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Heterogeneous Clutter Model for High-ResolutionPolarimetric SAR Data Processing

Gabriel Vasile, Frédéric Pascal, Jean-Philippe Ovarlez, Pierre Formont

To cite this version:Gabriel Vasile, Frédéric Pascal, Jean-Philippe Ovarlez, Pierre Formont. Heterogeneous ClutterModel for High-Resolution Polarimetric SAR Data Processing. IGARSS 2011 - IEEE Interna-tional Geoscience and Remote Sensing Symposium, Jul 2011, Vancouver, Canada. pp.3744-3747,�10.1109/IGARSS.2011.6050039�. �hal-00638874�

HETEROGENEOUS CLUTTER MODEL FOR HIGH RESOLUTION POLARIMETRIC SARDATA PROCESSING

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

1 : Grenoble-Image-sPeach-Signal-Automatics Lab, CNRS / Grenoble-INP, FRANCE,gabriel.vasile@gipsa-lab.grenoble-inp.fr2 : SONDRA, Supelec, FRANCE,frederic.pascal@supelec.fr

3 : French Aerospace Lab, ONERA DEMR/TSI, FRANCE,ovarlez@onera.fr

ABSTRACT

This paper presents a new estimation scheme for optimallyderiving clutter parameters with high resolution POLSARdata. The heterogeneous clutter in POLSAR data is describedby the Spherically Invariant Random Vectors model. Threeparameters are introduced for the high resolution POLSARdata clutter: the span, the normalized texture and the specklenormalized covariance matrix. The asymptotic distributionof the novel span estimator is investigated. A novel hetero-geneity test for the POLSAR clutter is also discussed. Theproposed method is tested with airborne POLSAR imagesprovided by the ONERA RAMSES system.

1. INTRODUCTION

The recently launched polarimetric SAR (POLSAR) sys-tems are now capable of producing high quality images ofthe Earth’s surface with meter resolution. The goal of theestimation process is to derive the scene signature from theobserved data set. In the case of spatially changing surfaces(”heterogeneous” or ”textured” scenes) the first step is to de-fine an appropriate model describing the dependency betweenthe polarimetric signature and the observable as a functionof the speckle. In general, the multiplicative model has beenemployed for POLSAR data processing as a product betweenthe square root of a scalar positive quantity (texture) and thedescription of an equivalent homogeneous surface (speckle)[1], [2].

This paper is organized as follows. The POLSAR pa-rameter estimation strategy for SIRV clutter model both withnormalized texture, and with normalized covariance matrixis presented in Sect. 2 and Sect. 3, respectively. Then, thenovel span estimator is introduced in Sect. 4. Next, some es-timation results are shown in Sect. 5 on a real high-resolutionPOLSAR dataset acquired by the ONERA RAMSES system.Eventually, in Sect. 6, some conclusions are presented.

2. SIRV CLUTTER MODEL WITH NORMALIZEDTEXTURE

The SIRV is a class of non-homogeneous Gaussian pro-cesses with random variance [3], [4]. The complexm-dimensional measurementk is defined as the product be-tween the independent complex circular Gaussian vectorζ ∼ N (0, [T ]) (speckle) with zero mean and covariancematrix [T ] = E{ζζ†} and the square root of the positiverandom variableξ (representing the texture):k =

√ξ · ζ. It

is important to notice that in the SIRV definition, the proba-bility density function (PDF) of the texture random variableis not explicitly specified. As a consequence, SIRVs describea whole class of stochastic processes [5].

For POLSAR clutter, the SIRV product model is theproduct of two separate random processes operating acrosstwo different statistical axes [6]. The polarimetric diversityis modeled by the multidimensional Gaussian kernel. Therandomness of spatial variations in the radar backscatteringfrom cell to cell is characterized byξ. Relatively to the po-larimetric axis, the texture random variableξ can be viewedas a unknown deterministic parameter from cell to cell.

The texture and the covariance matrix unknown param-eters can be estimated from the ML theory. ForN i.i.d.(independent and identically distributed) secondary data, letLk(k1, ...,kN |[T ], ξ1, ..., ξN ) be the likelihood function tomaximize with respect to[T ] andξi.

