Mirauta, Dumitru
(2025)
A general approach to handling complex basis variation in unmixing problems
PhD thesis, University of Liverpool.
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Abstract
Characterisation data, often in the form of spectra, can be used to liken, contrast, infer properties of, and generally further an understanding of, various objects of study across application domains. Examples include powder X-ray diffraction data for the study of crystal structures in materials discovery; hyperspectral absorbance data for the characterisation of ground objects in remote sensing; and Fourier transforms for characterisation of periodic signals, such as notes or chords in the transcription of music. Even within a single application domain, multiple methods of characterisation may be of interest, as they may communicate different aspects of an object. Spectral data is often "mixed". The unmixing problem is that of separating the individual constituents that comprise "mixed data". This thesis is concerned with a broad class of "mixed data" generated by not only combining a set of basis patterns in various proportions, through an arbitrary mechanism; but first generating variations of these, again through an arbitrary mechanism, and then combining these. This thesis broadly characterises past methods as "reconstruction" based, which are parametric, or "geometric methods", which are non-parametric. By focusing on properties of a model (expressed as geometric features), rather than parameterised expressions, geometric methods can be more broadly applicable, across data models (found across application disciplines) with similar properties. This thesis proposes a new non-parametric unmixing method, explored largely within the context of an arbitrary choice of metric space. The method can be specialised by picking a specific metric space. Criteria are provided for compatibility between the metric and a variation function of interest. The choice of Wasserstein space is explored in more detail. This is motivated by the ability to encode many kinds of data as distributions, and the Wasserstein metrics compatibility with variation mechanisms that can be interpreted as a warping of the domain the spectra is defined on. For example, this can capture non-uniform shifting and broadening of peak patterns in powder X-ray diffraction. While the abstract exploration forms solid foundations, much work has also been done to convert this into practical computational routines, to be used in exploratory data analysis, in the lab. So that the method developed is accessible to application domain experts, in turn these computational routines are condensed into a powerful graphical user interface.
| Item Type: | Thesis (PhD) |
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| Uncontrolled Keywords: | Characterisation, Inverse problem, Nonparametric, Optimal transport, Spectra, Unmixing |
| Divisions: | Faculty of Science & Engineering > School of Electrical Engineering, Electronics and Computer Science Faculty of Science & Engineering |
| Depositing User: | Symplectic Admin |
| Date Deposited: | 20 Aug 2025 13:43 |
| Last Modified: | 01 Aug 2026 01:31 |
| DOI: | 10.17638/03190542 |
| Supervisors: |
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| URI: | https://livrepository.liverpool.ac.uk/id/eprint/3190542 |
| Disclaimer: | The University of Liverpool is not responsible for content contained on other websites from links within repository metadata. Please contact us if you notice anything that appears incorrect or inappropriate. |
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