# A Convex Variational Model for Restoring SAR Images Corrupted by Multiplicative Noise

Authors: Hanmei Yang; Jiachang Li; Lixin Shen; Jian Lu
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

This paper studies a new convex variational model for denoising and deblurring images with multiplicative noise. Considering the statistical property of the multiplicative noise following Nakagami distribution, the denoising model consists of a data fidelity term, a quadratic penalty term, and a total variation regularization term. Here, the quadratic penalty term is mainly designed to guarantee the model to be strictly convex under a mild condition. Furthermore, the model is extended for the simultaneous denoising and deblurring case by introducing a blurring operator. We also study some mathematical properties of the proposed model. In addition, the model is solved by applying the primal-dual algorithm. The experimental results show that the proposed method is promising in restoring (blurred) images with multiplicative noise.


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## Body

## 1. Introduction

Image restoration problems [1–9] have attracted a great amount of attention in real applications. Noise in images is often roughly grouped into additive noise and multiplicative noise. In this paper, we mainly focus on the multiplicative noise removal problem [10], which can be formulated as(1)f=Kuv,where f:Ω⟶ℝ+ is the observed image (Ω⊂ℝ2 is a connected bounded open subset with compact Lipschitz boundary), v∈L2Ω denotes multiplicative noise with mean 1, and K∈ℒL2Ω is a known linear and continuous blurring operator. The goal is to find the unknown true image u from the degraded image f as well as possible.Multiplicative noise, also known as speckle [11–15], often occurs in active imaging systems, such as synthetic aperture radar (SAR), laser images, and ultrasound imaging. In this paper, the characteristics of speckle which follow a Nakagami distribution in SAR images are considered. In SAR imaging system which are widely used in military reconnaissance, marine monitoring, and other fields, a radar sends a coherent wave. Then, it is reflected on the ground and “recorded” by the radar sensor to capture SAR images. The fading of radar echo signal often causes that the SAR images are degraded by speckle, which limits their interpretation and processing.Over the past decades, there is a great deal of research that has been done on solving the speckled image problem [16–19]. Early filtering techniques are exploited under the minimum mean square error (MMSE) criterion in the spatial domain [20]. And the bilateral filtering [21], the meshless method [22], the low-rank method [23], and the wavelet-based [24, 25] and nonlocal-based [26–28] approaches have also been explored for despeckling. Particularly, in some statistics-based methods, the observed image intensity is considered as a random variable following a negative exponential law, and then the intensity of the L-look image which is obtained as the average of L different independent images follows a gamma law with mean equal to one. And thus, the image to be recovered is assumed to be degraded by multiplicative gamma noise. Based on this assumption, a series of variational models, including the ZWM model [29], the AA model, and its variants and convexity improvements (e.g., the exponent-based models [30–33], the mV model [34], the TwL-mV model [35], and DZ model [36]), are proposed, where the regularization terms generally play a key role to preserve the image structure such as edge detail. On the contrary, in some other statistics-based methods, the image amplitude which is the square root of the reflectivity (intensity) image is taken into consideration. And the pixel amplitudes are modeled as independent and identically distributed according to the Nakagami distribution. Related work includes the probabilistic patch-based (PPB) method [27], the SAR block-matching 3D (SAR-BM3D) method [28], DD-SRAD method [37], a variation-based model by using fields of experts (FoEs) [38] proposed by Chen et al. [11], and the references cited therein.Based on the works [11, 36, 39], we investigate a new convex variational model in this paper, where the image amplitude is considered, and the fidelity term is deduced from the probability density function of the Nakagami distribution and a quadratic penalty term is derived from the statistical property of Nakagami distribution, which guarantees the proposed model to be strictly convex under a mild condition. The convex model guarantees the uniqueness of the solution. Under the framework of continuous functional, we study the existence and uniqueness of the proposed model to the image denoising and deblurring problem. Furthermore, we employ the primal-dual method [36, 40] to solve the proposed model. Finally, we arrange some numerical comparisons on simulated images and real SAR images to illustrate the efficiency of the model.


## 2. Methods

To recoveru for the degraded image f in (1), based on the statistical properties of Nakagami noise, in Section 2.1, we first propose a new variational model (6) for image denoising (i.e., K=I). We analyze the existence and uniqueness of the solution of the new model and discuss the properties of the solution of the model. Then, we extend the denoising model (6) to simultaneous deblurring and denoising case and also show the properties of the solution of the model in Section 2.2. And on these bases, we introduce a primal-dual algorithm to solve the proposed model in Section 2.3. By adopting the similar ideas in [36, 39], we can get Propositions 1, 2–5, and Theorem 1. And the proof of Propositions 1,2, and 4 is given in the Appendices A, C, and D, respectively; and the proof of Theorem 1 can be found in Appendix B.


