From boundaries to bumps: When closed (extremal) contours are critical

Abstract

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 and, more accurately, the rim on the object that projects to it, to define closed extremal curves. This new shape descriptor is invariant over different renderings. It exists at the topological level, which guarantees an image-based counterpart. It surrounds bumps and dents, as well as common interior shape components, and formalizes the qualitative nature of bump perception. The invariants are biologically computable, unify shape inferences from shading and specular materials, and predict new phenomena in bump and dent perception. Most important, working at the topological level allows us to capture the elusive aspect of bump boundaries.

The Role of Specular Reflections and Illumination in the Perception of Thickness in Solid Transparent Objects

“…For instance, humans appear to use common image features when perceiving the shape of matte and velvet objects (Wijntjes et al, 2012 ; Sawayama and Nishida, 2018 ). Some studies have attempted to devise generic models for the perceived shape of opaque objects (Fleming et al, 2004 , 2011 ; Kunsberg and Zucker, 2021 ). However, neurophysiological research further supports the view that there are generic mechanisms of 3D shape recovery in the brain.…”

On Qualitative Shape Inferences: a journey from geometry to topology

“…This leads to a novel type of prediction: if critical contours are the basis on which shape inferences are built, what about the space between them? A number of experiments are beginning to show that this space is not of primary importance; rather it is the arrangement that matters [46,50].…”

“…The requirement is that they be Morse functions: all critical points must be non-degenerate (i.e., the Hessian at those points is non-singular) and no two critical points have the same function value. This introduction is from [50]; for additional material, see [59,26,7,58]. For a smooth Morse surface, the gradient ∇σ = (∂f /∂x, ∂f /∂y) exists at every point.…”

“…Bumps are a special case in which the saddles merge. This case is examined in detail in [50], where closed critical contours in the slant domain are called extremal curves, since they are a relaxation, in a technical sense, of the occluding contour. The template for a bump illustrates a kind of definition for bumps that takes in the global nature of shape features; such global properties are not really in the domain of differential geometry.…”

The many facets of shape

“…This results in a cellular graph structure of local surface patches that captures the qualitative aspects of a surface, without any information about its metric structure. A particularly clear discussion of Morse theory and its application to vision can be found in Kunsberg and Zucker (in press) . They use a more advanced version of that theory called the Morse–Smale complex to identify bumps on a surface and to create a graph of its topographic features.…”