Duality of multi-point and multi-frame geometry: Fundamental shape matrices and tensors

Abstract

We provide a complete analysis of the geometry of N points in 1 image, employing a formalism in which multi-frame and multi-point geometries appear in symmetry: points and projections are interchangeable. We derive bilinear equations for 6 points, trilinear equations for 7 points, and quadrilinear equations for 8 points. The new equations are used to design new algorithms for the reconstruction of projective shape from many frames. Shape is represented by shape descriptors, which are sufficient for object recognition, and for the simulation of new images of the object. We further propose a linear shape reconstruction scheme which uses all the available data- all points and all frames - simultaneously. Unlike previous approaches, the equations developed here lead to direct and linear computation of shape, without going through the cameras' geometry.