A Differential Approach to Photometric Stereo with Perspective Viewing

Shape from Shading and Photometric Stereo are two fundamental problems in Computer Vision aimed at reconstructing surface depth given either a single image taken under a known light source or multiple images taken under different illuminations from the same viewing angle. Whereas the former uses partial differential equation (PDE) techniques to solve the image irradiance equation, the latter can be expressed as a linear system of equations in surface derivatives when 3 or more images are given. Therefore, it seems that current photometric stereo techniques do not extract all possible depth information from each image by itself. Extending our previous results on this problem, we consider the more realistic perspective projection of surfaces during the photographic process. Under this assumption, there is a unique weak solution (Lipschitz continuous) to the problem at hand, solving the well known convex/concave ambiguity of the Shape from Shading problem. The main contribution of this paper is based on a new differential approach for multi-image Photometric Stereo. Most of the existing works on this topic do not directly address this problem. The common approach is to estimate the gradient field of the surface by minimizing some functional and integrate it afterwards to find the depth and hence the geometry of the object. Our new differential approach allows to solve the problem directly, whilst dealing with images having missing parts. The mathematical well-posedness of the new formulation allows fast numerical algorithm based on a combination of fast-marching and fast-sweeping methods.