Reconstruction of Axisymmetric Objects from One Silhouette

Abstract

Silhouettes of opaque convex objects in orthographic projections were studied in an earlier paper, using a representation on the Gaussian Sphere [1]. A similar argument is applied here to solve for the direct and inverse relations between axisymmetric objects and their silhouettes. Furthermore, a new method for determining object orientation in three dimensions is proposed for such objects, using our analysis of silhouette curvature [1]. Axisymmetric objects are completely determined by a section through their axis of symmetry, which we refer to as the object generator; their silhouettes are symmetric curves in the projection plane. The projection of the object into its silhouette reduces to a transformation between the object generator and the silhouette, which are both planar curves. This paper investigates the conditions under which inversion of this transformation is possible; see Fig. 1. Two cases arise, depending on whether or not the tilt is known between the axis of the object and the projection plane. The derivations summarized here apply to a large class of axisymmetric objects, including non-convex objects. The presentation will be accompanied by a large number of computer generated examples which illustrate various aspects of the theories we developed.