HETEROGENEOUS CLUTTER MODEL FOR HIGH RESOLUTION POLARIMETRIC SAR DATA PROCESSING

HETEROGENEOUS CLUTTER MODEL FOR HIGH RESOLUTION POLARIMETRIC SAR DATA

PROCESSING

G. Vasile1, F. Pascal2, J.P. Ovarlez2,3, P. Formont2,3

1 : GIPSA LAB & CNRS, France, [email protected] 2 : SONDRA, SUPELEC, France, [email protected], [email protected] : FRENCH AEROSPACE LAB, ONERA, France, [email protected]

mailto:[email protected]








/ 18 IEEE IGARSS‘2011 Vancouver, Canada

COMMENTS, REMARKS AND OPEN PROBLEMS IN SAR PROCESSING

The SAR images are more and more complex, detailed, heterogeneous, The spatial statistic of SAR images is not at all Gaussian !

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Many Non Coherent Polarimetric Decomposition techniques and speckle filtering techniques (PWF, MPWF) generally use an estimate <T> of the local covariance matrix (coherency matrix), typically the Sample Covariance Matrix (SCM),

All these techniques may give very different results when using an estimate that fits better to the reality! Are they more physically valid? Which one to choose?

res < 0.5m

@ONERA SETHI


SPAN Gamma SCM-Wishart

I1/I3/I2

Classification on intensity only and H/a classification seem to be the same!

The Gaussian SCM is contaminated by the power.3

EXAMPLE WITH THE UNSUPERVISED H/ POLARIMETRIC CLASSIFICATION

K-means H/α classification

(8 classes)

(a) (b)

(c) (d)

(e) (f)

Figure 3. RAMSES POLSAR data, X-band (501 ⇥ 501 pixels), Fixed Point normalized coherency estimator within theSDAN neighborhood SDAN-FP. (a) LLMMSE span and (b) unsupervized scalar Gamma span segmentation. (c) Colorcomposition of the normalized coherency diagonal elements [M ]11-[M ]33-[M ]22 and (d) unsupervized H/↵ SIRV normal-ized coherency segmentation. (e) Color composition of the diagonal elements of the diagonal elements [T ]11-[T ]33-[T ]22and (f) unsupervized H/↵ Wishart coherency segmentation.

(a) (b)

(c) (d)

(e) (f)


(a) (b)

(c) (d)

(e) (f)


(a) (b)

(c) (d)

(e) (f)


FP-Wishart

[Vasile et al. 2008][Formont et al. 2011]


OUTLINE OF THE TALK

Conventional Gaussian clutter model for Polarimetric SAR Analysis,

SIRV clutter model for Polarimetric SAR Analysis,

Estimation of the texture and speckle Covariance Matrix parameters - Link with the PWF,

New SPAN estimator and new test of heterogeneity,

Results on Polarimetric SAR images,

Conclusion.

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Advantages

easy to manage, well known, distribution of T is known.

Drawbacks

very sensitive and not robust to power or strong scatterers, does not allow to take into account the transition, heterogeneity of the background, not robust to small change in the statistic.

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GAUSSIAN MODEL FOR POLARIMETRIC SAR CLUTTER MODELLING

For a given set of spatial pixels of the Polarimetric SAR image, T characterizes the correlation existing within the polarimetric channels,

The Maximum Likelihood Estimate (MLE) of the Clutter Covariance Matrix is the Sample Covariance Matrix over a given set of K spatial pixels:

pm(c) =1

⇡m |T| exp��cH T�1 c

�Gaussian clutter model:

T =1

K

KX

k=1

ck cHk

where the polarimetric target descriptor c =[SHH+SVV SHH-SVV 2SHV]T is a multivariate complex circular zero-mean Gaussian m-vector (here m=3) with covariance matrix T.

^


x is a multivariate complex circular zero-mean Gaussian m-vector (speckle) with covariance matrix M with identifiability consideration tr(M)=m, is a positive scalar random variable (texture) well defined by its pdf p().