### 2.1. A Convex Variational Denoising Model

We now introduce a new variational model for image denoising, based on the statistical properties of Nakagami noise. For the multiplicative modelf=uv, we assume that v follows a Nakagami distribution, the square root of the reflectivity [11, 13]:(2)pfu=2LLΓLu2Lf2L−1e−Lf2/u2,where L is the number of looks of the image (i.e., number of independent values averaged) and Γ⋅ is the classical Gamma function.Based on (2), by using the MAP estimator and combining the resulting data term with the total variation (TV) prior model, we obtain the following AA-like variational model:(3)infu∈S¯Ω∫Ω2logu+f2u2dx+λ∫ΩDu,where λ>0 and S¯Ω:=u∈BVΩ:u≥0. And ∫ΩDu is the TV of image u.Letz=1/v and z denote the instance of random variable Z. It is not hard to get that its PDF is(4)pZz=2LLΓLz−2L−1e−L/z2,∀z≥0.A direct calculation shows that(5)EZ=LΓL−1/2ΓL>0,limL⟶+∞EZ−12=0.Based on this observation, we therefore introduce a quadratic penalty term∫Ωu/f−β12dx into the AA-like model (3), yielding the following optimization problem:(6)infu∈S¯ΩEu≔∫Ω2logu+f2u2dx+α∫Ωuf−β12dx+λ∫ΩDu,where α>0, λ>0, and β≥1 vary with the level of the noise. We now show that Eu in (6) is convex if the parameter α satisfies certain condition.Proposition 1.
Ifα is chosen satisfying(7)α≥112,then the model (6) is strictly convex.Next, we study the existence and uniqueness of the solution to (6) and the minimum-maximum principle. To this end, we make the following notations. Given a function f∈L∞Ω, we denote by supΩf (resp., infΩf) the supremum (resp., infimum) of f, i.e., supΩf=infC∈ℝ;fx≤Ca.e. (resp., infΩf=supC∈ℝ;fx≥Ca.e.).Theorem 1.
Iff∈L∞Ω and infΩf>0, then the problem (6) has a solution u∗∈BVΩ such that(8)0<infΩf≤u∗≤βsupΩf.
Moreover, the solution of (6) is unique when α is chosen to satisfy (7).We now introduce a comparison principle that characterizes the relationship of solutions of problem (6) with different f values.Proposition 2.
Supposef1 and f2 are in L∞Ω with infΩf1>0 and infΩf2>0. Let a1=infΩf1, a2=infΩf2, b1=supΩf1, and b2=supΩf2. Assume that f1<f2. We denote by u1∗ (resp., u2∗) a solution of (6) for f=f1(resp., f=f2). Then, we have u1∗≤u2∗ a.e. in Ω if α and β satisfy αβ/1+α<a12a22/b12b22.


### 2.2. A Simultaneous Deblurring and Denoising Model

Given a nonnegative, linear, and continuous blurring operatorK∈ℒL2Ω, the degraded image can be expressed as f=Kuv. Here, f is the observed image, u is the image to be recovered, and v is the noise. Similar to (6), the deblurring and denoising model can be formulated as(9)infu∈S¯ΩEKu≔∫Ω2logKu+f2Ku2dx+α∫ΩKuf−β12dx+λ∫ΩDu.Note that (6) is a special case of (9) with K being the identity operator.We can show that the above model (9) is convex if α satisfies some certain conditions.Proposition 3.
Ifα is chosen satisfying(10)α≥112,then the model (9) is convex.In view of Propositionconvexdesp and the fact that K is linear, it is not hard to show that the conclusion of Proposition 3holds.We next show thatKu∗ is positive; i.e., x∈Ω:Ku∗x=0 has a zero Lebesgue measure when ε approaches to zero. And furthermore, we can obtain that Ku∗ is strictly positive in the discrete situation.Proposition 4.
Supposef∈L∞Ω and infΩf>0. Let K∈ℒL2Ω be a linear and continuous blurring operator. Assume that u∗ is the solution of (9). Then, there exists a constant C such that, for any 0<ε<1,(11)x∈Ω:Ku∗x≤εfx≤ε21+2ε2logεC−2∫Ωlogfdx.The following Proposition5 implies that, in general, model (9) cannot automatically preserve the mean of the original image.Proposition 5.
Letu∗ be a solution of (7). Suppose that K1=1 and infΩKu∗>0. The following statements hold:(i)
∫Ωf2/Ku∗3−αKu∗/f2−β1/fdx=∫Ω1/Ku∗dx.(ii)
If there exists a solution of (10) for α=0, then(12)∫ΩKu∗f2dx≥∫Ω1Ku∗dx.We can enhance the model (9) to reduce the influence from the bias by keeping the same mean of the recovered image as f by the following model:(13)infu∈S¯Ω:EΩu=EΩf∫Ω2logKu+f2Ku2dx+α∫ΩKuf−β12dx+λ∫ΩDu,where EΩf is the mean value of f. Due to the fact that u∈S¯Ω:EΩu=EΩf is closed and convex, it is easy to see that (13) has exactly one solution.


### 2.3. An Iterative Algorithm

For the sake of developing numerical algorithms in real-world applications, in this section, we discuss a discrete version of model (6). For simplicity, we use the same notation as the abovementioned continuous version. Here, let K∈ℝd×d, Q=u∈ℝd:ui≥0,i=1,…,d,and∑i=1dui=∑i=1dfi, and the images u,f∈ℝd, and ∇∈ℝ2d×d be the discrete gradient operator with ∇x and ∇y being, respectively, the discrete derivatives in the x-direction and y-direction. With the above notations, the discrete version of model (6) is given as follows:(14)minu∈QEKu≔FKu+λuTV,where(15)FKu≔2logKu,1+f2Ku2,1+αKuf−β122,uTV=∑i=1d∇xui2+∇yui2.The minimization problem (14) can be solved by many numerical algorithms, such as the split-Bregman algorithm [41], the ADMM [42], and proximity algorithm [43]. In this section, we adopt the primal-dual algorithm to solve the problem (14) by formulating it as(16)maxp∈Pminu∈QFKu−λu,divp,where div is the divergence operator and P=p∈ℝ2d:maxi∈1,…,dpi2+pi+d21/2≤1 with p being the dual variable.Now, similar to [36], we present the primal-dual algorithm in Algorithm 1. In order to ensure the iterates un,pn of Algorithm 1 converge to a saddle point of (14), the parameters μ,ρ,andλ should satisfy μρλ2<1/8 [36, 44]. And for simplification, we set μ=ρ=3 in our experiments.Algorithm 1: Primal-dual iterative algorithm for solving model (14).
Given: a noisy imagef∈ℝ+d. The positive parametersλ,μ,ρ satisfyingρμλ2<1/8.Initialization:u0=u¯0=f and p0=0,…,0T∈ℝ2d.repeatStep 1:pn+1=argmaxp∈Pλu¯n,divp−1/2μp−pn22.Step 2:un+1=argminu∈QFKu−λu,divpn+1+1/2ρu−un22Step 3:u¯n+1=2un+1−un.untilun+1 converges or satisfies a stopping criterion.Output:u∗=un+1.


