SAR Image Despeckling Based on Combination of Fractional … · 2019. 12. 27. · SAR Image...

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IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING 1

SAR Image Despeckling Based on Combination ofFractional-Order Total Variation and Nonlocal

Low Rank RegularizationGao Chen , Gang Li, Senior Member, IEEE, Yu Liu , Xiao-Ping Zhang , Senior Member, IEEE, and Li Zhang

Abstract— Regularization method is an effective tool forsynthetic aperture radar (SAR) image despeckling. Design of theeffective regularization terms describing the image priors playsa vital role in this kind of method. In this article, a new combi-national regularization model for speckle reduction (CRM-SR) isproposed, in which a regularization term is elaborately designedto contain both a fractional-order total variation (FrTV) regular-ization and a nonlocal low rank (NLR) regularization. The newregularization model inherits both the advantages of FrTV andNLR and improves the performance of SAR despeckling and,therefore, better preserves the edges and geometrical features ofthe images during the despeckling process. An efficient algorithmbased on alternating direction optimization is derived to solvethe proposed combinational regularization model. Experimentalresults show that the proposed model can effectively remove SARimage speckle and preserve the geometrical features of imagesaccording to both subjective visual assessment of image qualityand objective evaluation.

Index Terms— Alternating direction, fractional-order totalvariation (FrTV), nonlocal low rank (NLR), synthetic apertureradar (SAR) image despeckling.

I. INTRODUCTION

SYNTHETIC aperture radar (SAR) has been widely usedin various remote sensing applications, such as oil spill

surveillance [1] and topographic mapping [2]. Owing to thecoherent interference of waves reflected from many elementaryscatterers, the raw SAR images suffer from speckle, whichhampers the analysis and interpretation of SAR images and

Manuscript received April 29, 2017; revised December 5, 2017,January 5, 2019, and September 5, 2019; accepted November 6, 2019. Thiswork was supported in part by the National Natural Science Foundation ofChina under Grant 61790551 and in part by the Civil Space Advance ResearchProgram of China under Grant D010305. (Corresponding author: Gang Li.)

G. Chen is with the School of Electrical Engineering and Intelligentization,Dongguan University of Technology, Dongguan 523808, China, and alsowith the Department of Electronic Engineering, Tsinghua University, Beijing10084, China (e-mail: [email protected]).

G. Li and L. Zhang are with the Department of Electronic Engineer-ing, Tsinghua University, Beijing 100084, China (e-mail: [email protected]).

Y. Liu is with the Research Institute of Information Fusion, Naval Aero-nautical and Astronautical University, Yantai 264001, China, and also withthe School of Electronic and Information Engineering, Beihang University,Beijing 100083, China.

X.-P. Zhang is with the Department of Electrical and Computer Engineering,Ryerson University, Toronto, ON M5B 2K3 Canada.

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

Digital Object Identifier 10.1109/TGRS.2019.2952662

reduces the accuracy of target detection and classification[3]–[5]. Therefore, developing efficient despeckling algorithmsis an essential procedure before using SAR images to obtainland-cover information [6].

In the 1980s, SAR despeckling was often achieved byspatial domain adaptive filtering, such as the Lee filter [7],Kuan filter [8], and the Frost filter [9]. These filters areeasy to implement and have low computational complexity,but they show limitations on either preserving image sharp-ness or suppressing speckle [10]. With the development ofdespeckling techniques, many transform domain techniqueshave been widely [11]–[16] applied to remove the specklenoise of SAR images, such as wavelet shrinkage [11], [12],principal component analysis [13], [14], and sparse represen-tation [15], [16]. These transform domain filtering methodscan deliver better performance than spatial domain filteringmethods in terms of edge preservation, but they generate someartifacts [17], [18].

Besides the filter-based techniques in the spatial domain andin the transform domain, the regularization method is anotherimportant branch of SAR image despeckling. The problemof SAR image despeckling is converted into an optimiza-tion problem that usually consists of a fidelity term, whichdescribes the difference between the noisy image and thedesired image, and a regularization term, which includes priorinformation about the original image [6]. In this framework,image prior knowledge plays a vital role. Therefore, designingeffective regularization terms to describe the image priors isat the core of SAR image despeckling.

Edge is a frequently used feature of SAR images, and itconsists of the pixels that are significantly different from theirneighboring pixels. The total variation (TV)-based regulariza-tion model, proposed by Rudin et al. [19], can effectivelyreflect the edge feature. By using the maximum a posteriori(MAP) estimator for multiplicative gamma noise, Aubert andAujol (AA) [20] proposed a nonconvex AA model for SARimage despeckling, which consists of a nonconvex fidelityterm and a TV regularization term. Although the AA modelshows good properties in theory, the optimization problem inthis model is difficult to solve due to the nonconvexity ofthe cost function. By using the logarithmic transformation,Shi and Osher (SO) [21] proposed a strictly convex SO model,which is easily solved as demonstrated in [22]–[24]. It hasbeen shown that the SO model has better performances than

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2 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING

the AA model, but its capability of preserving image featuresis still unsatisfactory [6]. In addition, the TV regularizedterm often favors piecewise constant solution and, therefore,fails to preserve the textures and often causes a staircaseeffect [25], [26]. To overcome the staircase effect, the totalgeneralized variation (TGV) model combining the first andhigher order TV functions was proposed in [27]. In orderto improve the ability of texture preservation and reduce thestaircase effect, a class of nonlocal fractional-order differentialtechniques has been widely used in recent years. Unlike theinteger order derivative, the fractional order derivative takesmore neighboring pixel information into account [28]. Theexperimental results in [29] and [30] have demonstrated thatthe fractional-order derivative performs well in eliminatingstaircase effect and preserving textures. In [25], a reweightedresidual-feedback iterative algorithm, which provides a generalframework to solve the fractional-order total variation (FrTV)regularized models with different fidelity terms, is proposed tosuppress multiplicative noise of SAR images, and the exper-imental results show that it works well for preserving detailsand eliminating the staircase effect. However, as shown in [16],some point-like and fine structures are still not accuratelyrecovered by the FrTV regularization algorithms.

