Counting and detecting figures using persistent homology

Abstract

We apply persistent homology to figure counting/detecting. This paper includes the theoretical construction, the numerical experiments, and the applications to real images. In the theoretical construction, we define some geometric concepts, formulate the counting problem, and prove the theorem that gives the threshold value for distinguishing noise and figures. We use the Rips and Čech complexes and compare the two complexes. In the numerical experiments, we count several model images. Finally, we apply our method to real images.