Shape-from-Shading with discontinuous image brightness

The Shape-from-Shading models in image analysis lead to first order Hamilton–Jacobi equations which may have several weak solutions (in the viscosity sense). Moreover, for real images, these equations are highly discontinuous in the space variable. The lack of uniqueness and the irregularity of the coefficients involve some troubles when we try to compute a solution. In order to avoid these difficulties, here we use recent results in the theory of viscosity solutions to characterize the maximal solution of these equations. Moreover we describe an approximation procedure via smooth equations with a unique viscosity solution.