi Ramamoorthi Eitan Grinspun Columbia University

  2. Normal Mapping • (Blinn 78)

  3. Normal Mapping • (Blinn 78) • Specify surface normals

  4. Normal Mapping

  5. ? A Problem… • Multiple normals per pixel • Undersampling • Filtering needed

  6. Supersampling • Correct results • Too slow

  7. MIP mapping • Pre-filter • Normals do not interpolate linearly • Blurring of details

  8. Comparison supersampled MIP mapped

  9. a single vector is not enough how do we represent multiple surface normals? Representation

  10. no general solution Previous Work • Gaussian Distributions • (Olano and North 97) • (Schilling 97) • (Toksvig 05) • Mixture Models • (Fournier 92) • (Tan, et.al. 05) 3D Gaussian 2D covariance matrix 1D Gaussian mixture of Phong lobes mixture of 2D Gaussians

  11. Our Contributions • Theoretical Framework • Normal Distribution Function (NDF) • Linear averaging for filtering • Convolution for rendering • Unifies previous works • New normal map representations • Spherical harmonics • von Mises-Fisher Distribution • Simple, efficient rendering algorithms

  12. Normal Distribution Function (NDF) • Describes normals within region • Defined on the unit sphere • Integrates to one • Extended Gaussian Image (Horn 84)

  13. Normal Distribution Function normal map NDF

  14. Normal Distribution Function normal map NDF

  15. Normal Distribution Function normal map NDF

  16. Normal Distribution Function normal map NDF

  17. NDF Filtering normal map

  18. NDF Filtering normal map

  19. NDF Filtering • NDF averaging is linear • Store NDFs in MIP map

  20. normal, lights BRDF pixel value Rendering Radially symmetric BRDFs • Lambertian: • Blinn-Phong: • Torrance-Sparrow: • Factored: rendered image

  21. samples Supersampling supersampled image Effective BRDF

  22. samples NDF, Effective BRDF

  23. Spherical Convolution • Form studied in lighting • (Basri and Jacobs 01) • (Ramamoorthi and Hanrahan 01) • Effective BRDF = convolution of NDF & BRDF

  24. Spherical Convolution BRDF NDF Effective BRDF

  25. NDF representations Previous Work • Gaussian Distributions • Olano and North (97) • Schilling (97) • Toksvig (05) • Mixture Models • Fournier (92) • Tan, et.al. (05) • Our Work 3D Gaussian 2D covariance matrix 1D Gaussian mixture of Phong lobes mixture of 2D Gaussians spherical harmonics von Mises-Fisher mixtures

  26. Spherical Harmonics • Analogous to Fourier basis • Convolution formula:

  27. BRDF Coefficients • Arbitrary BRDFs • Cheaply represented • Analytic: compute in shader • Measured: store on GPU • Easily changed at runtime

  28. NDF Coefficients • Store in MIP mapped textures • Finest-level NDFs are delta functions, so: • Use standard linear filtering

  29. Effective BRDF Coefficients • Product of NDF, BRDF coefficients • Proceed as usual

  30. Limitations • Storage cost of NDF • One texture per coefficient • O( ) cost • Limited to low frequencies

  31. more concentrated less concentrated von Mises-Fisher Distribution (vMF) • Spherical analogue to Gaussian • Desirable properties • Spherical domain • Distribution function • Radially symmetric

  32. 2 1 3 4 5 6 Mixtures of vMFs NDF number of vMFs

  33. data model NDF vMF Mixture Expectation Maximization (EM) • From machine learning • Used in (Tan et.al. 05) • Fit model parameters to data EM

  34. NDF rendered image Rendering • Convolution • Spherical harmonic coefficients • Analytic convolution formula • Extensions to EM • Aligned lobes (Tan et.al. 05) • Colored lobes

  35. Conclusion • Summary • Theoretical Framework • New NDF representations • Practical rendering algorithms • Future directions • Offline rendering, PRT • Further applications for vMFs • Shadows, parallax, inter-reflections, etc.

  36. Thanks! Tony Jebara, Aner Ben-Artzi, Peter Belhumeur, Pat Hanrahan, Shree Nayar, Evgueni Parilov, Makiko Yasui, Denis Zorin, and nVidia. http://www.cs.columbia.edu/cg/normalmap

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