This paper proposes the mixture of Alpha-stable (MAS) distributions for modeling statistical property of Synthetic Aperture Radar (SAR) images in a supervised Markovian classification algorithm. Our work is motivated by the fact that natural scenes consist of various reflectors with different types that are typically concentrated within a small area, and SAR images generally exhibit sharp peaks, heavy tails, and even multimodal statistical property, especially at high resolution. Unimodal distributions do not fit such statistical property well, and thus a multimodal approach is necessary. Driven by the multimodality and impulsiveness of high resolution SAR images histogram, we utilize the mixture of Alpha-stable distributions to describe such characteristics. A pseudo-simulated annealing (PSA) estimator based on Markov chain Monte Carlo (MCMC) is present to efficiently estimate model parameters of the mixture of Alpha-stable distributions. To validate the proposed PSA estimator, we apply it to simulated data and compare its performance to that of a state-of-the-art estimator. Finally, we exploit the MAS distributions and a Markovian context for SAR images classification. The effectiveness of the proposed classifier is demonstrated by experiments on TerraSAR-X images, which verifies the validity of the MAS distributions for modeling and classification of SAR images.
References
[1]
Perko, R.; Raggam, H.; Deutscher, J.; Gutjahr, K.; Schardt, M. Forest assessment using high resolution SAR data in X-band. Remote Sens. 2011, 3, 792–815.
[2]
Alberga, V. Similarity measures of remotely sensed multi-sensor images for change detection applications. Remote Sens. 2009, 1, 122–143.
[3]
Hasager, C.B.; Badger, M.; Pena, A.; Larsén, X.G.; Bingol, F. SAR-based wind resource statistics ′ in the Baltic Sea. Remote Sens. 2011, 3, 117–144.
[4]
Tison, C.; Nicolas, J.M.; Tupin, F.; Maitre, H. A new statistical model for Markovian classification of urban areas in high-resolution SAR images. IEEE Trans. Geosci. Remote Sens. 2004, 42, 2046–2057.
[5]
Jacob, A.M.; Hemmerly, E.M.; Fernandes, D. SAR Image Classification Using a Neural Classifier Based on Fisher Criterion. In Proceedings of the 7th Brasilian Symposium on Neural Networks (SBRN02), Recife, Brazil, 11–14 November 2002; pp. 168–172.
[6]
Chamundeeswari, V.V.; Singh, D.; Singh, K. Unsupervised Land Cover Classification of SAR Images by Contour Tracing. In Proceedings of Geoscience and Remote Sensing Symposium (IGARSS 2007), Barcelona, Spain, 23–27 July 2007; pp. 547–550.
Oliver, C. Optimum texture estimators for SAR clutter. J. Phys. D Appl.Phys. 1993, 26, 1824–1835.
[9]
Menon, M. Estimation of the shape and scale parameters of the Weibull distribution. Technometrics 1963, 5, 175–182.
[10]
Szajnowski, W. Estimators of log-normal distribution parameters. IEEE Trans. Aerosp Electron. Syst. 1977, AES-13, 533–536.
[11]
Jakeman, E.; Pusey, P. A model for non-Rayleigh sea echo. IEEE Trans. Antennas Propagat. 1976, 24, 806–814.
[12]
Jao, J.K. Amplitude distribution of composite terrain radar clutter and the K-distribution. IEEE Trans. Antennas Propag. 1984, AP-32, 1049–1062.
[13]
Li, H.C.; Hong, W.; Wu, Y.R.; Fan, P.Z. On the empirical-statistical modeling of SAR images with generalized gamma distribution. IEEE J. Sel. Top. Signal Process. 2011, 5, 386–397.
[14]
Galland, F.; Nicolas, J.M.; Sportouche, H.; Roche, M.; Tupin, F.; Refregier, P. Unsupervised synthetic aperture radar image segmentation using fisher distributions. IEEE Trans. Geosci. Remote Sens. 2009, 47, 2966–2972.
[15]
Frery, A.; Muller, H.-J.; Yanasse, C.; Sant’Anna, S. A model for extremely heterogeneous clutter. IEEE Trans. Geosci. Remote Sens. 1997, 35, 648–659.
[16]
Kuruoglu, E.E.; Zerubia, J. Modeling SAR images with a generalization of the Rayleigh distribution. IEEE Trans. Image Process. 2004, 13, 527–533.
[17]
Achim, A.; Tsakalides, P.; Bezerianos, A. SAR image denoising via bayesian wavelet shrinkage based on heavy-tailed modeling. IEEE Trans. Geosci. Remote Sens. 2003, 41, 1773–1784.
