FR1-T08-2.pdf

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FR1-T08-2.pdf

  1. 1. Bivariate Pearson Distributions for Radar and Optical Remote Sensing Images Bivariate Pearson Distributions for Radar and Optical Remote Sensing Images. M. Chabert and J.-Y. Tourneret University of Toulouse, IRIT-ENSEEIHT-T´SA, Toulouse, France e { marie.chabert,jean-yves.tourneret }@enseeiht.fr IGARSS 2011 1 / 25
  2. 2. Bivariate Pearson Distributions for Radar and Optical Remote Sensing Images Outline Problem formulation The univariate Pearson system The multivariate Pearson system Method of moments Performance analysis Conclusion 2 / 25
  3. 3. Bivariate Pearson Distributions for Radar and Optical Remote Sensing Images Problem formulation Problem Formulation Images provided by the CNES, Toulouse, France Optical image Synthetic Aperture Radar (SAR) Airborne PELICAN image TerraSAR-X sensor 3 / 25
  4. 4. Bivariate Pearson Distributions for Radar and Optical Remote Sensing Images Problem formulation Multivariate Distribution Applications change detection image registration Extraction of relevant parameters correlation coefficient [Tourneret et al. IGARSS09], mutual information [Chatelain et al. IEEE Trans. IP 2007], Kullback divergence [Inglada IGARSS03]. Previous work on multi-date SAR images multivariate Gamma distributions [Chatelain et al. IEEE Trans. IP 2007]. 4 / 25
  5. 5. Bivariate Pearson Distributions for Radar and Optical Remote Sensing Images Problem formulation Multivariate Distribution for Heterogeneous Data Previous works on optical image and database Bivariate distribution for Gaussian and thresholded Gaussian random variables [Tourneret et al. IGARSS09]. Previous works on optical and/or SAR image and database: logistic regression model [Chabert et al. IGARSS10]. Optical image SAR image Database 5 / 25
  6. 6. Bivariate Pearson Distributions for Radar and Optical Remote Sensing Images Problem formulation Marginal Distributions Optical images Gaussian distribution for the residual noise [Tupin, Wiley 2010]. SAR images [Oliver and Quegan, Artech House 1998] Single look At low resolution: Gaussian complex field with Rayleigh amplitude and negative exponential intensity At higher resolution: log-normal distribution for the intensity of build-up areas, Weibull distribution for ocean, land and sea-ice clutters... Multi-look Gamma distribution for intensity images. Flexible model Univariate Pearson system [Inglada IGARSS03], [Delignon et al. IEE Proc. Radar, Sonar, Nav. 1997]. 6 / 25
  7. 7. Bivariate Pearson Distributions for Radar and Optical Remote Sensing Images Univariate Pearson system Univariate Pearson System Probability density function defined by the following differential equation [Nagahara 2004] p (x) b0 + b1 x − = p(x) c0 + c1 x + c2 x2 8 types defined by β1 = E[X 3 ]2 (squared skewness) and β2 = E[X 4 ] (kurtosis) type 0: Gaussian type I: Beta with β1 = 0 type II: Beta with β1 = 0 type III: Gamma type IV: non standard type V: Inverse-gamma type VI: F type VII: Student 7 / 25
  8. 8. Bivariate Pearson Distributions for Radar and Optical Remote Sensing Images Univariate Pearson system Univariate Pearson System 8 / 25
  9. 9. Bivariate Pearson Distributions for Radar and Optical Remote Sensing Images Multivariate Pearson system Multivariate Pearson System General definition [Nagahara 2004] The random vector X defined by X = Mξ with ξ a random vector with independent Pearson components 2 3 4 ξ1 ,...,ξm (E(ξj ) = 0, E(ξj ) = 1, E(ξj ) = ζj , E(ξj ) = κj ), M a deterministic mixing matrix follows a multivariate Pearson distribution with covariance matrix Σ = MMT . Bivariate Pearson system X = (X1 , X2 )T with m11 m12 M= m21 m22 ξ = (ξ1 , ξ2 )T with κ = (κ1 , κ2 ), ζ = (ζ1 , ζ2 ). 9 / 25
  10. 10. Bivariate Pearson Distributions for Radar and Optical Remote Sensing Images Multivariate Pearson system Moments of the Bivariate Pearson Distribution f1 = E(X1 ) = m3 ζ1 + m3 ζ2 3 11 12 f2 = E(X2 ) = m3 ζ1 + m3 ζ2 3 21 22 f3 = E(X1 X2 ) = m2 m2 ζ1 + m2 m22 ζ2 2 11 21 12 f4 = E(X1 X2 ) = m11 m2 ζ1 + m12 m2 ζ2 2 21 22 f5 = E(X1 ) = m4 κ1 + 6m2 m2 + m4 κ2 4 11 11 21 12 f6 = E(X2 ) = m4 κ1 + 6m2 m2 + m4 κ2 4 21 22 12 22 f7 = E(X1 X2 ) = m3 m21 κ1 + 3(m11 m3 + m2 m22 m21 ) + m3 m22 κ2 3 11 21 11 21 2 2 f8 = E(X1 X2 ) = m2 m2 κ1 + (m4 + 4m11 m2 m22 + m2 m2 ) + m2 m2 κ2 11 21 12 21 11 22 21 22 f9 = E(X1 X2 ) = m11 m3 κ1 + 3(m22 m3 + m2 m11 m21 ) + m21 m3 κ2 3 21 12 22 22 f10 = E(X1 ) = m2 + m2 2 11 12 f11 = E(X2 ) = m2 + m2 2 22 12 f12 = E(X1 X2 ) = m12 (m11 + m22 ) m11 m12 with M = , κ = (κ1 , κ2 ) and ζ = (ζ1 , ζ2 ). m21 m22 10 / 25