## 2.1. A Convex Variational Denoising Model

We now introduce a new variational model for image denoising, based on the statistical properties of Nakagami noise. For the multiplicative modelf=uv, we assume that v follows a Nakagami distribution, the square root of the reflectivity [11, 13]:(2)pfu=2LLΓLu2Lf2L−1e−Lf2/u2,where L is the number of looks of the image (i.e., number of independent values averaged) and Γ⋅ is the classical Gamma function.Based on (2), by using the MAP estimator and combining the resulting data term with the total variation (TV) prior model, we obtain the following AA-like variational model:(3)infu∈S¯Ω∫Ω2logu+f2u2dx+λ∫ΩDu,where λ>0 and S¯Ω:=u∈BVΩ:u≥0. And ∫ΩDu is the TV of image u.Letz=1/v and z denote the instance of random variable Z. It is not hard to get that its PDF is(4)pZz=2LLΓLz−2L−1e−L/z2,∀z≥0.A direct calculation shows that(5)EZ=LΓL−1/2ΓL>0,limL⟶+∞EZ−12=0.Based on this observation, we therefore introduce a quadratic penalty term∫Ωu/f−β12dx into the AA-like model (3), yielding the following optimization problem:(6)infu∈S¯ΩEu≔∫Ω2logu+f2u2dx+α∫Ωuf−β12dx+λ∫ΩDu,where α>0, λ>0, and β≥1 vary with the level of the noise. We now show that Eu in (6) is convex if the parameter α satisfies certain condition.Proposition 1.
Ifα is chosen satisfying(7)α≥112,then the model (6) is strictly convex.Next, we study the existence and uniqueness of the solution to (6) and the minimum-maximum principle. To this end, we make the following notations. Given a function f∈L∞Ω, we denote by supΩf (resp., infΩf) the supremum (resp., infimum) of f, i.e., supΩf=infC∈ℝ;fx≤Ca.e. (resp., infΩf=supC∈ℝ;fx≥Ca.e.).Theorem 1.
Iff∈L∞Ω and infΩf>0, then the problem (6) has a solution u∗∈BVΩ such that(8)0<infΩf≤u∗≤βsupΩf.
Moreover, the solution of (6) is unique when α is chosen to satisfy (7).We now introduce a comparison principle that characterizes the relationship of solutions of problem (6) with different f values.Proposition 2.
Supposef1 and f2 are in L∞Ω with infΩf1>0 and infΩf2>0. Let a1=infΩf1, a2=infΩf2, b1=supΩf1, and b2=supΩf2. Assume that f1<f2. We denote by u1∗ (resp., u2∗) a solution of (6) for f=f1(resp., f=f2). Then, we have u1∗≤u2∗ a.e. in Ω if α and β satisfy αβ/1+α<a12a22/b12b22.


## 2.2. A Simultaneous Deblurring and Denoising Model

Given a nonnegative, linear, and continuous blurring operatorK∈ℒL2Ω, the degraded image can be expressed as f=Kuv. Here, f is the observed image, u is the image to be recovered, and v is the noise. Similar to (6), the deblurring and denoising model can be formulated as(9)infu∈S¯ΩEKu≔∫Ω2logKu+f2Ku2dx+α∫ΩKuf−β12dx+λ∫ΩDu.Note that (6) is a special case of (9) with K being the identity operator.We can show that the above model (9) is convex if α satisfies some certain conditions.Proposition 3.
Ifα is chosen satisfying(10)α≥112,then the model (9) is convex.In view of Propositionconvexdesp and the fact that K is linear, it is not hard to show that the conclusion of Proposition 3holds.We next show thatKu∗ is positive; i.e., x∈Ω:Ku∗x=0 has a zero Lebesgue measure when ε approaches to zero. And furthermore, we can obtain that Ku∗ is strictly positive in the discrete situation.Proposition 4.
Supposef∈L∞Ω and infΩf>0. Let K∈ℒL2Ω be a linear and continuous blurring operator. Assume that u∗ is the solution of (9). Then, there exists a constant C such that, for any 0<ε<1,(11)x∈Ω:Ku∗x≤εfx≤ε21+2ε2logεC−2∫Ωlogfdx.The following Proposition5 implies that, in general, model (9) cannot automatically preserve the mean of the original image.Proposition 5.
Letu∗ be a solution of (7). Suppose that K1=1 and infΩKu∗>0. The following statements hold:(i)
∫Ωf2/Ku∗3−αKu∗/f2−β1/fdx=∫Ω1/Ku∗dx.(ii)
If there exists a solution of (10) for α=0, then(12)∫ΩKu∗f2dx≥∫Ω1Ku∗dx.We can enhance the model (9) to reduce the influence from the bias by keeping the same mean of the recovered image as f by the following model:(13)infu∈S¯Ω:EΩu=EΩf∫Ω2logKu+f2Ku2dx+α∫ΩKuf−β12dx+λ∫ΩDu,where EΩf is the mean value of f. Due to the fact that u∈S¯Ω:EΩu=EΩf is closed and convex, it is easy to see that (13) has exactly one solution.


## 2.3. An Iterative Algorithm

For the sake of developing numerical algorithms in real-world applications, in this section, we discuss a discrete version of model (6). For simplicity, we use the same notation as the abovementioned continuous version. Here, let K∈ℝd×d, Q=u∈ℝd:ui≥0,i=1,…,d,and∑i=1dui=∑i=1dfi, and the images u,f∈ℝd, and ∇∈ℝ2d×d be the discrete gradient operator with ∇x and ∇y being, respectively, the discrete derivatives in the x-direction and y-direction. With the above notations, the discrete version of model (6) is given as follows:(14)minu∈QEKu≔FKu+λuTV,where(15)FKu≔2logKu,1+f2Ku2,1+αKuf−β122,uTV=∑i=1d∇xui2+∇yui2.The minimization problem (14) can be solved by many numerical algorithms, such as the split-Bregman algorithm [41], the ADMM [42], and proximity algorithm [43]. In this section, we adopt the primal-dual algorithm to solve the problem (14) by formulating it as(16)maxp∈Pminu∈QFKu−λu,divp,where div is the divergence operator and P=p∈ℝ2d:maxi∈1,…,dpi2+pi+d21/2≤1 with p being the dual variable.Now, similar to [36], we present the primal-dual algorithm in Algorithm 1. In order to ensure the iterates un,pn of Algorithm 1 converge to a saddle point of (14), the parameters μ,ρ,andλ should satisfy μρλ2<1/8 [36, 44]. And for simplification, we set μ=ρ=3 in our experiments.Algorithm 1: Primal-dual iterative algorithm for solving model (14).
Given: a noisy imagef∈ℝ+d. The positive parametersλ,μ,ρ satisfyingρμλ2<1/8.Initialization:u0=u¯0=f and p0=0,…,0T∈ℝ2d.repeatStep 1:pn+1=argmaxp∈Pλu¯n,divp−1/2μp−pn22.Step 2:un+1=argminu∈QFKu−λu,divpn+1+1/2ρu−un22Step 3:u¯n+1=2un+1−un.untilun+1 converges or satisfies a stopping criterion.Output:u∗=un+1.


