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 Markov Random Field Modeling in Image Analysis (Advances in Pattern Recognition), 9781848002784 (1848002785), Springer, 2009 Modeling problems in this book are addressed mainly from the computational viewpoint. The primary concerns are how to define an objective function for the optimal solution to a image analysis or computer vision problem and how to find the optimal solution. The solution is defined in an optimization sense because the perfect solution is difficult to find due to various uncertainties in the process, so we usually look for an optimal one that optimizes an objective in which constraints are encoded. Contextual constraints are ultimately necessary in the interpretation of visual information. A scene is understood in the spatial and visual contexts; the objects are recognized in the context of object features in a lower-level representation; the object features are identified based on the context of primitives at an even lower-level; and the primitives are extracted in the context of image pixels at the lowest level of abstraction. The use of contextual constraints is indispensable for a capable vision system. Markov random field (MRF) theory provides a convenient and consistent way of modeling context-dependent entities such as image pixels and correlated features. This is achieved through characterizing mutual influences among such entities using conditional MRF distributions. The practical use of MRF models is largely ascribed to a theorem stating the equivalence between MRF’s and Gibbs distributions that was established by Hammersley and Clifford (1971) and further developed by Besag (1974). This is because the joint distribution is required in most applications but deriving the joint distribution from conditional distributions turns out to be very difficult for MRF’s. The MRF-Gibbs equivalence theorem points out that the joint distribution of an MRF is a Gibbs distribution, the latter taking a simple form. This gives us not only a mathematically sound but also mathematically tractable means for statistical image analysis (Grenander 1983; Geman and Geman 1984). From the computational perspective, the local property of MRF’s leads to algorithms that can be implemented in a local and massively parallel manner.