Multilook Polarimetric SAR Change Detection Using Stochastic Distances Between Matrix-Variate Gd⁰ Distributions

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Stochastic Distances Between Matrix-Variate Gd�

Distributions

Nizar Bouhlel, Stephane Meric

To cite this version:

Nizar Bouhlel, Stephane Meric. Multilook Polarimetric SAR Change Detection Using Stochas- tic Distances Between Matrix-Variate Gd� Distributions. IEEE Transactions on Geoscience and Remote Sensing, Institute of Electrical and Electronics Engineers, 2020, 58 (10), pp.6823-6843.

�10.1109/TGRS.2020.2976766�. �hal-02948490�

Multilook Polarimetric SAR Change Detection Using Stochastic Distances Between

Matrix-Variate G

Distributions

Nizar Bouhlel , Member, IEEE, and Stéphane Méric , Member, IEEE

Abstract— In this article, we propose an efficient heterogeneous change detection algorithm based on stochastic distance measure between twoG_d⁰distributions. Due to its flexibility and simplicity, the matrix-variate G⁰_d distribution has been successfully used to model the multilook polarimetric synthetic aperture radar (PolSAR) data and has been tested for classification, segmen- tation, and image analysis. Concretely, closed-form expressions for the Kullback–Leibler, Rényi of order β, Bhattacharyya, and Hellinger distances are provided to compute the stochastic distance between Gd⁰ distributions. In this context, we resort to the expectation–maximization (EM) to estimate accurately with low complexity the parameters of the probability distribution of the two multilook polarimetric covariance matrices to be compared. Finally, the performance of the method is compared firstly to the performance of other known distributions, such as the scaled complex Wishart distribution, and secondly to other known statistical tests using simulated and real multilook PolSAR data.

Index Terms— Bhattacharyya and Hellinger distances, change detection, expectation–maximization (EM) algorithm, multilook polarimetric synthetic aperture radar (PolSAR) data, Rényi of order β, stochastic distances: Kullback–Leibler.

I. INTRODUCTION

T

Manuscript received September 15, 2019; revised January 13, 2020;

accepted February 21, 2020.(Corresponding author: Nizar Bouhlel.) Nizar Bouhlel is with the Signal, Communications, Embedded Electronics (SCEE) Team, Institut d’Electronique et de Telecommunications de Rennes (IETR), CentraleSupelec, Compus de Rennes, 35510 Cesson-Sévigné, France (e-mail: nizar.bouhlel@centralesupelec.fr).

Stéphane Méric is with the SAR and Hyperspectral Multi-Modal Imaging and Signal Processing, Electromagnetic Modeling (SHINE) Team, IETR, Institut National des Sciences Appliquées de Rennes (INSA), 35708 Rennes, France.

Color versions of one or more of the figures in this article are available online at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TGRS.2020.2976766

threshold is applied to produce a final binary change detection map with two classes associated with changed and unchanged pixels [3]. Several unsupervised change detection methods have been proposed and can be categorized into two groups:

one is based on pixel intensity and the neighboring of the pixel, whereas the other is based on local statistics and statistical information theory.

The first group included image differencing, mean ratio/log- ratio measures [4], [5], Gauss log-ratio [6], etc. The simplest one is to compute the ratio of SAR amplitudes or intensities observed at different times and test the hypotheses of change versus no-change. This kind of ratio is well-known test sta- tistic in single-polarization SAR-based change detection. The generalization of these methods to multilook polarimetric SAR (PolSAR) images has been introduced by Conradsenet al.[7].

The authors proposed a likelihood ratio test (LRT) for the equality of two complex covariance matrices and gave the approximated distribution of the LRT statistic. The LRT approach was extended to the multitemporal case [8], [9] and multifrequency data [10]. Recently, Akbariet al.[11] proposed a simpler test statistic to detect changes in many scenarios.

The test assumes the scaled complex Wishart distribution for the covariance matrix data. It is based on the complex-kind Hotelling–Lawley trace (HLT) which is used as a covariance equality test.

