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
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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
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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
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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].
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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
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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].
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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
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8. Bivariate Pearson Distributions for Radar and Optical Remote Sensing Images
Univariate Pearson system
Univariate Pearson System
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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 ).
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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
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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)
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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
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13. Bivariate Pearson Distributions for Radar and Optical Remote Sensing Images
Performance study
Estimation Performance - Simulated Data
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14. Bivariate Pearson Distributions for Radar and Optical Remote Sensing Images
Performance study
Estimation Performance - Simulated Data
Mean square errors of the estimates
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15. Bivariate Pearson Distributions for Radar and Optical Remote Sensing Images
Performance study
Real Data - Toulouse (France)
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16. Bivariate Pearson Distributions for Radar and Optical Remote Sensing Images
Performance study
Real Data - Ha¨
ıti
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17. Bivariate Pearson Distributions for Radar and Optical Remote Sensing Images
Performance study
Results on Real Images (Toulouse)
Window size: n = 1050
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18. Bivariate Pearson Distributions for Radar and Optical Remote Sensing Images
Performance study
Results on Real Images (Toulouse)
Window size: n = 1050
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19. Bivariate Pearson Distributions for Radar and Optical Remote Sensing Images
Performance study
Results on Real Images (Toulouse)
Window size: n = 338
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20. Bivariate Pearson Distributions for Radar and Optical Remote Sensing Images
Performance study
Results on Real Images (Toulouse)
Window size: n = 338
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21. Bivariate Pearson Distributions for Radar and Optical Remote Sensing Images
Performance study
Results on Real Images (Ha¨
ıti)
Window size: n = 26576
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22. Bivariate Pearson Distributions for Radar and Optical Remote Sensing Images
Performance study
Results on Real Images (Ha¨
ıti)
Window size: n = 26576
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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
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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
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25. Bivariate Pearson Distributions for Radar and Optical Remote Sensing Images
Conclusion and future works
Real Images (Goma)
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