#
Physically Based Lighting Calculations for Computer Graphics

by Peter Shirley

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Committee

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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.

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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)