Docstoc

S2007-FDNMF8

Document Sample
S2007-FDNMF8 Powered By Docstoc
					Frequency Domain
Normal Map Filtering


                      Charles Han
                           Bo Sun
               Ravi Ramamoorthi
                   Eitan Grinspun

                Columbia University
               Normal Mapping

• (Blinn 78)

                                QuickTime™ an d a
                               H.264 de compressor
                          are need ed to see this p icture .
             Normal Mapping

• (Blinn 78)
• Specify surface
  normals                     QuickTime™ an d a
                             H.264 de compressor
                        are need ed to see this p icture .
Normal Mapping
                A Problem…

• Multiple normals
  per pixel
• Undersampling                QuickTime™ and a
                              H.264 decomp resso r


• Filtering needed
                         are neede d to see this picture.




                     ?
                Supersampling

• Correct results
• Too slow
                                QuickTime™ and a
                               H.264 decomp resso r
                          are neede d to see this picture.
                 MIP mapping

• Pre-filter
• Normals do not
  interpolate linearly         QuickTime™ and a
                              H.264 decomp resso r


• Blurring of details
                         are neede d to see this picture.
          Comparison




supersampled      MIP mapped
          Representation


                   a single vector is not enough




how do we represent multiple surface normals?
               Previous Work

• Gaussian Distributions
  – (Olano and North 97)             3D Gaussian
  – (Schilling 97)           2D covariance matrix
  – (Toksvig 05)                     1D Gaussian
• Mixture Models
  – (Fournier 92)          mixture of Phong lobes
  – (Tan, et.al. 05)       mixture of 2D Gaussians



           no general solution
             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
Normal Distribution Function (NDF)

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



         QuickTime™ an d a
        H.264 de compressor
   are need ed to see this p icture .




 normal map



                                        NDF
Normal Distribution Function



         QuickTime™ an d a
        H.264 de compressor
   are need ed to see this p icture .




 normal map



                                        NDF
Normal Distribution Function



         QuickTime™ an d a
        H.264 de compressor
   are need ed to see this p icture .




 normal map



                                        NDF
Normal Distribution Function




 normal map



                        NDF
       NDF Filtering




normal map
       NDF Filtering




normal map
               NDF Filtering




• NDF averaging is linear
• Store NDFs in MIP map
                  Rendering


                      Radially symmetric BRDFs
                      • Lambertian: (in  n)
                      • Blinn-Phong: (h  n)S
                      • Torrance-Sparrow:
                       normal, n                exp h 
                                                   2
rendered image        • Factored: f (h )g(d )
                              
                                 

     B(out )       
                      L(               
                               )(  n) din
                            in 
    pixel value       lights     BRDF
               Supersampling
                                               1
                                               N    L(    )(  n1 ) d
                                                           in
                                                             
                                               1
                                               N    L( in )(  n2 ) d
                                                             
                                                             
                                                             
                            samples
                                               1
                                              
                                               N    L( in )(  nN ) d
supersampled image
                        1
            B( out )    L( in )(  n i ) d
                        N i           
                                  1            
            B( out )   L( in )  (  ni ) 
                                                  d
                                  N i          
                            Effective BRDF
     Effective BRDF




              NDF,  (n)
              samples

                  1
      ( )   (  n i )
      eff

                  N i
         
      eff ( )    (n) (  n) dn

     
          Spherical Convolution

              eff ( )     (n) (  n) dn

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

             eff ( )     (n) (  n) dn

                             
                   

                                      


Effective BRDF               NDF                BRDF
                           
                Previous Work

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

                             NDF representations
        Spherical Harmonics

• Analogous to Fourier basis
• Convolution formula:

           eff ( )     (n) (  n) dn

                             
                
                       l  lm
                         eff
                         lm
           BRDF Coefficients
                    l  lm
                   eff
                   lm


• Arbitrary BRDFs
• Cheaply represented
      
  – Analytic: compute in shader
  – Measured: store on GPU
• Easily changed at runtime
            NDF Coefficients
                     l  lm
                     eff
                     lm

• Store in MIP mapped textures
• Finest-level NDFs are delta functions, so:
         lm   (n)Ylm ()d  Ylm (n)

• Use standard linear filtering

  
    Effective BRDF Coefficients
                  l  lm
                  eff
                  lm


• Product of NDF, BRDF coefficients
• Proceed as usual
      
      QuickTime™ an d a
     H.264 de compressor
are need ed to see this p icture .
                   Limitations

 • Storage cost of NDF
     – One texture per coefficient
     – O(l 2 ) cost
 • Limited to low frequencies


  von Mises-Fisher Distribution (vMF)




more concentrated                    less concentrated

  • Spherical analogue to Gaussian
  • Desirable properties
      – Spherical domain
      – Distribution function
      – Radially symmetric
        Mixtures of vMFs




                NDF




1   2       3         4    5   6
          number of vMFs
  Expectation Maximization (EM)
• From machine learning
• Used in (Tan et.al. 05)
• Fit model parameters to data

  data                             model



                  EM


  NDF                            vMF Mixture
                 Rendering
• Convolution
  – Spherical harmonic coefficients   l
  – Analytic convolution formula
• Extensions to EM
  – Aligned lobes (Tan et.al. 05)
                           
  – Colored lobes




             NDF          rendered image
      QuickTime™ and a
     H.264 decompressor
are neede d to see this picture.
      QuickTime™ and a
     H.264 decompressor
are neede d to see this picture.
      QuickTime™ and a
     H.264 decompressor
are neede d to see this picture.
      QuickTime™ and a
     H.264 decompressor
are neede d to see this picture.
                 Conclusion
• Summary
  – Theoretical Framework
  – New NDF representations
  – Practical rendering algorithms
• Future directions
  – Offline rendering, PRT
  – Further applications for vMFs
  – Shadows, parallax, inter-reflections, etc.
                  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

				
DOCUMENT INFO
Shared By:
Categories:
Stats:
views:29
posted:3/6/2010
language:English
pages:41