The problem of recovering a Lambertian surface from a single two-dimensional image may be written as a first-order nonlinear equation which presents the disadvantage of having several continuous and even smooth solutions. A new approach based on Hamilton-Jacobi-Bellman equations and viscosity solutions theories enables one to study non-uniqueness phenomenon and thus to characterize the surface among the various solutions.A consistent and monotone scheme approximating the surface is constructed thanks to the dynamic programming principle, and numerical results are presented.

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This article deals with the so-called Shape-from-Shading problem which arises when recovering a shape from a single image. The general case of a distribution of light sources illuminating a Lambertian surface is considered. This involves original definitions of three types of edges, mainly the apparent contours, the grazing light edges and the shadow edges. The elevation of the shape is expressed in terms of viscosity solution of a first-order Hamilton-Jacobi equation with various boundary conditions on these edges. Various existence and uniqueness results are presented.Mathematics Subject Classification (1991): 65Y25; 65P05