A MULTIGRID METHOD FOR SHAPE FROM SHADING

The shape-from-shading problem has received much attention in the Computer Vision literature in recent years. The basic problem is to recover the shape z(x, y) of a surface from a given map of its shading, i.e., its variation of brightness over a given domain. Mathematically, one has to solve approximately the image irradiance equation R(p, q) (x, y) = E(x, y) relating a given image irradiance E(x, y) to the radiance of the surface at each point (x, y), with R(p, q) a given reflectance map which is a usually nonlinear function of p = z(x) and q = z(y). A possible presence of noise and lack of adequate boundary conditions adds to the difficulty of this problem. A number of different approaches towards its solution have been proposed in the Vision literature, including various regularization models. However, a reliable, efficient solution method for practical instances has remained elusive so far. In this paper the various solution models proposed with the aim of applying an efficient multigrid solver are analyzed. A combination of an FMG-continuation technique with an appropriate discretization of one such solution model proposed by B. Horn yields an efficient solver. The results are demonstrated by examples.