Besides edge feature, nonlocal self-similarity is anotherimportant property of SAR images. It characterizes the repet-itiveness of the textures or structures embodied by SARimages within nonlocal area. The nonlocal means (NLM)[16], [31], [32] is an effective method that exploits imageself-similarity for image denoising. It calculates the similar-ity between the patches surrounding the selected pixels andthen performs weighted averaging in a nonlocal region. Thisweighted approach is quite effective in generating sharperimage edges and preserving more image details and finestructures. Inspired by the success of NLM denoising filter,a series of nonlocal regularization terms for SAR imagedespeckling were proposed by exploiting the nonlocal self-similarity property of SAR images, such as the probabilisticpatch-based (PPB) algorithm [33], the SAR-oriented versionof block-matching 3-D (SAR-BM3-D) algorithm [12], andthe nonlocal framework for SAR denoising (NL-SAR) [34].All these methods adopt the NLM framework but withdifferent filtering modules. Owing to the exploitation of self-similarity prior of SAR images, nonlocal regularization algo-rithms have better despeckling performance than the localregularization algorithms, with sharper image edges and moreimage details [35]. However, the nonlocal filtering methodsalso present some artifacts in the form of structured signal-likepatches in flat areas, which are originated from the randomnessof speckle and reinforced through the patch selection process,as pointed out in [36].

In order to preserve both the edge and nonlocal self-similarity features of SAR images, variational and sparserepresentations have been combined by defining jointregularization terms [32], [37]–[40]. Duran, Fadili, andNikolova (DFN) [37] adopted the sparsity prior of the imagein the curvelet transformed domain and recovered the framecoefficients via a TV regularized formulation. It achievedgood despeckling performance, but some artifacts were

also generated, as shown in [40]. In [38]–[40], dictionarylearning techniques were introduced to the TV regularizedmodel for multiplicative noise removal. These methods canpreserve details while reducing multiplicative noise, but theyalso generate some point-wise artifacts, as shown in thesimulations of this article.

In this article, by characterizing the edge feature and thenonlocal self-similarity property of SAR images simultane-ously, we propose a new combinational regularization modelfor speckle reduction (CRM-SR) consisting of the FrTV andthe nonlocal low rank (NLR) regularization terms. The FrTVregularization term is employed to better preserve edges andtexture details of SAR images, whereas the NLR regularizationterm, constructed based on a low rank constraint of thematrix generated by stacking similar SAR image patches,is employed to preserve more geometrical structures of SARimages. Besides, an alternating direction method (ADM) isalso derived to efficiently solve the optimization problem inthe proposed model. Experimental results demonstrate thesatisfactory performances of the proposed model, in terms ofremoving SAR image speckles and preserving image textureand details. Part of the ideas and experimental results has beenpublished in our recent conference paper [41]. The authors arenot aware of any other publications that attempt to apply thecombination of FrTV and NLR regularization to multiplicativenoise removal of SAR images. There are significant differencesbetween our work and existing joint regularization models.

1) Existing joint regularization models employ TV regu-larization to describe the edge feature of SAR images,while CRM-SR takes the advantages of FrTV to betterpreserve edge and detail information.

2) Existing joint regularization models employ the sparsityin the transformed domain to describe the self-similarityproperty of the SAR images, while CRM-SR employsthe low rank regularization to characterize the self-similarity property. As shown in [42], low rank regu-larization performs better in image denoising since it iscapable of recovering the principal components from thecontaminative data. Thus, CRM-SR has better denois-ing performance than the existing joint regularizationmodels.

The rest of this article is organized as follows. Section IIelaborates the design of the combinational regularizationmodel for SAR image speckle reduction. Section III proposesthe ADM algorithm for solving the optimization problemin the proposed model. Experiment results are presented inSection IV. The concluding remarks are given in Section V.

II. COMBINATIONAL REGULARIZATION MODEL

FOR SAR IMAGE DESPECKLING

It is well established that the speckle in SAR images canbe characterized by the multiplicative noise model [43]

f = un (1)

where f is the noisy image, u is the noise-free SAR image,and n is the speckle noise. The goal of despeckling isto recover u from f . Similar to the SO model, here themultiplicative model is transformed into the additive model


CHEN et al.: SAR IMAGE DESPECKLING BASED ON COMBINATION OF FrTV AND NLR REGULARIZATION 3

by logarithmic transformation w = log u, and the problem ofdespeckling is converted into recovering w = log u from thenoisy observation log f .

In the framework of regularization method, the problemof SAR image despeckling is converted into an optimizationproblem that usually consists of a fidelity term and a regular-ization term

minw

[H(w, f ) + ρ�(w)] (2)

where H(w, f ) is the data-fitting term depending on the noisyimage and the desired clean image, �(w) is the regularizationterm which includes prior information about the originalimage, ρ is the regularization parameter.

Designing effective regularization term �(w) to describethe image priors is a key factor that affects the quality ofthe recovered image. In the proposed new model CRM-SR,a combinational regularization term is elaborately designedwhich reflects the edge feature and the nonlocal self-similarityof the SAR image simultaneously, it is formulated as follows:

�(w) = �S(w) + κ�NLS(w) (3)

where �S(w) and �NLS(w) indicate the image edge featureand nonlocal self-similarity prior information, respectively,κ is the regularization parameter, which controls the trade-offbetween the two regularization terms.

More details on how to design �S(w) and �NLS(w) tocharacterize the edge feature and the nonlocal self-similarityproperties of SAR images are given in Sections II-A and II-B,respectively.

A. FrTV Model for Edge Feature of Logarithmic SAR Image

The TV regularization model can effectively describe edgefeatures in images. However, as shown in [29] and [30],the images recovered by the TV model will lose some texturedetails in those areas where gray-level values do not changeevidently. In order to preserve more image edge and texturedetails, the FrTV model [30] is exploited to describe the edgefeature of the logarithmic SAR image.