The second group based on statistical information measure has shown good performances for change detection. Indeed, the complex PolSAR images are well described by proba- bility density function (pdf) due to the random nature of speckle. Thus, the statistical information measure uses the local pdfs of the neighborhood of homologous pixels of the pair of data used for the change detection [12]. These pdfs have been chosen particularly to adequately model the statistics of the SAR data. Once the parameters of the dis- tributions are estimated, the information measure quantifies the distance between the two pdfs. The comparison can be performed using various information measures such as the mutual information [13], [14], the variational and mixed information [13], [15], and the stochastic distances such as the Kullback-Leibler [16], Rényi [17], Bhattacharyya [18], [19], and Hellinger distances [20]. We limit our focus to change detection based on a stochastic distance measure between two scenes acquired at different times.

Closed-form expressions of the distance measure of some known distributions exist for single-polarization SAR images.

Otherwise, certain approximations should be applied, like the kernel density estimation (KDE) [21] or the 1-D edge worth series expansions used to estimate the local pdfs [22].

Nascimento et al. [23] developed analytic expressions for the Kullback–Leibler, Rényi, Bhattacharyya, and Hellinger distances between univariate G⁰ distributions. Full PolSAR gives more scattering information than single polarization channel SAR data and then can be used to increase the quality of the change detection map. Stochastic distances between models for complex multilook PolSAR data require dealing with integrals whose domain is the set of all positive definite complex Hermitian matrices [24]. Unfortunately, it is not obvious to find closed-form expressions for all the stochastic distances. In this case, certain approximation techniques based on the numerical evaluation of the integral of the distance measure are commonly applied. The most popular method is the Monte-Carlo (MC) estimation technique [25]. It can estimate the stochastic distance provided that a large number of independent and identically distributed (i.i.d.) samples is provided. Nevertheless, the MC integration is a too slow process to be useful in change detection. The simplest way to have a closed-form expression for all the stochastic mea- sures is to use a simple model. The complex scaled Wishart distribution is widely used as a statistical simple model for low-resolution multilook PolSAR data due to its ease of imple- mentation and low computational cost [26]. Freryet al. [24]

obtained analytic expressions for the Kullback–Leibler, Rényi, Bhattacharyya, and Hellinger distances between scaled com- plex Wishart distributions in their most general form and in important particular cases.

However, one critical issue in applying Wishart distribution is that when the resolution increases, the homogeneous hypoth- esis of the PolSAR is not valid any more, and non-Gaussian statistics are observed. Consequently, many heterogeneous models have been proposed based on the scalar texture product model which is commonly assumed to be an appropriate statis- tical model for PolSAR data [27]. Based on the product model, the speckle is modeled with the complex scaled Wishart distribution, and the texture is modeled as gamma (γ), inverse gamma (γ⁻¹), Fisher (F), and generalized inverse Gaussian distributions (N⁻¹). The first two distributions depend on a single shape parameter, whereas the other ones depend on two shape parameters. The parameter estimation becomes more challenging when the number increases. With different texture distributions, the product model leads to different compound distributions given by Kd [28], G_d⁰ [29], [30], Kummer-Ud

[31], and Gd [30]. In general, unless the special function is complicated and the shape of the distribution is flexible, the better.

In this regard, theG_d⁰distribution has been extensively used by the community for its flexibility and has been reported excellent performance showing the suitability for describ- ing different kind of regions: homogeneous, heterogeneous, and extremely heterogeneous clutter with low computational costs. In addition, the G_d⁰ distribution is simpler than the Kd

distribution, which contains the modified Bessel function of the second kind. Recent works also validate its use as an

change detection [12], [23], and classification of PolSAR images [34], [35].

