A Maximum Entropy solution of the Covariance Extension Problem for Reciprocal Processes

TL;DR

This paper develops a maximum entropy approach to solve the covariance extension problem for stationary reciprocal processes, enabling better modeling and identification of signals in finite regions, with applications in signal and image processing.

Contribution

It introduces a maximum entropy solution for the covariance extension problem specific to stationary reciprocal processes, generalizing existing methods for stationary processes.

Findings

Maximum likelihood leads to a covariance extension problem.

The maximum entropy principle provides a complete solution.

Generalizes the covariance band extension problem.

Abstract

Stationary reciprocal processes defined on a finite interval of the integer line can be seen as a special class of Markov random fields restricted to one dimension. Non stationary reciprocal processes have been extensively studied in the past especially by Jamison, Krener, Levy and co-workers. The specialization of the non-stationary theory to the stationary case, however, does not seem to have been pursued in sufficient depth in the literature. Stationary reciprocal processes (and reciprocal stochastic models) are potentially useful for describing signals which naturally live in a finite region of the time (or space) line. Estimation or identification of these models starting from observed data seems still to be an open problem which can lead to many interesting applications in signal and image processing. In this paper, we discuss a class of reciprocal processes which is the acausal analog of auto-regressive (AR) processes, familiar in control and signal processing. We show that maximum likelihood identification of these processes leads to a covariance extension problem for block-circulant covariance matrices. This generalizes the famous covariance band extension problem for stationary processes on the integer line. As in the usual stationary setting on the integer line, the covariance extension problem turns out to be a basic conceptual and practical step in solving the identification problem. We show that the maximum entropy principle leads to a complete solution of the problem.