On Qualitative Shape Inferences: a journey from geometry to topology

Shape inference is classically ill-posed, because it involves a map from the (2D) image domain to the (3D) world. Standard approaches regularize this problem by either assuming a prior on lighting and rendering or restricting the domain, and develop differential equations or optimization solutions. While elegant, the solutions that emerge in these situations are remarkably fragile. We exploit the observation that people infer shape qualitatively; that there are quantitative differences between individuals. The consequence is a topological approach based on critical contours and the Morse-Smale complex. This paper provides a developmental review of that theory, emphasizing the motivation at different stages of the research.