Heterogeneous Clutter Model for High-Resolution Polarimetric SAR Data Parameter Estimation

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1 Heterogeneous Clutter Model for High-Resolution Polarietric SAR Data Paraeter Estiation Gabriel Vasile, Frédéric Pascal, Jean-Philippe Ovarlez To cite this version: Gabriel Vasile, Frédéric Pascal, Jean-Philippe Ovarlez. Heterogeneous Clutter Model for High- Resolution Polarietric SAR Data Paraeter Estiation. European Space Agency. 5th International Workshop on Science and Applications of SAR Polarietry and Polarietric Interferoetry (PolInsar 20), Jan 20, Frascati, Italy. 6 p., 20. <hal > HAL Id: hal Subitted on 3 Jan 20 HAL is a ulti-disciplinary open access archive for the deposit and disseination of scientific research docuents, whether they are published or not. The docuents ay coe fro teaching and research institutions in France or abroad, or fro public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de docuents scientifiques de niveau recherche, publiés ou non, éanant des établisseents d enseigneent et de recherche français ou étrangers, des laboratoires publics ou privés.

2 HETEROGEEOUS CLUTTER MODEL FOR HIGH RESOLUTIO POLARIMETRIC SAR DATA PARAMETER ESTIMATIO G. Vasile (), F. Pascal (2), and J.-P. Ovarlez (3) () GIPSA-lab, CRS - Grenoble IP, 96 rue de la Houille Blanche, BP 46, Grenoble Cedex, FRACE Eail: gabriel.vasile@gipsa-lab.grenoble-inp.fr (2) SODRA, Supélec - Plateau du Moulon, 3 rue Joliot-Curie, 992 Gif-sur-Yvette Cedex, FRACE Eail: frederic.pascal@supelec.fr (3) French Aerospace Lab, OERA DEMR/TSI - Chein de la Huniere, 976 Palaiseau Cedex, FRACE Eail: ovarlez@onera.fr ABSTRACT This paper presents a new estiation schee for optially deriving clutter paraeters with high resolution POLSAR data. The heterogeneous clutter in POLSAR data is described by the Spherically Invariant Rando Vectors odel. Three paraeters are introduced for the high resolution POLSAR data clutter: the span, the noralized texture and the speckle noralized covariance atrix. The asyptotic distribution of the novel span estiator is investigated. A novel heterogeneity test for the POLSAR clutter is also discussed. The proposed ethod is tested with airborne POLSAR iages provided by the OERA RAMSES syste. Key words: non-gaussian clutter; POLSAR; estiation; segentation; high resolution. polarietric covariance atrix to produce a single scalar iage [], [3]. Using the coplex Wishart distribution, the PWF for hoogeneous surfaces has been generalized to an Multi-look PWF (MPWF) in [2], [4]. The objective of this paper is to present a novel paraeter estiation technique based on the Spherically Invariant Rando Vectors (SIRV) odel. For a detailed review on the use of SIRV with POLSAR data refer to [5]. This paper is organized as follows. The POLSAR paraeter estiation strategy for SIRV clutter odel both with noralized texture, and with noralized covariance atrix is presented in Sect. 2 and Sect. 3, respectively. Then, the novel span estiator is introduced in Sect. 4. ext, soe estiation results are shown in Sect. 5 on a real high-resolution POLSAR dataset acquired by the OERA RAMSES syste. Eventually, in Sect. 6, soe conclusions are presented.. ITRODUCTIO The recently launched polarietric SAR (POLSAR) systes are now capable of producing high quality iages of the Earth s surface with eter resolution. The goal of the estiation process is to derive the scene signature fro the observed data set. In the case of spatially changing surfaces ( heterogeneous or textured scenes) the first step is to define an appropriate odel describing the dependency between the polarietric signature and the observable as a function of the speckle. In general, the ultiplicative odel has been eployed for POLSAR data processing as a product between the square root of a scalar positive quantity (texture) and the description of an equivalent hoogeneous surface (speckle) [], [2]. In the context of the non-gaussian polarietric clutter odels, several studies tackled POLSAR paraeter estiation using the product odel. For deterinistic texture, ovak and Burl derived the