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Enhanced Dictionary-Based SAR Amplitude Distribution Estimation and its Validation with Very High Resolution Data Vladimir A. Krylov, Gabriele Moser, Member, IEEE, Sebastiano B. Serpico, Fellow, IEEE, and Josiane Zerubia, Fellow, IEEE

Abstract—In this letter we address the problem of estimating the amplitude probability density function of single-channel synthetic aperture radar images. A novel flexible method is developed to solve this problem, basing on the recently proposed dictionarybased stochastic expectation maximization approach (developed for medium-resolution SAR) to very high resolution satellite imagery and enhanced by plugging a novel procedure for estimating the number of mixture components, in order to appreciably reduce its computational complexity. The specific interest is the estimation of heterogenous statistics and the developed method is validated in the case of very high resolution SAR, acquired by the last generation satellite SAR systems TerraSAR-X and COSMOSkyMed. This very high resolution imagery allows to appreciate various ground materials resulting in highly mixed distributions, thus posing a difficult estimation problem, that has not been addressed so far. We also conduct an experimental study of the extended dictionary of state-of-the-art SAR-specific probability density function models and consider dictionary refinements. Index Terms—Finite mixture models, parametric estimation, probability density function estimation, stochastic expectation maximization (SEM), synthetic aperture radar (SAR) images.

I. I NTRODUCTION CCURATE modeling of statistical information is a crucial problem in the context of synthetic aperture radar (SAR) image processing and its applications. Specifically, an accurate probability density function (pdf) estimate can effectively improve the performance of SAR image denoising [1] or classification [2].The very high resolution (VHR) imagery, provided by, e.g., the last generation SAR satellite systems, such as TerraSAR-X and COSMO-SkyMed, allows to appreciate various ground materials resulting in highly mixed distributions. The resulting spatial heterogeneity is a critical problem in applications to image classification (estimation of class-conditional statistics) or filtering (estimation of local

A

This research has been conducted within a collaboration between the Institut National de Recherche en Informatique et en Automatique (INRIA) Sophia Antipolis - M´editerran´ee Center, France, and the Dept. of Biophysical and Electronic Engineering (DIBE) of the University of Genoa, Italy. V.A. Krylov is with the Dept. of Mathematical Statistics, Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, 1, Leninskie Gory, 119991, Moscow, Russia, and also with the EPI Ariana, INRIA/CNRS/UNSA, France; e-mail: [email protected]. G. Moser, S.B. Serpico are with the DIBE, University of Genoa, Via Opera Pia 11a, I-16145, Genoa, Italy; e-mail: [email protected], [email protected]. J. Zerubia is with the EPI Ariana, INRIA/CNRS/UNSA, CR INRIA Sophia Antipolis - M´editerran´ee, 2004, Route des Lucioles, B.P. 93, FR-06902, Sophia Antipolis, France; e-mail: [email protected].

