Bivariate Pearson Distributions for Radar and Optical Remote Sensing Images

Bivariate Pearson Distributions for Radar andOptical Remote Sensing Images.

M. Chabert and J.-Y. Tourneret

University of Toulouse, IRIT-ENSEEIHT-TeSA, Toulouse, France

{ marie.chabert,jean-yves.tourneret }@enseeiht.fr

IGARSS 2011

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Bivariate Pearson Distributions for Radar and Optical Remote Sensing Images

Outline

I Problem formulation

I The univariate Pearson system

I The multivariate Pearson system

I Method of moments

I Performance analysis

I Conclusion

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Bivariate Pearson Distributions for Radar and Optical Remote Sensing Images

Problem formulation

Problem Formulation

Images provided by the CNES, Toulouse, France

Optical imageAirborne PELICAN

Synthetic Aperture Radar (SAR)image TerraSAR-X sensor

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Bivariate Pearson Distributions for Radar and Optical Remote Sensing Images

Problem formulation

Multivariate Distribution

ApplicationsI change detectionI image registration

Extraction of relevant parametersI correlation coefficient [Tourneret et al. IGARSS09],I mutual information [Chatelain et al. IEEE Trans. IP 2007],I Kullback divergence [Inglada IGARSS03].

Previous work on multi-date SAR imagesI multivariate Gamma distributions

[Chatelain et al. IEEE Trans. IP 2007].

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Bivariate Pearson Distributions for Radar and Optical Remote Sensing Images

Problem formulation

Multivariate Distribution for Heterogeneous Data

Previous works on optical image and databaseI Bivariate distribution for Gaussian and thresholded Gaussian

random variables [Tourneret et al. IGARSS09].

Previous works on optical and/or SAR image and database:I logistic regression model [Chabert et al. IGARSS10].

Optical image SAR image Database5 / 25

Bivariate Pearson Distributions for Radar and Optical Remote Sensing Images

Problem formulation

Marginal Distributions

Optical images

I Gaussian distribution for the residual noise [Tupin, Wiley 2010].

SAR images [Oliver and Quegan, Artech House 1998]

I Single lookI At low resolution: Gaussian complex field with Rayleigh

amplitude and negative exponential intensityI At higher resolution: log-normal distribution for the

intensity of build-up areas, Weibull distribution for ocean, landand sea-ice clutters...

I Multi-lookI Gamma distribution for intensity images.

I Flexible modelUnivariate Pearson system [Inglada IGARSS03],[Delignon et al. IEE Proc. Radar, Sonar, Nav. 1997].

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Bivariate Pearson Distributions for Radar and Optical Remote Sensing Images

Univariate Pearson system

Univariate Pearson System

Probability density function defined by the following differentialequation [Nagahara 2004]

−p′(x)

p(x)=

b0 + b1x

c0 + c1x+ c2x2

8 types defined by β1 = E[X3]2 (squared skewness) and β2 = E[X4](kurtosis)

I type 0: Gaussian

I type I: Beta with β1 6= 0

I type II: Beta with β1 = 0

I type III: Gamma

I type IV: non standard

I type V: Inverse-gamma

I type VI: F

I type VII: Student

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Bivariate Pearson Distributions for Radar and Optical Remote Sensing Images

Univariate Pearson system

Univariate Pearson System

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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

withI ξ a random vector with independent Pearson componentsξ1,...,ξm (E(ξj) = 0, E(ξ2

j ) = 1, E(ξ3j ) = ζj , E(ξ4

j ) = κj),I M a deterministic mixing matrix

follows a multivariate Pearson distribution with covariance matrixΣ = MMT .

Bivariate Pearson systemX = (X1, X2)T with

I M =

(m11 m12

m21 m22

)I ξ = (ξ1, ξ2)T with κ = (κ1, κ2), ζ = (ζ1, ζ2).

