A comparison theorem for a piecewise Lipschitz continuous Hamiltonian and application to Shape-from-Shading problems

Abstract

The reconstruction from a shaded image of a Lambertian and not self-shadowing surface illuminated by a single distant pointwise light source may be written as a first-order Hamilton-Jacobi equation. In this paper, we continue the investigation begun in E. Rouy and A. Tourin into the uniqueness of the solution of this equation; the approach is based on the viscosity solutions theory and the dynamic programming principle. More precisely, we concentrate here on the uniqueness of the viscosity solution of this equation in case the measured luminous intensity reflected by the surface is discontinuous along a smooth curve. We prove a general comparison result for a piecewise Lipschitz continuous Hamiltonian and illustrate it by numerical experiments.