[18]
Gao, G.; Shi, G.T.; Zou, H.X.; Zhou, S.L. Characterizing the statistical properties of SAR clutter by using an empirical distribution. Int. J. Antennas Propag. 2013, 109145:1–109145:8, doi:10.1155/2013/109145.
[19]
Gao, G. Statistical modeling of SAR images: A survey. Sensors 2010, 10, 775–795.
[20]
Liao, M.S.; Wang, C.C.; Wang, Y.; Jiang, L.M. Using SAR images to detect ships from sea clutter. IEEE Geosci. Remote Sens. Lett. 2008, 5, 194–198.
[21]
Chen, J.Y.; Sun, H. Multi-Resolution Edge Detection Based on Alpha-Stable Model in SAR Images Using Translation-Invariance Contourlet Transform. In Proceedings of Signal Processing and Information Technology (ISSPIT 2006), Vancouver, BC, Canada, 27–30 August 2006; pp. 264–270.
[22]
Peng, Y.J.; Xu, X.; Zhou, W.B.; Zhao, Y. SAR image classification based on alpha-stable distribution. Remote Sens. Lett. 2011, 2, 51–59.
[23]
Wan, T.; Canagarajah, N.; Achim, A. Segmentation-driven image fusion based on alpha-stable modeling of wavelet coefficients. IEEE Trans. Multimed. 2009, 11, 624–633.
[24]
Deng, C.Z.; Zhu, H.S.; Wang, S.Q. Curvelet Domain Watermark Detection Using Alpha-Stable Models. In Proceedings of Information Assurance and Security (ICIAS 2009), XI’an, China, 18–20 August 2009; pp. 313–316.
[25]
Ge, X.H.; Zhu, G.X.; Zhu, Y.T. On the testing for alpha-stable distributions of network traffic. Comput. Commun. 2004, 5, 447–457.
[26]
Xu, W.D.; Wu, C.F.; Dong, Y.C. Modeling Chinese stock returns with stable distribution. Math. Comput. Modell. 2011, 54, 610–617.
[27]
Salas-Gonzalez, D.; Kuruoglu, E.E.; Ruiz, D.P. Finite mixture of alpha-stable distributions. Digital Signal Process. 2009, 19, 250–264.
[28]
Salas-Gonzalez, D.; Kuruoglu, E.E.; Ruiz, D.P. Modelling with mixture of symmetric stable distributions using Gibbs sampling. Signal Process. 2009, 90, 774–783.
[29]
Salas-Gonzalez, D.; Kuruoglu, E.E.; Ruiz, D.P. Bayesian Estimation of Mixtures of Skewed Alpha-Stable Distributions with an Unknown Number of Components. In Proceedings of European Signal Processing, EUSIPCO 2006, Florence, Italy, 4–8 September 2006.
[30]
Buckle, D.J. Bayesian inference for stable distribution. J. Am. Stat. Assoc. 1995, 90, 605–613.
[31]
Casarin, R. Bayesian Inference for Mixtures of Stable Distributions. Working Paper No. 0428;; CEREMADE, University Paris IX: Paris, France, 2004.
[32]
Boykov, Y.; Veksler, O.; Zabih, R. Fast approximate energy minimization via Graph Cuts. IEEE Trans. Pattern Anal. Mach. Intell. 2001, 23, 1222–1239.
[33]
Kolmogorov, V.; Zabih, R. What energy functions can be minimized via Graph Cuts? IEEE Trans. Pattern Anal. Mach. Intell 2004, 26, 147–159.
[34]
Boykov, Y.; Kolmogorov, V. An experimental comparison of Min-Cut/Max-Flow algorithms for energy minimization in vision. IEEE Trans. Pattern Anal. Mach. Intell. 2004, 26, 1124–1137.
[35]
Levy, P. Theorie des erreurs : La loi de Gauss et les lois exceptionnelles. Bull. Socit Mathmatique France 1924, 52, 49–85.
[36]
Belov, I.A. On the Computation of the Probability Density Function of Alpha-Stable Distributions. In Proceedings of the 10th International Conference on Mathematical Modelling and Analysis (ICMMA 2005), Trakai, Lithuania, 1–5 June 2005; pp. 333–341.
[37]
Chambers, J.; Mallows, C.; Stuck, B. A method for simulating stable random variables. J. Am. Stat. Assoc. 1976, 71, 340–344.
[38]
Geman, S.; Geman, D. Stochastic relaxation, Gibbs distribution, and the Bayesian restauration of images. IEEE Trans. Pattern Anal. Mach. Intell. 1984, PAMI-6, 721–741.
[39]
Wu, F. The Potts model. Rev. Mod. Phys. 1982, 54, 235–268.
