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Invariance Principles with Logarithmic Averaging for Ergodic Simulations.- Technical Analysis Techniques versus Mathematical Models: Boundaries of Their Validity Domains.- Weak Approximation of Stopped Dffusions.- Approximation of Stochastic Programming Problems.- The Asymptotic Distribution of Quadratic Discrepancies.- Weighted Star Discrepancy of Digital Nets in Prime Bases.- Explaining Effective Low-Dimensionality.- Selection Criteria for (Random) Generation of Digital (0,s)-Sequences.- Imaging of a Dissipative Layer in a Random Medium Using a Time Reversal Method.- A Stochastic Numerical Method for Diffusion Equations and Applications to Spatially Inhomogeneous Coagulation Processes.- Non-Uniform Low-Discrepancy Sequence Generation and Integration of Singular Integrands.- Construction of Good Rank-1 Lattice Rules Based on the Weighted Star Discrepancy.- Probabilistic Approximation via Spatial Derivation of Some Nonlinear Parabolic Evolution Equations.- Myths of Computer Graphics.- Illumination in the Presence of Weak Singularities.- Irradiance Filtering for Monte Carlo Ray Tracing.- On the Star Discrepancy of Digital Nets and Sequences in Three Dimensions.- Lattice Rules for Multivariate Approximation in the Worst Case Setting.- Randomized Quasi-Monte Carlo Simulation of Markov Chains with an Ordered State Space.- Experimental Designs Using Digital Nets with Small Numbers of Points.- Concentration Inequalities for Euler Schemes.- Fast Component-by-Component Construction, a Reprise for Different Kernels.- A Reversible Jump MCMC Sampler for Object Detection in Image Processing.- Quasi-Monte Carlo for Integrands with Point Singularities at Unknown Locations.- Infinite-Dimensional Highly-Uniform Point Sets Defined via Linear Recurrences in $$\mathbb{F}_{2^w } $$ .-Monte Carlo Studies of Effective Diffusivities for Inertial Particles.- An Adaptive Importance Sampling Technique.- MinT: A Database for Optimal Net Parameters.- On Ergodic Measures for McKean-Vlasov Stochastic Equations.- On the Distribution of Some New Explicit Inversive Pseudorandom Numbers and Vectors.- Error Analysis of Splines for Periodic Problems Using Lattice Designs.
A Reversible Jump MCMC Sampler for Object Detection in Image Processing (p. 389)
Mathias Ortner, Xavier Descombes, and Josiane Zerubia
Ariana Research Group (INRIA/I3S), INRIA, 2004 route des Lucioles BP 93, 06902 Sophia Antipolis Cedex, France
Summary. To detect an unknown number of objects from high resolution images, we use spatial point processes models. The method is adapted to our image processing applications since it describes images as realizations of a point process whose points represent geometrical objects. We consider models made of two parts: a data term which quanti.es the relevance of a set of objects with respect to the image and a prior term, containing strong geometrical interactions between objects. We use the Maximum A Posteriori estimator, which is obtained by combining a reversible Markov chain monte carlo (RJMCMC) point process sampler with a simulated annealing procedure. The quality of the results and the speed of the algorithm strongly depend on the used sampler.We present here an adaptation of Geyer-Møller sampler for point processes and show that the resulting Markov Chain keeps the required convergence properties. In particular, we design an updating scheme which allows the generation of points in the neighborhood of some others, and check the relevance of such moves on a toy example. We present experimental results on the di.cult problem of the detection of buildings in a Digital Elevation Model of a dense urban area.
Key words: Spatial point process, RJMCMC, non homogeneous Poisson point process, image processing, building detection.
1 Introduction
A natural way to model images in a probabilistic framework is to consider Markov Random Fields [2]. Such models allow the use of smoothing priors like the Ising model, but require to consider images as a collection of pixels. In the early nineties, A. Baddeley and M.N.M. Van Lieshout [1] proposed to model images as realizations of spatial point processes. Since a spatial point process realization can be a random con.guration of disks, rectangles, lines or any other kind of geometric shapes, spatial point process models appear to be especially amenable to the introduction of a geometric knowledge in the prior term. Spatial point process models in image processing have been used by Green [11] and Rue [12] for object detection and have been applied to real applications in remote sensing [7, 9, 13] and medical imaging [3].
Point process models are especially adapted to object detection as they allow the introduction of a probability distribution on con.gurations of geometrical shapes. Such distributions are usually sub-divided into two parts: a prior term acting on the con.guration of objects to favor or penalize speci.c geometrical patterns, and a likelihood comparing con.gurations of objects to the data. The estimator commonly used is the Maximum a Posteriori (MAP) estimator. Simulated annealing algorithms provide a practical way to compute the MAP despite numerous local maxima. However, the algorithm performance strongly depends on the mixing ability of the sampler. In this paper we present a new updating scheme that improves the quality of the Markov Chain.
