  We develop a linear algebraic framework for the shape-from-shading problem,
because tensors arise when scalar (e.g. image) and vector (e.g. surface normal)
fields are differentiated multiple times. Using this framework, we first
investigate when image derivatives exhibit invariance to changing illumination
by calculating the statistics of image derivatives under general distributions
on the light source. Second, we apply that framework to develop Taylor-like
expansions, and build a boot-strapping algorithm to find the polynomial surface
solutions (under any light source) consistent with a given patch to arbitrary
order. A generic constraint on the light source restricts these solutions to a
2-D subspace, plus an unknown rotation matrix. It is this unknown matrix that
encapsulates the ambiguity in the problem. Finally, we use the framework to
computationally validate the hypothesis that image orientations (derivatives)
provide increased invariance to illumination by showing (for a Lambertian
model) that a shape-from-shading algorithm matching gradients instead of
intensities provides more accurate reconstructions when illumination is
incorrectly estimated under a flatness prior.
