×

Spatial ARMA models and its applications to image filtering. (English) Zbl 1298.62162

Summary: The objective of this review paper is to summarize the main properties of the spatial ARMA models and describe some of the well-known methods used in image filtering based on estimation of spatial autoregressive models. A new proposal based on robust RA estimation is also presented. Previous studies have shown that under additive outliers the RA estimator is resistant to a small percentage of contamination and behaves better than the LS, M, and GM estimators. A discussion about how well these models fit to a digital image is presented. Some applications using real images are presented to illustrate how an image is filtered in practice.

MSC:

62M30 Inference from spatial processes
62H11 Directional data; spatial statistics

Software:

spBayes

References:

[1] Allende, H. and Heiller, S. (1992). Recursive generalized M estimates for autoregressive moving-average models., Journal of Time Series Analysis 13 1-18. · Zbl 0850.62666 · doi:10.1111/j.1467-9892.1992.tb00091.x
[2] Allende, H., Galbiati, J. and Vallejos, R. (1998). Digital image restoration using autoregressive time series type models., Bulletin European Spatial Agency 434 53-59.
[3] Allende, H., Galbiati, J. and Vallejos, R. (2001). Robust image modeling on image processing., Pattern Recognition Letters 22 1219-1231.
[4] Banerjee, S., Carlin, B. and Gelfand, A. (2004)., Hierarchical Modeling and Analysis for Spatial Data . Chapman and Hall/CRC Press, Florida. · Zbl 1053.62105
[5] Banham, M. R. and Katsaggelos, A. K. (1997). Digital image restoration., IEEE Signal Processing Magazine 14 24-41.
[6] Basu, S. and Reinsel, G. (1993). Properties of the spatial unilateral first-order ARMA model., Advances in Applied Probbability 25 631-648. JSTOR: · Zbl 0780.62072 · doi:10.2307/1427527
[7] Bennet, J. and Khotanzad, A. (1999). Maximum likelihood estimation methods for multispectral random field image models., IEEE Transaction Pattern Analysis and Machine Intelligence 21 537-543.
[8] Besag, J. (1974). Spatial interaction and the statistical analysis of lattice systems (with discussion)., Journal of the Royal Statistical Society, Series B 55 192-236. JSTOR: · Zbl 0327.60067
[9] Bustos, O. and Yohai, V. (1986). Robust estimates for ARMA models., Journal of the American Statistical Association 81 55-68. JSTOR: · doi:10.2307/2287983
[10] Bustos, O., Ruiz, M., Ojeda, S., Vallejos, R. and Frery, A. (2008). Asymptotic behavior of RA- estimates in autoregressive processes., Submitted. · Zbl 1167.62074 · doi:10.1016/j.jspi.2009.04.016
[11] Chang, I., Tiao, G. C. and Chen, C. (1988). Estimation of time series parameters in the presence of outliers., Technometrics 3 193-204. JSTOR: · doi:10.2307/1270165
[12] Chen, C. and Lui, L. (1993). Joint estimation of model parameters and outliers in time series., Journal of the American Statistical Association 88 284-297. · Zbl 0775.62229 · doi:10.2307/2290724
[13] Cliff, A. and Ord, J. (1981)., Spatial Processes: Models and Applications . Pion Ltd., London. · Zbl 0598.62120
[14] Cullis, B. R. and Glesson, A. C. (1991). Spatial analysis of field experiments-an extension to two dimensions., Biometrics 47 1449-1460.
[15] Eunho, H. and Newton, H. J. (1993). The bias of estimators of causal spatial autoregressive models., Biometrika 80 242-245. · Zbl 0769.62065 · doi:10.1093/biomet/80.1.242
[16] Francos, J., Narasimhan, A. and Woods, J. W. (1995). Maximum likelihood parameter estimation of textures using a Wold-decomposition based model., IEEE Transactions on Image Processing 4 1655-1666.
[17] Francos, J. and Friedlander, B. (1998). Parameter estimation of two-dimensional moving average random fields., IEEE Transaction Signal Processing 46 2157-2165. · Zbl 0978.60053 · doi:10.1109/78.705427
[18] Francos, J. and Nehorai, A. (2003). Interference mitigation in STAP using the two-dimensional Wold decomposition model., IEEE Transactions on Signal Processing 51 2461-2470.
[19] Fox, A. J. (1972). Outliers in time series., Journal of the Royal Statistical Society, Series B 34 350-363. JSTOR: · Zbl 0249.62089
[20] Griffith, D. A. (1988)., Advanced Spatial Statistics . Kluver, Dordrecht, The Netherlands.
[21] Grondona, M. R., Crossa, J., Fox, P. N. and Pfeiffer, W. H. (1996). Analysis of variety yield trials using two-dimensional separable ARIMA processes., Biometrics 52 763-770. · Zbl 0875.62527 · doi:10.2307/2532916
