A Shape Representation Based on Geometric Topology: Bumps, Gaussian Curvature, and the Topological Zodiac

Abstract

We develop a discrete representation for smooth objects embedded in 3-space, which describes the nesting of bumps, depressions, saddles, and related features within each other. The representation is intrin1ic and .table under perturbation of the surface shape and embedding. The final structure is a graph of level set graphs. Each level set graph represents a constant topology of level sets for a region of the Gauuian projective plane ( obtained from the Gaussian sphere by identifying antipodal points) bounded by the images of parabolic curves, corresponding to a range of choices of height orientation. The graph of these graphs has the topology of the adjacency graph of the above regions on the Gaussian projective plane. We show what the topology of the graphs is, and specify what their bifurcations are. Subparts correspond to subgraphs or collapsed graphs in a simple way. Scale space transformations and smoothing correspond to simple bifurcations of the graph structure.