## 3. Results and Discussion

In this section, we conduct numerical experiments on simulated images and real SAR images to evaluate the approximation accuracy and computational efficiency of our proposed algorithm. We compare our method in the denoising case with the Lee filters [20], the speckle reducing anisotropic diffusion (SRAD) method [45], the probabilistic patch-based (PPB) method [27], and the model (13) with α=0 using the algorithm adopted in [39] (here, we call it AA-like method), while in the deblurring and denoising case, our method is compared with the AA-like method. In order to preserve the mean of the observed image, the bias correction technique [36] is utilized for the AA-like method and our method. All the experiments are performed in MATLAB R2017b on a laptop with Intel(R) Core(TM) i7-5500U central processing unit (CPU, 2.40 GHz), 8.0 G memory, and Windows 8.1 operating system.In our simulated test, we choose images of “Boat” and “Lena” of size512×512, “Cameraman” of size 256×256, “Remote1” of size 460×460, and “Remote2” of size 570×570, which are shown in Figure 1.Figure 1
The five test images: (a) Lena; (b) Cameraman; (c) Boat; (d) Remote1; (e) Remote2.
(a)(b)(c)(d)(e)The quality of recovered images obtained from various despeckling algorithms is evaluated by the structural similarity index (SSIM) [46] and the peak signal-to-noise ratio (PSNR):(17)PSNRu,u^=10log102552du−u^2,where u and u^ denote the original image and the recovered image, respectively; and d is the total number of pixels of u. All the methods except PPB and Lee filter are terminated by the following stopping criterion:(18)un−un+12un2<ε,where ε is a fixed threshold. We set ε=1.5×10−4 in the denoising experiment and ε=2×10−5 in the deblurring experiment.


