Discrete Morse Theory & Persistent Homotopy

Abstract

In this thesis, we present new theoretical tools in topological data analysis with applications in image analysis. We draw on a number of tools from Discrete Morse theory, presenting the relevant concepts from [7], [15] and [16] in Chapter 2. We prove an original result in Chapter 3 concerning the attaching maps of CW complex comprised of the critical cells of a discrete Morse function. We introduce the concept of a homotopy merge tree in Chapter 4 as an algebraic tool to summarise homotopical changes over a ltered space. The de nition is an extension of the work of [14], retaining the important properties of interleaving distance and stability. We show that the results of Chapter 2 can be used to simplify calculations of the homotopy merge tree. We also provide original, provably correct algorithms in Chapters 3 and 5 to demonstrate the computational viability of our tools.