15 Statistical algorithms for noncausal Gauss-Markov fields

Summary In this chapter, we have studied the use of a powerful class of noncausal statistical models, the Gauss-Markov random fields, in 2D signal processing. These models were motivated through intuitive arguments regarding the importance of noncausality and the primacy of local dependence or Markovianity, as well as through the structures arising from numerical integration of PDEs. Our main focus was on the smoothing problem since it is central to many applications. We provided a fast recursive approach to smoothing of noncausal fields as an alternative to the conventional iterative algorithms. We sought to emphasize the importance of using noncausal models over causal ones by illustrating through examples in image enhancement and image compression the kind of artifacts or spurious information that are introduced when causality constraints are imposed on noncausal phenomena. In conclusion, we emphasize that our recursive approach is applicable to 2D signal processing in areas apart from image processing. For example, in physical oceanography, we consider elsewhere the problem of interpolating sparse oceanographic data over 2D lattices.

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