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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 thisfirst part we investigate when image derivatives exhibit invariance to changingillumination by calculating the statistics of image derivatives under generaldistributions on the light source. We computationally validate the hypothesisthat image orientations (derivatives) provide increased invariance toillumination by showing (for a Lambertian model) that a shape-from-shadingalgorithm matching gradients instead of intensities provides more accuratereconstructions when illumination is incorrectly estimated under a flatnessprior.

What's In A Patch, I: Tensors, Differential Geometry and Statistical Shading Analysis