Existence and Uniqueness for Shape From Shading Around Critical Points: Theory and an Algorithm

Abstract

To approach the difficult question of existence and unique ness of solutions for the shape-from-shading problem, we have examined the image irradiance equation using notation and concepts from dynamic systems theory. The method of charac teristic strips used by Horn (1975) defines a dynamic system on a four-dimensional space. Using modern methods for analyzing the behavior of dynamic systems, general uniqueness results and a new shape-from-shading algorithm emerge based on the image dynamic system. Solution surfaces for the shape-from- shading problem are invariant manifolds of the flow generated by the image dynamic system. The stable and unstable man ifolds associated with certain critical points in the image determine locally unique solution surfaces. A theorem about unstable manifolds (the Lambda Lemma) suggests a class of computational methods for finding stable and unstable mani folds around these critical points. A simple example of such a method is described and found to be robust in the presence of image noise and errors in assumptions about the light source.

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