A fractional-order gradient can be viewed as a generaliza-tion of the integer-order gradient composed of the fractional-order derivative of a different direction. Without loss ofgenerality, let wi, j denote the pixel in the i th row and thej th column of a N × N logarithmic image w, the fractional-order discrete gradient transform ∇α : C N×N → C N×N×2

is defined in terms of the fractional-order derivatives in thedirections of the columns and rows

[∇αw]i, j = ((wαh

)i, j

,(wα

v

)i, j

)(4)

where ⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

(wα

h

)i, j

=K−1∑k=0

E (α)k w(i − k, j)

(wα

v

)i, j =

K−1∑k=0

E (α)k w(i, j − k)

(5)

and

E (α)k = (−1)k �(α + 1)

�(k + 1)�(α − k + 1)(6)

where α is the fractional order of the FrTV norm, �(x) is theGamma function, K ≥ 3 is the number of terms involved inthe computation of the fractional-order derivative. As shownin [30], more number of terms involving in the computationof the fractional-order derivative will lead to a higher com-putational cost, but it has negligible influence on the valuesof FrTV. In this article, we set K = 10 by considering thetradeoff of the performance and the computational cost. TheFrTV model used to characterize the edge feature of the SARimage is formulated as follows:

�S(w)=‖w‖FrTV =N∑

i=1

N∑j=1

√[(wα

h

)i, j

]2 + [(wαv

)i, j

]2. (7)

B. NLR Model for Self-Similarity Propertyof Logarithmic SAR Image

Different from the dictionary learning methods, an NLRmodel is designed to characterize the nonlocal self-similarityproperty of logarithmic SAR image, by means of the low rankproperty of the matrix which is obtained by vectorizing andstacking similar image patches.

Specifically, the implementation details of the design of theNLR model for self-similarity property of logarithmic SARimage are given as follows.

1) The image w (of size N × N) is divided into Poverlapped blocks wl with the same size of n × n,l = 1, . . . , P .

2) For each block denoted by wl , find the similar patchesin a searching window. Then the set that includes cbest matched blocks to Swl in the searching windowis defined as Swl = {wl1 ,wl2 , . . . ,wlc }. For each Swl ,the best matched blocks belonging to Swl are vector-ized and stacked into a matrix, which is denoted asDwl ∈ Rn2×c. Since the matrix Dwl contains the samestructural information, it is a low rank matrix.

It is important to note that the criterion for calculat-ing similarity between different blocks is crucial for blockmatching. Many works [12], [33], [34] commonly use theEuclidean distance as the similarity measurement. However,as pointed out in [44], the commonly used Euclidean dis-tance metric assumes that each feature of data point isequally important and independent of others. This assumptionmay not be always satisfied in real SAR images. Besides,as shown in [6], the statistical properties of speckle in realSAR images do not follow a Gaussian distribution, especiallywhen the noise level is high. The Mahalanobis distance hasbeen proven to be a preferable choice for block similaritymatching for SAR images in [42]. Therefore, in this arti-cle, we adopt the Mahalanobis distance as the similaritymeasurement.

1) The singular values of each matrix Dwl are obtainedby applying a singular value decomposition (SVD) tothe matrix, i.e., σwl = SV D(Dwl ). Then a vector isformed by arranging the singular values of all the imageblocks in a lexicographic order, which is denoted as �w.Since the matrix Dwl has a low-rank property, �w is



a sparse vector. The nuclear norm (sum of the singularvalues) is used to measure the low-rank property

�NLS(w) = ‖�w‖0 =P∑

l=1

‖Dwl ‖∗ =P∑

l=1

‖Gl(w)‖∗ (8)

where ‖ · ‖∗ denotes the nuclear norm

Gl(w) = [R1l vec(w), . . . , Rc

l vec(w)]

(9)

with the elements

[Gl(w)]i, j =N2∑

k=1

(R j

l

)i,k [vec(w)]k

=N∑

ii=1

N∑j j=1

(R j

l

)i,( j j−1)N+ii wii, j j (10)

and R jl ∈ Rn2×N2

is the matrix operator that extracts thej th block from the image w.

C. Proposed Combinational Regularization Model

The proposed new CRM-SR model is defined by combiningboth the FrTV constraint, which depicts the edge feature, andthe NLR regularization constraint, which depicts the nonlocalself-similarity prior. With the replacement of the regular-ization term �(w) in (2) with the proposed combinationalregularization model (3), a new formulation for SAR imagedespeckling can be expressed as follows:

�w = min

w

⎡⎣ N∑

i=1

N∑j=1

(wi, j + f i, j e−wi, j )

+ λ‖w‖FrTV + ξ

P∑l=1

‖Gl(w)‖∗

](11)

where λ and ξ are regularization parameters. Comparing (11)with (2) and (3) we have λ = ρ, ξ = ρκ . In Section IV, we willshow the influence of the values of λ and ξ to the despecklingperformance. Once the optimal

�w is obtained, the recovered

image�u can be easily calculated by

�u = exp(

�w). In (11),

the first term∑N

i=1∑N

j=1 (wi, j + f i, j e−wi, j ) represents thedata-fitting constraint [20], and the second term λ‖w‖FrTV andthe third term ξ

∑Pl=1 ‖Gl(w)‖∗ indicate the edge feature and

the nonlocal self-similarity prior information, respectively.

III. ADM FOR SAR IMAGE DESPECKLING

USING CRM-SR

The proposed model in (11) is nonsmooth because of thenuclear norm and the FrTV regularization term. In this section,we propose an effective algorithm to solve this optimizationproblem. First, a variable splitting technique is employed toconstruct a decomposable structure in the objective function,leading to the formulation of an equivalent optimization prob-lem. Then the ADM framework [45] is applied to solve thisequivalent problem.

With the use of the variable splitting technique, (11) can beconverted to the following constraint optimization problem:

�w = arg min

w

⎡⎣ N∑

i=1

N∑j=1


+ξ

P∑l=1

‖Zl‖∗ + λ‖Z P+1‖FrTV

]

s.t. Zl = Gl(w)(l = 1, . . . , P)

ZP+1 = w (12)

where Zl(l = 1, . . . , P + 1) are the auxiliary variables thatmake the objective function separable with respect to thevariables Zl and w.