Lk(k1, ...,kN ; [T ], ξ1, ..., ξN ) =1

πmNdet{[T ]}N×

×N∏

i=1

1

ξmi

exp

(−k

†i [T ]−1

ki

ξi

). (1)

The corresponding ML estimators are given by [7]:

∂lnLk(k1, ..., kN |[T ], ξ1, ..., ξN)

∂ξi

= 0 ⇔ bξi =k†i[T ]−1

ki

m, (2)

∂lnLk(k1, ..., kN |[T ], ξ1, ..., ξN)

∂[T ]= 0 ⇔ [ bT ] =

1

N

NX

i=1

kik†i

bξi

.

(3)

As the variablesξi are unknown, the following normal-ization constraint on the texture parameters assures that theML estimator of the speckle covariance matrix is the SampleCovariance Matrix (SCM):

[ bT ] =1

N

NX

i=1

kik†i

= [ bT ]SCM ⇔1

N

NX

i=1

kik†i

„

1 −1

bξi

«

= [0m].

(4)The generalized ML estimator forξi are obtained by intro-

ducing Eq. 4 in Eq. 2:

ξi =k†i [T ]−1

SCMki

m. (5)

Note theki primary data is the cell under study.The normalized texture estimator from Eq. 5 is known as

the Polarimetric Whitening Filter (PWF-SCM) introduced byNovak and Burl in [1].

3. SIRV CLUTTER MODEL WITH NORMALIZEDCOVARIANCE MATRIX

Let now the covariance matrix be of the form[T ] = σ0[M ],such that Tr{[M ]} = 1. The product model can be also writ-ten ask =

√τ · z, wherez ∼ N (0, [M ]). σ0 andξ are two

scalar positive random variables such thatτ = σ0 · ξ.Using the same procedure as in Sect. 2 and given the fact

that the covariance matrix is normalized, it is possible to com-pute the generalized ML estimator of[M ] as the solution ofthe following recursive equation:

[M ]FP = f([M ]FP ) =1

N

N∑

i=1

kik†i

k†i [M ]−1

FP ki

. (6)

This approach has been used in [8] by Conte et al. to derivea recursive algorithm for estimating the matrix[M ]. This al-gorithm consists in computing the Fixed Point off using thesequence([M ]i)i≥0 defined by:

[M ]i+1 = f([M ]i). (7)

This study has been completed by the work of Pascal et al. [9],[10], which recently established the existence and the unique-ness, up to a scalar factor, of the Fixed Point estimator of thenormalized covariance matrix, as well as the convergence ofthe recursive algorithm whatever the initialization. The al-gorithm can therefore be initialized with the identity matrix[M ]0 = [Im].

The generalized ML estimator (PWF-FP) for theτi tex-ture for the primary dataki is given by:

τi =k†i [M ]−1

FP ki

m. (8)

One can observe that the PWF-FP texture from Eq. 8 hasthe same form as the PWF-SCM. The only difference is theuse of the normalized covariance estimate given by the FPestimator instead of the conventional SCM [6].

4. MAIN RESULT

The span (total power)σ0 can be derived using the covariancematrix estimators presented in Sect. 2 and Sect. 3 as:

σ0 =k†[M ]−1

FPk

k†[T ]−1SCMk

. (9)

Note that Eq. 9 is valid when consideringN identically dis-tributed linearly independent secondary data and one primarydata. It can be seen as a double polarimetric whitening filterissued from two equivalent SIRV clutter models: with nor-malized texture variables and with normalized covariance ma-trix parameter.

The main advantage of the proposed estimation scheme isthat it can be directly applied with standard boxcar neighbor-hoods.

4.1. Asymptotic statistics of σ0

This section is dedicated to the study of large sample prop-erties and approximations of the span estimatorσ0 form Eq.9.

On one hand, the asymptotic distribution of the FP esti-mator from Eq. 6 has been derived in [10]. The FP estimatorcomputed withN secondary data converges in distributionto the normalized SCM computed withN [m/(m + 1)] sec-ondary data. Since the normalized SCM is the SCM up to ascale factor, we may conclude that, in problems invariant withrespect to a scale factor on the covariance matrix, the FP esti-mate is asymptotically equivalent to the SCM computed withN [m/(m+1)] secondary data. Hence one can set the degreesof freedom of FP normalized covariance matrix estimators as:

q1 = Nm

m + 1. (10)

On the other hand, Chatelain et al. established the multi-sensor bivariate gamma distribution PDF [11]:

PbΓ(y1, y2; p1, p2, p12, q1, q2).

