Frequency Domain Normal Map Filtering

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Frequency DomainNormal Map Filtering

• Charles Han
• Bo Sun
• Ravi Ramamoorthi
• Eitan Grinspun
• Columbia University

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Normal Mapping

• (Blinn 78)

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Normal Mapping

• (Blinn 78)
• Specify surface normals

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Normal Mapping
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A Problem

• Multiple normals per pixel
• Undersampling
• Filtering needed

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Supersampling

• Correct results
• Too slow

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MIP mapping

• Pre-filter
• Normals do not interpolate linearly
• Blurring of details

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Comparison
supersampled
MIP mapped
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Representation
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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

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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

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Normal Distribution Function (NDF)

• Describes normals within region
• Defined on the unit sphere
• Integrates to one
• Extended Gaussian Image (Horn 84)

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Normal Distribution Function
normal map
NDF
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Normal Distribution Function
normal map
NDF
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Normal Distribution Function
normal map
NDF
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Normal Distribution Function
normal map
NDF
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NDF Filtering
normal map
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NDF Filtering
normal map
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NDF Filtering

• NDF averaging is linear
• Store NDFs in MIP map

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Rendering

• Radially symmetric BRDFs
• Lambertian
• Blinn-Phong
• Torrance-Sparrow
• Factored

rendered image
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Supersampling
supersampled image
Effective BRDF
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Effective BRDF
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Spherical Convolution

• Form studied in lighting
• (Basri and Jacobs 01)
• (Ramamoorthi and Hanrahan 01)
• Effective BRDF convolution of NDF BRDF

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Spherical Convolution
BRDF
NDF
Effective BRDF
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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
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Spherical Harmonics

• Analogous to Fourier basis
• Convolution formula

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BRDF Coefficients

• Arbitrary BRDFs
• Cheaply represented
• Analytic compute in shader
• Measured store on GPU
• Easily changed at runtime

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NDF Coefficients

• Store in MIP mapped textures
• Finest-level NDFs are delta functions, so
• Use standard linear filtering

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Effective BRDF Coefficients

• Product of NDF, BRDF coefficients
• Proceed as usual

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Limitations

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

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von Mises-Fisher Distribution (vMF)

• Spherical analogue to Gaussian
• Desirable properties
• Spherical domain
• Distribution function
• Radially symmetric

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Mixtures of vMFs
NDF
number of vMFs
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Expectation Maximization (EM)

• From machine learning
• Used in (Tan et.al. 05)
• Fit model parameters to data

EM
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Rendering

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

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Conclusion

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

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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