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<p><b>Abstract</b>—A new class of Gibbs random fields (GRFs) is proposed capable of modeling geometrical constraints in images by means of mathematical morphology. The proposed models, known as morphologically constrained GRFs, model images by means of their size density. Since the size density is a multiresolution statistical summary, morphologically constrained GRFs explicitly incorporate multiresolution information into image modeling. Important properties are studied and their implication to texture synthesis and analysis is discussed. For morphologically constrained GRFs to be useful in practice, it is important that an efficient technique is available for fitting these models to real data. It is shown that, at low enough temperatures and under a natural condition, the maximum-likelihood estimator of the morphologically constrained GRF parameters can be approximated by means of an important tool of mathematical morphology known as the pattern spectrum. Therefore, statistical inference can be easily implemented by means of mathematical morphology. This allows the design of a computationally simple morphological Bayes classifier which produces excellent results in classifying natural textures.</p>
Gibbs random fields, mathematical morphology, Metropolis algorithm, Monte Carlo simulation, morphological constraints, size density, statistical inference, texture synthesis and analysis.

J. Goutsias and K. Sivakumar, "Morphologically Constrained GRFs: Applications to Texture Synthesis and Analysis," in IEEE Transactions on Pattern Analysis & Machine Intelligence, vol. 21, no. , pp. 99-113, 1999.
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