Powerful statistical model that allows:

to encompass the Gaussian model, to extend the Gaussian model (K, Weibull, Fisher, Cauchy, Alpha-Stable, ...), to take into account the heterogeneity of the background power with the texture, to take into account possible correlation existing on the m-channels of observation.

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SIRV FOR POLARIMETRIC SAR CLUTTER MODELLING

Spherically Invariant Random Vector : Compound Gaussian Process [Yao 1973]

c =p⌧ x

For a given set of spatial pixels of the Polarimetric SAR image, M characterizes the correlation existing within the polarimetric channels,

Conditionally to the pixel, the polarimetric vector is Gaussian. The texture variable characterizes here the variation of the norm of each vector from pixels to pixels. It acts for all the polarization channels [Eltoft et al. IGARSS’11].

pm(c) =

Z +1

0

1

(⇡ ⌧)m |M| exp✓�cH M�1 c

⌧

◆p(⌧) d⌧


CHOICE OF THE COVARIANCE MATRIX ESTIMATE

The Sample Covariance Matrix SCM may be a «poor» estimate of the SIRV Covariance Matrix M because of the texture contamination:

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MSCM =1

K

KX

k=1

ck cHk =

1

K

KX

k=1

�k xk xHk

6= 1

K

KX

k=1

xk xHk

The Normalized Sample Covariance Matrix (NSCM) may be a good candidate of the SIRV Covariance Matrix M:

MNSCM =m

K

KX

k=1

ck cHk

c

Hk ck

=m

K

KX

k=1

xk xHk

x

Hk xk

This estimate does not depend on the texture but it is biased ( and M share the same eigenvectors but have different eigenvalues, with the same ordering) [Bausson et al. 2006].

EhMNSCM

i


MFP =m

K

KX

k=1

ck cHkcHk M�1

FP ck

For an unknown but deterministic texture parameter, the Maximum Likelihood Estimate (MLE) of the Covariance M (approached MLE in the SIRV context), called the Fixed Point MFP (FP), is the solution of the following implicit equation [Conte-Gini 2002]:

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COVARIANCE MATRIX ESTIMATION IN SIRV BACKGROUND

Fixed Point (FP):

The Fixed Point belongs to the family of M-estimators (Robust Statistics [Huber 1964, Maronna 1976, Yohai 2006]) in the more general context of Elliptically Random Process:

Les di↵erents estimateurs etudies


Obtenus a partir des donnees {xi , i = 1...N} de taille m, ils verifient tous l’equation

bM =1N

NX

i=1

hu(xH

ibM�1xi)

ixix

Hi (1)

u est une fonction de ponderation des xixHi .

Exemples :

La SCM :u(r) = 1

L’estimateur d’Huber(M-estimateur) :

u(r) =

⇢K/e si r <= e

K/r si r > e

L’estimateur FP : u(r) = mr

M. Mahot (ONERA, SONDRA) JDDPHY 2011 6 / 16




bM =1N

NX

i=1

hu(xH

ibM�1xi)

ixix

Hi (1)


Exemples :

La SCM :u(r) = 1


u(r) =

⇢K/e si r <= e

K/r si r > e






bM =1N

NX

i=1

hu(xH

ibM�1xi)

ixix

Hi (1)


Exemples :

La SCM :u(r) = 1


u(r) =

⇢K/e si r <= e

K/r si r > e



FP SCMHuberM =1

K

KX

k=1

u⇣cHk M�1 ck

⌘ck c

Hk

u(.) choice

This estimate does not depend on the texture,The Fixed Point is consistent, unbiased, asymptotically Gaussian and is, at a fixed number K, Wishart distributed with mK/(m+1) degrees of freedom,Existence and unicity of the solution are proven. The solution can be reached by recurrence Mk=f(Mk-1) whatever the starting point M0 (ex: M0=I, M1=MNSCM),Robust to outliers, strong targets or scatterers in the reference cells.

