What's In A Patch, II: Visualizing generic surfaces
We continue the development of a linear algebraic framework for the shape-from-shading problem, exploiting the manner in which tensors arise when scalar (e.g. image) and vector (e.g. surface normal) fields are differentiated multiple times. In this paper we apply that framework to develop Taylor expansions of the normal field and build a boot-strapping algorithm to find these polynomial surface solutions (under any light source) consistent with a given patch to arbitrary order. A generic constraint on the image derivatives restricts these solutions to a 2-D subspace, plus an unknown rotation matrix. The parameters for the subspace and rotation matrix encapsulate the ambiguity in the shading problem.
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