Contributions to the Statistical Modelling of Image Data and Spatial Point Patterns 1 Statistics for Image Data 1 3 Markov Chain Monte Carlo 38 1.2 Markov Connected Component Elds 1.2.1 Deenitions

Markov connected component elds 46 Phase transition and simulation for a penalized Ising model with applications in Bayesian image analysis 46 Analysis of residuals from segmentation of noisy images 46 Log Gaussian Cox processes 46 ii Preface This survey paper and the accompanying papers MMller & Waagepetersen (1996), Waagepetersen (1997b), Waagepetersen (1997a), and MMller, Syversveen & Waa-gepetersen (1996) constitute my Ph.D.-thesis in mathematical statistics. The rst part of the survey paper is mainly concerned with prior modelling in Bayesian image analysis. The subject of the second part is modelling and inference for spatial point patterns. Markov chain Monte Carlo (MCMC) has been an indispensable tool in my work, and an account of some basic notions and methods of MCMC is given in the nal section of the survey paper. The papers MMller & Waagepetersen (1996) and MMller et al. (1996) are extensive, and I have therefore found it useful to let the survey paper contain sections with brief presentations of the main results of these papers. I owe many thanks to my supervisor Jesper MMller for his careful, inspiring and enthusiastic guidance. I am also indebted to my wife Katrine for her patience and encouragement, and to my colleagues at the Department of Theoretical Statistics for stimulating discussions and helpful comments. Finally, I wish to thank Proof Western Australia, for the hospitality which I enjoyed during my stays abroad. iii Summary In statistical image analysis much attention is devoted to construction of penalized likelihoods for point estimation of images. Parsimonious image parametriza-tions are rarely applicable, and likelihood estimation is consequently often encumbered with problems like overrtting and nonuniqueness of estimates. A penalizing term is therefore multiplied to the likelihood in order to move the maximum of the penalized likelihood away from the highly variable and \rough" likelihood estimates, towards more precise smoothed estimates. For many commonly applied penalizing terms there are not both objective and practically applicable methods for choosing the smoothing parameter in the penalizing term. It is therefore relevant to analyze the residuals to check whether a suitable degree of smoothing has been applied. Summary statistics and tests to analyze residuals from penalized likelihood image segmentation are proposed and tried out on synthetic data in Waagepetersen (1997a). Apart from the problem of choosing the smoothing parameter, it is typically diicult to assess uncertainty of penalized likelihood image estimates. These problems can in principle be solved within the Bayesian …