Proximal Cech Complexes in Approximating Digital Image Object Shapes. Theory and Application

Abstract

This article introduces proximal Cech complexes in approximating object shapes in digital images. The theoretical
framework is based on Cech complexes and proximity spaces. Several topological structures are defined for
the Cech nerve based covers of a finite region of Euclidean plane. We define k-petals and k-corollas which are the
generalizations of spokes and maximal nuclear clusters. We extend the classical notion of a proximity as a binary
relation, to arbitrary number of sets. A new shape signature based on the distribution of orders of Cech nerves is
defined. A practical application of this framework in approximating object shapes in digital images is given.