statistics, e.g., in moving-window approaches) [1]. Analysis and modeling of heterogenous VHR SAR data pose a difficult statistical problem, which, to the best of our knowledge, has not been addressed so far. Over the years, a number of methods has been proposed for modeling SAR amplitude pdfs. Nonparametric methods, e.g., Parzen window estimator [3], support vector machines [4], do not assume any specific analytical model for the unknown pdf, thus providing a higher flexibility, although usually involving manual specification of internal architecture parameters [3]. Parametric methods postulate a given mathematical model for each pdf and formulate the pdf estimation problem as a parameter estimation problem. Empirical pdf models, including lognormal [1], Weibull [1], Fisher [2] and, recently, the generalized Gamma distribution (GΓD) [5], have been reported to accurately model amplitude SAR images with different heterogenous surfaces. Several theoretical models, such as Rayleigh [1], Nakagami [1], generalized Gaussian Rayleigh (GGR) [6], symmetric-α-stable generalized Rayleigh (SαSGR) [7], K [8] (K-root for amplitudes), G [9], have been derived from specific physical hypotheses for SAR images with different properties. However, several parametric families turned out to be effective only for specific land cover typologies [1], making the choice of a single optimal SAR amplitude parametric pdf model a hard task, especially in case of heterogenous imagery. To solve this problem, the dictionary-based stochastic expectation maximization (DSEM) approach was developed in [10] for medium resolution SAR. It addresses the problem by adopting a finite mixture model [11] for the SAR amplitude pdf, i.e., by postulating the unknown amplitude pdf to be a linear combination of parametric components, each corresponding to a specific statistical population. In this letter, we address the general problem of modeling the statistics of single-channel SAR amplitude images and, specifically, VHR SAR. Given the variety of approaches above, we extend and enhance the DSEM technique proposed in [10] for coarser-resolution SAR. We expect DSEM to be an appropriate tool for this modeling problem, since it is a flexible method, intrinsically modeling SAR statistics as resulting from mixing several populations, and it is not constrained to a specific choice of a given parametric model allowing to benefit from many of them (dictionary approach). Thus, we extend the earlier DSEM approach to VHR satellite SAR and enhance it by a novel procedure for estimating the number of mixture components, which enables to appreciably

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TABLE I P DFS AND M O LC EQUATIONS FOR THE PARAMETRIC PDF FAMILIES IN D. H ERE Γ(·) IS THE G AMMA FUNCTION [12], Kα (·) THE αTH ORDER MODIFIED B ESSEL FUNCTION OF THE SECOND KIND [12], J0 (·) IS THE ZERO - TH ORDER B ESSEL FUNCTION OF THE FIRST KIND [12], Ψ(·) THE D IGAMMA FUNCTION [12], Ψ(ν, ·) THE ν TH ORDER POLYGAMMA FUNCTION [12] AND Gν (·) ARE THE SPECIFIC INTEGRAL FUNCTIONS FOR GGR [6] Family

Probability density function

MoLC equations 2

(ln r−m) 1 √ exp[− 2σ2 ], r > 0, σr 2π ( )η η η−1 r = µη r exp[− µ ], r > 0,

f1

Lognormal

f1 (r|m, σ) =

f2

Weibull

f2 (r|η, µ)

L

f3

f3 (r|L, M, µ) =

Fisher

[ar] Γ(L+M ) Γ(L)Γ(M ) r[1+ar]L+M

, r > 0,

f4

GΓD

f5

Nakagami

with a = L/(M µ), ( )κν−1 [ ( )ν ] ν exp − σr f4 (r|ν, κ, σ) = σΓ(κ) σr , r > 0, [ ] L 2L−1 2 2 exp −λLr , r > 0, f5 (r|L, λ) = Γ(L) (λL) r

f6

K-root

4 Γ(L)Γ(M )

f6 (r|L, M, µ) =

×r L+M −1 KM −L f7

GGR

f7 (r|λ, γ) =

γ2r λ2 Γ2 (λ)

π/2 ∫

[λLM ] ( ) 2r [λLM ]1/2 , r > 0,

[ ] exp −(γr)1/λ s(θ) dθ, r > 0,

0

f8

f8 (r|α, γ) = r

SαSGR

ρ

exp[−γρα ]J0 (rρ)dρ,

κ2 = σ 2 .

κ1 = ln µ + η −1 Ψ(1),

κ2 = η −2 Ψ(1, 1).

κ1 = ln µ + (Ψ(L) − ln L) − (Ψ(M ) − ln M ), κj = Ψ(j − 1, L) + (−1)j Ψ(j − 1, M ), j = 2, 3. κ1 = Ψ(κ)/ν + ln σ,

κj = Ψ(j − 1, κ)/ν j , j = 2, 3.

2κ1 = Ψ(L) − ln λL,

(L+M )/2

with s(θ) = | cos θ|1/λ + | sin θ|1/λ , +∞ ∫

κ1 = m,

4κ2 = Ψ(1, L).

