Computer Graphics Research Group - Educational Activities

Technical University of Budapest

Faculty of Electrical Eng. and Informatics

Monte-Carlo Global Illumination Methods

Script and textbook:
1. Szirmay-Kalos L.: Monte-Carlo Global Illumination Methods, (PDF file),
2. Szirmay-Kalos L. (editor): Theory of Three-dimensional Computer Graphics, Akadémia Kiadó, 1995

Topics:

Flux, solid angle, radiance. Fundamental law of photometry. Probabilistic derivation of the rendering equation. Bi-directional reflectance distribution Function (BRDF). Measuring the radiance. Potential, potential equation. The global illumination problem: surface, lightsource, BRDF and camera models. Numerical methods, error.

Physical plausibility. Helmholtz principle. Albedo and its probabilistic interpretation. Diffuse reflection. Ideal reflection, Fresnel equations. Ideal refraction. Specular reflection: Phong model and its variations. Probabilistic derivation of the Cook-Torrance model. BRDF classes in C++.

Simplifications of the rendering problem: local illumination, recursive ray-tracing. Global illumination solution: inversion, expansion and iteration and their comparison. Neumann series. Formulation of the solution as an integration problem. Numerical stability of iteration. Analytically solveable cases.

Finite-element representation of functions: piece-wise constant and piece-wise linear bases. Adjoint base. Finite-element solution of integral equations. Galerkin's method. Point collocation method. Diffuse case: Galerkin's method with piece-wise constant and point collocation method with piece-wise linear basis functions. Form factors.

Probability spaces. Random variables. Probability distribution function, expected value, variance. Transformation of random variables. Pseudo-random number generation. Congruential methods. What is random? Statistical tests.

Numerical integration. Exponential core of classical quadrature rules. Monte-Carlo integration. Pseudo-random number generation. Quasi-Monte Carlo integration. Uniform series. Variation of functions. Koksma-Hlawka inequality. Low-discrepancy series. Halton sequence. Low-discrepancy series in higher dimensions. k-uniform and infinite-uniform sequences. Variance and error reduction. Importance sampling for Monte-Carlo and for quasi-Monte Carlo quadratures. Metropolis sampling. VEGAS adaptive methods.

Gathering and shooting walks. BRDF sampling: diffuse and specular materials. Russian-roulette. Combined BRDF models. Global importance sampling. Metropolis sampling.

General structure of gathering algorithms. C++ classes for material models and for a ray-shooting based renderer. Ray-casting, visibility ray-tracing, distributed ray-tracing, path-tracing. Handling small lightsources in gathering. General structure of shooting algorithms. Photon-tracing. Light-tracing. Bi-directional techniques: bi-directional path tracing, photon-map, instant radiosity. Metropolis light transport. "Global" global illumination: global radiosity with lines, ray-bundle tracing.

Analysis of classical iteration. Stochastic iteration. Diffuse radiosity: stochastic radiosity, transillumination, stochastic ray-radiosity. Non-diffuse radiosity: light tracing, ray-bundle method. Iteration and quasi-Monte Carlo integration.