Polarietric Whitening Filter (PWF) by optially cobining the eleents of the 2. SIRV CLUTTER MODEL WITH ORMAL- IZED TEXTURE The SIRV is a class of non-hoogeneous Gaussian processes with rando variance [6], [7]. The coplex - diensional easureent k is defined as the product between the independent coplex circular Gaussian vector ζ (0, [T]) (speckle) with zero ean and covariance atrix [T] = E{ζζ } and the square root of the positive rando variable ξ (representing the texture): k = ξ ζ. () It is iportant to notice that in the SIRV definition, the probability density function (PDF) of the texture rando variable is not explicitly specified. As a consequence, SIRVs describe a whole class of stochastic processes [8]. For POLSAR clutter, the SIRV product odel is the product of two separate rando processes operating across two different statistical axes [5]. The polarietric diversity is odeled by the ultidiensional Gaussian ker-

3 nel. The randoness of spatial variations in the radar backscattering fro cell to cell is characterized by ξ. Relatively to the polarietric axis, the texture rando variable ξ can be viewed as a unknown deterinistic paraeter fro cell to cell. The texture and the covariance atrix unknown paraeters can be estiated fro the ML theory. For i.i.d. secondary data, let L k (k,...,k [T], ξ,..., ξ ) be the likelihood function to axiize with respect to [T] and ξ i. L k (k,...,k ; [T], ξ,..., ξ ) = ξ i= i exp ( k i [T] k i ξ i π det{[t]} ). (2) The corresponding ML estiators are given by [9]: lnl k (k,..., k [T], ξ,..., ξ ) ξ i lnl k (k,...,k [T], ξ,..., ξ ) [T] = 0 ξ b i = k i [T] k i, (3) = 0 [ b T] = X i= k ik i bξ i. As the variables ξ i are unknown, the following noralization constraint on the texture paraeters assures that the ML estiator of the speckle covariance atrix is the Saple Covariance Matrix (SCM): [ T] b = X k ik i = [ T] b SCM X «k ik i bξi = [0 ]. i= i= (5) The generalized ML estiator for ξ i are obtained by introducing Eq. 5 in Eq. 3: ξ i = k i [ T] SCM k i (4). (6) ote the k i priary data is the cell under study. The noralized texture estiator fro Eq. 6 is known as the Polarietric Whitening Filter (PWF-SCM) introduced by ovak and Burl in []. 3. SIRV CLUTTER MODEL WITH ORMAL- IZED COVARIACE MATRIX Let now the covariance atrix be of the for [T] = σ 0 [M], such that Tr{[M]} =. The product odel for Eq. can be also written as: k = τ z, (7) where z (0, [M]). σ 0 and ξ are two scalar positive rando variables such that τ = σ 0 ξ. The likelihood function is: L k (k,...,k ; [M], τ,..., τ ) = τ i= i exp ( k i [M] k i τ i π det{[m]} ). (8) Using the sae procedure as in Sect. 2, the corresponding texture and noralized covariance ML estiators are given by: lnl k (k,..., k ;[M], τ,..., τ ) τ i lnl k (k,..., k ;[M], τ,..., τ ) [M] = 0 bτ i = k i [M] k i, (9) = 0 [ c M] = X k ik i. bτ i (0) i= Given the fact that the covariance atrix is noralized, it is possible to copute the generalized ML estiator of [M] as the solution of the following recursive equation: [ M] FP = f([ M] FP ) = k i k i k i [ M] FP k. () i i= This approach has been used in [0] by Conte et al. to derive a recursive algorith for estiating the atrix [M]. This algorith consists in coputing the Fixed Point of f using the sequence ([M] i ) i 0 defined by: [M] i+ = f([m] i ). (2) This study has been copleted by the work of Pascal et al. [], [2], which recently established the existence and the uniqueness, up to a scalar factor, of the Fixed Point estiator of the noralized covariance atrix, as well as the convergence of the recursive algorith whatever the initialization. The algorith can therefore be initialized with the identity atrix [ M] 0 = [I ]. The generalized ML estiator (PWF-FP) for the τ i texture for the priary data k i is given by: τ i = k i [ M] FP k i. (3) One can observe that the PWF-FP texture fro Eq. 3 has the sae for as the PWF-SCM. The only difference is the use of the noralized covariance estiate given by the FP estiator instead of the conventional SCM [5]. 4. MAI RESULT The span (total power) σ 0 can be derived using the covariance atrix estiators presented in Sect. 2 and Sect.

