Perspective Shape from Shading for Phong-type Non-Lambertian Surfaces

The shape-from-shading (SfS) problem in computer vision is to compute at hand of the shading variation in a given 2-D image the 3-D structure of depicted objects. We introduce an efficient numerical method for a new perspective SfS model for general non-Lambertian surfaces. First, the modelling process is given in detail. The model is based on the perspective model for Lambertian surfaces recently studied by Prados et al., which we extend by use of the Phong reflection model incorporating ambient, diffuse and specular components. The arising partial differential equation (PDE) is a non-linear time-independent Hamilton-Jacobi equation. In order to compute the sought viscosity supersolution of the PDE, we introduce an artificial time into the equation and solve for the steady state. Based on a multi-scale analysis of the PDE, we construct a fully explicit numerical method and elaborate on its stability. In order to achieve fast convergence of the resulting iterative scheme, a coarse-to-fine strategy combined with a sweeping technique is employed. Numerical experiments show the benefits of our approach: While computational times stay reasonable even for quite large images, a substantial qualitative gain can be achieved by use of the new model. Moreover, the computational technique is relatively easy to implement compared to other approaches in the field.