### 3.1. Denoising Experiment

The first experiment focuses on the denoising case; i.e.,K in model (14) is the identity operator. We first choose the test image of “Boat” and degrade it by multiplicative Nakagami noise with L=10. With a large value of L, the value of β is 1. We denote Algorithm 2, Algorithm 3, and Algorithm 4 by the same algorithm framework of Algorithm 1 but with the initialization replaced by u0=u¯0=uPPB, uLee, uSRAD, and p0=0,…,0T∈ℝ2d. Here, uPPB, uLee, and uSRAD are the solutions of the PPB method [27], Lee filter [20], and SRAD method [45], respectively. The parameter settings of four algorithms are λ=0.01, α=2.7, and β=1. It can be seen from Figure 2 that although Algorithms 1, 2, 3, and 4 using the same parameters converge to the same stable PSNR value in accordance with the strict convexity and the uniqueness of the solution of the model, the curve of PSNR values associated with Algorithm 2 has a higher peak than that of Algorithms 1, 3, and 4. AA-like method with the same initial guess (named the PPB-AA method) is also added for making the experiments more persuading. Based on this observation, Algorithm 2, PPB-AA, together with Algorithm 1, Lee filter, SRAD method, PPB method, and AA-like method are listed in Table 1 to make a comparison. In Table 1, the images are corrupted by multiplicative noise (Nakagami distribution with L=10,L=7, and L=5). The smaller the L is, the noisier the images are. We tune the parameters to insure a better performance for every image. The parameter settings of proposed algorithms are shown in Table 2. For the Lee filter, the default window size of 5 is used. For the SRAD method, the parameters are set as suggested in [45]. For the PPB method, we use the codes provided by the work in [27]. And for PPB-AA and Algorithm 2, the computing time is expressed as the sum of the CPU cost of getting the initial solution uPPB and true CPU time of Algorithm 2 and AA-like method, respectively. We observe that the PSNR and SSIM values of the restored images by Algorithm 1 are always higher than those of AA-like method, which indicates that the choices of α and β have a significant impact on the quality of denoised images. Moreover, the PSNR and SSIM values of the restored images by Algorithm 2 are generally the highest among all those seven methods tested.Figure 2
The PSNR values against the number of iterations on Boat image using Algorithm 1, Algorithm 2, Algorithm 3, and Algorithm 4.Table 1
The comparison of different denoising methods.
ImageMethodL=10L=7L=5PSNRSSIMTime (s)PSNRSSIMTimePSNRSSIMTime (s)LenaLee29.520.8720.2228.580.8470.2227.750.8160.19SRAD29.750.8961.5529.420.8731.2229.020.8771.68AA-like30.010.8825029.010.8645828.120.84471Algorithm 130.990.9108929.950.89510528.910.874125PPB31.820.9355231.080.9245130.370.90950PPB-AA32.050.93252 + 631.090.92151 + 630.180.90850 + 7Algorithm 232.280.93852 + 2331.290.92851 + 2330.820.92250 + 32BoatLee27.000.8390.2526.290.8140.2425.570.7860.20SRAD27.540.8561.1127.280.8421.0926.840.8251.17AA-like28.300.8674127.420.8455326.550.81864Algorithm 128.900.8859228.020.86310927.110.839125PPB29.010.8845228.240.8585127.530.83551PPB-AA29.070.87552 + 528.190.84951 + 527.330.82651 + 6Algorithm 229.440.89152 + 2728.490.86551 + 2928.020.85151 + 40CameramanLee27.110.7570.1226.070.7190.1425.060.6860.13SRAD27.460.7530.3227.060.7400.2826.610.7600.35AA-like27.910.759926.850.7401125.930.72914Algorithm 128.460.7672627.770.7942326.840.77427PPB28.090.8331727.530.8111626.830.78815PPB-AA28.190.83217 + 227.630.82216 + 226.820.81315 + 1Algorithm 229.670.84817 + 1028.660.83216 + 1027.660.81715 + 10Remote1Lee26.120.9000.2625.230.8800.1924.460.8600.20SRAD27.460.9340.8327.130.9310.6426.600.9180.79AA-like27.980.9393026.910.9223625.910.89647Algorithm 128.670.9528527.520.9419626.690.92099PPB27.540.9434426.810.9294226.070.91242PPB-AA28.030.93544 + 927.220.92342 + 826.390.89842 + 12Algorithm 228.790.95344 + 3527.900.93742 + 3926.860.92242 + 38Remote2Lee26.120.8360.2225.390.8130.3124.680.7850.25SRAD26.930.8651.0326.590.8461.4226.300.8321.34AA-like27.810.8715026.840.8446425.910.81475Algorithm 128.430.88514327.570.86613126.640.839150PPB27.940.8686427.220.8446326.550.82162PPB-AA28.020.85864 + 727.230.83163 + 826.470.80962 + 8Algorithm 229.020.88964 + 4728.040.86563 + 5227.100.83862 + 49Table 2
The parameter values used in the denoising experiment.
ImageMethodL=10L=7L=5λαβλαβλαβLenaAA-like0.23——0.23——0.23——Algorithm 10.021.910.02110.020.41.1PPB-AA0.25——0.25——0.25——Algorithm 20.010.610.010.310.010.41.1BoatAA-like0.22——0.23——0.23——Algorithm 10.023.310.021.810.020.91.1PPB-AA0.22——0.22——0.22——Algorithm 20.012.710.011.710.011.41.1CameramanAA-like0.18——0.19——0.19——Algorithm 10.011/1210.022.210.021.11.1PPB-AA0.21——0.21——0.2——Algorithm 20.027.510.023.910.021.51.1Remote1AA-like0.17——0.17——0.18——Algorithm 10.011/1210.011/1210.021.51.1PPB-AA0.17——0.16——0.18——Algorithm 20.01310.010.910.0111.1Remote2AA-like0.17——0.18——0.18——Algorithm 10.010.110.02310.021.61.1PPB-AA0.17——0.18——0.18——Algorithm 20.013.210.01210.011.21.1To visually compare the image restoration performance of above seven methods, Figures3 and 4 report the denoising results of “Lena” and “Remote1” with L=5, and the certain corresponding partial enlarged denoising images of “Lena” and “Remote1” with L=10, L=7, and L=5 by these methods are shown in Figures 5–10, respectively. Firstly, we find that much noise remains in the results of Lee filter, SRAD method, and AA-like method (e.g., the hat of “Lena” shown in Figures 5–7 and roofs in “Remote1” shown in Figures 8–10). Lee filter leads to a blurred effect on restored images. Since the AA-like method is oversmoothing in the process of despeckling, details of images tend to be reduced. Conversely, more details are preserved by Algorithm 1 and SRAD method. Secondly, the PPB method in [27] can remove the multiplicative noise effectively because it is a patch-based method, which is a different framework from TV-based methods (e.g., AA-like method and Algorithm 1). The images obtained from the PPB method provide smoother regions and better shape preservation (e.g., “Remote1” shown in Figure 4 and the backgrounds in “Lena” shown in Figure 3). However, we still notice some artifacts (e.g., the edge of the homogeneous regions in “Lena” and “Remote1”), and the PPB method seems to attenuate the image sharpness (e.g., the hat of “Lena” and trees in “Remote1”) shown in Figures 5–10. Indeed, those spurious artifacts can be removed by the TV regularization in our method, but Algorithm 1 may suffer from staircasing effects (e.g., the skin of “Lena”). Finally, we observe that Algorithm 2, which combines Algorithm 1 and PPB method, can resolve the problems of PPB method and Algorithm 1 and performs better than the PPB-AA method in attenuating the unpleasant artifacts.Figure 3
Denoising results of different methods on Lena image withL=5: (a) PSNR = 18.73dB; (b) PSNR = 27.75dB; (c) PSNR = 29.02dB; (d) PSNR = 28.12dB; (e) PSNR = 28.91dB; (f) PSNR = 30.37dB; (g) PSNR = 30.18dB; (h) PSNR = 30.82dB.
(a)(b)(c)(d)(e)(f)(g)(h)Figure 4
Denoising results of different methods on Remote1 image withL=5: (a) PSNR = 20.44dB; (b) PSNR = 24.46dB; (c) PSNR = 26.60dB; (d) PSNR = 25.91dB; (e) PSNR = 26.69dB; (f) PSNR = 26.07dB; (g) PSNR = 26.39dB; (h) PSNR = 26.86dB.
(a)(b)(c)(d)(e)(f)(g)(h)Figure 5
Partial enlarged detail of the denoising results on Lena image withL=10: (a) PSNR = 21.73dB; (b) PSNR = 29.52dB; (c) PSNR = 29.75dB; (d) PSNR = 30.01dB; (e) PSNR = 30.99dB; (f) PSNR = 31.82dB; (g) PSNR = 32.05dB; (h) PSNR = 32.28dB.
(a)(b)(c)(d)(e)(f)(g)(h)Figure 6
Partial enlarged detail of the denoising results on Lena image withL=7: (a) PSNR = 20.16dB; (b) PSNR = 28.58dB; (c) PSNR = 29.42dB; (d) PSNR = 29.01dB; (e) PSNR = 29.95dB; (f) PSNR = 31.08dB; (g) PSNR = 31.09dB; (h) PSNR = 31.29dB.
(a)(b)(c)(d)(e)(f)(g)(h)Figure 7
Partial enlarged detail of the denoising results on Lena image withL=5: (a) PSNR = 18.73dB; (b) PSNR = 27.75dB; (c) PSNR = 29.02dB; (d) PSNR = 28.12dB; (e) PSNR = 28.91dB; (f) PSNR = 30.37dB; (g) PSNR = 30.18dB; (h) PSNR = 30.82dB.
(a)(b)(c)(d)(e)(f)(g)(h)Figure 8
Partial enlarged detail of the denoising results on Remote1 image withL=10: (a) PSNR = 23.45dB; (b) PSNR = 26.12dB; (c) PSNR = 27.46dB; (d) PSNR = 27.98dB; (e) PSNR = 28.67dB; (f) PSNR = 27.54dB; (g) PSNR = 28.03dB; (h) PSNR = 28.79dB.
(a)(b)(c)(d)(e)(f)(g)(h)Figure 9
Partial enlarged detail of the denoising results on Remote1 image withL=7: (a) PSNR = 21.90dB; (b) PSNR = 25.23dB; (c) PSNR = 27.13dB; (d) PSNR = 26.91dB; (e) PSNR = 27.52dB; (f) PSNR = 26.81dB; (g) PSNR = 27.22dB; (h) PSNR = 27.90dB.
(a)(b)(c)(d)(e)(f)(g)(h)Figure 10
Partial enlarged detail of the denoising results on Remote1 image withL=5: (a) PSNR = 20.44dB; (b) PSNR = 24.46dB; (c) PSNR = 26.60dB; (d) PSNR = 25.91dB; (e) PSNR = 26.69dB; (f) PSNR = 26.07dB; (g) PSNR = 26.39dB; (h) PSNR = 26.86dB.
(a)(b)(c)(d)(e)(f)(g)(h)


