The USC Andrew and Erna Viterbi School of Engineering USC Signal and Image Processing Institute USC Ming Hsieh Department of Electrical Engineering University of Southern California

Technical Report USC-IPI-1120

“Modern Two-Dimensional Spectrum Estimation Using Noncausal Spatial Models”

by Govind Sharma

December 1984

Two-dimensional (2-D) spectrum estimation from raw data is of interest in signal and image processing. In this thesis, parametric techniques using noncausal spatial models are given. In parametric techniques, the spectrum estimation problem can be divided into two parts, one to estimate the model order and the other to estimate the model parameters characterizing the model. Since there is no ordering property in 2-D, the generalization of causal models used in 1-D, to 2-D causal models may only be an approximate representation. The 1-D autoregressive model can be generalized in two non- equivalent ways to 2-D noncausal models. These two models are Spatial Auto regressive (SAR) models and Gaussian Markov Random Field (GMRF) models. These models characterize the statistical dependence of the observation at location at s on its neighbors in all directions.

The GMRF models possess a Markov property in 2-D, which is a generalization of the Markov property in 1-D. The model parameters are obtained by a method of maximum likelihood (ML). Existence and uniqueness properties of the ML estimates have the following interesting correlation matching property. The inverse Fourier transform of the spectrum obtained by using the ML estimate of parameters matches sample correlations exactly in a neighbor set N. Using this property of GMRF spectrum and the similarity of GMRF spectrum and maximum entropy power spectrum (MEPS), a model based approach for estimating MEPS is presented. Confidence regions are an important part of any point estimation scheme. Using the asymptotic normality property of the ML estimates, simultaneous confidence bands for estimated GMRF spectrum are derived. Some decision rules for choosing appropriate GMRF models are also discussed.

The driving noise in GMRF models is a correlated noise sequence. Some times it is more appropriate to model the driving noise as white noise sequence. In this case one obtains SAR model. The SAR model parameters are obtained by maximizing likelihood function. The existence and uniqueness properties of the ML estimates are established. Using the asymptotic normality property of the of the ML estimates, confidence band are derived for the estimated SAR spectrum. Some decision rules for choosing appropriate SAR model are also given.

The parametric techniques are very vulnerable to `outliers'. A small number of outlying observations can influence the spectrum estimates considerably. To get robust spectrum estimate an iterative algorithm is presented. Some simulation results are also given.

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