From Boundaries to Bumps: when closed contours are critical

Invariants underlying shape inference are elusive: a variety of shapes can give rise to the same image, and a variety of images can be rendered from the same shape. The occluding contour is a rare exception: it has both image salience, in terms of isophotes, and surface meaning, in terms of surface normal. We relax the notion of occluding contour to define closed extremal curves, a new shape invariant that exists at the topological level. They surround bumps, a common but ill-specified interior shape component, and formalize the qualitative nature of bump perception. Extremal curves are biologically computable, unify shape inferences from shading, texture, and specular materials, and predict new phenomena in bump perception.

Characterizing Ambiguity in Light Source Invariant Shape from Shading

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.