2κ1 = Ψ(L) + Ψ(M ) − ln LM λ, 2j κ

j

= Ψ(j − 1, L) + Ψ(j − 1, M ), j = 2, 3.

κ1 = λΨ(2λ) − ln γ − λG1 (λ)[G0 (λ)]−1 , ( ) G (λ) G (λ) 2 κ2 = λ2 [Ψ(1, 2λ) + G2 (λ) − G1 (λ) ]. 0

r > 0,

ακ1 = (α − 1)Ψ(1) +

ln γ2α ,

0

κ2 = α−2 Ψ(1, 1).

0

reduce its computational complexity (as much as 5 times in some cases), resulting in an Enhanced DSEM algorithm (EDSEM). We revise the dictionary content with respect to VHR data, by adding the Fisher pdf, which is a particular case of the Snedecor-Fisher distribution and was suggested for heterogenous HR SAR, and GΓD, a very flexible empirical model, including Weibull and Nakagami as particular cases and Lognormal as an asymptotic case. The main novel contribution of this letter is the highly accurate, flexible and computationally fast pdf estimation model for single-channel SAR amplitude images and its validation with VHR satellite SAR data. We also perform a comparative experimental study of parametric pdf models in the dictionary (most of the state-of-the-art models), suggesting a refinement of the dictionary for this problem. The scope of applications where the developed approach can be used include SAR image equalization, segmentation and supervised classification. This letter is organized as follows. In Section II we propose the EDSEM algorithm. Section III reports experiments of the developed technique on VHR satellite SAR images acquired by TerraSAR-X and COSMO-SkyMed, and on a VHR image, acquired by the airborne RAMSES sensor. Finally, conclusions are drawn in Section IV. II. M ETHODOLOGY To take into account the heterogenous scenario, when several distinct land-cover typologies are present in the same SAR image, a finite mixture model (FMM) [11] for the distribution of grey levels is assumed. An amplitude SAR image is modeled as a set I = {r1 , . . . , rN } of independent samples drawn from a mixture pdf with K components: pr (r) =

K ∑ i=1

Pi pi (r),

r > 0,

(1)

where pi (r) ∑Kare parametric pdfs and {Pi } are mixing proportions: i=1 Pi = 1, with 0 6 Pi 6 1, i = 1, . . . , K. Each component pi (r) in (1) is modeled by resorting to a finite dictionary D = {f1 , . . . , f8 } (see Table I) of 8 SARspecific distinct parametric pdfs fj (r|θj ), parameterized by θj ∈ Aj , j = 1, . . . , 8. Dealing with VHR heterogenous SAR, we include the Fisher model, suggested for heterogenous HR SAR imagery, and GΓD, highly flexible empirical model, including Lognormal, Weibull and Nakagami as special cases, into the considered dictionary D. As discussed in [10], considering the variety of estimation approaches for FMMs, an appropriate choice for this particular estimation problem is the stochastic expectation maximization (SEM) scheme [11]. SEM was developed as a stochastic modification of the classical EM algorithm, involving stochastic sampling on every iteration, and demonstrating higher chances of avoiding local maxima of the likelihood function [11]. SEM is an iterative estimation procedure dealing with the problem of data incompleteness. In case of FMMs the complete data is represented by the set {(ri , si ), i = 1, . . . , N }, where ri are the observations (SAR amplitudes) and si - the missing labels: given an FMM with K components, si ∈ {σ1 , σ2 , . . . , σK } denotes to which of the K components the i-th observation belongs. Instead of adopting the maximum likelihood (ML) estimates as the classical SEM scheme [11] suggests, in DSEM the Method of Log-Cumulants (MoLC) [2] for component parameter estimation is adopted, which has been demonstrated to be a feasible and effective estimation tool for all the pdfs in D [2], [5], [10]. MoLC has recently been proposed as a parametric pdf estimation technique suitable for distributions defined on [0, +∞), and has been widely applied in the context of SARspecific parametric families for amplitude and intensity data modeling. MoLC adopts the Mellin transform [12] by analogy