The scale parametersp2 andp1, the shape parametersq2 >q1 andp12 are linked to the mean parametersµ1, µ2, to thenumber of degrees of freedomn1, n2, and to the normalizedcorrelation coefficientρ such as:

q1 = n1, q2 = n2, p1 =µ1

q1, p2 =

µ2

q2, p12 =

µ1µ2

q1q2(1−ρ).

Using these results, we derived the PDF of the ratioR =y1/y2 of two correlated Gamma random variables:

PRΓ(R, p1, p2, p12, q1, q2) = Rq1−1

(p2

p12

)q1(

1

p2

)q2

×

×(

p12

p1 + Rp2

)q2+q1 Γ(q1 + q2)

Γ(q1)Γ(q2)× (11)

×H3

[q1 + q2, q2 − q1, q2; R

p1p2 − p12

(p1 + Rp2)2,

p1p2 − p12

p2(p1 + Rp2)

],

where H3(α, β, γ; x, y) =

∞∑

m,n=0

(α)2m+n(β)n

(γ)m+nm!n!xmyn is

one of the twenty convergent confluent hypergeometric seriesof order two (Horn function), and(α)n is the Pochhammersymbol such that(a)0 = 1 and(a)k+1 = (a + k)(a)k for anypositive integerk.

By taking into consideration both Eqs. 10, 11 and theCochran’s theorem, the PDF of the span estimator from Eq.9 converges asymptotically to the the ratio of two correlatedGamma random variables PDF (the ratio of two quadrat-ics). Moreover, the degrees of freedomn1 and n2 are setto N [m/(m + 1)] and N (the number of secondary data),respectively.

4.2. The σ0 test

In this section we propose to show how the estimator from Eq.9 is linked with a binary hypothesis testing problem, also:

• under the null hypothesisH0, the observed target vectork =

√ξ · ζ belongs to the SIRV clutterζ ∼ N (0, [T ])

with normalized texture,

• under the alternative hypothesisH1, the primary targetvectork =

√τ · z belongs to the SIRV clutterz ∼

N (0, [M ]) with normalized covariance matrix.

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

Λ (k) =pk(k/H1)

pk(k/H0)

H1

≷H0

λ. (12)

After expressing the PDF under each hypothesis, it resultsthat:

Λ (k) =

1

πmdet{[M ]}τm

exp(−k

†[M ]−1k

τ

)

1

πmdet{[T ]}ξm

exp(−k†[T ]−1k

ξ

)H1

≷H0

λ. (13)

By plugging into the LRT the ML texture estimators fromEqs. 5 and 8 we obtain:

Λ (k) =det{[T ]}det{[M ]}

(k†[T ]−1

k

k†[M ]−1k

)mH1

≷H0

λ. (14)

Next, we assume the ratio of determinants is a deterministicquantity and we denote it byα. This is an approximation,since in practice the ratio of determinants is also computedusing the ML estimators of the respective covariance matrixwith N secondary data. Finally, by replacing the known co-variances by their ML estimates the generalized LRT is:

Λ (k) = ασ0−m

H1

≷H0

λ. (15)

As α appears as a deterministic quantity only, it is possible touse the PDF derived in Sect. 4.1 to set the decision thresholdλ for a specific false alarm probability.

5. RESULTS AND DISCUSSIONS

The high resolution POLSAR data set, illustrated in Fig. 1,was acquired by the ONERA RAMSES system over Toulouse,France with a mean incidence angle of500. It represents afully polarimetric (monostatic mode) X-band acquisition witha spatial resolution of approximately50 cm in range and az-imuth. In the upper part of the image one can observe theCNES buildings.

Fig. 1. Toulouse, RAMSES POLSAR data, X-band,1500 ×2000 pixels: amplitude color composition of the target vectorelementsk1-k3-k2.

Fig. 2 presents the zoom image, where a narrow diplanetarget was previously detected. Fig. 2-(a),(b),(c) shows theFP-PWF texture, the SCM-PWF normalized texture, and theproposed span estimatorσ0, respectively. For comparison,the Multi-look PWF (MPWF) has been illustrated in Fig. 2-(d). The proposed estimator exhibits better performances interms of spatial resolution preservation than the MPWF spanestimator: the ring effect (two large dips on a spatial profilenear the boundaries of a pointwise target is reduced.

