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 ,  and  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...[Show more]
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