Abstract

Preprocessing a 2D image often produces a noisy cloud of interest points. We study the problem of counting holes in noisy clouds in the plane. The holes in a given cloud are quantifified by the topological persistence of their boundary contours when the cloud is analyzed at all possible scales. We design the algorithm to count holes that are most persistent in the fifiltration of offsets (neighborhoods) around given points. The input is a cloud of n points in the plane without any user-defifined parameters. The algorithm has O(n log n) time and O(n) space. The output is the array (number of holes, relative persistence in the fifiltration). We prove theoretical guarantees when the algorithm fifinds the correct number of holes (components in the complement) of an unknown shape approximated by a cloud