Title: What's In A Patch, I: Tensors, Differential Geometry and Statistical Shading Analysis,
Abstract: 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. The work is in two parts. In this
first part we investigate when image derivatives exhibit invariance to changing
illumination by calculating the statistics of image derivatives under general
distributions on the light source. We 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.