### 3.2. Deblurring Experiment

In the second experiment, we compare our proposed method with AA-like method for the simultaneous deblurring and denoising case. The two different blurring operators are the motion blur with length 5 and angle 30 and Gaussian blur with a window size7×7 and a standard deviation of 2. And after the test images are blurred, they are corrupted by multiplicative Nakagami noise with L=10, L=7, and L=5, respectively. We list the parameter values in Table 3, and the empirical choice of β is the same as denoising case. The numerical results are presented in Table 4. We can see that the PSNR and SSIM values of the restored images by Algorithm 1 are higher than that of the AA-like method. Moreover, we show the degraded images and the restored results of both methods for the images “Cameraman” in Figures 11–13. One can see Algorithm 1 preserves more details (e.g., the tripod in “Cameraman”). Due to oversmoothing, the AA-like method tends to lose details and attenuate the image contrast. Note that both methods cost more time than that in the denoising case because of the existence of the blurring operator.Table 3
The parameter values used in deblurring experiment.
LImageMethodMotion blurGaussian blurλαβλαβ10LenaAA-like0.03——0.04——Algorithm 10.013.510.017.11BoatAA-like0.04——0.04——Algorithm 10.015.510.0111.51CameramanAA-like0.03——0.03——Algorithm 10.01710.011217LenaAA-like0.03——0.04——Algorithm 10.011.910.0141BoatAA-like0.04——0.04——Algorithm 10.013.210.016.21CameramanAA-like0.03——0.03——Algorithm 10.013.610.01715LenaAA-like0.03——0.04——Algorithm 10.0111.10.0121.1BoatAA-like0.03——0.04——Algorithm 10.011.81.10.013.41.1CameramanAA-like0.03——0.03——Algorithm 10.011.81.10.013.71.1Table 4
The comparison of different deblurring and denoising methods.
LImageMethodMotion blurGaussian blurPSNR (dB)SSIMTime (s)PSNR (dB)SSIMTime (s)10LenaAA-like27.730.83252226.800.828554Algorithm 129.400.90394128.290.874869BoatAA-like25.500.80458724.470.751518Algorithm 126.480.840230225.540.7962349CameramanAA-like23.580.6748822.210.61497Algorithm 125.210.75048623.570.7246387LenaAA-like27.030.80660326.470.815713Algorithm 128.760.890104227.840.8631012BoatAA-like25.030.77777824.170.731648Algorithm 125.960.818221025.060.7692215CameramanAA-like23.310.64912022.090.594120Algorithm 124.660.74846023.370.7164915LenaAA-like26.340.77873825.960.805991Algorithm 127.990.874116427.150.8471131BoatAA-like24.510.74170823.800.710922Algorithm 125.430.798219824.560.7482319CameramanAA-like22.990.61914221.810.575143Algorithm 124.250.73652822.970.709562Figure 11
Deblurring results of different methods on Cameraman image withL=10: (a) degraded (motion); (b) AA-like (23.58 dB); (c) Algorithm 1 (25.21 dB); (d) degraded (Gaussian); (e) AA-like (22.21 dB); (f) Algorithm 1 (23.57 dB).
(a)(b)(c)(d)(e)(f)Figure 12
Deblurring results of different methods on Cameraman image withL=7: (a) degraded (motion); (b) AA-like (23.31 dB); (c) Algorithm 1 (24.66 dB); (d) degraded (Gaussian); (e) AA-like (22.09 dB); (f) Algorithm 1 (23.37 dB).
(a)(b)(c)(d)(e)(f)Figure 13
Deblurring results of different methods on Cameraman image withL=5: (a) degraded (motion); (b) AA-like (22.99 dB); (c) Algorithm 1 (24.25 dB); (d) degraded (Gaussian); (e) AA-like (21.81 dB); (f) Algorithm 1 (22.97 dB).
(a)(b)(c)(d)(e)(f)


