How Many 2D Silhouettes Does It Take to Reconstruct a 3D Object?

Abstract

A 2D silhouette of a 3D object O constrains O inside the volume obtained by back-projecting the silhouette from the viewpoint. A set of silhouettes specifies a boundary volume R, the intersection of the volumes due to each silhouette. This approach to the reconstruction of 3D objects is referred to as volume intersection (VI). Not every concave object O is exactly reconstructable from its silhouettes. The closest approximation of O that can be obtained with VI is its visual hull. Only objects coincident with their visual hulls are exactly reconstructable. In practice, to reconstruct an object or its visual hull we must also face computational problems. This paper addresses the problem of finding the theoretical minimum number of silhouettes necessary for the best possible reconstruction of an object. We have found that, in general, the optimal reconstruction of polyhedra with a bounded number n of faces may take an unbounded number of silhouettes. In the case of viewpoints lying also inside the convex hull of a polyhedron exactly reconstructable or with a polyhedral visual hull, we show that O(n5) silhouettes are sufficient, and we describe an algorithm for finding the viewpoints. © 1997 Academic Press.