Mathematical frameworks and Monte Carlo Algorithms for global illumination in computer graphics
Philip Dutré |
Contact: Philip Dutré
Ph.D. Thesis, Katholieke Universiteit Leuven, 191 p
ISBN 90-5682-049-4
Abstract
The title of this thesis `Mathematical Frameworks and Monte Carlo Algorithms for Global Illumination in Computer Graphics' refers to a domain in the field of computer graphics known as photo-realistic image rendering or global illumination. The goal of this domain is to compute realistic pictures of a three-dimensional scene, as could have been observed by a human observer or more precisely, a camera.
The first part of this work describes the physical and mathematical foundations which are needed in order to describe the global illumination problem. The fundamental physical measure needed to describe the distribution of light in an environment is radiance. The equation describing the transport of radiance is a recursive integral equation. The dual problem introduces potential as a basic measure, and the potential equation as the corresponding transport equation. Both dual formulations can be used in order to solve the global illumination problem.
Once the mathematical framework has been developed, the equations describing the transport of light or potential can be solved. Due to the high number of integrals and the complexity and unknown behaviour of the functions to be integrated, Monte Carlo integration provides a viable method of computing the global illumination in a three-dimensional scene. Depending on the choice of what transport equation to use, the radiance transport equation leads to distributed ray tracing or path tracing, and the potential transport equation leads to light tracing or particle tracing. The latter method generates particles at the light sources, which each carry a small amount of power. They carry out a random walk in the three-dimensional scene, and possibly contribute their power to the flux of a pixel on the screen. Mathematically, this algorithm can be considered as the dual algorithm of ray tracing.
The sampling functions used for the generation of the random walks can be based on reflective properties of the surfaces encountered. However, in diffuse environments, better results can be expected when they are based on the (unknown) potential distribution. Since the optimal sampling function is not known in advance, one solution is to use adaptive probability density functions. As more particles are being generated, the potential distribution can be approximated more accurately, and thus a better sampling function can be constructed. This technique requires a substantial amount of memory, but produces better results. The used sampling algorithms can also be extended to other Monte Carlo rendering algorithms, such as bidirectional path tracing.
Downloads
- Title Pages (PS.GZ, 25KB)
- Full Text (excl. title pages) (PS.GZ, 917KB)
- Table of Contents (PS.GZ, 17KB)
- Chapter 1 (PS.GZ, 44KB)
- Chapter 2 (PS.GZ, 180KB)
- Chapter 3 (PS.GZ, 74KB)
- Chapter 4 (PS.GZ, 98KB)
- Chapter 5 (PS.GZ, 251KB)
- Chapter 6 (PS.GZ, 305KB)
- Chapter 7 (PS.GZ, 25KB)
- Appendix (PS.GZ 17KB)
- Bibliography (PS.GZ 25KB)
- Dutch Summary (PS.GZ 366KB)