Physically Based Lighting Calculations for Computer Graphics

by Peter Shirley

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Abstract

Realistic image generation is presented in a theoretical formulation that builds from previous work on the rendering equation. Previous and new solution techniques for the global illumination are discussed in the context of this formulation. The basic physics of reflection and light transport are used to derive the rendering equation. The problem of generating an image is phrased in terms of evaluating the Global Radiance Function, which consists of radiance values for all points and directions. This formulation of the rendering equation differs from previous formulations by rigorously accounting for transparent surfaces. The physical rules governing reflection are used to make improvements in reflection models. In diffuse transmission it is shown that light is filtered to the same extent regardless of which side of the surface the light comes from. This eliminates one of the parameters from previous diffuse transmission models. The microscopic structure of polished surfaces was used to justify coupling the diffuse and specular coefficients according to the Fresnel Equations. The Fresnel Equations are commonly used to vary the reflectivity of metal and transparent dielectrics, but have not been used before to vary the reflectivity of the polish and underlying diffuse substrate. Image-based solution methods are phrased as a lazy evaluation of the Global Radiance Function; evaluation takes place for visible points. Several constraints were outlined for what part of the image function should contribute to each pixel, and a separable, symmetric filter is developed that satisfies these constraints. A stochastic shadow ray generation method is introduced that reduces the number of shadow rays needed for scenes with multiple light sources. The sampling distributions used for shadow rays and other dimensions of the integral are evaluated by introducing to computer graphics the notion of discrepancy from numerical integration theory. The use of discrepancy provided some insight not given by the signal processing theory traditionally used in computer graphics. As part of this discussion a new sampling scheme, N-rooks sampling, is introduced. N-rooks sampling is shown to be as efficient to generate as jittered sampling, while often outperforming Poisson disk sampling. It also can generate distributions for any positive integer number of samples, including primes. The peculiarities of the sampling spaces used in distributed ray tracing are shown to preclude naive hierarchical sampling. It is demonstrated that hierarchical sampling can greatly reduce noise, however, if we have sufficient knowledge of the sampling space. Zonal methods represent the opposite extreme of image methods, where all function values are computed and stored, and each evaluation is a table lookup. The zonal method is phrased as a transport simulation, similar to progressive refinement radiosity methods. Using this direct simulation model, it is straightforward to generate zonal methods for anisotropic reflection. This requires storing accumulated power in a directional table for each zone. A proof is given that, subject to certain constraints, only $O(N)$ rays are required for a zonal solution with $N$ zones. Simulation allows for surfaces which are not zoned to interact with those that are. This is a generalization of the diffuse and specular ray tracing transport work of Malley. This technique can be useful for highly complex or difficult to zone surfaces such as a human face. The zonal solution methods can be applied to participating media in a fairly natural manner. This zonal method has the benefit of not requiring as much computation time when the medium is sparse. This also applies to media with anisotropic scattering characteristics, but such a solution requires a large amount of storage. Wavelength dependent solutions introduce some complications, but can be handled by traditional point sampling techniques. Time dependent solutions are easily handled by image-based solution methods, but are very difficult to apply using zonal methods.

Comments

This thesis is a not bad introduction for some of the issues involving global illumination.

Section 3.2 (pages 32+) discusses the global radiance function. I think this is a good way to envision the light moving between surfaces. The Arvo/Torrance/Smits SIGGRAPH 94 paper uses better terminology however, so please read "surface radiance" instead of "outgoing radiance" and "field radiance" instead of incoming radiance.

Chapters 5 and 6 give some details on how to implement things like path tracing and ray tracing based radiosity. Chapter 6 speculates on how non-diffuse radiosity could be implemented, but this algorithm was not actually implemented until the Graphics Interface 91 paper I did with Kelvin Sung and Bill Brown.

Mitchell has noted that N-Rooks sampling performs poorly for sample sizes much large than 16, so that part of the thesis should be ignored.

The treatment of direct lighting is now outdated, and interested parties should look at more recent papers dealing with direct lighting on my Home Page.

Peter Shirley (shirley@cs.utah.edu)