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Wavelet Algorithms for Illumination Computations (Thesis)

Report ID:
October 1994
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One of the core problems of computer graphics is the computation of the
equilibrium distribution of light in a scene. This distribution is given as
the solution to a Fredholm integral equation of the second kind involving
an integral over all surfaces in the scene. In the general case such
solutions can only be numerically approximated, and are generally costly to
compute, due to the geometric complexity of typical computer graphics
scenes. For this computation both Monte Carlo and finite element techniques
(or hybrid approaches) are typically used.
A simplified version of the illumination problem is known as {em
radiosity}, which assumes that all surfaces are diffuse reflectors. For
this case hierarchical techniques, first introduced by Hanrahan {em et
al./}cite{hanr91}, have recently gained prominence. The hierarchical
approaches lead to an asymptotic improvement when only finite precision is
required. The resulting algorithms have cost proportional to $O(k^2+n)$
versus the usual $O(n^2)$ ($k$ is the number of input surfaces, $n$ the
number of finite elements into which the input surfaces are meshed).
Similarly a hierarchical technique has been introduced for the more general
radiance problem (which allows glossy reflectors) by Aupperle {em et
In this dissertation we show the equivalence of these hierarchical
techniques to the use of a Haar wavelet basis in a general Galerkin
framework. By so doing, we come to a deeper understanding of the properties
of the numerical approximations used and are able to extend the hierarchical
techniques to higher orders. In particular, we show the correspondence of
the geometric arguments underlying hierarchical methods to the theory of
Calderon-Zygmund operators and their sparse realization in wavelet bases.
The resulting wavelet algorithms for radiosity and radiance are analyzed
and numerical results achieved with our implementation are reported. We
find that the resulting algorithms achieve smaller and smoother errors at
equivalent work.

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