Exact and approximate computational geometry solutions of an unrestricted point set stereo matching problem

Abstract

In this paper we study the problem of computing an exact, or arbitrarily close to exact, solution of an unrestricted point set stereo matching problem. Within the context of classical approaches like the Marr-Poggio algorithm, this means that we study how to solve the unrestricted basic subproblems created within such approaches, possibly yielding an improved overall performance of such methods. We present an O( n 2 + 4 k ) time and O( n 4) space algorithm for exact unrestricted stereo matching, where n represents the number of points in each set and k the number of depth levels considered. We generalize the notion of a δ-approximate solution for point set congruence to the stereo matching problem and present an O(( ε δ ) kn 2 + 2k) time and O(( ε δ )n 2) space δ- approximate algorithm for unrestricted stereo matching (ε represents measurement inaccuracies in the image). We introduce new computational geometry tools for stereo matching: the translation square arrangement, approximate translation square arrangement and approximate stereo matching tree.