We discuss the uniqueness of 3-D shape reconstruction 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. An example, in which two convex and two concave shapes were obtained from a single shaded image for a trihedral corner, has been given by Horn. However, how many solutions exist for a general polyhedron wasn't described. 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, which were considered to be deleterious in conventional approaches, are used as a constraint to determine the shape solution in our approach.

In this paper, we extend two-image photometric stereo method to treat a concave polyhedron, and present an iterative algorithm to remove the influence of interreflections. By the method we can obtain the shape and reflectance of a concave polyhedron with perfectly diffuse (Lambertian) and unknown constant reflectance. Both simulation and experiment show the feasibility and accuracy of the method.