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 com-plex, 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 geo-metric 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.