Overview Last lecture Statistical sampling and Monte Carlo

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Overview Last lecture Statistical sampling and Monte Carlo integration Today Variance reduction Importance sampling

Overview Last lecture Statistical sampling and Monte Carlo integration Today Variance reduction Importance sampling Stratified sampling Multidimensional sampling patterns Discrepancy and Quasi-Monte Carlo Later Signal processing and sampling Path tracing for interreflection Density estimation University of Texas at Austin CS 395 T - Advanced Image Synthesis Spring 2007 Don Fussell

Cameras Motion Blur Source: Cook, Porter, Carpenter, 1984 University of Texas at Austin Depth

Cameras Motion Blur Source: Cook, Porter, Carpenter, 1984 University of Texas at Austin Depth of Field Source: Mitchell, 1991 CS 395 T - Advanced Image Synthesis Spring 2007 Don Fussell

Variance 1 shadow ray per eye ray University of Texas at Austin 16 shadow

Variance 1 shadow ray per eye ray University of Texas at Austin 16 shadow rays per eye ray CS 395 T - Advanced Image Synthesis Spring 2007 Don Fussell

Variance Definition Properties Variance decreases with sample size University of Texas at Austin CS

Variance Definition Properties Variance decreases with sample size University of Texas at Austin CS 395 T - Advanced Image Synthesis Spring 2007 Don Fussell

Variance Reduction Efficiency measure If one technique has twice the variance of another technique,

Variance Reduction Efficiency measure If one technique has twice the variance of another technique, then it takes twice as many samples to achieve the same variance If one technique has twice the cost of another technique with the same variance, then it takes twice as much time to achieve the same variance Techniques to increase efficiency Importance sampling Stratified sampling University of Texas at Austin CS 395 T - Advanced Image Synthesis Spring 2007 Don Fussell

Biasing Previously used a uniform probability distribution Can use another probability distribution But must

Biasing Previously used a uniform probability distribution Can use another probability distribution But must change the estimator University of Texas at Austin CS 395 T - Advanced Image Synthesis Spring 2007 Don Fussell

Unbiased Estimate Probability Estimator University of Texas at Austin CS 395 T - Advanced

Unbiased Estimate Probability Estimator University of Texas at Austin CS 395 T - Advanced Image Synthesis Spring 2007 Don Fussell

Importance Sampling Sample according to f University of Texas at Austin CS 395 T

Importance Sampling Sample according to f University of Texas at Austin CS 395 T - Advanced Image Synthesis Spring 2007 Don Fussell

Importance Sampling Variance Sample according to f Zero variance! University of Texas at Austin

Importance Sampling Variance Sample according to f Zero variance! University of Texas at Austin CS 395 T - Advanced Image Synthesis Spring 2007 Don Fussell

Example method Sampling function variance importance (6 -x)/16 56. 8 N-1 Samples needed for

Example method Sampling function variance importance (6 -x)/16 56. 8 N-1 Samples needed for standard error of 0. 008 887, 500 importance 1/4 21. 3 N-1 332, 812 importance (x+2)/16 6. 4 N-1 98, 432 importance x/8 0 1 stratified 1/4 21. 3 N-3 70 Peter Shirley – Realistic Ray Tracing University of Texas at Austin CS 395 T - Advanced Image Synthesis Spring 2007 Don Fussell

Examples Projected solid angle Area 4 eye rays per pixel 100 shadow rays University

Examples Projected solid angle Area 4 eye rays per pixel 100 shadow rays University of Texas at Austin CS 395 T - Advanced Image Synthesis Spring 2007 Don Fussell

Irradiance Generate cosine weighted distribution University of Texas at Austin CS 395 T -

Irradiance Generate cosine weighted distribution University of Texas at Austin CS 395 T - Advanced Image Synthesis Spring 2007 Don Fussell

Cosine Weighted Distribution University of Texas at Austin CS 395 T - Advanced Image

Cosine Weighted Distribution University of Texas at Austin CS 395 T - Advanced Image Synthesis Spring 2007 Don Fussell

Sampling a Circle Equi-Areal University of Texas at Austin CS 395 T - Advanced

Sampling a Circle Equi-Areal University of Texas at Austin CS 395 T - Advanced Image Synthesis Spring 2007 Don Fussell

Shirley’s Mapping University of Texas at Austin CS 395 T - Advanced Image Synthesis

Shirley’s Mapping University of Texas at Austin CS 395 T - Advanced Image Synthesis Spring 2007 Don Fussell

Stratified Sampling Stratified sampling is like jittered sampling Allocate samples per region New variance

Stratified Sampling Stratified sampling is like jittered sampling Allocate samples per region New variance Thus, if the variance in regions is less than the overall variance, there will be a reduction in resulting variance For example: An edge through a pixel University of Texas at Austin CS 395 T - Advanced Image Synthesis Spring 2007 Don Fussell

