On a Vietoris-Rips complex and its persistent homology, which extracts complexity of images

Abstract

In a broad area of data analysis including image analysis and sound analysis, it is a pivotal problem to determine position and quantity of salient objects in data, and is expected for several applications as the object detection and area detection. Further, as the foundation of that, it is an efficient problem to formulate and determine importance of data and objects in data. However, because of the enormity of feature values with which we deal and that of areas to which they are applied, it is difficult to solve this problem integrally and generally. Hence it is our urgent subject to construct a unified theory on the importance of data and objects in data. In this paper, we thoroughly deal with our problem mathematically, and we apply methods from Topological Data Analysis to take on the challenge to this problem. More specifically, we construct a metric space from images, and we consider the persistent homology group of the Vietoris-Rips complex of it. The persistent homology group with which we deal in this paper is a generalization of the `` depth of images" (Asao-Sakamoto JSAI2019), and we can argue "uniformity" of images toward studying quantity and position of salient objects.