Approximation scheme for the maximal solution of the shape-from-shading model

The shape-from-shading model leads to a first order Hamilton-Jacobi equation coupled with a boundary condition, f.e. of Dirichlet type. The analytical characterization of the solution presents some difficulties since this is an eikonal type equation which has several weak solutions (in the viscosity sense). The lack of uniqueness is also a big trouble when we try to compute a solution. In order to avoid those difficulties the problem is usually solved adding some additional informations such as the height at points where the brightness has a maximum, or the complete knowledge of a level curve. Here we use recent results in the theory of viscosity solutions to characterize the maximal solution without extra informations besides the equation and we construct an algorithm which converges to that solution. Some examples show the accuracy of the algorithm.