[22] Guyon, X. (1995)., Random Fields on a Network. Modeling, Statistics and Applications . Springer, Berlin. · Zbl 0839.60003
[23] Haining, R. P. (1978). The moving average model for spatial interaction., Transactions and Papers, Institute of British Geographers, New Series 3 202-225.
[24] Isaksson, A. J. (1993). Analysis of identified 2-D noncausal models., IEEE Transactions on Information Theory 39 525-534. · Zbl 0776.93021 · doi:10.1109/18.212282
[25] Jain, A. K. (1989)., Fundamentals of Digital Image Processing . Prentice Hall, Lugar. · Zbl 0744.68134
[26] Kashyap, R. L. and Chellappa, R. (1983). Estimation and choice of neighboors in spatial-interaction models of images., IEEE Transactions on Information Theory 19 60-72. · Zbl 0507.62082 · doi:10.1109/TIT.1983.1056610
[27] Kashyap, R. and Eom, K. (1988). Robust images techniques with an image restoration application., IEEE Transactions on Acoustics and Speech Signal Processing 36 1313-1325. · Zbl 0663.62105 · doi:10.1109/29.1659
[28] Katsaggelos, A. K. (1989). Iterative image restoration algorithms., Optical Engineering 28 735-748.
[29] Krishnamurthy, R., Woods, J. and Francos, J. (1996). Adaptive restoration of textures images with mixed spectra using a generalized Wiener filter., IEEE Transactions on Image Processing 5 648-652.
[30] Kokaram, A. (2004). A statistical framework for picture reconstruction using 2D AR models., Image and Vision Computing 22 165-171.
[31] Liu, F. and Piccard, R. (1996). Periodicity, directionality and randomness: Wold features for image modeling and retrieval., IEEE Transactions on Pattern Analysis and Machine Intelligence 18 722-733.
[32] Martin, R. J. (1979). A subclass of lattice processes applied to a problem in planar sampling., Biometrika 66 209-217. JSTOR: · Zbl 0404.62017 · doi:10.1093/biomet/66.2.209
[33] Martin, R. D. (1980). Robust estimation of autoregressive models. In, Direction in Time Series (D. R. Brillinger and G. C. Tiao, eds.). Institute of Mathematical Statistics, Haywood, CA. · Zbl 0531.62038
[34] Martin, R. J. (1990). The use of time-series models and methods in the analysis of agricultural field trials., Communications in Statistics Theory Methods 19 55-81. · doi:10.1080/03610929008830187
[35] Martin, R. J. (1996). Some results on unilateral ARMA lattice processes., Journal of Statistical Planning and Inference 50 395-411. · Zbl 0848.62051 · doi:10.1016/0378-3758(95)00066-6
[36] Ojeda, S. M. (1999). Robust RA estimators for bidimensional autoregressive models. Ph.D. dissertation, Facultad de Matemáticas, Astronomía y Física, Universidad Nacional de Córdoba, Argentina.
[37] Ojeda, S. M., Vallejos, R. O. and Lucini, M. (2002). Performance of RA estimator for bidimensional autoregressive models., Journal of Statistical Simulation and Computation 72 47-62. · Zbl 1091.62536 · doi:10.1080/00949650211426
[38] Quenouille, M. H. (1949). Problems in plane sampling., Annals of Mathematical Statistics 20 355-375. · Zbl 0035.09103 · doi:10.1214/aoms/1177729989
[39] Rukhin, A. and Vallejos, R. (2008). Codispersion coefficient for spatial and temporal series., Statistics and Probability Letters 78 1290-1300. · Zbl 1144.62081 · doi:10.1016/j.spl.2007.11.017
[40] Tsay, R. S., Peña, D. and Pankratz, A. E. (2000). Outliers in multivariate time series., Biometrics 87 789-804. · Zbl 1028.62073 · doi:10.1093/biomet/87.4.789
[41] Tsay, R. S. (1988). Outliers, level shifts, and variance change in time series., Journal of Forecasting 7 1-20.
[42] Tjostheim, D. (1978). Statistical spatial series modelling., Advances in Applied Probability 10 130-154. JSTOR: · Zbl 0383.62060 · doi:10.2307/1426722
[43] Vallejos, R. and Mardesic, T. (2004). A recursive algorithm to restore images based on robust estimation of NSHP autoregressive models., Journal of Computational and Graphical Statistics 13 674-682. · doi:10.1198/106186004X2183
[44] Vallejos, R. and Garcia-Donato, G. (2006). Bayesian analysis of contaminated quarter plane moving average models., Journal of Statistical Computation and Simulation 76 131-147. · Zbl 1088.62043 · doi:10.1080/00949650412331321133
[45] Vallejos, R., Ojeda, S. and Bustos, O. (2008). On RA and GM estimates for spatial autoregressive models., Submitted.
[46] Whittle, P. (1954). On stationary processes on the plane., Biometrika 41 434-449. JSTOR: · Zbl 0058.35601 · doi:10.1093/biomet/41.3-4.434
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.