Elke Thönnes

Perfect simulation in stochastic geometry

Simulation plays an important role in stochastic geometry and related fields, because all but the simplest random set models tend to be intractable to analysis. Many simulation algorithms deliver (approximate) samples of such random set models, for example by simulating the equilibrium distribution of a Markov chain such as a spatial birth-and-death process. The samples usually fail to be exact because the algorithm simulates the Markov chain for a long but finite time, and thus convergence to equilibrium is only approximate. The seminal work by Propp and Wilson made an important contribution to simulation by proposing a coupling method, coupling from the past (CFTP), which delivers perfect, that is to say exact, simulations of Markov chains. In this paper we introduce this new idea of perfect simulation and illustrate it using two common models in stochastic geometry: the dead leaves model and a Boolean model conditioned to cover a finite set of points.

Inferring Vascular Structure from 2D and 3D Imagery

We describe a method for inferring vascular (tree-like) structures from 2D and 3D imagery. A Bayesian formulation is used to make effective use of prior knowledge of likely tree structures with the observed being modelled locally with intensity profiles as being Gaussian. The local feature models are estimated by combination of a multiresolution, windowed Fourier approach followed by an iterative, minimum mean-square estimation, which is both computationally efficient and robust. A Markov Chain Monte Carlo (MCMC)algorit hm is employed to produce approximate samples from the posterior distribution given the feature model estimates. We present results of the multiresolution parameter estimation on representative 2D and 3D data, and show preliminary results of our implementation of the MCMC algorithm.

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.

Perfect simulation and inference for point processes given noisy observations

SummaryThe paper is concerned with the exact simulation of an unobserved true point process conditional on a noisy observation. We use dominated coupling from the past (CFTP) on an augmented state space to produce perfect samples of the target marked point process. An optimized coupling of the target chains makes the algorithm considerable faster than with the standard coupling used in dominated CFTP for point processes. The perfect simulations are used for inference and the results are compared to an ordinary Metropolis-Hastings sampler.

Competition systems in a random environment: A convergence result

In this article, a competition system in a random environment is considered. There are two species of particles and each will propagate as follows. An individual particle will move according to a Poisson jump process on the lattice Z d , split into two at a rate which is random, depending on the environment and die off at a rate which is random, depending on the environment. The main result is that, under the mass/speed rescaling (the particles are of mass {\rm ε } while the reproduction and death rate are rescaled accordingly), as the mass of an individual particle tends to zero, the densities of the species are given precisely by the pair of coupled stochastic partial differential equations where Δ is the ‘Lattice Laplacian’. Here the {\rm κ } i are the diffusion rates of each species (which are assumed to be constant) and the {\rm γ } i are the parameters measuring the competitive effects of one species on the other. The quantities u and v denote the densities of the first and second species respectively. {\rm ξ } (1) and {\rm ξ } (2) denote ‘noise’ terms and are the rescaled differences between the natural birth rates and death rates respectively (i.e. the differences between the birth rates and the death rates in the absence of any other species). In the mass/speed rescaling, the variance of the densities of each species has vanished, so that these equations give the precise evolution of the zero-mass limit. *The work is based substantially upon the dissertation for the Master of Science degree written by Elke Thönnes under the supervision of John Noble while both authors were at the Department of Statistics, University College Cork, Ireland.