[Pascal et al. 2006]


TEXTURE ESTIMATION IN SIRV BACKGROUND

⌧k =cHk M�1

FP ckm

For an unknown but deterministic texture parameter, the Maximum Likelihood Estimate of the texture at pixel k is given by:

This quantity is an extended version of the well known Polarimetric Whitening Filter [Novak and Burl 1990, Vasile et al. 2010] for the speckle filtering in Polarimetric SAR images:

PWF = cHk M�1SCM ck

FP-PWF texture SCM-PWF9


NEW SPAN ESTIMATOR

contains all the power

x has a normalized covariance matrix

y contains all the power

New span estimator

primary datasecondary data

Equivalence of SIRV Models

�0 =cH M

�1

FP c

cH T�1

SCM c

c =p⌧ x =

p�0 ⇠ x =

p⇠ y

E⇥yyH

⇤= �0 M

⌧ = cH M�1

FP c

⇠ = cH T�1

SCM c

normalized texture

The new span estimator can be viewed as a double Polarimetric Whitening Filter

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Where to put the power ?

E⇥xx

H⇤= M with tr(M) = 1


RESULTS ON EXPERIMENTAL DATA

Toulouse, RAMSES X-band, R=0.5m

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Pauli Representation



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SCM-PWF textureToulouse, RAMSES X-band, R=0.5m



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FP-PWF textureToulouse, RAMSES X-band, R=0.5m



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0-PWF textureToulouse, RAMSES X-band, R=0.5m


Toulouse, RAMSES X-band, R=0.5m

FP-PWF SCM-PWF σ0-estimate FP-MPWF

5x5 boxcar


The 0-PWF PDF, ratio of two correlated

gamma distributions [Chatelain 2008] has

been established

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This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination.

VASILE et al.: PARAMETER ESTIMATION IN HETEROGENEOUS CLUTTER FOR POLARIMETRIC SAR DATA 3

Fig. 1. Ratio pdf of two correlated Gamma random variables (11) for different! values and the empirical pdf of simulated "0 in Gaussian clutter.

univariate gamma distributions with different shape param-eters [13]

Pb!(y1, y2; p1, p2, p12, q1, q2).

The scale parameters p2 and p1 and the shape parameters q2 >q1 and p12 are linked to the mean parameters µ1 and µ2, to thenumber of degrees of freedom n1 and n2, and to the normalizedcorrelation coefficient ! such as

q1 = n1, q2 = n2, p1 =µ1

q1, p2 =

µ2

q2, p12 =

µ1µ2

q1q2(1 ! !).

Using these results, we derived the pdf of the ratio R = y1/y2

of two correlated Gamma random variables

PR!(R, p1, p2, p12, q1, q2)

=Rq1!1

!p2

p12

"q1!

1

p2

"q2!

p12

p1 + Rp2

"q2+q1 !(q1 + q2)

!(q1)!(q2)

" H3

#q1 + q2, q2 ! q1, q2;

Rp1p2 ! p12

(p1 + Rp2)2,

p1p2 ! p12

p2(p1 + Rp2)

$(11)

where H3(",#, $; x, y) =%#

m,n=0((")2m+n(#)n/($)m+n

m!n!)xmyn is one of the 20 convergent confluenthypergeometric series of order two (Horn function) and(")n is the Pochhammer symbol such that (a)0 = 1 and(a)k+1 = (a + k)(a)k for any positive integer k [14].

By taking into consideration both (10) and (11) andCochran’s theorem [15], the pdf of the span estimator from(9) converges asymptotically to the the ratio of two corre-lated Gamma random variable pdfs (the ratio of two quadrat-ics). Moreover, the degrees of freedom n1 and n2 are setto N [m/(m + 1)] and N (the number of secondary data),respectively.

Fig. 1 shows the behavior of the %0 pdf with respect to thenormalized correlation coefficient !. The pdf parameters are setaccording to the processing illustrated in Section IV, namely,N = 24, m = 3, µ1 = 10, and µ2 = 1. Notice that, when thenormalized correlation coefficient approaches to one, the pdftends to a Dirac.

