Optical computational geometry

We apply opticaJ computing techniques to basic problems in computational geometry. We present an abstract computational model for optical computing and show its applicability to the efficient solution of geometric problems. Many such problems can be solved with a constant number of optical operations. We also discuss the advantage of optical computing over classical techniques in computational geometry and study issues in the design and analysis of geometric algorithms based on optical computing. 1 Introduction In the last several years there has been significant progress in the design of optical devices that can perform computational tasks. The input to such devices is a 2-dimensional function, ~(z, g) (or several such functions), encoded in some optical form. For example, it can be encoded as a plane light wave with variable amplitude, or with variable phase, or as a planar slab made of optical material with variable transmit-tance coefficient, or in several other methods. The optical device then takes the input " frames " , and pass them through various lenses, mirrors, filters, holograms, possibly also interacting several input frames with each other. In this manner a variety of output frames can be generated, some of which are fairly complex functions of the input frames. In particular , in constant time, one can compute various pointwise *Work on this paper by the first author was supported by a fellowship for new immigrants. Permission to copy without fee all or part of thk material is granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Association for Computing Machine~. To copy otherwise , or to republish, requires a fee and/or specific permission. arithmetic operations on the input frames, compute Fourier transforms, convolutions, conformal transformations of the coordinate system, thresholding, and many other operations (see below for more details). The output frames can then be stored in various ways, similar to those in which the input frames are represented, and then be used as inputs for further optical operations. The resolution of an optical frame is getting progressively smaller, and nowadays it is possible to store several million " pixels " on a single frame. The power of optical computing can be further magnified by multiplexing optical frames, each with a different light frequency. This …