- Title
- Estimating the topological features of manifolds that can exist in simulated four-dimensional image-type data
- Creator
- Hannouch, Khalil Mathieu
- Relation
- University of Newcastle Research Higher Degree Thesis
- Resource Type
- thesis
- Date
- 2025
- Description
- Research Doctorate - Doctor of philosophy (PhD)
- Description
- Four-dimensional (4D) spaces are particularly interesting because they can require specific study that is separate from higher- and lower-dimensional spaces to fully understand. Unfortunately, human intuition begins to fail beyond even three dimensions. Methods for the analysis of 4D data become relevant in cases of data with four dimensions that would lose essential information if reduced to 3D or 2D. Hence, 4D data analysis is a domain that may open a range of research avenues, which may lead to improvements in applications that were previously restricted to 2D and 3D methods. This thesis will explore this problem in the context of an important class of topological spaces known as manifolds, with a particular focus on manifolds that can be represented in 4D image-type data and the tools that can be used to study them. For instance, persistent homology is a computational approach that employs algebraic topology in order to estimate topological and geometrical properties of data, and forms part of a larger tool set that is known as topological data analysis. While 3D and 4D image-type data arise in many areas of science, they can quickly become so large that directly applying tools such as persistent homology can become infeasible. Convolutional neural networks (CNNs) were inspired by our understanding of the neuronal organisation of the visual cortex and have demonstrated SOA{} performance in many computer vision problems; they have also shown some promise in topological applications. Broadly speaking, this project endeavours to explore how synthetic data, CNNs, and image preprocessing techniques can be used to complement persistent homology methods in the analysis of low-dimensional image-type data, with a focus on 4D data. This thesis will detail the implementation of a 4D CNN and offer a mathematically-founded approach to acquiring labelled synthetic 4D data with which it is possible to train a CNN that can `see' topological properties of these data; this approach is compared with a representative persistent homology approach. The deep learning results are supported with a demonstration that utilises a real-world derived 3D dataset and a discussion is provided to motivate several potential future lines of research.
- Subject
- topology; manifold; betti numbers; persistent homology; machine learning; computer vision; 4D
- Identifier
- http://hdl.handle.net/1959.13/1520250
- Identifier
- uon:57455
- Rights
- Copyright 2025 Khalil Mathieu Hannouch
- Language
- eng
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