Projective shape analysis

Emulating human vision, computer vision systems aim to recognize object sha pe from images. The main difficulty in recognizing objects from images is t hat 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 indep endent cross-ratios, Using the six-fold set of symmetries of the cross-rati o, corresponding to six permutations of the points, we introduce an inverse stereographic projection of the linear cross-ratio (c) to a stereographic cross-ratio (xi). To exploit this symmetry, we study the distribution of co s 3 xi when the four points are randomly distributed under appropriate dist ributions and find the mapping of the cross-ratio so that the distribution of xi is uniform. These mappings provide a link between projective invarian ts and directional statistics so that well-established techniques of direct ional statistics can be used. For example, the goal in object recognition c ould be to determine whether there is a "false" alarm. We show that the tes ting 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. I n such cases the cross-ratio will be expected not to be too variable-that i s, concentrated. We show that the analysis of variance type identity is sho wn to hold for concentrated cross-ratios under maps up to angles. We apply our results to an object recognition problem involving both collinear and c oplanar sets of points. A characterization of projective shape spaces is al so given.