IEEE GEOSCIENCE AND REMOTE SENSING LETTERS

(a) CSK1 image

(d) CSK1 histograms

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(b) CSK2 image

(c) RAMS image

(e) CSK2 histograms

(f) RAMS histograms

Fig. 1. CSK1 (a), CSK2 (b) and RAMS (c) images with corresponding plots of pdf estimates (d), (e), (f). The plots contain: normalized image histogram, EDSEM pdf estimate with plots of K ∗ components in the estimated mixture, and the plot of the best fitting pdf model from D.

to the Laplace transform in moment generating function [2]. Given a non-negative random variable u, the second-kind characteristic function ϕu of u is defined as the Mellin transform [12] M of the pdf of u, i.e.: ∫ +∞ ϕu (s) = M(pu )(s) = pu (u)us−1 du, s ∈ C. 0

The derivatives κν = [ln ϕu ](ν) (1) are the νth order logcumulants, where (ν) stands for the νth derivative, ν = 1, 2, . . .. In case of the transform convergence, the following MoLC equations take place [2]: { κ1 = E{ln u} , j = 2, 3. κj = E{(ln u − κ1 )j } Analytically expressing κj as functions of unknown parameters and estimating these log-cumulants in terms of sample logmoments one derives a system of nonlinear equations. These equations have one solution for any observed values of logcumulants for all pdfs in D (see Table I), except for, in some cases, GGR, K-root, see [10], and GΓD due to a complicated parameter estimation procedure. For the purpose of K estimation, [10] suggested running DSEM for all values of K from 1 to predefined Kmax , and then choosing the number K ∗ that provides the highest loglikelihood estimate. In this letter, in order to avoid this computationally costly procedure, we adopt a K estimation procedure similar to the one suggested in [11], that consists of initializing SEM with K0 = Kmax , and then allowing components to be eliminated from the mixture during the iterative process, once their priors Pi become too small, thus decreasing K. This strategy provides efficient K ∗ estimates consistent with DSEM estimates, allowing to significantly reduce the DSEM computation complexity, especially for high values of Kmax .

Thus, each iteration of EDSEM goes as follows: • E-step: compute, for each greylevel z and i-th component, the posterior probability estimates corresponding to the current pdf estimates, i.e. z = 0, . . . , Z − 1: P t pt (z) , i = 1, . . . , Kt , τit (z) = ∑Kti i t t j=1 Pj pj (z) where pti (·) is the σi -conditional pdf estimate on the tth step. • S-step: sample the label st (z) of each greylevel z according to the current estimated posterior probability distribution {τit (z) : i = 1, . . . , Kt }, z = 0, . . . , Z − 1. • MoLC-step: for the i-th mixture component, compute the following histogram-based estimates of the mixture proportions and the first three log-cumulants: ∑ ∑ z∈Qit h(z) z∈Q h(z) ln z t+1 t Pi = ∑Z−1 , κ1i = ∑ it , z∈Qit h(z) z=0 h(z) ∑ t b z∈Qit h(z)(ln z − κ1i ) t ∑ κbi = , i = 1, . . . , Kt , z∈Qit h(z) where b = 2, 3; h(z) is the image histogram; Qit = {z : st (z) = σi } is the set of grey levels assigned to the i-th component; then, solve the corresponding MoLC equations (see Table I) for each parametric family fj (·|θj ) (θj ∈ Aj ) in the dictionary, thus computing the resulting MoLC estimate t θij , j = 1, . . . , M . • K-step: ∀i, i = 1, . . . , Kt : if Pit+1 < γ, eliminate the i-th component, update Kt+1 . The choice of threshold γ does not appreciably affect EDSEM, provided it is small, e.g. 0.005. • Model Selection-step: for each mixture component i, t compute the log-likelihood of each estimated pdf fj (·|θij ) according to the data assigned to the i-th component: ∑ t Ltij = h(z) ln fj (z|θij ), i = 1, . . . , Kt+1 , z∈Qit