ADM can be viewed as an attempt to combine the decom-posable benefit of dual-ascent and the superior convergenceproperty of the augmented Lagrangian methods for constrainedoptimization [40]. With two penalty terms and two extraquadratic terms related to the constraints added, the augmentedLagrangian function in (12) can be formulated as follows:

L A(w, Zl, Y l)

=N∑

i=1

N∑j=1

(wi, j + f i, j e−wi, j ) + ξ

P∑l=1

‖Zl‖∗ + λ‖Z P+1‖FrTV

+P∑

l=1

[〈Y l , Zl − Gl(w)〉 + β

2‖Zl − Gl(w)‖2

F

]

+ 〈Y P+1, ZP+1 − w〉 + β

2‖ZP+1 − w‖2

F (13)

where Y l(l = 1, . . . , P + 1) are the Lagrangian multipliers,β is the penalty coefficient. ADM is employed to alternatelyupdate each of the variables {w, Zl , Y l}, while keeping theothers fixed. In the k-th iteration, the following steps arecarried out:w(k+1) = arg min

wL A(w, Z(k)

l , Y (k)l

)(14)

Z(k+1)l = arg min

Zl

L A(w(k+1), Zl , Y (k)

l

)l =1,. . .P+1 (15)

Y (k+1)l = Y (k)

l +γβ[Z(k+1)

l −Gl(w(k+1))

]l = 1, . . . P (16)

Y (k+1)P+1 = Y (k)

P+1 + γβ[Z(k+1)

P+1 − w(k+1)]. (17)

The ADM iterations (14)–(17) are performed until themaximum number of iterations is reached. In what follows,we elaborate how to solve (14)–(17).

A. w Subproblem in (14)

For w optimization subproblem, it can be reformulated asfollows:

w(k+1) = arg minw

⎡⎣ N∑

i=1

N∑j=1


+P∑

l=1

β

2

∥∥∥∥∥Z(k)l − Gl(w) + Y (k)

l

β

∥∥∥∥∥2

F

+ β

2

∥∥∥∥∥Z(k)P+1 − w + Y (k)

P+1

β

∥∥∥∥∥2

F

⎤⎦ . (18)



This convex quadratic problem can be solved by

[F(w)]i, j = ∂L A(Zl ,w)

∂wi, j=1− f i, j e−wi, j − Ai, j − Bi, j = 0

(19)

where the value of Ai, j is given in (20) shown at the bottomof this page, and

Bi, j = β

⎡⎣(Z(k)

P+1

)i, j − wi, j +

(Y (k)

P+1

β

)i, j

⎤⎦ . (21)

Here the Newton iteration method [46] is employed toefficiently solve the nonlinear equation (19). Given asingle-variable nonlinear equation F(x) = 0, the iterationscheme of the Newton method is given by

xn+1 = xn − F(xn)

F ′(xn)(22)

where F ′(x) is the derivative of F(x). Hence, the iterativesolution of (19) is given by

(wi, j )n+1

= (wi, j )n

− 1 − f i, j e−(wi, j )n − Ai, j − Bi, j

f i, j e−(wi, j )n +P∑

l=1

n2∑ii=1

c∑j j=1

β((

R j jl

)ii,( j−1)N+i

)2 + β

.

(23)

B. Zl(l = 1, . . . , P) Subproblem in (15)

For Zl(l = 1, . . . , P) optimization subproblem, it can bereformulated as follows:

Z(k+1)l

= arg minZl

[ξ‖Zl‖∗ + ⟨Y (k)

l , Zl − Gl(w(k+1))

⟩+ β

2‖Zl − Gl(w

(k+1))‖2F

]

= arg minZl

⎡⎣ξ‖Zl‖∗ + β

2

∥∥∥∥∥Zl − Gl(w(k+1)) + Y (k)

l

β

∥∥∥∥∥2

F

⎤⎦

= arg minZl

⎡⎣‖Zl‖∗ + β

2ξ

∥∥∥∥∥Zl −(

Gl(w(k+1)) − Y (k)

l

β

)∥∥∥∥∥2

F

⎤⎦ .

(24)

This is a nuclear norm-regularized least squares problem,which can be solved by the singular value thresholding(SVT) [47]

Z(k+1)l = Dξ/β

(Gl(w

(k+1)) − Y (k)l

β

)(25)

where Dτ (·) denotes the SVT operator

Dτ (X)U · diag(max(σ − τ, 0)) · V T . (26)

X = U · diag(σ ) · V T is the SVD of the input matrix, σ isthe singular value of the input matrix and τ is a thresholdvalue.

C. Z p+1 Subproblem in (15)

For ZP+1 optimization subproblem, it can be reformulatedas follows:

Z(k+1)P = arg min

ZP

[‖ZP+1‖FrTV + β

2λ‖ZP+1

−(

w(k+1) − Y (k)P+1

β

)∥∥∥∥∥2

F

⎤⎦ . (27)

This is a FrTV-regularized least squares problem, whichcan be solved by the majorization–minimization (MM)algorithm [26].

After updating w(k+1) and Z(k+1)l (l = 1, . . . , P + 1),

Y (k+1)l (l = 1, . . . , P + 1) can be easily calculated according

to (16) and (17).

Algorithm 1 ADM Iterative Algorithm for Solving CRM-SRModel

� Input: the noisy SAR image f , the parameters α, β, ξ , λ,γ , No

� Initialization: set k = 0, w(k) = log( f ) and Y (k), k (Z =1, . . . , P + 1) as zero matrix

� While k < N0 do• Block matching and compute Gl(w(k))(l = 1, · · · , P)

• Compute w(k+1)i, j using Newton iteration method based

on (23)• Compute Z(k+1)

l (l = 1, . . . , P) based on (25)• Compute Z(k+1)

P+1 using MM algorithm based on (27)• Compute Y(k+1)

P+1 based on (16) and (17)• k = k + 1

� End while� Output

�u = exp(w(k))

In summary, the iterative algorithm for solving the problemin (11) is described in Algorithm 1. Since the proposedalgorithm is derived based on the ADM framework, from theconvergence analysis of ADM in [45] it can be concluded thatthe proposed algorithm is convergent.