For these reasons and without losing generality, we assume that multilook PolSAR data follow a G_d⁰ distribution. We present analytic expressions for the Kullback–Leibler, Rényi, Bhattacharyya, and Hellinger distances between two matrix- variate G_d⁰ distributions in general forms and in particular cases. We believe that this is the first work that establishes the closed-form expressions for these stochastic distances. In order to apply these stochastic distance measures to change detection, the parameters of matrix-variate G_d⁰ distributions need to be estimated first. In particular, we estimate the equivalent number of looks (ENLs), the covariance matrix of the speckle component, and the texture distribution parameters by the polarimetric expectation–maximization (EM) estimator [36]. For that end, the first contribution of this article is to present closed-form expressions of the Kullback–Leibler, Rényi, Bhattacharyya, and Hellinger distances between two matrix-variateG_d⁰ distributions. The second one is to provide a numerical analysis of the closed-form expression and provid- ing benchmark resources. The last contribution is to present a parameter estimation procedure for compound distribution and specifically forG⁰_d distribution.

This article is organized as follows: Section II introduces the statistical product model for multilook PolSAR data and the matrix-variateG_d⁰distribution. The closed-form expressions for four stochastic distances betweenG_d⁰ distributions are derived in Section III. They are the Kullback–Leibler, Rényi of order β, Bhattacharyya, and Hellinger distances. Section IV presents the parameter estimation method needed for the computation of the previous stochastic distances. The polarimetric EM esti- mator is used for the multilook PolSAR data distribution. The content of Section V is related to the performance evaluation.

Simulated PolSAR data and real PolSAR data are used for evaluation. Finally, Section VI concludes this article.

II. PRODUCT MODEL FORMULTILOOKPOLARIMETRIC

RADARDATA

The backscattering of a full PolSAR system is characterized by the polarimetric scattering vector and defined as

s= [sHH, sHV, sVH, sVV]^T ∈C^d. (1) The elements represent the complex backscattering coefficients in the four combinations of the linear receive and transmit polarizations. [.]^T means transposition, and d = dim(s) is the vector dimension. The polarimetric scattering vector s represents a complex single-look. To reduce the speckle, a multilook operation is used by averaging the covariance matrix. The averaged multilook covariance matrix is obtained as follows:

C= 1 L

L

=1

ss^H , L≥d (2) whereL is the number of looks,(.)^H denotes the Hermitian operator, and C∈Ω₊ ⊂C^d×d is the multilook polarimetric

Ω

definite complex Hermitian matrices. According to the multi- look polarimetric product model, the covariance matrix C is considered as the product of a positive scalar texture random variable, denotedτ, and a random speckle matrix, denotedX, such that

C=τX. (3)

Matrix X follows a scaled complex Wishart distribution sW_d^C(L,Σ)with a pdf given by

fX(X) = L^Ld|X|^L−d

Γ_d(L)|Σ|^Letr(−LΣ⁻¹X) (4) where Σ is the covariance matrix of the speckle, etr(.) = exp(tr(.))is the exponential trace operator,|.|is the determi- nant operator, andΓ_d(L)is the multivariate gamma function of the complex kind defined as Γ_d(L) = π^d(d−1)/2

d−1 i=0 Γ (L −i), where Γ(L) is the standard Euler gamma function.

The pdf of Cusing the Bayes’ theorem becomes as follows:

f^C(C) =

+∞

0 fC|τ(C|τ)fτ(τ)dτ (5) where fC|τ(C|τ) is the pdf of C with a specific value of τ and is given by

fC|τ(C|τ) = L^Ld|C|^L−d Γ_d(L)|Σ|^L

1 τ^dLetr

−L τΣ⁻¹C

. (6)

Substituting (6) in (5), the pdf ofCis obtained by fC(C) =L^Ld|C|^L−d

Γ_d(L)|Σ|^L

+∞

0

1 τ^dLetr

−L τΣ⁻¹C

fτ(τ)dτ. (7) The variable texture τ follows a normalized inverse gamma distribution to the unit mean, denoted¯γ⁻¹(λ) as follows:

fτ(τ) =(λ−1)^λ Γ(λ)

