Shape from shading and viscosity solutions

Abstract

This article presents an approach to the shape from shading problem which is based upon the notion of viscosity solutions to the shading partial differential equation, in effect a Hamilton-Jacobi equation. The power of this approach is twofolds: 1) it allows nonsmooth, i.e. nondifferentiable, solutions which allows to recover objects with sharp troughs and creases and 2) it provides a framework for deriving a numerical scheme for computing approximations on a discrete grid of these solutions as well as for proving its correctness, i.e. the convergence of these approximations to the solution when the grid size vanishes. Our work extends previous work in the area in three aspects. First, it deals with the case of a general illumination in a simpler and a more general way (since they assume that the solutions are continuously differentiable) than in the work of Dupuis and Oliensis [9]. Second, it allows us to prove the existence and uniqueness of”continuous” solutions to the shading equation in a more general setting (general direction of illumination) than in the work of Rouy and Tourin [24], thereby extending the applicability of shape from shading methods to more realistic scenes. Third, it allows us to produce an approximation scheme for computing approximations of the”continuous” solution on a discrete grid as well as a proof of their convergence toward that solution.