IV. EXPERIMENTAL RESULTS

In this section, some experimental results based on sim-ulated and real SAR images are provided to demonstratethe performance of the proposed model and algorithm.

Ai, j =P∑

l=1

n2∑ii=1

c∑j j=1

β

⎛⎝(

z(k)i

)ii, j j −

N∑i=1

N∑j=1

(R j j

i

)ii,(t−1)N+s

wt,s +(Y (k)

i

)ii, j j

β

⎞⎠(R j j

i

)ii,( j−1)N+i (20)



Fig. 1. Test images used in the experiments. The pixels in the white box are used for ENL estimation, and the pixels in the blue box are used for TCRestimation. The ENL values of real SAR image 1 and real SAR image 2 are 29.37 and 18.55, respectively. The TCR values of real SAR image 1 and realSAR image 2 are 7.16 and 8.25 dB, respectively.

The following classical and state-of-the-art algorithms areselected as the comparison:

1) AA [20], a classical multiplicative noise removal algo-rithm based on TV regularization;

2) SAR-BM3-D [12], a state-of-the-art algorithm of SARimage despeckling based on the nonlocal sparse regu-larization;

3) NL-SAR [34], a resolution-preserving algorithm basedon the nonlocal-means regularization;

4) DFN [37], a classical multiplicative noise removal algo-rithm based on the combination of the curvelet sparseregularization and the TV regularization;

5) Multiplicative noise removal using analysis dictionarylearning and a smoothness regularizer (MNR-ADL-SR)[40], a recent state-of-the-art multiplicative noiseremoval algorithm based on the combination of dictio-nary learning and smoothness regularization.

For these algorithms, we used the default parameter setssuggested in their respective papers. In order to show thebenefit of employing FrTV, we also compare the proposedCRM-SR algorithm with the same algorithm framework butusing the regular TV instead of FrTV, this algorithm is referredto as total variation and nonlocal low rank (TVNLR).

We use two simulated optical images (Lena and Barbara)with size 512 × 512 and two single-look Ku-band SAR imagesin amplitude format taken over Luoyang in China, and theresolution of the two real SAR images is 0.3 m × 0.3 m(along-track and cross-track). These test images are shownin Fig. 1 and the detailed information of the real SAR imagesis summarized in Table I.

All experiments are performed in MATLAB 2016a environ-ment on a computer with Intel(R) Core (TM) processor, 16Gmemory, and under Microsoft Windows 7 operating system.

TABLE I

DETAILED INFORMATION OF THE REAL SAR IMAGESUSED IN THE EXPERIMENTS

The size of each block is set to be 8 × 8 with 4-pixel-widthbetween the adjacent blocks, and the size of window forsearching matched blocks is set to be 40 × 40.

A. Performance Metrics

Two quality measures are taken to evaluate the performanceof the synthetic optical images. The first one is the peak signal-to-noise ratio (PSNR) [48], which is useful to obtain meansquare error (MSE) performance assessments on the wholeimage. It is calculated by

PSNR(d B) = 10 log10R2

MSE(28)

with

MSE =N∑

i=1

N∑j=1

(�ui, j − ui, j )

2 (29)



TABLE II

MEAN VALUES AND THE STANDARD DEVIATIONS THEENL OF THE ORIGINAL REAL SAR IMAGES

where u is the original clean image and�u is the recovered

image, respectively. R > 0 is the maximum value of the imagegray level range.

The second one is the structural similarity index(SSIM) [48], which measures the quality of structural infor-mation of the recovered image

�u in comparison to the original

clean image u. It is defined as follows:

SSIM(u,�u) = (2μXμY + C1)(2σXY + C2)(

μ2X + μ2

Y + C1)(

σ 2X + σ 2

Y + C2) (30)

where μX and μY are the mean values of image u and�u,

σX and σY are the standard deviations of image u and u,respectively, the constant C1 and C2 are included to avoidinstability when μ2

X +μ2Y and σ 2

X +σ 2Y are very close to zero,

and

σXY =(

1

N − 1

)2 N∑i=1

N∑j=1

(ui, j − μX)(�ui, j − μY ). (31)

For real SAR images, due to lacking of ground truth, thereis no way to accurately calculate PSNR and SSIM. Thus otherfour quality metrics are used to evaluate the performance ofimage despeckling. The first one is the equivalent number oflooks (ENL) [49], which can assess the speckle removingperformance in homogeneous (smooth) regions. For a givenhomogeneous region X reg, the ENL is defined as follows:

ENL =[

μXreg

σXreg

]2

(32)

where μXreg and σXreg denote the mean and the standarddeviation of the pixel values in the region X reg. In Fig. 1,two homogeneous areas in the white rectangles are selectedto calculate the ENL. Table II gives the corresponding meanand standard deviations collected from these regions as wellas ENL values for both SAR images.

The second performance metric is the edge saving index(ESI) [50], which is used to evaluate the edge retainingcapability of the methods of SAR image despeckling. It isdefined as follows:

ESI =∑N−1

i=1∑N−1

j=1

√(ui, j − ui+1, j )2 + (ui, j − ui, j+1)2

∑N−1i=1

∑N−1j=1

√( f i, j − f i+1, j )

2 + ( f i, j − f i, j+1)2.

