A Linear Prediction Approach to Two-dimensional Spectral Factorization

This thesis is concerned with the extension of the theory and computational techniques of time-series linear prediction to two-dimensional (2-D) random processes. 2-D random processes are encountered in image processing, array processing, and generally wherever data is spatially dependent. The fundamental problem of linear prediction is to determine a causal and causally invertible (minimumphase), linear, shift-invariant whitening filter for a given random process. In some cases, the exact power density spectrum of the process is known (or is assumed to be known) and finding the minimum-phase whitening filter is a deterministic problem. In other cases, only a finite set of samples from the random process is available, and the minimum-phase whitening filter must be estimated. Some potential applications of 2-D linear prediction are Wiener filtering, the design of recursive digital filters, highresolution spectral estimation, and linear predictive coding of images. 2-D linear prediction has been an active area of research in recent years, but very little progress has been made on the problem. The principal difficulty has been the lack of computationally useful ways to represent 2-D minimum-phase filters. In this thesis research, a general theory of 2-D linear prediction has been developed. The theory is based on a particular definition for 2-D causality which totally orders the points in the plane. By paying strict attention to the ordering property, all of the major results of 1-D linear prediction theory are extended to the 2-D case. Among other things, a particular class of 2-D, least-squares, linear, prediction error filters are shown to be minimum-phase, a 2-D version of the Levinson algorithm is derived, and a very simple interpretation for the failure of Shanks' conjecture is obtained. From a practical standpoint, the most important result of this thesis is a new canonical representation for 2-D minimum-phase filters. The representation is an extension of the reflection coefficient (or partial correlation coefficient) representation for 1-D minimum-phase filters to the 2-D case. It is shown that associated with any 2-D minimum-phase filter, analytic in some neighborhood of the unit circles, is a generally infinite 2-D sequence of numbers, called reflection coefficients, whose magnitudes are less than one, and which decay exponentially to zero away from the origin. Conversely, associated with any such 2-D reflection coefficient sequence is a unique 2-D minimum-phase filter. The 2-D reflection coefficient representation is the basis for a new approach to 2-D linear prediction. An approximate whitening filter is designed in the reflection coefficient domain, by representing it in terms of a finite number of reflection coefficients. The difficult minimum-phase requirement is automatically satisfied if the reflection coefficient magnitudes are constrained to be less than one. A remaining question is how to choose the reflection coefficients optimally; this question has only been partially addressed. Attention was directed towards one convenient, but generally suboptimal method in which the reflection coefficients are chosen sequentially in a finite raster scan fashion according to a least-squares prediction error criterion. Numerical results are presented for this approach as applied to the spectral factorization problem. The numerical results indicate that, while this suboptimal, sequential algorithm may be useful in some cases, more sophisticated algorithms for choosing the reflection coefficients must be developed if the full potential of the 2-D reflection coefficient representation is to be realized. Thesis Supervisor: Arthur B. Baggeroer Title: Associate Professor of Electrical Engineering Associate Professor of Ocean Engineering