4 3 as: FP k T] SCM σ 0 = k [ M] (4) k [ k. ote that Eq. 4 is valid when considering identically distributed linearly independent secondary data and one priary data. It can be seen as a double polarietric whitening filter issued fro two equivalent SIRV clutter odels: with noralized texture variables and with noralized covariance atrix paraeter. The ain advantage of the proposed estiation schee is that it can be directly applied with standard boxcar neighborhoods. where H 3 (α, β, γ; x, y) =,n=0 (α) 2+n (β) n (γ) +n!n! x y n is one of the twenty convergent confluent hypergeoetric series of order two (Horn function), and (α) n is the Pochhaer sybol such that (a) 0 = and (a) k+ = (a + k)(a) k for any positive integer k [4]. By taking into consideration both Eqs. 5, 6 and the Cochran s theore, the PDF of the span estiator fro Eq. 4 converges asyptotically to the the ratio of two correlated Gaa rando variables PDF (the ratio of two quadratics). Moreover, the degrees of freedo n and n 2 are set to [/( + )] and (the nuber of secondary data), respectively. 4.. Asyptotic statistics of σ 0 This section is dedicated to the study of large saple properties and approxiations of the span estiator σ 0 for Eq. 4. On one hand, the asyptotic distribution of the FP estiator fro Eq. has been derived in [2]. The FP estiator coputed with secondary data converges in distribution to the noralized SCM coputed with [/( + )] secondary data. Since the noralized SCM is the SCM up to a scale factor, we ay conclude that, in probles invariant with respect to a scale factor on the covariance atrix, the FP estiate is asyptotically equivalent to the SCM coputed with [/( + )] secondary data. Hence one can set the degrees of freedo of FP noralized covariance atrix estiators as: q = +. (5) On the other hand, the bivariate Gaa PDF has been established by Chatelain et al. in [3]: P bγ (y, y 2 ; p, p 2, p 2, q, q 2 ). The scale paraeters p 2 and p, the shape paraeters q 2 > q and p 2 are linked to the ean paraeters µ, µ 2, to the nuber of degrees of freedo n, n 2, and to the noralized correlation coefficient ρ such as: q = n, q 2 = n 2, p = µ q, p 2 = µ 2 q 2, p 2 = µ µ 2 q q 2 ( ρ). Using these results, we derived the PDF of the ratio R = y /y 2 of two correlated Gaa rando variables: ( ) q ( ) q2 P RΓ (R, p, p 2, p 2, q, q 2 ) = R q p2 p 2 p 2 ( ) q2+q p2 Γ(q + q 2 ) p + Rp 2 Γ(q )Γ(q 2 ) (6) H 3 [ q + q 2, q 2 q, q 2 ; R p p 2 p 2 (p + Rp 2 ) 2, p p 2 p 2 p 2 (p + Rp 2 ) ], Epirical PDF P(σ 0 ; µ=0, =24, =3, ρ=0.05) P(σ 0 ; µ=0, =24, =3, ρ=0.5) P(σ 0 ; µ=0, =24, =3, ρ=0.95) Figure. Ratio PDF of two correlated Gaa rando variables (Eq. 6) for different ρ and the epirical PDF of siulated σ 0 in Gaussian clutter Fig. illustrates the behavior of the σ 0 PDF with respect to the noralized correlation coefficient ρ. The PDF paraeters are set according to the processing illustrated in Sect. 4, naely = 24, = 3, µ = 0, µ 2 =. otice that when the noralized correlation coefficient approaches to, the PDF tends to a Dirac Saple ean Expected ean Saple variance Expected variance Figure 2. Epirical ean and variance of the σ 0 estiator fro Eq. 4 and the their expected values for siulated Gaussian clutter. A Monte Carlo siulation has been represented in Fig., also saples of σ 0 were obtained by coputing saples drown fro a zero-ean ultivariate circular coplex Gaussian distribution with a covariance atrix selected fro the real POLSAR data. The span of the selected covariance atrix equal 0. One can observe the good correspondence between the epirical PDF of siulated σ 0 and the PDF derived in Eq. 6 for ρ = 0.95.

