Reduction techniques for the persistent homology transform on digital images

Abstract

A digital image can be naturally represented as a set of lattice cubes defined by elementary intervals. This thesis presents a new object to represent digital images, the generalized cubical complex. This combinatorial object is also composed of lattice cubes, but does not restrict its cubes to be defined by elementary length intervals. The union of these cubes represents a space in R^n. We later show how to construct a generalized cubical complex’s persistent homology transform (PHT) and prove two generalized cubical complexes representing the same space in Rn have the same PHT. Using image segmentation, we show a digital image can be represented by an object requiring less information with respect to its traditional representation which retains its PHT.