Geometric Distributions for Catadioptric Sensor Design

Abstract

Catadioptric sensors are visual sensors that employ lenses (dioptrics) and mirrors (catoptrics). We present a general method of catadioptric sensor design for realizing prescribed projections. Our method makes use of geometric distributions in. 3-dimensional space, which are generalizations of vector fields. The main idea is this: if one desires a reflective surface that will image the world in a certain way, then this condition determines the orientation of the tangent planes to the surface. Analytically, this means that the surface will then be determined by a pair of partial differential equations, which may or may not have a common solution. We show how to check if a common solution exists. If no common solution exists, we describe a method for obtaining optimal approximate solutions in a least-squares sense. As an example application, we construct a mirror that will give a panoramic view of a scene without any digital unwarping.