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UBC CS TR-91-02 Summary

A Multigrid Method for Shape from Shading, April 1991 Uri M. Ascher and Paul M. Carter, 17 pages

The shape-from-shading problem has received much attention in the Computer Vision literature in recent years. The basic problem is to recover the {\em shape z(x,y)} of a surface from a given map of its {\em shading}, i.e. its variation of brightness over a given domain. Mathematically, one has to solve approximately the {\em image irradiance equation}\\ \begin{center} E(x,y) \end{center} relating a given image irradiance {\em E(x,y)} to the radiance of the surface at each point {\em (x,y)}, with {\em R(p,q)} a given {\em reflectance z_ {x}$ and $q = z_ {y}$. \nA 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. \nIn this paper we analyze the various solution models proposed with the aim of applying an efficient multigrid solver. A combination of an FMG-continuation technique with an appropriate discretization of one such solution model proposed by B. Horn yields an efficient solver. Our results are demonstrated by examples.

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