Shape from shading is a classical inverse problem in computer vision. It is inherently ill-defined and, in different formulations, it depends on the assumed light source direction. In contrast to these mathematical difficulties, we introduce a novel mathematical formulation for calculating local surface shape based on covariant derivatives of the shading flow field, rather than the customary integral minimization or P.D.E approaches.Working with the shading flow field rather than the original image intensities is important to both neurogeometry and neurophysiology. To make the calculations concrete we assume a Lambertian model for image formation, butwe do not make global light source positional assumptions. In particular, for smooth surfaces in generic position, we show that second derivatives of brightness are independent of the light sources and can be directly related to surface properties. We use these measurements to define the matching local family of surfaces that could result from a given shading patch. In total our results change the emphasis from seeking a single, well-define solution to an ill-posed problem to characterizing the ambiguity in possible solutions to this problem. The result is relevant both mathematically and perceptually, because we then show how the equations simplify and the ambiguity reduces are certain critical points of intensity. We conclude with a discussion of image reconstruction at these critical points.
Kunsberg, B. , Zucker, S. (2014)., Why shading matters along contours, in G. Citti & A. Sarti (eds.), Neuromathematics of vision, Dordrecht, Springer, pp. 107-129.
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