On the classification of singular points for the global shape from shading problem: A study of the constraints imposed by isophotes

This paper concerns with the global surface reconstruction from a shaded image. It is known that the key to the global reconstruction lies in classifying singular points in the image into three categories--convex, concave or saddle classes. Several methods have been proposed for this problem. All of them require explicit solution of the shape from shading equation. Therefore the results may suffer from image noise and modeling errors of the surface reflecting properties and illumination. If the classification of singular points without such an explicit solution of the equation would have been possible, then the global reconstruction could have been made much more precise. The present work aims towards developing a classification technique where such a explicit solution is not required. Our first step is to show that isophotes (lines of equal brightness) provide some information on the types of singular points; if two isophotes exist which evolve from two singular points and meet each other, then the one of the two singular points is elliptic and the other is hyperbolic. This is derived from intrinsic properties of a smooth (twice differentiable) surface.