### 3.3. Real SAR Images Restoration Experiment

Furthermore, we evaluate the performance of a newly proposed model for despeckling based on three real SAR images. The stopping criterion is the same as that of the denoising experiment. Since there are no original noise-free images to be compared, PSNR and SSIM cannot be used to evaluate the quality of recovered images. AssumingL=5, we tune the parameters until the recovered images show the best results. The parameters settings are reported in Table 5. Figures 14–16 present the denoising results on real SAR images. It is obvious that all of their objective function values are monotonically decreasing. The less speckle is eliminated by Lee filter, SRAD method, and AA-like method. Lee filter does not enhance edges. There are staircasing effects existing in the results of AA-like method. Algorithm 1 can remove noise more effectively than AA-like method. PPB method performs well in restoring images, such as smoother regions and better shape preservation. But there are some spurious artifacts in the edge of the images. Comparing the experimental results, we can find that Algorithm 2 reduces the artifacts more effectively than PPB-AA method. At the same time, the staircasing effects also decrease in these two methods.Table 5
The parameter values used in SAR images restoration experiment.
ImagesMethodsλαβSAR image 1AA-like0.12——Algorithm 10.021/121.1PPB-AA0.12——Algorithm 20.020.11.1SAR image 2AA-like0.15——Algorithm 10.021/121.1PPB-AA0.17——Algorithm 20.011/121.1SAR image 3AA-like0.15——Algorithm 10.021/121.1PPB-AA0.15——Algorithm 20.021/121.1Figure 14
Denoising results on SAR image 1: (a) degraded image; (b) Lee; (c) SRAD; (d) PPB; (e) AA-like; (f) Algorithm 1; (g) PPB-AA; (h) Algorithm 2.
(a)(b)(c)(d)(e)(f)(g)(h)Figure 15
Denoising results on SAR image 2: (a) degraded image; (b) Lee; (c) SRAD; (d) PPB; (e) AA-like; (f) Algorithm 1; (g) PPB-AA; (h) Algorithm 2.
(a)(b)(c)(d)(e)(f)(g)(h)Figure 16
Denoising results on SAR image 3: (a) degraded image; (b) Lee; (c) SRAD; (d) PPB; (e) AA-like; (f) Algorithm 1; (g) PPB-AA; (h) Algorithm 2.
(a)(b)(c)(d)(e)(f)(g)(h)


### 3.4. The Influence of Parameters and Mean

Here, we study the sensitivities of parameters and the effect of mean of our restoration model. According to the analysis in Section2.1, the value of parameter β varies with the level of the noise. Experimentally, β is chosen as 1 when L>5. When L=5, β is set as 1.1. In the proposed model, parameter α needs to satisfy condition α≥1/12. Parameter λ is set as λ2<1/72 based on the description in Section 2.3, which is changing from 0.01 to 0.11 with an interval 0.01 in experiment. In the denoising case, we take the noisy images “Lena” and “Cameraman” as examples. For each image, we vary the values of α with three levels of the noise, L=10,7,5. The PSNR values of restored images using Algorithm 1 are presented in Figure 17. We observe that there is an upmost point on each curve. With the increasing α, the performance of “Lena” with L=5 and “Cameraman” with L=10 decreases rapidly, which means that the denoising results are sensitive to value of α for each L. In the simultaneous deblurring and denoising case, we take the noisy image of “Cameraman” blurred by the motion blur with length 5 and angle 30 and then corrupted by multiplicative Nakagami noise with L=10 as an example. The parameter settings are λ=0.01, α=7, and β=1 in this case known from Table 3. At first, for the case of λ=0.01, we study the sensitivity of the parameter α in terms of solution quality and computation time. The PSNR (dB), SSIM, and CPU time (s) used for various values of α are listed in Table 6. We observe that if α is less than 7, the PSNR value of the restored image by Algorithm 1 increases with the increase in α. However, the PSNR and SSIM values decrease as the value of α increases when α is more than 7. The numerical results indicate that there is a certain value of α to make the model get the best quality of recovered image for a fixed value of λ. Then, for the case of α=7, we further analyze the sensitivity of the parameter λ. The PSNR (dB), SSIM, and CPU time (s) used for various values of λ are listed in Table 7. From Table 7, we observe that λ=0.01 is the best choice for obtaining the highest PSNR value of denoised image.Figure 17
The PSNR values against the values ofα under three noise levels: (a) Lena; (b) Cameraman.
(a)(b)Table 6
Numerical results of Algorithm 1 with parameterλ=0.01 on Cameraman image.
α0.13789PSNR (dB)24.1524.9925.2125.2025.17SSIM0.7620.7760.7500.7380.726Time (s)274395486528529Table 7
Numerical results of Algorithm 1 with parameterα=7 on Cameraman image.
λ0.010.020.050.080.11PSNR (dB)25.2124.8523.7723.1022.62SSIM0.7500.7740.7510.7340.725Time (s)486367246231229Finally, in order to discuss the influence of preserving the mean of the original image, there are four assumptions, including recovered images obtained from model (9) and model (9) with preserving 1 times, 1.2 times, 0.8 times the mean of the original image, respectively. Table 8 reports the PSNR (dB) and SSIM values on Cameraman image with different mean. We find the fact the PSNR (dB) and SSIM values obtained from model (9) with preserving mean of the original image are almost higher than the others. As shown in Figure 18, when the mean of recovered image is larger than that of the original image, the restored image by Algorithm 1 looks lighter. In contrast, the restored image is darker. PSNR and SSIM values of both decrease significantly. Although there is not much difference between the recovered images obtained from preserving the mean and not preserving the mean, the PSNR value is higher when the mean is maintained as the original. These indicate that bias correction technique [36] is helpful to enhance the performance of the model (9).Table 8
Numerical results on Cameraman image with different mean.
MeanPreservedNot preservedIncreasedDecreasedPSNR (dB)25.2124.3419.1118.22SSIM0.7500.7500.7260.732Figure 18
Results on Cameraman image with different mean: (a) preserved (25.21 dB); (b) not preserved (24.34 dB); (c) increased (19.11 dB); (d) decreased (18.22 dB).
(a)(b)(c)(d)


