Computational and Algorithmic Representations of Geometric Objects - CARGO: The Geometry of Optical Paths: Intrinsic Properties, Complexity of Approximation, and Applications

California Institute Of Technology, Pasadena CA

Investigators

Abstract

DMS-0138440 James R. Arvo The most familiar principle of geometrical optics asserts that a ray of light impinging on an ideal mirror will emerge in such a way that the angle of reflection equals the angle of incidence. The trajectory of a photon (or a billiard ball, using another common metaphor), is thus completely determined by its initial conditions and the geometry of the mirrors it subsequently encounters. The geometry of such trajectories has been of interest in numerous fields, including plane geometry, computational geometry, and computer graphics, yet they comprise a vanishingly small subset (a set of measure zero) within the class of all optical paths that fall within the purview of geometrical optics. In particular, optical paths resulting from non-specular reflections constitute a vastly larger class. Moreover, they are of far greater importance to image synthesis as they result from physically realizable models of reflection. Nonetheless, non-specular paths have thus far been largely overlooked as a source of interesting geometrical problems. The main objective of this research is therefore to launch an initial investigation into the basic geometrical properties of non-specular optical paths, both in the traditional combinatorial sense, such as finding a optimal paths connecting two points, and in the continuous sense, such s finding extremal paths, or computing the measure of all k-segment paths connecting two regions. A secondary objective is to explore connections with probabilistic methods, such as standard Monte Carlo visibility techniques and Metropolis light transport, which will likely be the first direct beneficiaries of this work. This work is expected to contribute primarily to the mathematical foundations of image synthesis by identifying optical paths as interesting geometrical entities in themselves, and by exposing some of their fundamental properties in terms of density, measure, and computational complexity. Moreover, it is expected that this new perspective will ultimately be instrumental in studying the accuracy and computational complexity of realistic image synthesis in general, about which very little is known currently. While the problems investigated here will invariably have much in common with previous work on direct and indirect illumination problems in computational geometry, the approaches taken will have a distinctly more continuous flavor, drawing heavily from fields such as measure theory, differential geometry, and geometric probability. Finally, it is expected that this work will serve as a segue into a longer-term investigation of computational complexity in computer graphics by establishing basic tools and connections with other disciplines.