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Abstract. The aim of this work is to obtain explicit conditions (i.e., conditions on the transition rates) for the stochastic comparison of Markov Processes. A general coupling technique is used to obtain necessary and sufficient conditions for the construction of a coupling Markov Process which stays in a fixed set K for all times and with given marginal processes. The strong stochastic comparison-or, more generally, the stochastic comparison through states functions-appears as a particular case. An example in the Reliability Theory is developed and proves the efficiency of the method. Systems with multiple component types and redundant units are stochastically compared directly or through particular functions.

This paper proposes the important issues in signal segmentation. The signal is disturb ed b y multiplicative noise where the numb er of segments is unknown. A Bayesian approach is proposed to estimate the parameter. The parameter includes the numb er of segments, the location of the segment, and the amplitude. The posterior distrib ution for the parameter does not have a simple equation so that the Bayes estimator is not easily determined. Reversib le Jump Markov chain Monte Carlo (MCMC) method is adopted to overcome the prob lem. The Reversib le Jump MCMC m ethod creates a Markov chain whose distrib ution is close to the posterior distrib ution. The performance of the algorithm is shown b y simulation data. The result of this simulation shows that the algorithm works well. As an application, the algorithm is used to segment a Synthetic Aperture Radar (SAR) signal. The advantage of this method is that the numb er of segments, the position of the segment change, and the amplitude are estimated simultaneously.

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This paper addresses the problem of SAR image segmentation by using reversible jump MCMC sampling.The SAR image segmentation problem is formulated &8 a Bayesian estimation problem. The reversible jump MCMC algorithm is then used to generate samples distributed according to the joint posterior distribution of the unknown parameters.These samples allow to compute marginal maximum a posteriori estimates for the interesting features. The performance of the proposed methodology is illustrated via several simulation results. 1 •. INTRODUCTION The problem of Synthetic Aperture Radar (SAR) image segmentation h&8 received considerable interest in the signal and image processing literature (see for instance (1)[2) and referenCe/! therein). Indeed\ because of the multiplicative speckle noise, most edge. detectors such &8 gradient-based detectors perform poorly· when used for SAR images. This paper studies a new off-line· segmentation algorithm, which operates line-by-line and column-by-column &8 in (3). As explained in [3J, in image segmentation, a retrospective scheme is more attractive as it reflect, the global rather than local aspects 0/ the edge detection problem. The proposed algorithm estimates the number of changepoints, the changepoint locations and the image intensities (usually referred to &8 reflectivities) using a fully Bayesian approach. More precisely, samples distributed according to the a posteriori distribution of the unknown parameters are generated from the reversible jump MCMC algorithm [4). Note that the proposed methodology differs from the off-line procedure studied in [5J. Indeed, instead of generating samples fro� the marginal change location distribution, the proposed MCMC algorithm generates samples· distributed according to the joint distribution of all unknown parameters (i.e. number of changepoints, change locations and image refiectivities). This strategy does not require any marginalization step. As a consequence. the choice of parameter priors is not driven by computational requirements (when nuisance parameters are integrated out. the parameter priers have to be chosen in order to obtain closed form expressions of the interesting marginal distributions). Section 2 formulates the SAR image segmentation problem &8 a Bayesian estimation problem. The likelihood function and the parameter priors necessary to determine the joint posterior distribution of unknown parameters are derived in section 3. Section 4 summarizes the reversible jump MCMC algorithm. which allows to sample from this joint posterior distribution. A simulated annealing algorithm is described in section 5. Simulation results and conclusions are reported in sections 6 and 7.

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