Persistent Homology and Shape Description – I

In this paper, we initiate a study of shape description and classification through the use of persistent homology and three tangential constructions. The homology of our first construction, the tangent complex, can distinguish between topologically identical shapes with different “hard” features, such as sharp corners. To capture “soft” curvature-dependent features, we define two other complexes, the filtered and tame complex. The first is a parametrized family of increasing subcomplexes of the tangent complex. Applying persistent homology, we obtain a shape descriptor in terms of a finite union of intervals. We define a metric over the space of such intervals, arriving at a continuous invariant that reflects the geometric properties of shapes. We illustrate the power of our methods through numerous detailed studies of parametrized families of mathematical shapes. In a later paper, we shall apply our techniques to point cloud data to obtain a computational method of shape recognition based on persistent homology.