Finally, Fig. 3 illustrates the detection map obtain usingthe LRT from Eq. 15 with25 secondary and one primarydata. The detection threshold has been obtained by MonteCarlo integration of the PDF from Eq. 11 with a false alarmprobability set toPfa = 10−3 in each pixel. Note that thePDF integration for such a smallPfa is quite time consumingand fast numerical approximations need to be investigated inthe future for going to an operational level.

6. CONCLUSIONS

This paper presented a new estimation scheme for optimallyderiving clutter parameters with high resolution POLSAR im-ages. The heterogeneous clutter in POLSAR data was de-scribed by the SIRV model. Three estimators were introducedfor describing the high resolution POLSAR data set: the span,the normalized texture and the speckle normalized covariancematrix. The asymptotic distribution of the new span estimatorhas been established. The estimation bias on homogeneousregions have been assessed also by Monte Carlo simulations.Based on these issues, a novel test has been introduced forselecting the most appropriate model for POLSAR heteroge-neous clutter described by SIRVs.

(a) (b)

(c) (d)

Fig. 2. Toulouse, RAMSES POLSAR data, X-band,50 ×50 pixels, zoom image: (a) FP-PWF texture, (b) SCM-PWFnormalized texture, (c) span estimated usingσ0 from Eq. 9and (d) SCM-MPWF span.

7. REFERENCES

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[2] A. Lopes and F. Sery, “Optimal speckle reduction for theproduct model in multilook polarimetric SAR imagery and theWishart distribution,” IEEE Transactions on Geoscience andRemote Sensing, vol. 35, no. 3, pp. 632–647, 1997.

[3] B. Picinbono, “Spherically invariant and compound Gaussianstochastic processes,”IEEE Transactions on Information The-ory, vol. 16, no. 1, pp. 77–79, 1970.

Fig. 3. Toulouse, RAMSES POLSAR data, X-band,1500 ×2000 pixels: LRT detection map atPfa = 10−3 (SIRV withnormalized texture inblueand SIRV with normalized covari-ance inred).

[4] K. Yao, “A representation theorem and its applications tospherically-invariant random processes,”IEEE Transactionson Information Theory, vol. 19, no. 5, pp. 600–608, 1973.

[5] S. Zozor and C. Vignat, “Some results on the denoising prob-lem in the elliptically distributed context,”IEEE Transactionson Signal Processing, vol. 58, no. 1, pp. 134–150, 2010.

[6] G. Vasile, J.-P. Ovarlez, F. Pascal, and C. Tison, “Coherencymatrix estimation of heterogeneous clutter in high resolutionpolarimetric SAR images,”IEEE Transactions on Geoscienceand Remote Sensing, vol. 48, no. 4, pp. 1809–1826, 2010.

[7] F. Gini and M. V. Greco, “Covariance matrix estimation forCFAR detection in correlated heavy tailed clutter,”Signal Pro-cessing, vol. 82, no. 12, pp. 1847–1859, 2002.

[8] E. Conte, A. DeMaio, and G. Ricci, “Recursive estimationofthe covariance matrix of a compound-Gaussian process and itsapplication to adaptive CFAR detection,”IEEE Transactionson Image Processing, vol. 50, no. 8, pp. 1908–1915, 2002.

[9] F. Pascal, Y. Chitour, J.-P. Ovarlez, P. Forster, and P. Larza-bal, “Covariance structure maximum-likelihood estimatesincompound Gaussian noise: existence and algorithm analysis,”IEEE Transactions on Signal Processing, vol. 56, no. 1, pp.34–48, 2008.

[10] F. Pascal, P. Forster, J.-P. Ovarlez, and P. Larzabal, “Perfor-mance analysis of covariance matrix estimates in impulsivenoise,” IEEE Transactions on Signal Processing, vol. 56, no.6, pp. 2206–2216, 2008.

[11] F. Chatelain, J. Y. Tourneret, and J. Inglada, “Change detec-tion in multisensor SAR images using bivariate gamma distri-butions,” IEEE Transactions on Image Processing, vol. 17, no.3, pp. 249–258, 2008.