A Multigrid Method for Shape from Shading

ID
TR-91-02
Authors
Uri M. Ascher and Paul M. Carter
Publishing date
April 1991
Length
17 pages
Abstract

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 usual1y nonlinear function of p = Zx and q = Zy.

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 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.