Projective Shape Analysis

Abstract

Emulating human vision, computer vision systems aim to recognize object shape from images. The main difficulty in recognizing objects from images is that the shape depends on the viewpoint. This difficulty can be resolved by using projective invariants to describe the shape. For four colinear points the cross-ratio is the simplest statistic that is invariant to projective transformations. Five coplanar sets of points can be described by two independent cross-ratios. Using the six-fold set of symmetries of the cross-ratio, corresponding to six permutations of the points, we introduce an inverse stereographic projection of the linear cross-ratio (c) to a stereographic cross-ratio (ξ). To exploit this symmetry, we study the distribution of cos 3ξ when the four points are randomly distributed under appropriate distributions and find the mapping of the cross-ratio so that the distribution of ξ is uniform. These mappings provide a link between projective invariants and directional statistics so that well-established techniques of directional statistics can be used. For example, the goal in object recognition could be to determine whether there is a “false” alarm. We show that the testing of false alarm with a possible specific alternative can be reduced to testing the hypotheses of uniformity on the circle. Furthermore, the goals of the analysis in machine vision might be to discriminate among objects. In such cases the cross-ratio will be expected not to be too variable—that is, concentrated. We show that the analysis of variance type identity is shown to hold for concentrated cross-ratios under maps up to angles. We apply our results to an object recognition problem involving both collinear and coplanar sets of points. A characterization of projective shape spaces is also given.