Recovering Three Dimensional Shape from a Single Image of Curved Objects

Abstract

We propose an algorithm to recover three-dimensional shape i.e. surface orientation and relative depth from a single segmented image of scenes containing opaque curved objects bounded by piecewise smooth surfaces. It is assumed that the surfaces have no markings or texture and that the reflectance map E=R(n) is known. Solutions for simpler versions of this problem have been presented by Sugihara for polyhedra, and by Hornet al for smooth surface patches. We first analyze the constraints from the line drawing and the image brightness values on the faces, edges and vertices in the scene. For a face this is done by Horn's image irradiance equation. To set up the other constraints we need to know line labels--which can be found using the algorithm developed in [Malik '85]. At a limb, the direction of the surface normal is known directly. We develop a variational formulation of the constraints at an edgeboth from the known direction of the image curve corresponding to the edge and the shading. The associated Euler-Lagrange equation completely captures the local information. At a vertex, the constraints are modelled by a set of non-linear equations. An algorithm has been developed to solve this system of constraints.