Date of Award

5-1-2022

Language

English

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

College/School/Department

Department of Mathematics and Statistics

Content Description

1 online resource (v, 53 pages) : illustrations (some color)

Dissertation/Thesis Chair

Boris Goldfarb

Committee Members

Matthew Zaremsky, Justin Curry

Keywords

Algorithms, Discrete Morse Theory, Geometric Topology, Topological Data Analysis, Morse theory, Vector fields, Digital images, Computer algorithms

Subject Categories

Physical Sciences and Mathematics

Abstract

We address the basic question in discrete Morse theory of combining discrete gradientfields that are partially defined on subsets of the given complex. This is a well-posed question when the discrete gradient field V is generated using a fixed algorithm which has a local nature. One example is ProcessLowerStars, a widely used algorithm for computing persistent homology associated to a grayscale image in 2D or 3D. While the algorithm for V may be inherently local, being computed within stars of vertices and so embarrassingly parallelizable, in practical use it is natural to want to distribute the computation over patches Pi, apply the chosen algorithm to compute the fields Vi associated to each patch, and then assemble the ambient field V from these. Simply merging the fields from the patches, even when that makes sense, gives a wrong answer. We develop both very general merging procedures and leaner versions designed for specific, easy to arrange covering patterns.

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