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(a) TSX1 image

(d) TSX1 histograms

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(b) TSX2 image

(c) TSXH image

(e) TSX2 histograms

(f) TSXH histograms

Fig. 2. TSX1 (a), TSX2 (b) and TSXH (c) images with corresponding plots of pdf estimates (d), (e), (f). The plots contain: normalized image histogram, EDSEM pdf estimate with plots of K ∗ components in the estimated mixture, and the plot of the best fitting pdf model from D.

t and define pt+1 (·) as the estimated pdf fj (·|θij ) yielding the i t highest value of Lij , j = 1, . . . , M . The sequence of estimates Θt generated by SEM converges to a unique stationary distribution, and the maximum likelihood estimate of the mixture parameters is asymptotically equivalent to the mathematical expectation of this stationary distribution. This behavior has been proved under suitable assumptions [11], which do not hold strictly for all the pdfs in D. However, we recall that SEM, compared to the classical EM or other deterministic variants for FMM, was specifically designed to improve the exploratory properties of EM in case of multimodal likelihood function [11].

III. E XPERIMENTAL RESULTS A. Datasets for experiments EDSEM was tested on four VHR SAR datasets: • Single-look COSMO-SkyMed, Stripmap, approx. 2.5 m ground resolution, HH-polarized image acquired over c Piemonte, Italy (⃝ASI, 2008). We present experiments with two portions CSK1, CSK2 (see Fig. 1). • Single-look image, acquired by the RAMSES airborne c sensor, approx. 0.5 m res., Toulouse, France (⃝ONERACNES, 2004), hereafter denoted as RAMS (see Fig. 1). • Two-look TerraSAR-X, Stripmap, approx. 6 m res., HH-polarized image acquired over Sanchagang, China c (⃝Infoterra, 2008). We present experiments with two portions TSX1, TSX2 (see Fig. 2). • Single-look TerraSAR-X, HR Spotlight, approx. 1 m res., VV-polarized image acquired over Barkedji, Senegal c (⃝CNES, 2007) TSXH (see Fig. 2). All the employed images present heterogenous scenes and, except for CSK2 and TSXH, exhibit multimodal histograms. Size of all test images is 500x500 pixels.

TABLE II EDSEM RESULTS : K ∗ ESTIMATE , THE ESTIMATED MIXTURE (fi S AS IN TABLE I), KOLMOGOROV-S MIRNOV DISTANCE KS, p- VALUE FOR THE KOLMOGOROV-S MIRNOV TEST, THE COMPUTATION TIME t ( IN SECONDS ), ALONG WITH THE BEST FITTING MODEL IN D WITH KS AND p- VALUE Image K∗ CSK1 CSK2 RAMS TSX1 TSX2 TSXH

EDSEM estimate Mixture KS p-value

3 f4 , f 4 , f 5 4 f 5 , f4 , f 1 , f 5 5 f 4 , f2 , f 1 , f 2 , f2 3 f2 , f 1 , f 7 3 f4 , f 3 , f 2 2 f6 , f4

t

0.011 0.9989 121s 0.007 1.0 192s 0.008 1.0 253s 0.006 1.0 155s 0.004 1.0 148s 0.008 1.0 97s

