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Polygonal random fields and the reconstruction of piecewise continuous functions

Polygonal random fields (PRFs) are random fields which map a region of the plane into a finite set such that the mapping's edges divide the region into polygons. Because PRFs provide a probability measure for the set of polygonal colorings of a region, they can be used to form a model for a Bayesian approach to the image segmentation problem. Previously, Markov random fields have been used in Bayesian formulations of the segmentation problem, but such formulations confine the problem to a discrete lattice. In this thesis, PRFs are defined and their properties are discussed. Models are formulated to reconstruct piecewise constant functions and piecewise smooth functions. In each case, an algorithm is presented to generate sample realizations from the posterior distributions of the models. The algorithms are then implemented to demonstrate the performance of each model. Thesis Supervisor: Sanjoy K. Mitter Title: Professor of Electrical Engineering