  Shape from shading is a classical inverse problem in computer vision. This
shape reconstruction problem is inherently ill-defined; it depends on the
assumed light source direction. 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. On smooth surfaces, we show 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 can result from any given shading patch, changing the emphasis to
characterizing ambiguity in the problem. We give an example of how these local
surface ambiguities collapse along certain image contours and how this can be
used for the reconstruction problem.
