  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.
