2D Shape Modeling using Skeletal Graphs in a Morse Theoretic Framework

Abstract

Topology and geometry are the attributes that uniquely define a shape. Two objects are said to have the same topological structure if one can be morphed into the other without tearing and gluing, whereas geometry describes the relative position of points on a surface. Existing shape descriptors pay little attention to the topology of shapes and instead operate on a smaller subset, where all shapes are assumed to have a genus of one. In this chapter, we will describe a novel 2D shape modeling method that keeps track of the topology of a shape in combination with its geometry for a robust shape representation. Using a Morse theoretic approach and the 3D shape modeling technique in [2] as an inspiration, we focus on representing planar shapes of arbitrary topology. The proposed approach extends existing modeling techniques in the sense that it encompasses a larger class of shapes.

In short, we represent a shape in the form of a topo-geometric graph, which encodes both of its attributes. We show that the model is rigid transformation invariant, and demonstrate that the original shape may be recovered from the model. While classification is beyond the scope of this chapter and hence not explicitly addressed, note that it may greatly benefit from such models.