An orientation field approach to modelling fibre-generated spatial point processes

Abstract

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