A rigorous and realistic Shape From Shading method and some of its applications

This article proposes a rigorous and realistic solution of the Lambertian Shape From Shading (SFS) problem. The power of our approach is threefolds. First, our work is based on a rigorous mathematical method: we define a new notion of weak solutions (in the viscosity sense) which does not necessarily requires boundary data (contrary to the work of [rouy-tourin:92,prados-faugeras-etal:02,prados-faugeras:03,camilli-falcone:96,falcone-sagona-etal:01]) and which allows to define a solution as soon as the image is (Lipschitz) continuous (contrary to the work of [oliensis:91,dupuis-oliensis:94]). We prove the existence and uniqueness of this (new) solution and we approximate it by using a provably convergent algorithm. Second, it improves the applicability of the SFS to real images: we complete the realistic work of [prados-faugeras:03,tankus-sochen-etal:03], by modeling the problem with a pinhole camera and with a single point light source located at the optical center. This new modelization appears very relevant for applications. Moreover, our algorithm can deal with images containing discontinuities and black shadows. It is very robust to pixel noise and to errors on parameters. It is also generic: i.e. we propose a unique algorithm which can compute numerical solutions of the various perspective and orthographic SFS models. Finally, our algorithm seems to be the most efficient iterative algorithm of the SFS literature. Third, we propose three applications (in three different areas) based on our SFS method

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

Shape From Shading

Shape From Shading is the process of computing the threedimensional shape of a surface from one image of that surface. Contrary to most of the other three-dimensional reconstruction problems (for example, stereo and photometric stereo), in the Shape From Shading problem, data are minimal (we use a single image!). As a consequence, this inverse problem is intrinsically a difficult one. In this chapter we describe the main difficulties of the problem and the most recent theoretical results. We also give some examples of realistic modelings and of rigorous numerical methods

A PDE approach to Shape from Shading via Photometric Stereo

We present a new analytic and numerical approach to the shape from shading using photometric stereo technique. That is, we solve the problem to find the 3D surface of an object starting from its several 2D pictures taken from the same point of view, but changing, for every image, the direction of the light source