1 τ^1+λ exp

−λ−1 τ

τ∈R+, λ>1. (8) As a consequence, the multilook polarimetric covariance matrix follows the matrix-variate G⁰_d distributions [30], denoted as G_d⁰(Σ, L,λ), and is given as follows:

f^C(C) = L^Ld|C|^L−d Γ_d(L)|Σ|^L

(λ−1)^λΓ(dL+λ)

Γ(λ)(Ltr(Σ⁻¹C) +λ−1)^dL+λ. (9) III. STOCHASTICDISTANCES

In this section, the closed-form expressions of the stochas- tic distances such as the Kullback–Leibler, Rényi of order β, Bhattacharyya, and Hellinger between two matrix-variate G_d⁰(Σ1, L1,λ1) andG_d⁰(Σ2, L2,λ2) distributions are derived.

The analytic expressions are available for different ENLs, various covariance matrices, and different texture parameters.

A particular case is examined when the ENL is the same for the two distributions. This case is likely to be the most frequently used in practice since it allows the comparison of two possibly different areas from the same image [24].

Let C¹ and C² be two random matrices that follow G_d⁰ dis- tribution with pdfsfC¹(C|Σ₁, L1,λ₁)andfC²(C|Σ₂, L2,λ₂)

A. Kullback–Leibler Distance

The Kullback–Leibler divergence [16] betweenC¹ andC² is given by

DKL(C¹||C²) =

Ω+

ln

fC¹(C) fC²(C)

fC¹(C)dC

=EC¹{lnfC¹(C)} −EC¹{lnfC²(C)}. (10) Jeffreys [37] used a symmetric version of (10) as a measure of divergence between two distributions. In this case, the measure is called “distance” and is given by

dKL(C¹,C²) =DKL(C¹||C²) +DKL(C²||C¹). (11) The expressions ofEC¹{lnfC¹(C)} andEC¹{lnfC²(C)} are given as follows:

EC¹{lnfC¹(C)}

= ln

Γ(dL1+λ1)(λ1−1)^λ1 Γ(λ1)Γ_d(L1)

L1Σ⁻¹₁ ^−L1

+ (L1−d)EC¹{ln|C|}

−(dL1+λ1)EC¹{ln[tr(L1Σ⁻¹₁ C) +λ1−1]} (12) and

EC¹{lnfC²(C)}

= ln Γ(dL2+λ₂)(λ₂−1)^λ2 Γ(λ2)Γ_d(L2)|L1Σ⁻¹₂ |^−L2

+ (L2−d)EC¹{ln|C|}

−(dL2+λ₂)EC¹

ln

tr

L2Σ⁻¹₂ C

+λ₂−1. (13) These expectations depend on EC¹{ln|C|}which is given as follows (see Appendix B-A for details):

EC¹{ln|C|}=ψd(L1)−dlnL1+ ln|Σ₁|

+dln(λ₁−1)−dψ(λ₁) (14) where function ψd(L) is the multivariate digamma function defined as ψd(L) = ∂ln Γ_d(L)/∂L =

d−1

j=0ψ(L−j), and ψ(.)is the digamma function. In fact, the last expectation can be computed in another way by using the product model. Then it is equivalent toEC¹{ln|C|}=EX¹{ln|X|}+ dEτ¹{lnτ}.

The second expectation EC¹{ln[tr(L1Σ⁻¹₁ C) +λ1−1]} is given as follows (see Appendix B-B for details):

EC¹{ln[tr(L1Σ⁻¹₁ C) +λ1−1]}

= ln(λ1−1)−ψ(λ1)+ψ(dL1+λ1). (15) Finally, to completely compute EC¹{lnfC²(C)}, the third expectation EC¹{ln[tr(L2Σ⁻¹₂ C) +λ2−1]} is needed and it is given as follows (see Appendix B-C for details):

EC¹

ln

tr

L2Σ⁻¹₂ C

+λ2−1

= ln(λ₂−1)−ψ(λ₁) +ψ(λ₁+dL1)−

d

i=1

Λ^−L_i ¹

×∂

∂aF_D^(d)

