Ambiguous Shape from Shading with Critical Points

Abstract

The eikonal equation can have an infinite number of solutions when the image has critical points. We exhibit a family, indexed by a continuous parameter, of non isomorphic surfaces with one critical point, which give the same simple image. Hence, shape from shading can be an ill-posed problem when no additional condition on the shape is imposed, even when the image has critical points. Also, deformations without distortion are possible, i.e., there can exist a continuous deformation of the surface which does not modify its image.