Persistent homology in the cubical setting: theory, implementations and applications

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The Theory of persistent homology, as developed by Gunnar Carlsson at Stanford and others, is based on simplicial homology. In this thesis we explore the possibility of basing persistent homology on cubical homology. We managed to achieve this to some extent and have created a working set of prototype procedures able to calculate the persistent homology of a filtered cubical complex in 2D, and in part 3D, with mod 2 coefficients. We also propose a path that should transform our embryo to a set of procedures capable of handling real applications, in e.g. digital image processing, involving large amounts of data. Extensions to arbitrary finite dimension, orientation, spaces with tor...