  11. 11. Bivariate Pearson Distributions for Radar and Optical Remote Sensing Images Parameter estimation Parameter Estimation using the Method of Moments Problem Estimation of the mixing matrix M and the parameters ζ and κ Method of moments Principle: Matching the theoretical fi and empirical moments fi of the distribution by minimization of 12 2 J(M , ζ, κ) = wi fi − fi i=1 Linear solution estimation of M from the covariance matrix estimate Σ = n n X(l)X T (l) and Σ = M M T 1 l=1 estimation of ζ and κ by solving a linear system leading to the usual least-squares estimators. Nonlinear optimization procedure use the unconstrained Nelder-Mead simplex method (starting value obtained by the linear method) 11 / 25
  12. 12. Bivariate Pearson Distributions for Radar and Optical Remote Sensing Images Performance study Estimation Performance - Simulated Data Simulated data 10000 realizations of independent Pearson variables ξ = (ξ1 , ξ2 )T (generated using pearsrnd.m) with κ = (3, 3) and ζ = (0, 1) 0.8 0.6 M= 0.6 0.8 Parameter estimation with the method of moments Generation of 100000 realizations of the estimated bivariate Pearson random vector 12 / 25
  13. 13. Bivariate Pearson Distributions for Radar and Optical Remote Sensing Images Performance study Estimation Performance - Simulated Data 13 / 25
  14. 14. Bivariate Pearson Distributions for Radar and Optical Remote Sensing Images Performance study Estimation Performance - Simulated Data Mean square errors of the estimates 14 / 25
  15. 15. Bivariate Pearson Distributions for Radar and Optical Remote Sensing Images Performance study Real Data - Toulouse (France) 15 / 25
  16. 16. Bivariate Pearson Distributions for Radar and Optical Remote Sensing Images Performance study Real Data - Ha¨ ıti 16 / 25
  17. 17. Bivariate Pearson Distributions for Radar and Optical Remote Sensing Images Performance study Results on Real Images (Toulouse) Window size: n = 1050 17 / 25
  18. 18. Bivariate Pearson Distributions for Radar and Optical Remote Sensing Images Performance study Results on Real Images (Toulouse) Window size: n = 1050 18 / 25
  19. 19. Bivariate Pearson Distributions for Radar and Optical Remote Sensing Images Performance study Results on Real Images (Toulouse) Window size: n = 338 19 / 25
  20. 20. Bivariate Pearson Distributions for Radar and Optical Remote Sensing Images Performance study Results on Real Images (Toulouse) Window size: n = 338 20 / 25
  21. 21. Bivariate Pearson Distributions for Radar and Optical Remote Sensing Images Performance study Results on Real Images (Ha¨ ıti) Window size: n = 26576 21 / 25
  22. 22. Bivariate Pearson Distributions for Radar and Optical Remote Sensing Images Performance study Results on Real Images (Ha¨ ıti) Window size: n = 26576 22 / 25
  23. 23. Bivariate Pearson Distributions for Radar and Optical Remote Sensing Images Conclusion and future works Conclusion and Future Works Conclusion High flexibility of the multivariate Pearson system for heterogeneous data Parameter estimation with the method of moments Performance studied on synthetic and real data Future works Generalized method of moments Method of log-moments Application to change detection and image registration 23 / 25
  24. 24. Bivariate Pearson Distributions for Radar and Optical Remote Sensing Images Conclusion and future works Bivariate Pearson Distributions for Radar and Optical Remote Sensing Images. M. Chabert and J.-Y. Tourneret University of Toulouse, IRIT-ENSEEIHT-T´SA, Toulouse, France e { marie.chabert,jean-yves.tourneret }@enseeiht.fr IGARSS 2011 24 / 25
  25. 25. Bivariate Pearson Distributions for Radar and Optical Remote Sensing Images Conclusion and future works Real Images (Goma) 25 / 25

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