λ1+dL1, L1, . . . , L1

d

;

λ₁+dL1+a; 1− 1

Λ₁,1− 1

Λ₂, . . . ,1− 1 Λ_d

a=0

(16)

TABLE I

ANALYTICEXPRESSIONS OFKULLBACK–LEIBLER ANDRÉNYI OFORDERβDISTANCESBETWEENTWOMATRIX-VARIATE G⁰_d(Σ1, L1,λ1)^ANDG_d⁰(Σ2, L2,λ2)DISTRIBUTIONS

whereF_D^(d)(.) is the Lauricella D-hypergeometric series ofd variables (see Appendix G for more details), andΛ₁, . . . ,Λ_d are the eigenvalues of the positive definite complex Hermitian matrix Λ= [L2(λ₁−1)/L1(λ₂−1)]Σ⁻¹₂ Σ₁.

The analytic expressions for the stochastic divergence DKL(C¹||C²) and distance dKL(C¹,C²) between G_d⁰(Σ1, L1,λ1) and G⁰_d(Σ2, L2,λ2) distributions are given, respectively, by (20) and (21) in Table I.

1) Case L¹ = L² = L: The positive definite complex Hermitian matrix becomes Λ = (λ1−1)(λ2−1)⁻¹Σ⁻¹₂ Σ1

and the symmetric Kullback–Leibler divergence is as follows:

dKL(C¹,C²)

= (λ −λ (λ − (λ λ − λ

−(dL+λ2)

d

i=1

Λ^−L_i

∂

∂aF_D^(d)

λ1+dL, L, . . . , L; λ₁+dL+a; 1−Λ⁻¹₁ , . . . ,1−Λ⁻¹_d

a=0

−(dL+λ1)

d

i=1

Λ^L_i

∂

∂aF_D^(d)(λ2+dL, L, . . . , L; λ2+dL+a; 1−Λ₁, . . . ,1−Λ_d)

a=0. (17) 2) CaseL¹=L²=LandΣ1=Σ2: The positive definite complex Hermitian matrix becomesΛ= (λ1−1)(λ2−1)⁻¹I_d and the Lauricella function becomes equivalent to the Gauss hypergeometric function as follows:

F_D^(d)(λ1+dL, L, . . . , L;λ1+dL+a; 1−Λ⁻¹, . . . ,1−Λ⁻¹)

(λ λ − ⁻¹

where Λ = (λ1 − 1)(λ2 − 1)⁻¹. As a consequence, the Kullback–Leibler distance is as follows:

dKL(C¹,C²)

= (λ1−λ2)[ψ(λ1)−ψ(λ2) +ψ(dL+λ2)−ψ(dL+λ1)]

−(dL+λ2)Λ^−dL ∂

∂a ²F1(λ1+dL, dL; λ1+dL+a; 1−Λ⁻¹)

a=0− (dL+λ1)Λ^dL

∂

∂a ²F1(λ2+dL, dL;λ2+dL+a; 1−Λ)

a=0.

(19) B. Rényi Distance of Orderβ

Rényi [17] presented the first parametric generalization of (10) given by

D^β_R(C¹||C²) = 1 β−1ln

Ω+

fC^β1(C)fC^1−β2 (C)dC (25)

= 1

β−1lnEC¹

fC¹(C) fC²(C)

β−1

(26) where0< β <1. The Rényi distance is given by

d^β_R(C¹,C²) = 1 2

D^β_R(C¹||C²) +D^β_R(C²||C¹)

. (27) It is clear that the KL divergence is a particular case of the Rényi divergence when β → 1. The analytic expres- sion for the stochastic distance D^β_R(C¹||C²) between two matrix-variate G⁰_d(Σ1, L1,λ1) and G_d⁰(Σ2, L2,λ2) distribu- tions is given by (22) in Table I. The computation details are given in Appendix C. The symmetric expression of the Rényi divergence is then given by (23). Using pro- priety (116), d^β_R(C¹,C²) can also be written otherwise by (24).

