Ill-posed problems in surface and surface shape recovery

We present new theoretical results which have implications in answering one of the fundamental questions in computer vision: recognition of surfaces and surface shapes. We state the conditions under which: (i) a surface can be recovered, uniquely, from the tangent plane map, in particular from the Gauss map; (ii) a surface shape can be recovered from the metric and the deforming forces. In case where such conditions are not satisfied we classify all exceptions, i.e. the surfaces and surface shapes for which the recovery and registration problems are ill-posed