Computing Geometrical Measures of Topological Holes

Abstract

In algebraic topology, persistent homology is a method that computes the homology of an object growing in time. Intuitively, this technique detects holes and provides information about their importance. By combining this topological approach to a notion of distance, it is possible to define geometric relevant measures associated with these holes. This paper introduces two theoretical methods for computing hole measures in volumetric objects defined by surface meshes. Our approach combines the geometrical and topological properties of the medial axis with the efficiency of persistent homology. We present a practical implementation and results on 3D meshed objects.