Optimal Parameter Estimation in Heterogeneous Clutter for High Resolution Polarimetric SAR Data

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1 Optial Paraeter Estiation in Heterogeneous Clutter for High Resolution Polarietric SAR Data Gabriel Vasile, Frédéric Pascal, Jean-Philippe Ovarlez, Pierre Foront, Michel Gay To cite this version: Gabriel Vasile, Frédéric Pascal, Jean-Philippe Ovarlez, Pierre Foront, Michel Gay. Optial Paraeter Estiation in Heterogeneous Clutter for High Resolution Polarietric SAR Data. IEEE Geoscience and Reote Sensing Letters, IEEE - Institute of Electrical and Electronics Engineers, 20, 8 (6), pp <hal > HAL Id: hal Subitted on 7 Nov 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 Optial Paraeter Estiation in Heterogeneous Clutter for High Resolution Polarietric SAR Data Gabriel Vasile, Meber, IEEE, Frédéric Pascal, Meber, IEEE, Jean-Philippe Ovarlez, Meber, IEEE, Pierre Foront, Student Meber, IEEE Michel Gay, Meber, IEEE 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 ONERA RAMSES syste. Index Ters Estiation, detection, polarietry, SAR. I. INTRODUCTION 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, Novak and Burl derived the Polarietric Whitening Filter (PWF) by optially cobining the eleents of the 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 G. Vasile and M. Gay are with the Grenoble-Iage-sPeech-Signal- Autoatics Lab (GIPSA-lab), CNRS, Grenoble, France (e-ail: gabriel.vasile@gipsa-lab.grenoble-inp.fr; ichel.gay@gipsa-lab.grenobleinp.fr) F. Pascal is with the Supélec, National University of Singapore, Defence Science and Technology Agency Research Alliance, SONDRA, Gif-sur- Yvette, France (e-ail: frederic.pascal@supelec.fr) J.-P. Ovarlez and P. Foront are with the French Aerospace Lab (ONERA), DEMR/TSI, Palaiseau, France and with the Supélec, National University of Singapore, Defence Science and Technology Agency Research Alliance, SONDRA, Gif-sur-Yvette, France (e-ail: ovarlez@onera.fr; pierre.foront@supelec.fr) 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. II and Sect. III, respectively. Then, the novel span estiator is introduced in Sect. IV. Next, soe estiation results are shown in Sect. V on a real high-resolution POLSAR dataset acquired by the ONERA RAMSES syste. Eventually, in Sect. VI, soe conclusions are presented. II. SIRV CLUTTER MODEL WITH NORMALIZED TEXTURE The SIRV is a class of non-hoogeneous Gaussian processes with rando variance [6], [7]. The coplex - diensional easureent k ( being the nuber of polarietric channels) is defined as the product between the independent coplex circular Gaussian vector ζ N (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 kernel. 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 N i.i.d. (independent and identically distributed) secondary data, let L k (k,...,k N [T], ξ,..., ξ N ) be the likelihood function to axiize with respect to [T] and ξ i. L k (k,...,k N ; [T], ξ,..., ξ N ) = π N det{[t]} N N ξ i exp ( k i [T] k i ξ i The corresponding ML estiators are given by [9]: lnl k (k,..., k N [T], ξ,..., ξ N) ξ i ). () = 0 ξ b i = k i [T] k i, (2)

3 2 lnl k (k,...,k N [T], ξ,..., ξ N) [T] = 0 [ b T] = N NX k ik i bξ i. (3) 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 = NX k ik i N = [ T] b SCM NX «k ik i bξi = [0 ]. N (4) The generalized ML estiator for ξ i are obtained by introducing Eq. 4 in Eq. 2: ξ i = k i [ T] SCM k i. (5) Note the k i priary data is the cell under study. The noralized texture estiator fro Eq. 5 is known as the Polarietric Whitening Filter (PWF-SCM) introduced by Novak and Burl in []. III. SIRV CLUTTER MODEL WITH NORMALIZED COVARIANCE MATRIX Let now the covariance atrix be of the for [T] = σ 0 [M], such that Tr{[M]} =. The product odel can be also written as k = τ z, where z N (0, [M]). σ 0 and ξ are two scalar positive rando variables such that τ = σ 0 ξ. Using the sae procedure as in Sect. II and 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 ) = N N k i k i k i [ M] FP k. (6) 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 ). (7) 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. (8) One can observe that the PWF-FP texture fro Eq. 8 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]. IV. MAIN RESULT The span (total power) σ 0 can be derived using the covariance atrix estiators presented in Sect. II and Sect. III as: FP k T] SCM σ 0 = k [ M] (9) k [ k. Note that Eq. 9 is valid when considering N 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. A. 