Numerical Shape-From-Shading for Discontinuous Photographic Images

Abstract

The height, u(x, y), of a continuous, Lambertian surface of known albedo (i.e., grayness) is related to u(x, y), information recoverable from a black and white flash photograph of the surface, by the partial differential equation \[\sqrt{u^2_x + u^2_y} - n\,{=}\,0.\] We review the notion of a unique viscosity solution for this equation when n is continuous and a recent unique extension of the viscosity solution when n is discontinuous. We prove convergence to this extension for a wide class of the numerical algorithms that converge when n is continuous. After discussing the properties of the extension and the order of error in the algorithms simulating the extension, we point out warning signs which, when observed in the numerical solution, usually indicate that the surface is not continuous or that the viscosity solution or its extension does not correspond to the actual surface. Finally, we discuss a method that, in some of these cases, allows us to correct the simulation and recover the actual surface again.