Two-dimensional spectrum estimation using noncausal autoregressive models

Two-dimensional (2-D) spectrum estimation from raw data is of interest in signal and image processing. A parametric technique for spectrum estimation using 2-D noncausal autoregressive (NCAR) models is given. The NCAR models characterize the statistical dependency of the observation at location s on its neighbors in all directions. This modeling assumption reduces the spectrum estimation problem to two subproblems: the choice of appropriate structure of the NCAR model and the estimation of parameters in NCAR models. By assuming that the true structure of the NCAR model is known, we first analyze the existence and uniqueness of Gaussian maximum likelihood (GML) estimates of NCAR model parameters. Due to the noncausal nature of the models, the computation of GML estimates is burdensome. By assuming specific boundary conditions, computationally tractable expressions are obtained for the likelihood function. Expressions for the asymptotic covariance matrix of the GML estimates as well as the simultaneous confidence bands for the estimated spectrum using GML estimates are derived. Finally, the usefulness of the method is illustrated by computer simulation results.

[1]  H. Künsch.  Thermodynamics and statistical analysis of Gaussian random fields , 1981 .

[2]  W. Larimore,et al.  Statistical inference on stationary random fields , 1977, Proceedings of the IEEE.

[3]  J. Cadzow,et al.  Two-dimensional spectral estimation , 1981 .

[4]  O. Frost.  High resolution 2-D spectral analysis at low SNR , 1980, ICASSP.

[5]  X. Guyon.  Parameter estimation for a stationary process on a d-dimensional lattice , 1982 .

[6]  R. Chellappa,et al.  Digital image restoration using spatial interaction models , 1982 .

[7]  M. Priestley,et al.  A Study of Autoregressive and Window Spectral Estimation , 1981 .

[8]  D.E. Dudgeon,et al.  Fundamentals of digital array processing , 1977, Proceedings of the IEEE.

[9]  Dag Tjostheim,et al.  Autoregressive Modeling And Spectral Analysis of Array Data in the Plane , 1981, IEEE Transactions on Geoscience and Remote Sensing.

[10]  A.K. Jain,et al.  Advances in mathematical models for image processing , 1981, Proceedings of the IEEE.

[11]  Azriel Rosenfeld,et al.  A Comparative Study of Texture Measures for Terrain Classification , 1975, IEEE Transactions on Systems, Man, and Cybernetics.

[12]  D. Thomson,et al.  Robust-resistant spectrum estimation , 1982, Proceedings of the IEEE.

[13]  Marcello Pagano,et al.  Simultaneous confidence bands for autoregressive spectra , 1984 .

[14]  P. Whittle.  ON STATIONARY PROCESSES IN THE PLANE , 1954 .

[15]  K. Ord.  Estimation Methods for Models of Spatial Interaction , 1975 .

[16]  Rama Chellappa,et al.  Two-dimensional spectral estimation using spatial autoregressive models , 1983, ICASSP.

[17]  J. Capon,et al.  Multidimensional maximum-likelihood processing of a large aperture seismic array , 1967 .

[18]  John W. Woods,et al.  Two-dimensional discrete Markovian fields , 1972, IEEE Trans. Inf. Theory.

[19]  Thomas M. Sullivan,et al.  High-resolution two-dimensional spectral analysis , 1979, ICASSP.

[20]  Rama Chellappa,et al.  A model-based approach for estimation of two-dimensional maximum entropy power spectra , 1985, IEEE Trans. Inf. Theory.

[21]  Rama Chellappa,et al.  Estimation and choice of neighbors in spatial-interaction models of images , 1983, IEEE Trans. Inf. Theory.

[22]  Yu. A. Rosanov.  On Gaussian Fields with Given Conditional Distributions , 1967 .

[23]  M. Ekstrom,et al.  Multidimensional spectral factorization and unilateral autoregressive models , 1980 .

[24]  D. Thomson,et al.  Robust Estimation of Power Spectra , 1979 .

[25]  P. Moran.  A Gaussian Markovian process on a square lattice , 1973, Journal of Applied Probability.