5 Using the sae paraeters as in the previous Monte Carlo siulation, Fig. 2 illustrates the behavior of the epirical ean and variance of the proposed σ0 in Gaussian clutter (e.g. in hoogeneous regions). By using 24 up to 48 secondary data, the estiation bias is negligible and the epirical variance is close to zero The σ0 test 5. RESULTS AD DISCUSSIOS The high resolution POLSAR data set, illustrated in Fig. 3, was acquired by the OERA RAMSES syste over Toulouse, France with a ean incidence angle of 500. It represents a fully polarietric (onostatic ode) X-band acquisition with a spatial resolution of approxiately 50 c in range and aziuth. In the upper part of the iage one can observe the CES headquarters. In this section we propose to show how the estiator fro Eq. 4 is linked with a binary hypothesis testing proble, also: under the null hypothesis H0, the observed target vector k = ξ ζ belongs to the SIRV clutter ζ (0, [T ]) with noralized texture, under the alternative hypothesis H, the priary vector target vector k = τ z belongs to the SIRV clutter z (0, [M ]) with noralized covariance atrix. The eyan-pearson optial detector is given by the following likelihood ratio test (LRT): Λ (k) = pk (k/h ) H λ. pk (k/h0 ) H0 (7) After expressing the PDF under each hypothesis, it results that: k [M] k exp H τ π det{[m]}τ λ. (8) Λ (k) = exp k [Tξ] k H0 π det{[t ]}ξ By plugging into the LRT the ML texture estiators fro Eqs. 6 and 3 we obtain: Λ (k) = det{[t ]} det{[m ]} k [T ] k k [M ] k H λ. (9) H0 ext, we assue the ratio of deterinants is a deterinistic quantity and we denote it by α. This is an approxiation, since in practice the ratio of deterinants is also coputed using the ML estiators of the respective covariance atrix with secondary data. Finally, by replacing the known covariances by their ML estiates the generalized LRT is: Λ (k) = αc σ0 H λ. Figure 3. Toulouse, RAMSES POLSAR data, X-band, pixels: aplitude color coposition of the target vector eleents k -k3 -k2. (20) H0 As α appears as a deterinistic quantity only, it is possible to use the PDF derived in Sect. 4. to set the decision threshold λ for a specific false alar probability. Fig. 4-(a),(b),(c) presents the three SIRV paraeters which copletely describe the POLSAR data set: the total power, the noralized texture and the noralized covariance atrix. The 55 boxcar neighborhood has been selected for illustration, hence 24 secondary saples and priary data. Fig. 5 presents the zoo over the red rectangle fro Fig. 4-(a), where a narrow diplane target was detected in [5]. Fig. 5-(a),(b),(c) shows the FP-PWF texture, the SCMPWF noralized texture, and the proposed span estiator σ c0, respectively. For coparison, the Multi-look PWF (MPWF) has been illustrated in Fig. 5-(d). The proposed estiator exhibits better perforances in ters of spatial resolution preservation than the MPWF span estiator: the ring effect (two large dips on a spatial profile near the boundaries of a pointwise target [6]) is reduced. Finally, Fig. 5 illustrates the detection ap obtain using the LRT fro Eq. 20 with 25 secondary and one priary data. The detection threshold has been obtained by Monte Carlo integration of the PDF fro Eq. 6 with a false alar probability set to Pf a = 0 3. This work has any interesting perspectives. We believe that this paper contributes toward the description and the analysis of heterogeneous clutter over scenes exhibiting coplex polarietric signatures. Firstly, the exact texture noralization condition for the PWF-SCM estiator has been derived in Sect. 2 under the SIRV clutter hypothesis. A novel estiation / detection strategy has

6 (a) (b) (a) (c) (d) Figure 5. Toulouse, RAMSES POLSAR data, X-band, pixels, zoo iage: (a) FP-PWF texture, (b) SCM-PWF noralized texture, (c) span estiated using σ 0 fro Eq. 4 and (d) SCM-MPWF span. (b) (c) Figure 4. Toulouse, RAMSES POLSAR data, X-band, pixels: (a) span estiated using σ 0 fro Eq. 4, (b) noralized texture ξ, and (c) color coposition of the noralized coherency diagonal eleents [M] - [M] 33 -[M] 22. Figure 6. Toulouse, RAMSES POLSAR data, X-band, pixels: LRT detection ap at P fa = 0 3 (SIRV with noralized texture in blue and SIRV with noralized covariance in red).