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. 9. On one hand, the asyptotic distribution of the FP estiator fro Eq. 6 has been derived in [2]. The FP estiator coputed with N secondary data converges in distribution to the noralized SCM coputed with N[/(+)] 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 N[/( + )] secondary data. Hence one can set the degrees of freedo of FP noralized covariance atrix estiators as: q = N +. (0) On the other hand, Chatelain et al. established the ultisensor bivariate gaa distribution PDF, whose argins are univariate gaa distributions with different shape paraeters [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 ) () 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 ) ],

4 3 (α) 2+n (β) n where H 3 (α, β, γ; x, y) = (γ),n=0 +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. 0, and the Cochran s theore [5], the PDF of the span estiator fro Eq. 9 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 N[/( + )] and N (the nuber of secondary data), respectively. 0.8 Epirical PDF P(σ 0 ; µ=3, N=24, =3, ρ=0.05) P(σ 0 ; µ=3, N=24, =3, ρ=0.5) P(σ 0 ; µ=3, N=24, =3, ρ=0.95) Fig.. Ratio PDF of two correlated Gaa rando variables (Eq. ) 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. IV, naely N = 24, = 3, µ = 0, µ 2 =. Notice that when the noralized correlation coefficient approaches to, the PDF tends to a Dirac. A Monte Carlo siulation has been represented in Fig., also saples of σ 0 were obtained by coputing saples draw 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 3. One can observe the good correspondence between the epirical PDF of siulated σ 0 and the PDF derived in Eq. for ρ = TABLE I EMPIRICAL MEAN AND VARIANCE OF THE σ 0 ESTIMATOR FROM EQ. 9 AND THE THEIR EXPECTED VALUES FOR SIMULATED GAUSSIAN CLUTTER. Expected Epirical Boxcar Mean Variance Mean Variance Using the sae paraeters as in the previous Monte Carlo siulation, Table I illustrates the behavior of the epirical ean and variance of the proposed σ 0 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. B. The σ 0 test In this section we propose to show how the estiator fro Eq. 9 is linked with a binary hypothesis testing proble, also: under the null hypothesis H 0, the observed target vector k = ξ ζ belongs to the SIRV clutter ζ N(0, [T]) with noralized texture, under the alternative hypothesis H, the priary target vector k = τ z belongs to the SIRV clutter z N(0, [M]) with noralized covariance atrix. Fro the operational point of view, the proposed detector is a classical constant false alar rate detector with current pixel as priary data, and with the local boxcar neighborhood around it as secondary data. The Neyan-Pearson optial detector is given by the following likelihood ratio test (LRT): Λ (k) = Λ (k) = p k(k/h ) H λ. (2) p k (k/h 0 ) H 0 After expressing the PDF under each hypothesis, it results that: ( ) exp k [M] k π det{[m]}τ τ ( exp π det{[t]}ξ k [T] k ξ ) H H 0 λ. (3) By plugging into the LRT the ML texture estiators fro Eqs. 5 and 8 we obtain: Λ (k) = det{[t]} ( k [T] ) k H det{[m]} k [M] λ. (4) k H 0 Next, 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 N secondary data. Finally, by replacing the known covariances by their ML estiates the generalized LRT is: Λ (k) = α σ 0 H H 0 λ. (5) As α appears as a deterinistic quantity only, it is possible to use the PDF derived in Sect. IV-A to set the decision threshold λ for a specific false alar probability. V. RESULTS AND DISCUSSIONS The high resolution POLSAR data set, illustrated in Fig. 2, was acquired by the ONERA RAMSES syste over Toulouse, France with a ean incidence angle of 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 CNES buildings. Fig. 5-(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 5 5 boxcar neighborhood has been selected for illustration, hence 24 secondary saples and priary data. Fig. 3 presents the zoo over the red rectangle fro Fig. 5-(a), where a narrow diplane target was previously detected. Fig. 3-(a),(b),(c) shows the FP-PWF texture, the SCM-PWF noralized texture, and the proposed span estiator σ 0, respectively. For coparison, the Multi-look PWF (MPWF) has been illustrated in Fig. 3-(d). The proposed estiator exhibits

