Monte Carlo Integration for Image Synthesis Adapted from Thomas - PowerPoint PPT Presentation

Monte Carlo Integration for Image Synthesis Adapted from… Thomas Funkhouser Princeton University C0S 526, Fall 2002

Main Sources • Books � Realistic Ray Tracing, Peter Shirley � Realistic Image Synthesis Using Photon Mapping, Henrik Wann Jensen � Advanced Global Illumination, Dutre, Bekaert & Bala • Theses � Robust Monte Carlo Methods for Light Transport Simulation, Eric Veach � Mathematical Models and Monte Carlo Methods for Physically Based Rendering, Eric La Fortune • Course Notes � Mathematical Models for Computer Graphics, Stanford, Fall 1997 � State of the Art in Monte Carlo Methods for Realistic Image Synthesis, Course 29, SIGGRAPH 2001

Outline • Motivation • Monte Carlo integration • Monte Carlo path tracing • Variance reduction techniques • Sampling techniques • Conclusion

Motivation • Rendering = integration � Antialiasing � Soft shadows � Indirect illumination � Caustics

Motivation • Rendering = integration � Antialiasing � Soft shadows Eye � Indirect illumination � Caustics Pixel x P � = → L L(x e) dA Surface S

Motivation • Rendering = integration Light � Antialiasing x ’ � Soft shadows Eye � Indirect illumination � Caustics x Surface � � ( , ′ , ) ′ ( , ′ ) ( , ′ ) = → + → → → L(x, w ) L (x,x e) f x x x x e L( x x)V x x G x x dA e r S

Motivation • Rendering = integration � Antialiasing � Soft shadows � Indirect illumination � Caustics Herf � � ( , ′ , ) ′ ( , ′ ) ( , ′ ) = → + → → → L(x, w ) L (x,x e) f x x x x e L( x x)V x x G x x dA e r S

Motivation • Rendering = integration Surface � Antialiasing Light � Soft shadows Eye � Indirect illumination � Caustics ω ω ’ x Surface � � � � � � � � � ( , ′ , ) ′ ( ′ ) = + • L (x, w ) L (x, w ) f x w w L (x, w ) w n d w o e r i Ω

Motivation • Rendering = integration � Antialiasing � Soft shadows � Indirect illumination � Caustics Debevec � � � � � � � � � ( , ′ , ) ′ ( ′ ) = + • L (x, w ) L (x, w ) f x w w L (x, w ) w n d w o e r i Ω

Motivation Specular • Rendering = integration Surface Light � Antialiasing � Soft shadows Eye � Indirect illumination � Caustics ω ω ’ x Diffuse Surface � � � � � � � � � ( , ′ , ) ′ ( ′ ) = + • L (x, w ) L (x, w ) f x w w L (x, w ) w n d w o e r i Ω

Motivation • Rendering = integration � Antialiasing � Soft shadows � Indirect illumination � Caustics Jensen � � � � � � � � � ( , ′ , ) ′ ( ′ ) = + • L (x, w ) L (x, w ) f x w w L (x, w ) w n d w o e r i Ω

Challenge • Rendering integrals are difficult to evaluate � Multiple dimensions � Discontinuities » Partial occluders » Highlights » Caustics Drettakis � � ( , ′ , ) ′ ( , ′ ) ( , ′ ) = → + → → → L(x, w ) L (x,x e) f x x x x e L( x x)V x x G x x dA e r S

Challenge • Rendering integrals are difficult to evaluate � Multiple dimensions � Discontinuities » Partial occluders » Highlights » Caustics Jensen � � ( , ′ , ) ′ ( , ′ ) ( , ′ ) = → + → → → L(x, w ) L (x,x e) f x x x x e L( x x)V x x G x x dA e r S

Outline • Motivation • Monte Carlo integration • Monte Carlo path tracing • Variance reduction techniques • Sampling techniques • Conclusion

Slide courtesy of Integration in 1D Peter Shirley 1 � ( ) ? = f x dx 0 f(x) x=1

Slide courtesy of We can approximate Peter Shirley 1 1 � � ( ) ( ) = f x dx g x dx 0 0 g(x) f(x) x=1

Slide courtesy of Or we can average Peter Shirley 1 � ( ) ( ( )) = f x dx E f x 0 f(x) E(f(x)) x=1

Slide courtesy of Estimating the average Peter Shirley 1 1 N � � ( ) ( ) = f x dx f x i N 1 = i 0 f(x) E(f(x)) x 1 x N

Slide courtesy of Other Domains Peter Shirley b − b a N � � ( ) ( ) = f x dx f x i N 1 = i a f(x) < f > ab x=a x=b

