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

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

Adam GoodenoughSampling: Strategies and ToolsApril 11, 2005 Derivation

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

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

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

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

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

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

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

Adam GoodenoughSampling: Strategies and ToolsApril 11, 2005 Integral Regular Numerical Integration 50 Samples L out = Samples L out = Samples L out =

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

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

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

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

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

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

Adam GoodenoughSampling: Strategies and ToolsApril 11, 2005 Samplers GridRandom

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

Adam GoodenoughSampling: Strategies and ToolsApril 11, 2005 Samplers StratifiedShirley

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

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

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”

Adam GoodenoughSampling: Strategies and ToolsApril 11, 2005 Projections NoneGaussian

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

Adam GoodenoughSampling: Strategies and ToolsApril 11, 2005 Projections SphereHemisphere

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

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

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

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

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

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

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

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

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)

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

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

Adam GoodenoughSampling: Strategies and ToolsApril 11, 2005 Questions