A Monte Carlo simulation has also been represented inFig. 1. Five thousand samples of %0 were obtained by comput-ing 5000 " 24 samples drawn from a zero-mean multivariatecircular complex Gaussian distribution with a covariance matrix

TABLE IEMPIRICAL MEAN AND VARIANCE OF THE "0 ESTIMATOR FROM (9) AND

THEIR EXPECTED VALUES FOR SIMULATED GAUSSIAN CLUTTER

selected from the real POLSAR data. The span of the selectedcovariance matrix equals three. One can observe the goodcorrespondence between the empirical pdf of simulated %0 andthe pdf derived in (11) for ! = 0.95.

Using the same parameters as in the previous Monte Carlosimulation, Table I illustrates the behavior of the empiricalmean and variance of the proposed %0 in Gaussian clutter (e.g.,in homogeneous regions). By using 24 up to 48 secondary data,the estimation bias is negligible, and the empirical variance isclose to zero.

B. %0 Test

In this section, we propose to show how the estimator from(9) is linked with a binary hypothesis testing problem, alsounder the following.

1) Null hypothesis H0: The observed target vector k =$& · ' belongs to the SIRV clutter ' % N (0, [T ]) with

normalized texture.2) Alternative hypothesis H1: The primary target vector k =$

( · z belongs to the SIRV clutter z % N (0, [M ]) withnormalized covariance matrix.

From the operational point of view, the proposed detector isa classical constant false-alarm-rate detector with current pixelas the primary data and with the local boxcar neighborhoodaround it as the secondary data.

The Neyman–Pearson optimal detector is given by the fol-lowing likelihood ratio test (LRT):

"(k) =pk(k/H1)

pk(k/H0)

H1

!H0

). (12)

After expressing the pdf under each hypothesis, it results in

"(k) =

1!m det{[M ]}"m exp

&!k†[M ]!1k

"

'

1!m det{[T ]}#m exp

&!k†[T ]!1k

#

'H1

!H0

). (13)

By plugging into the LRT the ML texture estimators from (5)and (8), we obtain

"(k) =det {[T ]}det {[M ]}

!k†[T ]!1k

k†[M ]!1k

"m H1

!H0

). (14)

Next, we assume that the ratio of determinants is a deterministicquantity, and we denote it by ". This is an approximation, sincein practice, the ratio of determinants is also computed using


INTERPRETATION OF THE NEW SPAN ESTIMATOR

Normalized texture hypothesis characterizes homogeneous areas,Normalized covariance matrix hypothesis characterizes heterogeneous areas.

Elaboration of a statistical hypothesis test

H0 :

H1 : {

⇤(c) = ↵ ��m0

H1

?H0

�

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c =

p⇠ y, where y ⇠ CN (0,T) and ⇠ normalized texture

c =

p⌧ y, where y ⇠ CN (0,M) and M normalized Covariance Matrix


RESULT ON POLARIMETRIC SAR DATA

Toulouse, RAMSES X-band, detection map

Pauli color composition

Pfa = 10-3

H0: Homogeneous areas - Normalized

texture

H1: Heterogeneous areas - Normalized Covariance Matrix

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/ 18 IEEE IGARSS‘2011 Vancouver, Canada18

CONCLUSION

• The SIRV model is a very powerful tool for the polarimetric SAR imaging that can handle the heterogeneity and non-Gaussianity of the images,

• A new SPAN estimator has been proposed. It acts as a double whitening process and can be used for detecting heterogeneous and homogeneous areas in polarimetric SAR imaging,

• The SIRV methodology has already been used successfully for other topics such as radar detection, STAP, hyperspectral imaging [Ovarlez et al. IGARSS’11], SAR classification [Formont et al. IGARSS’11], SAR change detection, target detection in SAR images, INSAR, POLINSAR. It can help in understanding very complex polarimetric SAR images behavior,

• Many theoretical works still remain to be done in these domains. The most important work is perhaps their validation through physical interpretation.

HETEROGENEOUS CLUTTER MODEL FOR HIGH RESOLUTION POLARIMETRIC SAR DATA PROCESSING

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