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Bivariate Pearson Distributions for Radar and Optical Remote Sensing Images

Multivariate Pearson system

Moments of the Bivariate Pearson Distribution

f1 = E(X31 ) = m3

11ζ1 +m312ζ2

f2 = E(X32 ) = m3

21ζ1 +m322ζ2

f3 = E(X21X2) = m2

11m221ζ1 +m2

12m22ζ2f4 = E(X1X

22 ) = m11m

221ζ1 +m12m

222ζ2

f5 = E(X41 ) = m4

11κ1 + 6m211m

221 +m4

12κ2

f6 = E(X42 ) = m4

21κ1 + 6m222m

212 +m4

22κ2

f7 = E(X31X2) = m3

11m21κ1 + 3(m11m321 +m2

11m22m21) +m321m22κ2

f8 = E(X21X

22 )

= m211m

221κ1 + (m4

12 + 4m11m221m22 +m2

11m222) +m2

21m222κ2

f9 = E(X1X32 ) = m11m

321κ1 + 3(m22m

312 +m2

22m11m21) +m21m322κ2

f10 = E(X21 ) = m2

11 +m212

f11 = E(X22 ) = m2

22 +m212

f12 = E(X1X2) = m12(m11 +m22)

with M =

(m11 m12

m21 m22

), κ = (κ1, κ2) and ζ = (ζ1, ζ2).

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Bivariate Pearson Distributions for Radar and Optical Remote Sensing Images

Parameter estimation

Parameter Estimation using the Method of Moments

ProblemEstimation of the mixing matrix M and the parameters ζ and κ

Method of momentsI Principle: Matching the theoretical fi and empirical momentsfi of the distribution by minimization of

J(M , ζ,κ) =

12∑i=1

wi

[fi − fi

]2I Linear solution

I estimation of M from the covariance matrix estimateΣ = 1

n

∑nl=1X(l)XT (l) and Σ = MMT

I estimation of ζ and κ by solving a linear system leading to theusual least-squares estimators.

I Nonlinear optimization procedure use the unconstrainedNelder-Mead simplex method (starting value obtained by thelinear method)

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Bivariate Pearson Distributions for Radar and Optical Remote Sensing Images

Performance study

Estimation Performance - Simulated Data

I Simulated dataI 10000 realizations of independent Pearson variables ξ = (ξ1, ξ2)T

(generated using pearsrnd.m) with κ = (3,3) and ζ = (0,1)

I M =

(0.8 0.60.6 0.8

)I Parameter estimation with the method of moments

I Generation of 100000 realizations of the estimated bivariatePearson random vector

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Bivariate Pearson Distributions for Radar and Optical Remote Sensing Images

Performance study

Estimation Performance - Simulated Data

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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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Bivariate Pearson Distributions for Radar and Optical Remote Sensing Images

Performance study

Real Data - Toulouse (France)

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Bivariate Pearson Distributions for Radar and Optical Remote Sensing Images

Performance study

Real Data - Haıti

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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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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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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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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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Bivariate Pearson Distributions for Radar and Optical Remote Sensing Images

Performance study

Results on Real Images (Haıti)

Window size: n = 26576

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Bivariate Pearson Distributions for Radar and Optical Remote Sensing Images

Performance study

Results on Real Images (Haıti)

Window size: n = 26576

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Bivariate Pearson Distributions for Radar and Optical Remote Sensing Images

Conclusion and future works

Conclusion and Future Works

Conclusion

I High flexibility of the multivariate Pearson system forheterogeneous data

I Parameter estimation with the method of moments

I Performance studied on synthetic and real data

Future works

I Generalized method of moments

I Method of log-moments

I Application to change detection and image registration

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Bivariate Pearson Distributions for Radar and Optical Remote Sensing Images

Conclusion and future works

Bivariate Pearson Distributions for Radar andOptical Remote Sensing Images.

M. Chabert and J.-Y. Tourneret

University of Toulouse, IRIT-ENSEEIHT-TeSA, Toulouse, France

{ marie.chabert,jean-yves.tourneret }@enseeiht.fr

IGARSS 2011

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Bivariate Pearson Distributions for Radar and Optical Remote Sensing Images

Conclusion and future works

Real Images (Goma)

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