Optimally Combining Sampling Techniques for Monte Carlo Rendering Eric Veach and Leonidas J. Guibas Computer Science Department Stanford University SIGGRAPH 1995

Variance Sampling the light sourcesSampling the BRDF Specular Roughness Highly specular surface and large light source Very rough surface and small light source

Multi-sample Model,where Weighted combination of all the sample values An example of w i (balance heuristic) c i : relative number of samples taken from p i

Theorem 1 w i : any non-negative functions with No choice of the w i can improve upon the variance of the balance heuristic by more than (1/min i n i – 1/N)F 2

Weighting Heuristics (a) balance (b) cutoff (c) power (d) maximum

variance roughness

Combine Sampling With power heuristic

A simple test scene (a) Sampling the light source (b) Sampling the hemisphere according to the projected solid angle (c) Combination of samples using the power heuristic (a)(b)(c) One area light source and an adjacent diffuse surface.

Bidirectional Path Tracing

Bidirectional Sampling Strategies (a)Standard MC path tracing. (b)Standard MC path tracing with direct lighting. (c)Depositing light on a visible surface (photomap). (d)Depositing light when a photon hits the camera lens.

Bidirectional Path Tracing Standard path tracing using the same amount of work (same computation time) Combines samples from all the bidirectional techniques 25 samples per pixel56 samples per pixel

Conclusion Power heuristic (with beta=2) gave the best results. These techniques are practical, and the additional cost is small – less than 10%

PBRT inline float BalanceHeuristic(int nf, float fPdf, int ng, float gPdf) { return (nf * fPdf) / (nf * fPdf + ng * gPdf); } inline float PowerHeuristic(int nf, float fPdf, int ng, float gPdf) { float f = nf * fPdf, g = ng * gPdf; return (f*f) / (f*f + g*g); } PBRT define balance and power heuristic in “mc.h”