Analysis and approximation of some Shape-from-Shading models for non-Lambertian surfaces

The reconstruction of a 3D object or a scene is a classical inverse problem in Computer Vision. In the case of a single image this is called the Shape-from-Shading (SfS) problem and it is known to be ill-posed even in a simplified version like the vertical light source case. A huge number of works deals with the orthographic SfS problem based on the Lambertian reflectance model, the most common and simplest model which leads to an eikonal type equation when the light source is on the vertical axis. In this paper we want to study non-Lambertian models since they are more realistic and suitable whenever one has to deal with different kind of surfaces, rough or specular. We will present a unified mathematical formulation of some popular orthographic non-Lambertian models, considering vertical and oblique light directions as well as different viewer positions. These models lead to more complex stationary nonlinear partial differential equations of Hamilton-Jacobi type which can be regarded as the generalization of the classical eikonal equation corresponding to the Lambertian case. However, all the equations corresponding to the models considered here (Oren-Nayar and Phong) have a similar structure so we can look for weak solutions to this class in the viscosity solution framework. Via this unified approach, we are able to develop a semi-Lagrangian approximation scheme for the Oren-Nayar and the Phong model and to prove a general convergence result. Numerical simulations on synthetic and real images will illustrate the effectiveness of this approach and the main features of the scheme, also comparing the results with previous results in the literature.Comment: Accepted version to Journal of Mathematical Imaging and Vision, 57 page

Fast Shape from Shading for Phong-Type Surfaces

Abstract. Shape from Shading (SfS) is one of the oldest problems in image analysis that is modelled by partial differential equations (PDEs). The goal of SfS is to compute from a single 2-D image a reconstruction of the depicted 3-D scene. To this end, the brightness variation in the image and the knowledge of illumination conditions are used. While the quality of models has reached maturity, there is still the need for efficient nu-merical methods that enable to compute sophisticated SfS processes for large images in reasonable time. In this paper we address this problem. We consider a so-called Fast Marching (FM) scheme,which is one of the most efficient numerical approaches available. However, the FM scheme is not trivial to use for modern non-linear SfS models. We show how this is done for a recent SfS model incorporating the non-Lambertian reflectance model of Phong. Numerical experiments demonstrate that – without compromising quality – our FM scheme is two orders of magni-tude faster than standard methods.