1. Adam GoodenoughSampling: Strategies and ToolsApril 11, 2005 Sampling: Strategies and Tools Adam Goodenough DIRSIG Meeting April 11, 2005

2. Adam GoodenoughSampling: Strategies and ToolsApril 11, 2005 DIRSIG = Integration Numerical integration is inefficient Each sample can lead to many other samples Monte Carlo integration uses knowledge of sample “importance” Techniques are essential to photon mapping Applicable in many other areas Introduction

3. Adam GoodenoughSampling: Strategies and ToolsApril 11, 2005 Derivation

4. Adam GoodenoughSampling: Strategies and ToolsApril 11, 2005 Outline Monte Carlo integration Monte Carlo integration of an example “scene” DIRSIG Tool: CDSampleGen Samplers and Projections Arbitrary BRDF Representation and Sampling Future Applications

5. Adam GoodenoughSampling: Strategies and ToolsApril 11, 2005 Integration Equation 4.31 in Schott (1997)

6. Adam GoodenoughSampling: Strategies and ToolsApril 11, 2005 Monte Carlo Monte Carlo integration for the same term

7. Adam GoodenoughSampling: Strategies and ToolsApril 11, 2005 Sky Dome Picture of the sky taken with a fisheye lens Data taken from Conesus collect

8. Adam GoodenoughSampling: Strategies and ToolsApril 11, 2005 Parameterized Sky dome unwrapped for easy sampling Only the green band was acquired Parameterized as θ versus Φ θ Φ 0 0 2π2π ½π½π

9. Adam GoodenoughSampling: Strategies and ToolsApril 11, 2005 Ward BRDF Models specular lobe + diffuse Anisotropic (brushed surfaces) Parameters are “physical” BRDF accuracy has been validated against measurements Lightly Brushed Aluminum

10. Adam GoodenoughSampling: Strategies and ToolsApril 11, 2005 Setup Using sky dome and Ward BRDF Viewer at 80 o zenith and 45 o azimuth Calculate the numerical integral first Calculate Monte Carlo integrals using different sampling methods View Direction

11. Adam GoodenoughSampling: Strategies and ToolsApril 11, 2005 Integral Regular Numerical Integration 50 Samples L out = 0.0629 200 Samples L out = 0.0551 5000 Samples L out = 0.0556

12. Adam GoodenoughSampling: Strategies and ToolsApril 11, 2005 Grid Monte Carlo Integration with Grid sampling 5000 Samples L out = 0.0559 Num: L out = 0.0556

13. Adam GoodenoughSampling: Strategies and ToolsApril 11, 2005 Random Monte Carlo Integration with Random sampling 5000 Samples L out = 0.0553 Num: L out = 0.0556

14. Adam GoodenoughSampling: Strategies and ToolsApril 11, 2005 Shirley Monte Carlo Integration with Shirley sampling 5000 Samples L out = 0.0558 Num: L out = 0.0556

15. Adam GoodenoughSampling: Strategies and ToolsApril 11, 2005 Cosine Integration with Cosine Weighted Sampling 5000 Samples L out = 0.0561 Num: L out = 0.0556

16. Adam GoodenoughSampling: Strategies and ToolsApril 11, 2005 Importance Cosine and BRDF Weighted Sampling 5000 Samples L out = 0.0550 Num: L out = 0.0556

17. Adam GoodenoughSampling: Strategies and ToolsApril 11, 2005 DIRSIG Tool CDSampleGen None Gaussian Sphere Hemisphere Sphere Section Grid Random N-Rooks Stratified Shirley Halton Hammersley Samplers Cosine Henyey- Greenstein Schlick Ward BRDF Factored GeometryWeighted Projections

18. Adam GoodenoughSampling: Strategies and ToolsApril 11, 2005 Samplers GridRandom

19. Adam GoodenoughSampling: Strategies and ToolsApril 11, 2005 Samplers N-Rooks (Latin Hypercube)

20. Adam GoodenoughSampling: Strategies and ToolsApril 11, 2005 Samplers StratifiedShirley

21. Adam GoodenoughSampling: Strategies and ToolsApril 11, 2005 Samplers Halton “Sequence”(Quasi-Monte Carlo)

22. Adam GoodenoughSampling: Strategies and ToolsApril 11, 2005 Set vs. SequenceHammersley Set Samplers Sets are defined by “n” and “m” and yield nxm samples Sets are randomly shuffled All samplers work in either mode

23. Adam GoodenoughSampling: Strategies and ToolsApril 11, 2005 Quasi-Monte Carlo Samplers Samplers Halton Sequence and Hammersley Set Deterministic Optimal uniformity and repeatability “Leaping” and “Scrambling”

24. Adam GoodenoughSampling: Strategies and ToolsApril 11, 2005 Projections NoneGaussian

25. Adam GoodenoughSampling: Strategies and ToolsApril 11, 2005 Projections Gaussian Revisited PDFRandomShirley

26. Adam GoodenoughSampling: Strategies and ToolsApril 11, 2005 Projections SphereHemisphere

27. Adam GoodenoughSampling: Strategies and ToolsApril 11, 2005 Projections Sphere Section

28. Adam GoodenoughSampling: Strategies and ToolsApril 11, 2005 Projections Henyey-Greenstein SPF (N-Terms)

29. Adam GoodenoughSampling: Strategies and ToolsApril 11, 2005 Projections Schlick SPF (N-Terms)

30. Adam GoodenoughSampling: Strategies and ToolsApril 11, 2005 Projections CosineWard BRDF

31. Adam GoodenoughSampling: Strategies and ToolsApril 11, 2005 Arbitrary BRDF Synthetic or Modeled 4-D Measurements Measurements Factorization Re-Creation

32. Adam GoodenoughSampling: Strategies and ToolsApril 11, 2005 Components Re-ParameterizationNon-Negative Matrix Factorization Most BRDFs consist of diffuse and specular components Variability exists primarily around specular direction Re-Parameterize into Half- Angle space Factor Y into two matrices F and G F and G are non-negative Iterative algorithm Gradient descent Rate of descent picked to ensure monotonically decreasing RSE or distortion

33. Adam GoodenoughSampling: Strategies and ToolsApril 11, 2005 Approach Re-Parameterization and Factorization From Lawrence et al. (2004) Re-Shuffling Re-ParameterizingRe-Shuffling and NMF NMF G F G u v

34. Adam GoodenoughSampling: Strategies and ToolsApril 11, 2005 Example Significance of u and v From Lawrence et al. (2004)

35. Adam GoodenoughSampling: Strategies and ToolsApril 11, 2005 Compression Large data sets represented by few terms Better accuracy than general basis functions (Zernike, Spherical Harmonics, LaFortune Lobes) Compression ratios of ~200:1 shown for measured data (measurements performed by Matusik [2003]) From Lawrence et al. (2004)

36. Adam GoodenoughSampling: Strategies and ToolsApril 11, 2005 Sampling Compression rates are sub-optimal! Factorization form maintains each sampling variable (incident direction) independently Factored forms are used directly to sample F- matrix maintains view direction indexing G (u and v) contains incident direction PDFs

37. Adam GoodenoughSampling: Strategies and ToolsApril 11, 2005 Future Apps Standard for storing BRDF measurements Addition of a spectral dimension Storage of modeled, spectral phase function data

38. Adam GoodenoughSampling: Strategies and ToolsApril 11, 2005 Questions