(33)

Larger ESI values indicate stronger edge retaining capability.The third metric is the detail preservation indexes

(DPIs) [51], which can assess the detail-preservation capacity

Fig. 2. PSNRs of the image Barbara with different values of the regularizationparameters.

of the methods. DPIs include DPI_M and DPI_V that aredefined as follows:⎧⎪⎪⎨

⎪⎪⎩DPI_M = mean

(f (D)�u(D)

)

DPI_V = var

(f (D)�u(D)

) (34)

where D is the region to be preserved and can be obtainedaccording to [51]. f (D) and

�u(D) denote the pixel valuesin region D of f and

�u, respectively. If DPI_V is small and

DPI_M is close to one, the details are retained well.The fourth metric is the target-to-clutter ratio (TCR) [52],

which indicates how much a despeckling algorithm preservesthe point-like structures in the patch. It is defined as follows:

TCR(dB) = 20 log10max(Ro)

mean(Ro)(35)

where Ro is the image patch containing a point-like dominanttarget. In Fig. 1, two blue rectangles that contain point-likestructures are selected to calculate TCR and the TCR valuesin these regions are 7.16 and 8.25 dB, respectively.

B. Parameter Setting

There are several parameters in the proposed model. Theparameter α is the fractional order of the FrTV norm.According to the experimental results of [26], α can beselected between 1.2 and 2. In our experiments, α is set tobe 1.4. The parameters γ and β control the convergence ofthe ADM algorithm, where γ is the step-length and β isthe penalty parameter. As stated in [53], ADM algorithm hasglobal convergence if β > 0 and γ < (

√5 + 1)/2. In this

article, we set γ = 1.5 and β = 0.025, respectively. In ourexperiments, the maximum number of iterations,N0, is set tobe 100. Note that our additional experiments show that theperformance of the proposed algorithm is robust to the changeof the parameters α, β, γ within the suggested ranges.

The parameters λ and ξ balance the data fidelity termand the two regularization terms in (11). In our experi-ments, the values of these two regularization parameters



Fig. 3. Despeckling results for Lena corrupted by 4-look speckle, the SNR is 6.02 dB. (a) Noisy image. (b) Local enlargement of noisy image. (c) AA.(d) SAR-BM3-D. (e) NL-SAR. (f) DFN. (g) MNR-ADL-SR. (h) TVNLR. (i) CRM-SR.

are empirically selected by searching roughly to get thehighest metric [54]. Likewise, the parameters of the algo-rithms in [40] and [55] were also determined in this way.Taking the image Barbara with 4-look speckle as an example,Fig. 2 shows that the PSNR values vary with the regularizationparameters λ and ξ . It can be seen that, when λ is setas a relatively small value, the increase of ξ leads to animprovement in PSNR to some point followed by a reduction.When the value of λ is relatively large, the PSNR will decreasewith the increase of ξ . Note that although the optimal choicesfor λ and ξ may be different in different images, the variationtrend of the PSNR values is similar with respect to the changeof the λ and ξ , which has been verified in our additionalexperiments. The parameters used in the experiments aresummarized in Table III.

TABLE III

PARAMETER SETTING OF THE PROPOSED ALGORITHM

C. Experimental Results on Synthetic Images

Speckle in SAR images is generally modeled as the mul-tiplicative random noise, hence in this experiment SAR-likeimages are obtained by multiplying original clean optical



Fig. 4. Despeckling results for Barbara corrupted by 4-look speckle, the SNR is 6.04 dB. (a) Noisy image. (b) Local enlargement of noisy image. (c) AA.(d) SAR-BM3-D. (e) NL-SAR. (f) DFN. (g) MNR-ADL-SR. (h) TVNLR. (i) CRM-SR.

images by simulated speckle noise. Experiments are carriedout on two different noise levels, i.e., the number of looks Lis set equal to be 4 and 10. For image Lena and Barbara, whenL = 4, the signal-to-noise-ratios (SNRs) are 6.02 and 6.04 dB,respectively, and when L = 10, the SNRs are 9.98 and10.02 dB, respectively. The regularization parameters λ and ξis set to be λ = 0.05 and ξ = 0.27, respectively.

Figs. 3 and 4 show the local enlargement of the imagesobtained by different despeckling methods on the simulatedoptical images contaminated by the 4-look speckle. Table IVlists the corresponding PSNR and SSIM of the two syntheticimages produced by different algorithms. The best resultsare in boldface and the results of the proposed algorithm

are in italics. From Figs. 3 and 4 and Table IV, we canfind that, all of these methods achieve positive despecklingperformance. Note that the TV-based method AA producesthe staircase-like effect [see Figs. 3(c) and 4(c)], thus ithas the lowest PSNR and SSIM values. The TVNLR modelalleviates the staircase effect and achieves a better performancein reducing speckle than the AA model. Therefore, it hashigher PSNR and SSIM values than AA model, but it losessome texture details in smooth areas [see Figs. 3(h) and 4(h)].SAR-BM3-D has good despeckling performance. Edges arewell preserved without significant spreading, whereas homo-geneous areas are strongly smoothed [see Figs. 3(d) and 4(d)].Thus, it has the highest PSNR values. NL-SAR has strong



Fig. 5. Despeckling results of different algorithms on real SAR image 1 (single-look, 512 × 512). (a) Noisy image. (b) Local enlargement of noisy image.(c) AA. (d) SAR-BM3-D. (e) NL-SAR. (f) DFN. (g) MNR-ADL-SR. (h) TVNLR. (i) CRM-SR.

speckle-reduction ability, but it generates some artifacts inflat regions [see Figs. 3(e) and 4(e)]. Some visible specklenoise still exists in the despeckled image obtained via DFNand MNR-ADL-SR. Besides, MNR-ADL-SR generates somepoint-wise artifacts [see Figs. 3(g) and 4(g)]. The PSNR valuesof DFN are lower than any other algorithms except AA. Forimage Barbara, AA, DFN, MNR-ADL-SR, and TVNLR losesome texture details. Thus, their SSIM values are lower thanCRM-SR, NL-SAR, and SAR-BM3-D. Compared with the

other algorithms, the proposed CRM-SR algorithm has strongspeckle-reduction and detail-preserving capability with highestSSIM values.

From Figs. 3 and 4 and Table IV, we have the followingconclusions.

1) In terms of the PSNR, for the two simulated opti-cal images SAR-BM3-D has the best PSNR, followedby CRM-SR. The average PSNR difference betweenSAR-BM3-D and CRM-SR is about 0.6 dB.



