Stability and Likelihood of Views of Three Dimensional Objects

Abstract

Can we say anything general about the distribution of two dimensional views of general three dimensional objects? In this paper we present a first formal analysis of the stability and likelihood of two dimensional views (under weak perspective projection) of three dimensional objects. This analysis is useful for various aspects of object recognition and database indexing. Examples are Bayesian recognition; indexing to a three dimensional database by invariants of two dimensional images; the selection of “good” templates that may reduce the complexity of correspondence between images and three dimensional objects; and ambiguity resolution using generic views. We show the following results: (1) Both the stability and likelihood of views do not depend on the particular distribution of points inside the object; they both depend on only three numbers, the three second moments of the object. (2) The most stable and the most likely views are the same view, which is the “flattest” view of the object. Under orthographic projection, we also show: (3) the distance between one image to another does not depend on the position of its viewpoint with respect to the object, but only on the (geodesic) distance between the view-points on the viewing sphere. We demonstrate these results with real and simulated data.