5 4 Based on these issues, a novel test has been introduced for selecting the ost appropriate odel for POLSAR heterogeneous clutter described by SIRVs. 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. II under the SIRV clutter hypothesis. A novel estiation / detection strategy has 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. Fig. 2. Toulouse, RAMSES POLSAR data, X-band, pixels: aplitude color coposition of the target vector eleents k -k 3 -k 2. 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. 4 illustrates the detection ap obtain using the LRT fro Eq. 5 with 25 secondary and one priary data. The detection threshold has been obtained by Monte Carlo integration of the PDF fro Eq. with a false alar probability set to P fa = 0 3 in each pixel. Note that the PDF integration for such a sall P fa is quite tie consuing and fast nuerical approxiations need to be investigated in the future for going to an operational level. This detection ap can be interpreted as follows: heterogeneous clutter areas, represented in red, revel dense urban areas, which exhibit fewer doinant scatterers within the resolution cell. Over these areas, according to the hypotheses test fro Sect. IV, it is better to estiate clutter paraeters using the noralized covariance SIRV odel. hoogeneous clutter areas, represented in blue, where the noralized texture odel is better. Concerning the validation of our results, the generalized LRT is known to be asyptotically uniforly ost powerful according to the Neyan-Pearson lea [7]. This optiality holds provided the ML estiators plugged into the LRT are consistent, which is the case for our study [], [2]. VI. CONCLUSIONS 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 POLSAR 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 have been assessed also by Monte Carlo siulations. ACKNOWLEDGMENT 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 (CNES, France) for providing the high-resolution POLSAR iages over Toulouse. REFERENCES [] L. M. Novak and M. C. Burl, Optial speckle reduction in polarietric SAR iagery, IEEE Transactions on Aerospace and Electronic Systes, vol. 26, no. 2, pp , 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, vol. 35, no. 3, pp , 997. [3] L. M. Novak, M. C. Burl, and W. W. Irving, Optial polarietric processing for enhanced target detection, IEEE Transactions on Aerospace and Electronic Systes, vol. 29, no., pp , 993. [4] G. Liu, S. Huang, A. Torre, and F. 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6 5 [4] A. Erdlyi, W. Magnus, F. Oberhettinger, and F. Tricoi, Higher Transcendental Functions. New York: Krieger, 98, vol.. [5] W. G. Cochran, The distribution of quadratic fors in a noral syste, with applications to the analysis of covariance, Matheatical Proceedings of the Cabridge Philosophical Society, vol. 30, no. 2, pp. 78 9, 934. [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, vol. 4, no. 0, pp , [7] L. L. Scharf, Statistical Signal Processing: Detection, Estiation, and Tie Series Analysis. Addison-Wesley, Inc., 99. (a) (a) (b) (c) Fig. 3. Toulouse, RAMSES POLSAR data, X-band, pixels, zoo iage: (a) FP-PWF texture, (b) SCM-PWF noralized texture, (c) span estiated using cσ 0 fro Eq. 9 and (d) SCM-MPWF span. (d) (b) Fig. 4. 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). (c) Fig. 5. Toulouse, RAMSES POLSAR data, X-band, pixels: (a) span estiated using cσ 0 fro Eq. 9, (b) noralized texture ξ, and (c) color coposition of the noralized coherency diagonal eleents [M] -[M] 33 - [M] 22.