Multidimensional Domains • Same ideas apply for integration over … � Pixel areas 1 � Surfaces N � � ( ) ( ) = f x dx f x � Projected areas i N = 1 i UGLY � Directions Eye � Camera apertures � Time Pixel � Paths x Surface

Outline • Motivation • Monte Carlo integration • Monte Carlo path tracing • Variance reduction techniques • Sampling techniques • Conclusion

Monte Carlo Path Tracing Specular • Integrate radiance Surface for each pixel Light by sampling paths randomly Eye Pixel x Diffuse Surface � � � � � � � � � ′ ′ ′ ( , , ) ( ) = + • L (x, w ) L (x, w ) f x w w L (x, w ) w n d w o e r i Ω

Pixel Sampling

Simple Stochastic Ray Tracing

Radiance Sampling • Monte Carlo approximation • but RHS has unknowns

Simple Monte Carlo Path Tracer Step 1: Choose a ray (x, y), (u, v), t; weight = 1 • Step 2: Trace ray to find intersection with nearest surface • Step 3: Randomly decide whether to • compute emitted or reflected light Step 3a: If emitted, • return weight * Le Step 3b: If reflected, • weight *= reflectance • Generate ray in random direction • Go to step 2 •

A Simple Algorithm

Monte Carlo Path Tracing • Advantages � Any type of geometry (procedural, curved, ...) � Any type of BRDF (specular, glossy, diffuse, ...) � Samples all types of paths (L(SD)*E) � Accuracy controlled at pixel level � Low memory consumption � Unbiased - error appears as noise in final image • Disadvantages � Slow convergence � Noise in final image

Monte Carlo Path Tracing Big diffuse light source, 20 minutes Jensen

Monte Carlo Path Tracing 1000 paths/pixel Jensen

Variance N [ ] = � ( ( )) [ ( ) ( ( ))] 2 i − Var E f x f x E f x = 1 i E(f(x)) x 1 x N

Variance 1 [ ] [ ] ( ( )) ( ) = Var E f x Var f x N Variance decreases with 1/N Error decreases with 1/sqrt(N) E(f(x)) x 1 x N

Outline • Motivation • Monte Carlo integration • Monte Carlo path tracing • Variance reduction techniques • Sampling techniques • Conclusion

Variance • Problem: variance decreases with 1/N More samples More samples removes noise removes noise SLOWLY SLOWLY E(f(x)) x 1 x N

Variance Reduction Techniques • Importance sampling • Stratified sampling • Metropolis sampling • Quasi-random 1 1 N � � ( ) ( ) = f x dx f x i N = 1 i 0

Importance Sampling • Put more samples where f(x) is bigger 1 N � � ( ) = f x dx Y i N 1 = i Ω ( ) f x = Y i E(f(x)) i ( ) p x i x 1 x N

Importance Sampling • This is still “unbiased” [ ] � ( ) ( ) = E Y Y x p x dx i Ω ( ) f x � ( ) = p x dx ( ) p x E(f(x)) Ω � ( ) = f x dx Ω for all N x 1 x N

Importance Sampling • Zero variance if p(x) ~ f(x) ( ) ( ) = p x cf x ( ) 1 f x = = Y i i ( ) p x c i E(f(x)) ( ) 0 = Var Y Less variance with better x 1 x N importance sampling

Stratified Sampling • Estimate subdomains separately Arvo E k (f(x)) x 1 x N

Stratified Sampling • This is still unbiased 1 N � ( ) = F f x N i N = 1 i 1 M � = N F i i N 1 = k E k (f(x)) x 1 x N

Stratified Sampling • Less overall variance if less variance in subdomains 1 M [ ] [ ] � = Var F N Var F N i i 2 N 1 = k E k (f(x)) x 1 x N

Outline • Motivation • Monte Carlo integration • Monte Carlo path tracing • Variance reduction techniques • Sampling techniques • Conclusion

Basic Monte Carlo Path Tracer Step 1: Choose a ray (x, y), (u, v), t • Step 2: Trace ray to find intersection with nearest surface • Step 3: Randomly decide whether to • compute emitted or reflected light Step 3a: If emitted, • return weight * Le Step 3b: If reflected, • weight *= reflectance • Generate ray in random direction • Go to step 2 •

Sampling Techniques • Problem: how do we generate random points/directions during path tracing? � Non-rectilinear domains � Importance (BRDF) � Stratified Eye x Surface

Generating Random Points • Uniform distribution: � Use random number generator 1 Probability 0 Ω

Generating Random Points • Specific probability distribution: � Function inversion � Rejection � Metropolis 1 Probability 0 Ω

Generating Random Points • Specific probability distribution: � Function inversion � Rejection � Metropolis 1 Cumulative Probability 0 Ω