Best fit in D Model KS p-value f1 f5 f2 f5 f1 f7

0.062 0.020 0.059 0.037 0.058 0.016

0.2787 0.9869 0.3349 0.8747 0.3552 0.9936

B. PDF estimation results The obtained EDSEM pdf estimates have been assessed quantitatively by computing Kolmogorov-Smirnov distances (KS) between estimates and normalized image histograms. The estimates were also assessed by the Kolmogorov-Smirnov goodness-of-fit test [13] and the corresponding p-values are reported in Table II. Here p can be interpreted as a confidence level at which the corresponding fit hypothesis will be rejected. The following EDSEM parameters were used: initial number of mixture components K0 = 6, number of iterations T = 200. Both were set manually high enough to give sufficient burn-in for SEM and their further increase did not affect the results. For comparison, for all images we provide the best fitting pdf (with the smallest KS), with parameters estimated by MoLC, among 8 models in D (see Table II). For the considered images the accuracy improvement granted by EDSEM is significant (∆KS up to 0.05), somewhat smaller for almost unimodal histograms (CSK2 and TSXH). Visual analysis of EDSEM estimate plots (see Fig. 1, Fig. 2) confirms an important improvement in the estimation accuracy.

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TABLE III N UMBERS OF TIMES THE MODELS IN D WERE CHOSEN FOR MIXTURE COMPONENTS DURING EDSEM FOR CSK1, RAMS AND TSXH IMAGES . Image

f1

f2

f3

f4

f5

f6

f7

f8

Total

CSK1 RAMS TSXH

192 145 45

155 265 69

47 7 5

209 340 224

194 172 107

9 13 36

53 94 61

25 10 17

793 1046 564

The computation time of EDSEM is presented in Table II, experiments were conducted on Intel Core 2 Duo 1.83GHz, 1Gb RAM, WinXP system. Compared to DSEM, EDSEM allowed the following improvements: it worked 3.85 times faster for CSK1, 2.63 for RAMS and 5.88 for TSXH (with the same Kmax and T ). We stress here the consistency of results reported by both techniques, in terms of K ∗ , KS and p-value. Finally, we address the issue of dictionary content. We would like to mention, that given a new model not contained in D provided with corresponding MoLC-equations and solution, the process of adding this model into the dictionary is straightforward. We study the number of times the pdfs in D were chosen in the Model Selection-step for mixture components along the EDSEM process (see Table III). The least frequently chosen models (in bold) were Fisher, K-root and SαSGR. Additional experiments with dictionary of 5 models (without f3 , f6 and f8 ) reported a negligible decline in accuracy (KS increase of less than 0.005), suggesting a smaller significance of these models in D. The computational speed-up, however, was significant due to cumbersome numerical procedures for K-root and SαSGR. A further refinement of D can be achieved by removing the Weibull pdf which is a particular case of GΓD. Thus, very accurate results can be achieved with dicb = {f1 , f4 , f5 , f7 }: the same estimate was obtained tionary D for CSK1 in 78 s; compared to the result in Table II, f2 was replaced by f4 in the estimate for RAMS, which achieved KS = 0.01 and took 157 s; similarly, f6 was replaced by f5 in the estimate for TSXH with KS = 0.012 and 62 s. Thus, this dictionary refinement further reduces the computation time of EDSEM by 35% on the average, only marginally affecting the accuracy (increase in KS always below 0.007). IV. C ONCLUSIONS In this letter we propose a general flexible pdf-estimation method for SAR images and specifically validate it on heterogeneous VHR SAR data, which represent a very current and relevant case in which heterogenous behavior is expected. The developed model is based on the dictionarybased stochastic expectation maximization (DSEM) approach recently developed in [10] for medium resolution SAR. The extension of the proposed method to intensity data (instead of amplitudes) is straightforward, by replacing the amplitude pdfs in the dictionary with the corresponding intensity pdfs. The contribution of this letter is twofold. First, the developed Enhanced DSEM extended DSEM to the novel type of imagery (heterogenous VHR), enhanced by a novel efficient procedure for estimating the number of mixture components, reported very accurate and computationally fast estimation results in experiments with VHR SAR images acquired by the

