|Technical University of Budapest
Faculty of Electrical Eng. and Informatics
Monte-Carlo Global Illumination
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,
Rendering problem (PDF) (PPT)
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.
Reflectance models (BRDFs) (PDF) (PPT)
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++.
Solution strategies (PDF)
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
Finite element methods (PDF)
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 theory and random number generation revisited
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 quadratures for high dimensional integrals
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.
Random walk solution of the global illumination (PDF)
Gathering and shooting walks. BRDF sampling: diffuse and specular materials.
Russian-roulette. Combined BRDF models. Global importance sampling. Metropolis
Review of random walk global illumination algorithms
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.
Iteration solution of the global illumination problem
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