From Boundaries to Bumps: when closed (extremal) 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.