modern satellite systems TerraSAR-X and COSMO-SkyMed, and the RAMSES airborne sensor. We stress here that the problem of modeling the statistics of VHR satellite SAR amplitude images has not been addressed so far, and the proposed EDSEM technique looks very promising for this type of images, since it is based on a finite mixture approach, and intrinsically takes into account heterogeneity, which is an inherent VHR image property. Second, we studied the extended dictionary of SAR-specific pdfs (including Fisher and GΓD models) and analyzed refinement options. This study suggested that, at least on the considered datasets, dictionary reduction from 8 models to 4 affects the estimation accuracy only marginally, reducing the computation burden of EDSEM by 35%. The reported results suggest EDSEM to be an attractive approach for various application problems, e.g., it can be efficiently used for SAR image segmentation (to discriminate the resulting mixture components) [10] and to class-conditional pdf modeling in supervised VHR SAR classification [14]. ACKNOWLEDGMENT The first author would like to thank the French Space Agency (CNES), Poncelet Lab. in Moscow and INRIA for the partial funding of his internships at INRIA. The authors would like to thank the Italian Space Agency (ASI) for providing the COSMO-SkyMed image and CNES for providing the RAMSES and the TerraSAR-X Barkedji image. The TerraSAR-X image of Sanchagang was taken from http://www.infoterra.de/. R EFERENCES [1] C. Oliver and S. Quegan, Understanding Synthetic Aperture Radar Images. Norwood: Artech House, 1998. [2] C. Tison, J.-M. Nicolas, F. Tupin, and H. Maitre, “A new statistical model for Markovian classification of urban areas in high-resolution SAR images,” IEEE Trans. Geosci. Remote Sens., vol. 42, no. 10, pp. 2046–2057, 2004. [3] R. O. Duda, P. E. Hart, and D. G. Stork, Pattern classification. NY: Wiley Interscience, 2001. [4] P. Mantero, G. Moser, and S. B. Serpico, “Partially supervised classification of remote sensing images using SVM-based probability density estimation,” IEEE Trans. Geosci. Remote Sens., vol. 43, no. 3, pp. 559– 570, 2005. [5] H.-C. Li, W. Hong, and Y.-R. Wu, “Generalized Gamma distribution with MoLC estimation for statistical modeling of SAR images,” in Proceedings of APSAR, Huangshan, China, 2007, pp. 525–528. [6] G. Moser, J. Zerubia, and S. B. Serpico, “SAR amplitude probability density function estimation based on a generalized Gaussian model,” IEEE Trans. Image Process., vol. 15, no. 6, pp. 1429–1442, 2006. [7] E. E. Kuruoglu and J. Zerubia, “Modelling SAR images with a generalization of the Rayleigh distribution,” IEEE Trans. Image Process., vol. 13, no. 4, pp. 527–533, 2004. [8] E. Jakeman and P. N. Pusey, “A model for non-Rayleigh sea echo,” IEEE Trans. Antennas Propagat., vol. 24, pp. 806–814, 1976. [9] A. C. Frery, H.-J. Muller, C. C. F. Yanasse, and S. Sant’Anna, “A model for extremely heterogeneous clutter,” IEEE Trans. Geosci. Remote Sens., vol. 35, no. 3, pp. 648–659, 1997. [10] G. Moser, S. B. Serpico, and J. Zerubia, “Dictionary-based Stochastic Expectation Maximization for SAR amplitude probability density function estimation,” IEEE Trans. Geosci. Remote Sens., vol. 44, no. 1, pp. 188–199, 2006. [11] G. Celeux, D. Chauveau, and J. Diebolt, “On stochastic versions of the EM algorithm,” INRIA, Research Report 2514, 1995. [12] I. Sneddon, The use of integral transforms. NY: McGraw-Hill, 1972. [13] M. Stephens, “Test of fit for the logistic distribution based on the empirical distribution function,” Biometrika, vol. 66, pp. 591–595, 1979. [14] G. Moser, V. Krylov, S. B. Serpico, and J. Zerubia, “High resolution SAR-image classification by Markov random fields and finite mixtures,” in Proceedings of SPIE, vol. 7533, San Jose, USA, 2010, p. 753308.