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 we...

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