Classification of lung disease in HRCT scans using integral geometry measures and functional data analysis

A framework for classification of chronic lung disease from high-resolution CT scans is presented. We use a set of features which measure the local morphology and topology of the 3D voxels within the lung parenchyma and apply functional data classification to the extracted features. We introduce the measures, Minkowski functionals, which derive from integral geometry and show results of classification on lungs containing various stages of chronic lung disease: emphysema, fibrosis and honey-combing. Once trained, the presented method is shown to be efficient and specific at characterising the distribution of disease in HRCT slices

N-fold way simulated tempering for pairwise interaction point processes

Pairwise interaction point processes with strong interaction are usually difficult to sample. We discuss how Besag lattice processes can be used in a simulated tempering MCMC scheme to help with the simulation of such processes. We show how the N-fold way algorithm can be used to sample the lattice processes efficiently and introduce the N-fold way algorithm into our simulated tempering scheme. To calibrate the simulated tempering scheme we use the Wang-Landau algorithm

Perfect and imperfect simulations in stochastic geometry

This thesis presents new developments and applications of simulation methods in stochastic geometry. Simulation is a useful tool for the statistical analysis of spatial point patterns. We use simulation to investigate the power of tests based on the J-function, a new measure of spatial interaction in point patterns. The power of tests based on J is compared to the power of tests based on alternative measures of spatial interaction. Many models in stochastic geometry can only be sampled using Markov chain Monte Carlo methods. We present and extend a new generation of Markov chain Monte Carlo methods, the perfect simulation algorithms. In contrast to conventional Markov chain Monte Carlo methods perfect simulation methods are able to check whether the sampled Markov chain has reached equilibrium yet, thus ensuring that the exact equilibrium distribution is sampled. There are two types of perfect simulation algorithms. Coupling from the Past and Fill’s interruptible algorithm. We present Coupling from the Past in the most general form available and provide a classification of Coupling from the Past algorithms. Coupling from the Past is then extended to produce exact samples of a Boolean model which is conditioned to cover a set of locations with grains. Finally we discuss Fill’s interruptible algorithm and show how to extend the original algorithm to continuous distributions by applying it to a point process example

A gradient field approach to modelling fibre-generated spatial point processes

A new non-parametric model is introduced for point processes that are clustered along curves or fibres, with additional background noise. The model identifies random curves as integral lines of a gradient field. In principle this enables the inclusion of all possible non-self-intersecting curves with one underlying smoothness constraint. Markov chain Monte Carlo is combined with Empirical Bayes to provide a practical estimation procedure for properties of the underlying fibre distribution, based on the observed point pattern data. Comparisons are made with other techniques in the literature. Illustrations of the methodology include applications to fingerprints, earthquakes and galaxies

Fibre-generated point processes and fields of orientations

This paper introduces a new approach to analyzing spatial point data clustered along or around a system of curves or "fibres." 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. We then sample from the posterior distribution of fibres, exploring models with different numbers of clusters, fitting fibres to the clusters as we proceed. The Bayesian approach permits inference on various properties of the clusters and associated fibres, and the results perform well on a number of very different curvilinear structures

A Primer on Perfect Simulation

Markov Chain Monte Carlo has long become a very useful, established tool in statistical physics and spatial statistics. Recent years have seen the development of a new and exciting generation of Markov Chain Monte Carlo methods: perfect simulation algorithms. In contrast to conventional Markov Chain Monte Carlo, perfect simulation produces samples which are guaranteed to have the exact equilibrium distribution. In the following we provide an example-based introduction into perfect simulation focussed on the method called Coupling From The Past. 1 Introduction A model that is suciently realistic and exible often leads to a distribution over a high-dimensional or even innite-dimensional space. Examples for such complex distributions include Markov random elds in statistical physics and Markov point processes in stochastic geometry. For many of these complex distributions direct sampling is not feasible. However, there is a very useful tool which may produce (approximate) samples, Mar..

A Bayesian approach to inferring vascular tree structure from 2D imagery

We describe a method for inferring tree-like vascular structures from 2D imagery. A Markov Chain Monte Carlo (MCMC) algorithm is employed to sample from the posterior distribution given local feature estimates, derived from likelihood maximisation for a Gaussian intensity profile. A multiresolution scheme, in which coarse scale estimates are used to initialise the algorithm for finer scales, has been implemented and used to model retinal images. Results are presented to show the effectiveness of the method

Inference on point processes with unobserved one-dimensional reference structure

We present a novel approach to examining local anisotropy in planar point processes. Our method is based on a kernel Principal Component Analysis and produces a tensor field that describes local orientation. The approach is illustrated on an example examining pore patterns in ink fingerprints