A Multigrid Method for Shape from Shading
Uri M. Ascher, Paul Carter · SIAM Journal on Numerical Analysis · 1993
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.