Generalized Statistical Complexity of SAR Imagery

  • Eliana S. de Almeida
  • Antonio Carlos de Medeiros
  • Osvaldo A. Rosso
  • Alejandro C. Frery
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7441)


A new generalized Statistical Complexity Measure (SCM) was proposed by Rosso et al in 2010. It is a functional that captures the notions of order/disorder and of distance to an equilibrium distribution. The former is computed by a measure of entropy, while the latter depends on the definition of a stochastic divergence. When the scene is illuminated by coherent radiation, image data is corrupted by speckle noise, as is the case of ultrasound-B, sonar, laser and Synthetic Aperture Radar (SAR) sensors. In the amplitude and intensity formats, this noise is multiplicative and non-Gaussian requiring, thus, specialized techniques for image processing and understanding. One of the most successful family of models for describing these images is the Multiplicative Model which leads, among other probability distributions, to the \(\mathcal G^0\) law. This distribution has been validated in the literature as an expressive and tractable model, deserving the “universal” denomination for its ability to describe most types of targets. In order to compute the statistical complexity of a site in an image corrupted by speckle noise, we assume that the equilibrium distribution is that of fully developed speckle, namely the Gamma law in intensity format, which appears in areas with little or no texture. We use the Shannon entropy along with the Hellinger distance to measure the statistical complexity of intensity SAR images, and we show that it is an expressive feature capable of identifying many types of targets.


information theory speckle feature extraction 


  1. 1.
    Allende, H., Frery, A.C., Galbiati, J., Pizarro, L.: M-estimators with asymmetric influence functions: the GA0 distribution case. Journal of Statistical Computation and Simulation 76(11), 941–956 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Almeida, E.S., Medeiros, A.C., Frery, A.C.: Are Octave, Scilab and Matlab reliable? Computational and Applied Mathematics (in press)Google Scholar
  3. 3.
    Almiron, M., Almeida, E.S., Miranda, M.: The reliability of statistical functions in four software packages freely used in numerical computation. Brazilian Journal of Probability and Statistics Special Issue on Statistical Image and Signal Processing, 107–119 (2009),
  4. 4.
    Feldman, D.P., Crutchfield, J.P.: Measures of statistical complexity: Why? Physics Letters A 238(4-5), 244–252 (1998), MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Feldman, D.P., McTague, C.S., Crutchfield, J.P.: The organization of intrinsic computation: Complexity-entropy diagrams and the diversity of natural information processing. Chaos 18, 043106 (2008), MathSciNetCrossRefGoogle Scholar
  6. 6.
    Frery, A.C., Cribari-Neto, F., Souza, M.O.: Analysis of minute features in speckled imagery with maximum likelihood estimation. EURASIP Journal on Applied Signal Processing (16), 2476–2491 (2004)Google Scholar
  7. 7.
    Frery, A.C., Müller, H.J., Yanasse, C.C.F., Sant’Anna, S.J.S.: A model for extremely heterogeneous clutter. IEEE Transactions on Geoscience and Remote Sensing 35(3), 648–659 (1997)CrossRefGoogle Scholar
  8. 8.
    Gao, G.: Statistical modeling of SAR images: A survey. Sensors 10, 775–795 (2010)CrossRefGoogle Scholar
  9. 9.
    Goodman, J.W.: Some fundamental properties of speckle. Journal of the Optical Society of America 66, 1145–1150 (1976)CrossRefGoogle Scholar
  10. 10.
    Jakeman, E., Pusey, P.N.: A model for non-Rayleigh sea echo. IEEE Transactions on Antennas and Propagation 24(6), 806–814 (1976)CrossRefGoogle Scholar
  11. 11.
    Kowalski, A.M., Martín, M.T., Plastino, A., Rosso, O.A., Casas, M.: Distances in probability space and the statistical complexity setup. Entropy 13(6), 1055–1075 (2011), MathSciNetCrossRefGoogle Scholar
  12. 12.
    Lamberti, P.W., Martín, M.T., Plastino, A., Rosso, O.A.: Intensive entropic non-triviality measure. Physica A: Statistical Mechanics and its Applications 334(1-2), 119–131 (2004), MathSciNetCrossRefGoogle Scholar
  13. 13.
    López-Ruiz, R., Mancini, H., Calbet, X.: A statistical measure of complexity. Physics Letters A 209(5-6), 321–326 (1995), CrossRefGoogle Scholar
  14. 14.
    Martin, M.T., Plastino, A., Rosso, O.A.: Generalized statistical complexity measures: Geometrical and analytical properties. Physica A 369, 439–462 (2006)CrossRefGoogle Scholar
  15. 15.
    Mejail, M.E., Frery, A.C., Jacobo-Berlles, J., Bustos, O.H.: Approximation of distributions for SAR images: proposal, evaluation and practical consequences. Latin American Applied Research 31, 83–92 (2001)Google Scholar
  16. 16.
    Mejail, M.E., Jacobo-Berlles, J., Frery, A.C., Bustos, O.H.: Classification of SAR images using a general and tractable multiplicative model. International Journal of Remote Sensing 24(18), 3565–3582 (2003)CrossRefGoogle Scholar
  17. 17.
    Rosso, O.A., De Micco, L., Larrondo, H.A., Martín, M.T., Plastino, A.: Generalized statistical complexity measure. International Journal of Bifurcation and Chaos 20(3), 775–785 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Rosso, O.A., Larrondo, H.A., Martín, M.T., Plastino, A., Fuentes, M.A.: Distinguishing noise from chaos. Physical Review Letters 99, 154102 (2007), CrossRefGoogle Scholar
  19. 19.
    Salicrú, M., Mendéndez, M.L., Pardo, L.: Asymptotic distribution of (h,φ)-entropy. Communications in Statistics - Theory Methods 22(7), 2015–2031 (1993)zbMATHCrossRefGoogle Scholar
  20. 20.
    Salicrú, M., Morales, D., Menéndez, M.L.: On the application of divergence type measures in testing statistical hypothesis. Journal of Multivariate Analysis 51, 372–391 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Shannon, C., Weaver, W.: The Mathematical Theory of Communication. University of Illinois Press (1949)Google Scholar
  22. 22.
    Team, R.D.C.: R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria (2011) ISBN 3-900051-07-0,
  23. 23.
    Yueh, S.H., Kong, J.A., Jao, J.K., Shin, R.T., Novak, L.M.: K-distribution and polarimetric terrain radar clutter. Journal of Electromagnetic Waves and Applications 3(8), 747–768 (1989)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Eliana S. de Almeida
    • 1
  • Antonio Carlos de Medeiros
    • 1
  • Osvaldo A. Rosso
    • 1
    • 2
  • Alejandro C. Frery
    • 1
  1. 1.Laboratório de Computação Científica e Análise Numérca – LaCCANUniversidade Federal de Alagoas – UFALMaceióBrazil
  2. 2.Laboratorio de Sistemas Complejos, Facultad de IngenieríaUniversidad de Buenos AiresCiudad Autónoma de Buenos AiresArgentina

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