Determining a polyhedral shape using interreflections

Abstract

We discuss the uniqueness of 2-D shape recovery of a polyhedron from a single shading image. First, we analytically show that multiple convex (and concave) shape solutions usually exist for a simple polyhedron if interreflections are not considered. Then we propose a new approach to uniquely determine the concave shape solution using interreflections as a constraint. A numerical example, in which two convex shapes and two concave shapes exist for a trihedral corner, has been shown by Horn. However, it is difficult to prove the uniqueness using constraint equations. We analytically show that multiple convex (and concave) shape solutions usually exist for a pyramid using a reflectance map, if interreflection distribution is not considered. However, if interreflection distribution is used as a constraint that limits the shape solution (for a concave polyhedron), the polyhedral shape can be uniquely determined. Interreflections are used as a constraint to determine the shape solution in our approach.