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Hill, Bryony J.
Languages: English
Types: Doctoral thesis
Subjects: QA
This thesis introduces a new approach to analysing spatial point data clustered along\ud or around a system of curves or fibres with additional background noise. Such data\ud arise in catalogues of galaxy locations, recorded locations of earthquakes, aerial\ud images of minefields, and pore patterns on fingerprints. Finding the underlying\ud curvilinear structure of these point-pattern data sets may not only facilitate a better\ud understanding of how they arise but also aid reconstruction of missing data.\ud We base the space of fibres on the set of integral lines of an orientation field. Using\ud an empirical Bayes approach, we estimate the field of orientations from anisotropic\ud features of the data. The orientation field estimation draws on ideas from tensor\ud field theory (an area recently motivated by the study of magnetic resonance imaging\ud scans), using symmetric positive-definite matrices to estimate local anisotropies in\ud the point pattern through the tensor method. We also propose a new measure of\ud anisotropy, the modified square Fractional Anisotropy, whose statistical properties\ud are estimated for tensors calculated via the tensor method.\ud A continuous-time Markov chain Monte Carlo algorithm is used to draw samples\ud from the posterior distribution of fibres, exploring models with different numbers\ud of clusters, and fitting fibres to the clusters as it proceeds. The Bayesian approach\ud permits inference on various properties of the clusters and associated fibres, and the\ud resulting algorithm performs well on a number of very different curvilinear structures.

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