TABLE IV

PSNR AND SSIM OF THE TWO SYNTHETIC IMAGES PRODUCED BY DIFFERENT DESPECKLING ALGORITHMS: AA [20], SAR-BM3-D [12],NL-SAR [34], DFN [37], MNR-ADL-SR [40], AND TVNLR. THE BEST RESULT IN EACH COLUMN IS IN BOLDFACE

TABLE V

ENL, ESI, DPIS, AND TCR OF THE TWO REAL SAR IMAGES PRODUCED BY DIFFERENT DESPECKLING ALGORITHMS: AA [20], SAR-BM3-D [12],NL-SAR [34], DFN [37], MNR-ADL-SR [40], AND TVNLR. THE BEST RESULT IN EACH COLUMN IS IN BOLDFACE

2) By taking advantages of FrTV, CRM-SR generates ahigher PSNR than the TV-based methods, such as AA,TVNLR, and DFN for all test images. The averagePSNR difference between CRM-SR and TV-basedmethods is about 2 dB.

3) In terms of SSIM, the best performance is provided byCRM-SR, indicating its powerful geometry preservingcapability.

D. Experimental Results on Real SAR Images

In this experiment we use real SAR images to evaluatethe despeckling performance of the proposed algorithm. Theregularization parameters λ and ξ is set to be λ = 0.07 andξ = 0.31, respectively.

Figs. 5 and 6 present the local enlargement of the filteredimages obtained by different despeckling methods on the real

SAR images. Table V gives the numerical evaluation results ofdifferent algorithms on two real SAR images. The best resultsare in boldface and the results of the proposed algorithm arein italics. Visually, all of the methods can remove the specklein homogeneous regions. Note that in the regions where thepiecewise constant assumption is violated, the result of the AAmethod is significantly degraded by the staircase artifacts [seeFigs. 5(c) and 6(c)], thus it has the lowest ESI and DPIs values.The TVNLR over-smoothens out some texture details in flatareas [see Figs. 5(h) and 6(h)]. The DPIs values of TVNLRare lower than any other algorithms except AA. The specklereduction ability of SAR-BM3-D in flat areas is not as goodas that in the simulated case [see Figs. 5(d) and 6(d)], sinceit is developed under the hypothesis of uncorrelated speckle.The speckle-reduction ability of DFN is not good enough, andsome low value speckles still exist [see Figs. 5(f) and 6(f)].



Fig. 6. Despeckling results of different algorithms on real SAR image 2 (single-look, 512 × 512). (a) Noisy image. (b) Local enlargement of noisy image.(c) AA. (d) SAR-BM3-D. (e) NL-SAR. (f) DFN. (g) MNR-ADL-SR. (h) TVNLR. (i) CRM-SR.

Therefore, the ENL values of SAR-BM3-D and DFN arelower than any other algorithms and the DFN has the lowestENL values. NL-SAR has strong detail-preserving ability inflat regions, thus it has the highest DPIs values. However,it blurs some edges and strong targets [see Figs. 5(e) and 6(e)]and the TCR values of NL-SAR are lower than any otheralgorithms except AA. MNR-ADL-SR has strong speckle-reduction ability in flat regions. However, it generates some

point-wise artifacts [see Figs. 5(g) and 6(g)]. The proposedCRM-SR method performs better in removing speckle noiseand preserving details [see Figs. 5(i) and 6(i)], and it has thehighest ENL and TCR values.

Figs. 5 and 6 also show visual examples corresponding todifferent numerical metrics of various despeckling algorithms.From Figs. 5 and 6, we can see the visual differences corre-sponding to all four different metrics. For example, comparing



TABLE VI

COMPUTATIONAL TIME OF DIFFERENT METHODS

Fig. 5(i) with (d), we can see that, SAR-BM3-D preserve moreedge features than CRM-SR, and it has higher ESI values thanCRM-SR. It is corresponding to the results in Table V.

From Table V, we have the following conclusions.1) ENL of CRM-SR, NL-SAR, and MNR-ADL-SR, cal-

culated with the pixels in the white rectangles in thesetwo SAR images, are much higher than those of othermethods. CRM-SR has the highest ENL. It means thatthe speckle-restraining ability of CRM-SR outperformsthose of other methods.

2) In terms of ESI, SAR-BM3-D produces the strongestedge preservation performance, and CRM-SR is slightlyinferior to SAR-BM3-D. The average ESI differencebetween CRM-SR and SAR-BM3-D is about 0.02.

3) In terms of DPIs, the best DPIs is guaranteed byNL-SAR, followed by CRM-SR and SAR-BM3-D,it is consistent with the visual observations in Fig. 6,in which the proposed CRM-SR also smoothens outsome details that the NL-SAR is able to preserve. Theaverage DPIs difference between CRM-SR and NL-SARis about 0.1.

4) TCR of CRM-SR, calculated with the pixels in the bluerectangles in the SAR images, is higher than that of theother methods, indicating that our method has the bestperformance in point-like and fine structures preserving.

5) It is important to note that while CRM-SR has thebest performance in ENL and TCR'Cits performancein ESI, DPI_M, and DPI_V is only slightly inferior tothe best ones. For example, for real SAR, SAR-BM3-Dhas the highest ESI values 0.710, and the ESI valuesof CRM-SR is 0.683. However, the ENL, the DPIs,and the TCR of CRM-SR are all higher than those ofSAR-BM3-D. Overall the proposed CRM-SR algorithmis superior to the other algorithms.

E. About the Computational Complexity

In addition to the performance in reducing speckle andpreserving edges and details, the computation complexity isalso an important index to evaluate the performance of thedespeckling algorithms. The computation time of the pro-posed algorithm and the other algorithms with the real SARimage 1 is shown in Table VI, in which the slowest one isin boldface. It can be seen that the NL-SAR algorithm isfaster than all other algorithms due to its parallel implementa-tion. Compared with the other models, SARBM3D, CRM-SRand MNR-ADL-SR need more running time, since bothSARBM3D and the proposed CRM-SR algorithm need a lotof time to group similar patches together and MNR-ADL-SRneeds much running time to train the dictionary. Besides,additional computational overhead is added to CRM-SR andMNR-ADL-SR with the inclusion of two regularization terms.