1) CaseL¹=L²=L: The distance,d^β_R(C¹,C²), given by (23) becomes in this case as follows:

d^β_R(C¹,C²)

= 1

2(β−1)

lnΓ(dL+λ₁)Γ(dL+λ₂) Γ(λ1)Γ(λ2) − L1

d i=1

ln Λ_i + ln Γ(β(λ₁−λ₂) +λ₂)Γ(β(λ₂−λ₁) +λ₁)

Γ (β(λ1−λ2) +dL+λ2) Γ (β(λ2−λ1) +dL+λ1) + lnF_D^(d)

β(dL+λ1), L, . . . , L;β(λ1−λ2) +dL+λ2; 1−Λ⁻¹₁ , . . . ,1−Λ⁻¹_d

+ lnF_D^(d)

(1−β)(dL+λ1), L, . . . , L;β(λ2−λ1) +dL+λ₁; 1−Λ⁻¹₁ , . . . ,1−Λ⁻¹_d

. (28)

C. Hellinger Distance

The Hellinger distance [38] is defined as follows:

dH(C¹,C²) = 1−

Ω+

fC¹(C)fC²(C)dC (29)

= 1−exp

−1

2d^1/2_R (C¹,C²)

(30) with

Ω+

fC¹(C)fC²(C)dC

=

Γ(dL1+λ1)Γ(dL2+λ2) Γ_d(L1)Γ_d(L2)Γ(λ1)Γ(λ2)

1 2

× Γ_d(L1/2 +L2/2)Γ(λ₁/2 +λ₂/2) Γ(dL1/2 +dL2/2 +λ₁/2 +λ₂/2)

d

i=1

Λ⁻

L1

i 2

×F_D^(d)

dL1+λ1

2 ,L1+L2

2 , . . . ,L1+L2

2 ; d(L1+L2) +λ1+λ2

2 ;

1− 1

Λ₁,1− 1

Λ₂, . . . ,1− 1 Λ_d

. (31)

The details of this derivation are given in Appendix D. This expression can be written otherwise by using the following relation deduced from (116):

d

i=1

Λ⁻

L1 2

i F_D^(d)

dL1+λ₁

2 ,L1+L2

2 , . . . ,L1+L2

2 ; d(L1+L2) +λ1+λ2

2 ; 1− 1

Λ₁,1− 1

Λ₂, . . . ,1− 1 Λ_d

=

d

i=1

Λ

L2

i2 F_D^(d)

dL2+λ₂

2 ,L1+L2

2 , . . . ,L1+L2

2 ; d(L1+L2) +λ1+λ2

2 ; 1−Λ₁,1−Λ₂, . . . ,1−Λd

. (32) As a result, both expressions (39) and (40) can be estab- lished for the Hellinger distance between two matrix-variate G_d⁰(Σ1, L1,λ1)andG_d⁰(Σ2, L2,λ2)distributions. The analytic expressions are given in Table II.

1) CaseL¹ =L² =L: The symmetric Hellinger distance dH(C¹,C²)given by (39) becomes as follows:

dH(C¹,C²)

= 1−

Γ(dL+λ₁)Γ(dL+λ₂) Γ(λ1)Γ(λ2)

1

2 Γ

λ1

2 +^λ₂²

Γ

dL+^λ₂¹ +^λ₂²

×

d

i=1

Λ⁻_i ^L2

F_D^(d)

dL+λ₁

2 , L, . . . , L

d

;dL+λ₁+λ₂ 2 ; 1− 1

Λ₁,1− 1

Λ₂, . . . ,1− 1 Λ_d

. (33) D. Bhattacharyya Distance

The Bhattacharyya distance measure [18] is given by dB(C¹,C²) =−ln

Ω+

fC¹(C)fC²(C)dC (34)

= 1

2d^1/2_R (C¹,C²). (35) The developed expression of dB(C¹,C²) is given by (37) in Table II. Using relation (32) deduced from (116), one can deduce a second expression of the Bhattacharyya distance