Mitchell 91 Uniform random University of Texas at Austin Spectrally optimized CS 395 T

Mitchell 91 Uniform random University of Texas at Austin Spectrally optimized CS 395 T - Advanced Image Synthesis Spring 2007 Don Fussell

Discrepancy University of Texas at Austin CS 395 T - Advanced Image Synthesis Spring

Discrepancy University of Texas at Austin CS 395 T - Advanced Image Synthesis Spring 2007 Don Fussell

Theorem on Total Variation Theorem: Proof: Integrate by parts University of Texas at Austin

Theorem on Total Variation Theorem: Proof: Integrate by parts University of Texas at Austin CS 395 T - Advanced Image Synthesis Spring 2007 Don Fussell

Quasi-Monte Carlo Patterns Radical inverse (digit reverse) of integer i in integer base b

Quasi-Monte Carlo Patterns Radical inverse (digit reverse) of integer i in integer base b Hammersley points 1 2 3 4 1 10 11 100 . 1. 01. 11. 001 1/2 1/4 3/8 5 101 5/8 Halton points (sequential) University of Texas at Austin CS 395 T - Advanced Image Synthesis Spring 2007 Don Fussell

Hammersley Points University of Texas at Austin CS 395 T - Advanced Image Synthesis

Hammersley Points University of Texas at Austin CS 395 T - Advanced Image Synthesis Spring 2007 Don Fussell

Edge Discrepancy Note: SGI IR Multisampling extension: 8 x 8 subpixel grid; 1, 2,

Edge Discrepancy Note: SGI IR Multisampling extension: 8 x 8 subpixel grid; 1, 2, 4, 8 samples University of Texas at Austin CS 395 T - Advanced Image Synthesis Spring 2007 Don Fussell

Low-Discrepancy Patterns Process 16 points 256 points 1600 points Zaremba 0. 0504 0. 00478

Low-Discrepancy Patterns Process 16 points 256 points 1600 points Zaremba 0. 0504 0. 00478 0. 00111 Jittered 0. 0538 0. 00595 0. 00146 Poisson-Disk 0. 0613 0. 00767 0. 00241 N-Rooks 0. 0637 0. 0123 0. 00488 Random 0. 0924 0. 0224 0. 00866 Discrepancy of random edges, From Mitchell (1992) Random sampling converges as N-1/2 Zaremba converges faster and has lower discrepancy Zaremba has a relatively poor blue noise spectra Jittered and Poisson-Disk recommended University of Texas at Austin CS 395 T - Advanced Image Synthesis Spring 2007 Don Fussell

High-dimensional Sampling Numerical quadrature For a given error … Random sampling For a given

High-dimensional Sampling Numerical quadrature For a given error … Random sampling For a given variance … Monte Carlo requires fewer samples for the same error in high dimensional spaces University of Texas at Austin CS 395 T - Advanced Image Synthesis Spring 2007 Don Fussell

Block Design Latin Square Alphabet of size n Each symbol appears exactly once in

Block Design Latin Square Alphabet of size n Each symbol appears exactly once in each row and column Rows and columns are stratified University of Texas at Austin CS 395 T - Advanced Image Synthesis Spring 2007 Don Fussell

Block Design N-Rook Pattern Incomplete block design Replaced n 2 samples with n samples

Block Design N-Rook Pattern Incomplete block design Replaced n 2 samples with n samples Permutations: Generalizations: N-queens, 2 D projection University of Texas at Austin CS 395 T - Advanced Image Synthesis Spring 2007 Don Fussell

Space-time Patterns Distribute samples in time Complete in space Samples in space should have

Space-time Patterns Distribute samples in time Complete in space Samples in space should have blue-noise spectrum Incomplete in time Decorrelate space and time Nearby samples in space should differ greatly in time Cook Pattern Pan-diagonal Magic Square University of Texas at Austin CS 395 T - Advanced Image Synthesis Spring 2007 Don Fussell

Path Tracing 4 eye rays per pixel 16 shadow rays per eye ray 64

Path Tracing 4 eye rays per pixel 16 shadow rays per eye ray 64 eye rays per pixel 1 shadow ray per eye ray Complete Incomplete University of Texas at Austin CS 395 T - Advanced Image Synthesis Spring 2007 Don Fussell

Views of Integration 1. Signal processing Sampling and reconstruction, aliasing and antialiasing Blue noise

Views of Integration 1. Signal processing Sampling and reconstruction, aliasing and antialiasing Blue noise good 2. Statistical sampling (Monte Carlo) Sampling like polling Variance High dimensional sampling: 1/N 1/2 3. Quasi Monte Carlo Discrepancy Asymptotic efficiency in high dimensions 4. Numerical Quadrature/Integration rules Smooth functions University of Texas at Austin CS 395 T - Advanced Image Synthesis Spring 2007 Don Fussell