The computational complexity of CRM-SR is the highestamong these algorithms, which is the cost of the performanceimprovement.

V. CONCLUSION

In this article, a novel regularization model for SARimage despeckling is proposed by simultaneously character-izing the two SAR image properties: the edge feature andthe nonlocal self-similarity. The combination of the FrTVregularization and the NLR regularization can produce a satis-factory despeckling performance. An efficient algorithm basedon ADM is derived to solve this new regularization model.Experimental results show that the proposed model and thecorresponding algorithm can effectively remove SAR imagespeckles and preserve texture and details according to bothsubjective visual assessment of image quality and objectiveevaluation. Future work includes development of adaptivemethods to select the optimal value of the regularizationparameters.

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Gao Chen received the master’s degree from Xia-men University, Xiamen, China, in 2009, and thePh.D. degree from Southwest Jiaotong University,Chengdu, China, in 2016.

From October 2016 to October 2018, he wasa Post-Doctoral Researcher with the Departmentof Electronic Engineering, Tsinghua University,Beijing, China. He is currently a Lecturer withthe School of Electrical Engineering and Intelli-gentization, Dongguan University of Technology,Dongguan, China. His research interests include

remote sensing image processing and multimedia image processing.



Gang Li (M’08–SM’13) received the B.S. and Ph.D.degrees in electronic engineering from TsinghuaUniversity, Beijing, China, in 2002 and 2007,respectively.

Since July 2007, he has been a Faculty Memberwith Tsinghua University, where he is currently aProfessor with the Department of Electronic Engi-neering. From 2012 to 2014, he visited Ohio StateUniversity, Columbus, OH, USA, and Syracuse Uni-versity, Syracuse, NY, USA. He has authored orcoauthored more than 120 journals and conference

papers. His research interests include radar imaging, distributed signal process-ing, sparse signal processing, micro-Doppler analysis, and information fusion.

Dr. Li is an Associate Editor of the IEEE TRANSACTIONS ON SIGNALPROCESSING.

Yu Liu received the B.S. and Ph.D. degrees ininformation and communication engineering fromNaval Aeronautical and Astronautical University,Yantai, China, in 2008 and 2014, respectively.

From 2016 to 2017, he was a Post-DoctoralResearcher with the Department of Information andCommunication Engineering, Beihang University,Beijing, China. Since 2014, he has been a FacultyMember with Naval Aeronautical and AstronauticalUniversity, where he is currently an Associate Pro-fessor with the Institute of Information Fusion. His

research interests include multisensor fusion, state estimation, and situationawareness.

Xiao-Ping Zhang (M’97–SM’02) received the B.S.and Ph.D. degrees in electronic engineering fromTsinghua University, Beijing, China, in 1992 and1996, respectively, and the MBA degree (Hons.) infinance, economics, and entrepreneurship from TheUniversity of Chicago Booth School of Business,Chicago, IL, USA, in 2008.

Since 2000, he has been with the Departmentof Electrical and Computer Engineering, RyersonUniversity, Toronto, ON, Canada, where he is cur-rently a Professor and the Director of the Com-

munication and Signal Processing Applications Laboratory. He has servedas the Program Director of graduate studies. He is cross-appointed to theFinance Department, Ted Rogers School of Management, Ryerson University.He was a Visiting Scientist with the Research Laboratory of Electronics,Massachusetts Institute of Technology, Cambridge, MA, USA, in 2015 and2017. He is a frequent Consultant for biotech companies and investment firms.He is the Co-Founder and CEO for EidoSearch, Toronto, an Ontario-basedcompany offering a content-based search and analysis engine for financialbig data. His research interests include statistical signal processing, sensornetworks and electronic systems, image and multimedia content analysis,machine learning, and applications in big data, finance, and marketing.

Dr. Zhang is an Elected Member of the ICME Steering Committee.He is a registered Professional Engineer in Ontario, Canada, and a memberof the Beta Gamma Sigma Honor Society. He is the General Co-Chairfor the IEEE International Conference on Acoustics, Speech, and SignalProcessing, 2021. He was the General Co-Chair for the 2017 GlobalSIPSymposium on Signal and Information Processing for Finance and Businessand is the General Co-Chair for the 2019 GlobalSIP Symposium on Signal,Information Processing and AI for Finance and Business. He was the GeneralChair for the IEEE International Workshop on Multimedia Signal Processing,in 2015. He was the Publicity Chair for the International Conference onMultimedia and Expo in 2006, and the Program Chair for the InternationalConference on Intelligent Computing in 2005 and 2010. He served as theGuest Editor for Multimedia Tools and Applications and the InternationalJournal of Semantic Computing. He was a tutorial speaker at ACMMM2011,ISCAS2013, ICIP2013, ICASSP2014, IJCNN2017, and ISCAS2019. He is aSenior Area Editor of the IEEE TRANSACTIONS ON SIGNAL PROCESSING

and the IEEE TRANSACTIONS ON IMAGE PROCESSING. He was an AssociateEditor of the IEEE TRANSACTIONS ON IMAGE PROCESSING, the IEEETRANSACTIONS ON MULTIMEDIA, the IEEE TRANSACTIONS ON CIR-CUITS AND SYSTEMS FOR VIDEO TECHNOLOGY, the IEEE TRANSACTIONS

ON SIGNAL PROCESSING, and the IEEE SIGNAL PROCESSING LETTERS.He was awarded as a Distinguished Lecturer for the term from January 2020 toDecember 2021 by the IEEE Signal Processing Society.

Li Zhang received the B.S., M.S., and Ph.D. degreesin signal and information processing from TsinghuaUniversity, Beijing, China, in 1987, 1992, and 2008,respectively.

In 1992, he joined the Department of Elec-tronic Engineering, Tsinghua University, as a Fac-ulty Member, where he is currently a Professor.His research interests include image processing,computer vision, pattern recognition, and computergraphics.

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