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Some basic results on the use of Gaussian Markov random fields in image modelling
Abstract
This dissertation addresses three basic issues that arise in the use of Gaussian Markov random fields (GMRFs) in image modelling: the multi-resolution properties, the valid parameter space, and the existence of Maximum Likelihood (ML) and Maximum Entropy (ME) parameter estimates. For the multi-resolution properties, we study GMRFs under two types of resolution transformations, Sampling and Block-to-Point. We show that under both these transformations the coarser level fields are non-Markov, and obtain exact descriptions for their covariances and power spectra. To approximate the coarser level non-Markov fields as GMRFs, we propose a new methodology called the Covariance Invariance Approximation (CIA) and study its measure-theoretic properties. We argue that CIA is better suited to image processing than the free-energy based approximations used in renormalization group studies. On the valid parameter space issue, for both 1-D infinite-length GM processes and 2-D infinite-lattice GMRFs, we present a complete procedure for verifying the validity of a given set of parameters. We illustrate this result by applying it to second-order fields in both 1-D and 2-D, and obtain an explicit and simple description of the respective parameter spaces. We observe that in both these examples, the valid parameter space is considerably larger than the space implied by the previously known sufficient condition. For both 1-D and 2-D finite-lattice fields, we show that the valid parameter space does not admit a simple description. The infinite-lattice conditions, however, provide a tight lower-bound approximation to the valid parameter space of finite-lattice fields. Finally, we consider the existence of the ML and the ME estimates for GMRF parameters. The existence of ME estimates is closely related to the extendibility of covariance sequences. Using this fact in conjunction with our results on the valid parameter space of GMRFs, we obtain analytical and computational solutions to the existence problem. For several examples, we obtain an explicit set of conditions that ascertain extendibility and hence existence. For the general case, we propose a cutting-plane algorithm as an alternative to the two numerical procedures that already exist for determining extendibility, namely, the linear programming algorithm and expanding-hull algorithm. Next, we explore the duality between the valid parameter space of GMRFs and the space of extendible covariances, and their relationships with the space of admissible covariances for finite-size data sequences. Using duality, we also relate the existence of ML estimates to extendibility and show that the existence of ML estimates would have to be ascertained through a computationally intensive linear programming procedure. Finally, we present some results regarding the extendibility of covariances over increasing window sizes.
Subject Area
Electrical engineering|Statistics|Mathematics
Recommended Citation
Lakshmanan, Sridhar, "Some basic results on the use of Gaussian Markov random fields in image modelling" (1991). Doctoral Dissertations Available from Proquest. AAI9207421.
https://scholarworks.umass.edu/dissertations/AAI9207421