7 been proposed which can be used with conventional boxcar neighborhoods directly. Finally, the proposed estiation schee can be extended to other ultidiensional SAR techniques using the covariance atrix descriptor, such as the following: repeat-pass interferoetry, polarietric interferoetry, or ultifrequency polarietry. 6. COCLUSIOS This paper presented a new estiation schee for optially deriving clutter paraeters with high resolution POLSAR iages. The heterogeneous clutter in POLSAR data was described by the SIRV odel. Three estiators were introduced for describing the high resolution POL- SAR data set: the span, the noralized texture and the speckle noralized covariance atrix. The asyptotic distribution of the new span estiator has been established. The estiation bias on hoogeneous regions has been assessed also by Monte Carlo siulations. Based on these issues, a novel test has been introduced for selecting the ost appropriate odel for POLSAR heterogeneous clutter described by SIRVs. ACKOWLEDGMET The authors would like to thank Dr. S. Zozor and Dr. F. Chatelain (GIPSA-lab, France) for the very fruitful discussions and advices. The authors would also like to thank Dr. C. Tison (CES, France) for providing the high-resolution POLSAR iages over Toulouse. REFERECES [] L. M. ovak and M. C. Burl. Optial speckle reduction in polarietric SAR iagery. IEEE Transactions on Aerospace and Electronic Systes, 26(2): , 990. [2] A. Lopes and F. Sery. Optial speckle reduction for the product odel in ultilook polarietric SAR iagery and the Wishart distribution. IEEE Transactions on Geoscience and Reote Sensing, 35(3): , 997. [3] L. M. ovak, M. C. Burl, and W. W. Irving. Optial polarietric processing for enhanced target detection. IEEE Transactions on Aerospace and Electronic Systes, 29(): , 993. [4] G. Liu, S. Huang, A. Torre, and F. Rubertone. The Multilook Polarietric Whitening Filter (MPWF) for intensity speckle reduction in polarietric SAR iages. IEEE Transactions on Geoscience and Reote Sensing, 36(3):06 020, 998. [5] G. Vasile, J.-P. Ovarlez, F. Pascal, and C. Tison. Coherency atrix estiation of heterogeneous clutter in high resolution polarietric SAR iages. IEEE Transactions on Geoscience and Reote Sensing, 48(4): , 200. [6] B. Picinbono. Spherically invariant and copound Gaussian stochastic processes. IEEE Transactions on Inforation Theory, 6():77 79, 970. [7] K. Yao. A representation theore and its applications to spherically-invariant rando processes. IEEE Transactions on Inforation Theory, 9(5): , 973. [8] S. Zozor and C. Vignat. Soe results on the denoising proble in the elliptically distributed context. IEEE Transactions on Signal Processing, 58():34 50, 200. [9] F. Gini and M. V. Greco. Covariance atrix estiation for CFAR detection in correlated heavy tailed clutter. Signal Processing, 82(2): , [0] E. Conte, A. DeMaio, and G. Ricci. Recursive estiation of the covariance atrix of a copound- Gaussian process and its application to adaptive CFAR detection. IEEE Transactions on Iage Processing, 50(8):908 95, [] F. Pascal, Y. Chitour, J.-P. Ovarlez, P. Forster, and P. Larzabal. Covariance structure axiulikelihood estiates in copound Gaussian noise: existence and algorith analysis. IEEE Transactions on Signal Processing, 56():34 48, [2] F. Pascal, P. Forster, J.-P. Ovarlez, and P. Larzabal. Perforance analysis of covariance atrix estiates in ipulsive noise. IEEE Transactions on Signal Processing, 56(6): , [3] F. Chatelain, J. Y. Tourneret, and J. Inglada. Change detection in ultisensor SAR iages using bivariate gaa distributions. IEEE Transactions on Iage Processing, 7(3): , [4] A. Erdlyi, W. Magnus, F. Oberhettinger, and F. Tricoi. Higher Transcendental Functions, volue. ew York: Krieger, 98. [5] L. Bobrun, G. Vasile, M. Gay, J.-P. Ovarlez, and F. Pascal. Roll invariant target detection based on POLSAR clutter odels,. In Proceedings of the IEEE International Geoscience and Reote Sensing Syposiu, Honolulu, Hawaii, USA, pages , 200. [6] J. S. Lee, S. R. Cloude, K. P. Papathanassiou, M. R. Grunes, and I. H. Woodhouse. Speckle filtering and coherence estiation of polarietric SAR interferoetry data for forest applications. IEEE Transactions on Geoscience and Reote Sensing, 4(0): , 2003.