#### Technical Report USC-SIPI-102

“Semi-Markov Random Field Models for Nonstationary Signal Processing”

by John Goutsias

August 1986

With the increasing availability and decreasing cost of digital computers, it is desirable to develop sophisticated and accurate parametric models for the mathematical description of different measured signals. Often, a major characteristic of such signals is their nonstationary nature, a characteristic that is usually ignored in signal modeling. In this dissertation we focus our attention on modeling the nonstationary signal behavior, thus resulting in an accurate signal description.

We develop a new class of nonstationary signals, the class of semi-Markov random fields. The likelihood function, which completely describes the statistical behavior of this class of random fields, is derived. Necessary and sufficient conditions are examined, for the statistical equivalence between a Markov random field and a semi-Markov random field. We argue that the semi-Markov random fields are appropriate for the statistical description of the structural representation of a two-dimensional lattice.

We introduce the class of doubly stochastic Gaussian random fields and we examine their statistical properties. We develop a Bayesian procedure for the solution of the problem of detecting and estimating an observed, noisy and filtered version of a doubly stochastic Gaussian random field. We derive a separation principle indicating that the detection and estimation problem, for an observed doubly stochastic Gaussian random field, can be solved by first solving a detection problem, second solving an estimation problem, and, finally, combining their solutions in a natural fashion. A new likelihood detector, the integer most-likely search detector, is designed for the efficient solution of the signal detection problem. We develop an adaptive parameter estimation/signal detection algorithm for the solution of the previous problem. We argue that this algorithm is advantageous, since it allows the real time estimation of the unknown underlying parameters.

We illustrate our models, and the performance of the proposed algorithms, by considering some synthetic and real data examples. The problems of signal segmentation (for the one-dimensional and the two-dimensional cases) and signal restoration (for the one-dimensional case) are illustrated. Finally, texture analysis and synthesis is considered based on the proposed model.

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