A Semi-Lagrangian Approximation of the Oren-Nayar PDE for the Orthographic Shape–from–Shading Problem

Several advances have been made in the last ten years to improve the Shape–from–Shading model in order to allow its use on real images. The classic Lambertian model, suitable to reconstruct 3D surfaces with uniform reflection properties has shown to be unsuitable for other types of surfaces, for example for rough objects consisting of materials such as clay. Other models have been proposed but it is still unclear what would be the best model. For this reason, we start our analysis for non-Lambertian surfaces. The goal being to find a unique model which should be flexible enough to deal with many kinds of real images. As a starting point for this big project, we consider the non-Lambertian Oren–Nayar reflectance model. In this paper we construct a semi-Lagrangian approximation scheme for its nonlinear partial differential equation and we compare its performances with the classical model in terms of some error indicators on series of benchmarks images.