## 3.1. Denoising Experiment

The first experiment focuses on the denoising case; i.e.,K in model (14) is the identity operator. We first choose the test image of “Boat” and degrade it by multiplicative Nakagami noise with L=10. With a large value of L, the value of β is 1. We denote Algorithm 2, Algorithm 3, and Algorithm 4 by the same algorithm framework of Algorithm 1 but with the initialization replaced by u0=u¯0=uPPB, uLee, uSRAD, and p0=0,…,0T∈ℝ2d. Here, uPPB, uLee, and uSRAD are the solutions of the PPB method [27], Lee filter [20], and SRAD method [45], respectively. The parameter settings of four algorithms are λ=0.01, α=2.7, and β=1. It can be seen from Figure 2 that although Algorithms 1, 2, 3, and 4 using the same parameters converge to the same stable PSNR value in accordance with the strict convexity and the uniqueness of the solution of the model, the curve of PSNR values associated with Algorithm 2 has a higher peak than that of Algorithms 1, 3, and 4. AA-like method with the same initial guess (named the PPB-AA method) is also added for making the experiments more persuading. Based on this observation, Algorithm 2, PPB-AA, together with Algorithm 1, Lee filter, SRAD method, PPB method, and AA-like method are listed in Table 1 to make a comparison. In Table 1, the images are corrupted by multiplicative noise (Nakagami distribution with L=10,L=7, and L=5). The smaller the L is, the noisier the images are. We tune the parameters to insure a better performance for every image. The parameter settings of proposed algorithms are shown in Table 2. For the Lee filter, the default window size of 5 is used. For the SRAD method, the parameters are set as suggested in [45]. For the PPB method, we use the codes provided by the work in [27]. And for PPB-AA and Algorithm 2, the computing time is expressed as the sum of the CPU cost of getting the initial solution uPPB and true CPU time of Algorithm 2 and AA-like method, respectively. We observe that the PSNR and SSIM values of the restored images by Algorithm 1 are always higher than those of AA-like method, which indicates that the choices of α and β have a significant impact on the quality of denoised images. Moreover, the PSNR and SSIM values of the restored images by Algorithm 2 are generally the highest among all those seven methods tested.Figure 2
The PSNR values against the number of iterations on Boat image using Algorithm 1, Algorithm 2, Algorithm 3, and Algorithm 4.Table 1
The comparison of different denoising methods.
ImageMethodL=10L=7L=5PSNRSSIMTime (s)PSNRSSIMTimePSNRSSIMTime (s)LenaLee29.520.8720.2228.580.8470.2227.750.8160.19SRAD29.750.8961.5529.420.8731.2229.020.8771.68AA-like30.010.8825029.010.8645828.120.84471Algorithm 130.990.9108929.950.89510528.910.874125PPB31.820.9355231.080.9245130.370.90950PPB-AA32.050.93252 + 631.090.92151 + 630.180.90850 + 7Algorithm 232.280.93852 + 2331.290.92851 + 2330.820.92250 + 32BoatLee27.000.8390.2526.290.8140.2425.570.7860.20SRAD27.540.8561.1127.280.8421.0926.840.8251.17AA-like28.300.8674127.420.8455326.550.81864Algorithm 128.900.8859228.020.86310927.110.839125PPB29.010.8845228.240.8585127.530.83551PPB-AA29.070.87552 + 528.190.84951 + 527.330.82651 + 6Algorithm 229.440.89152 + 2728.490.86551 + 2928.020.85151 + 40CameramanLee27.110.7570.1226.070.7190.1425.060.6860.13SRAD27.460.7530.3227.060.7400.2826.610.7600.35AA-like27.910.759926.850.7401125.930.72914Algorithm 128.460.7672627.770.7942326.840.77427PPB28.090.8331727.530.8111626.830.78815PPB-AA28.190.83217 + 227.630.82216 + 226.820.81315 + 1Algorithm 229.670.84817 + 1028.660.83216 + 1027.660.81715 + 10Remote1Lee26.120.9000.2625.230.8800.1924.460.8600.20SRAD27.460.9340.8327.130.9310.6426.600.9180.79AA-like27.980.9393026.910.9223625.910.89647Algorithm 128.670.9528527.520.9419626.690.92099PPB27.540.9434426.810.9294226.070.91242PPB-AA28.030.93544 + 927.220.92342 + 826.390.89842 + 12Algorithm 228.790.95344 + 3527.900.93742 + 3926.860.92242 + 38Remote2Lee26.120.8360.2225.390.8130.3124.680.7850.25SRAD26.930.8651.0326.590.8461.4226.300.8321.34AA-like27.810.8715026.840.8446425.910.81475Algorithm 128.430.88514327.570.86613126.640.839150PPB27.940.8686427.220.8446326.550.82162PPB-AA28.020.85864 + 727.230.83163 + 826.470.80962 + 8Algorithm 229.020.88964 + 4728.040.86563 + 5227.100.83862 + 49Table 2
The parameter values used in the denoising experiment.
ImageMethodL=10L=7L=5λαβλαβλαβLenaAA-like0.23——0.23——0.23——Algorithm 10.021.910.02110.020.41.1PPB-AA0.25——0.25——0.25——Algorithm 20.010.610.010.310.010.41.1BoatAA-like0.22——0.23——0.23——Algorithm 10.023.310.021.810.020.91.1PPB-AA0.22——0.22——0.22——Algorithm 20.012.710.011.710.011.41.1CameramanAA-like0.18——0.19——0.19——Algorithm 10.011/1210.022.210.021.11.1PPB-AA0.21——0.21——0.2——Algorithm 20.027.510.023.910.021.51.1Remote1AA-like0.17——0.17——0.18——Algorithm 10.011/1210.011/1210.021.51.1PPB-AA0.17——0.16——0.18——Algorithm 20.01310.010.910.0111.1Remote2AA-like0.17——0.18——0.18——Algorithm 10.010.110.02310.021.61.1PPB-AA0.17——0.18——0.18——Algorithm 20.013.210.01210.011.21.1To visually compare the image restoration performance of above seven methods, Figures3 and 4 report the denoising results of “Lena” and “Remote1” with L=5, and the certain corresponding partial enlarged denoising images of “Lena” and “Remote1” with L=10, L=7, and L=5 by these methods are shown in Figures 5–10, respectively. Firstly, we find that much noise remains in the results of Lee filter, SRAD method, and AA-like method (e.g., the hat of “Lena” shown in Figures 5–7 and roofs in “Remote1” shown in Figures 8–10). Lee filter leads to a blurred effect on restored images. Since the AA-like method is oversmoothing in the process of despeckling, details of images tend to be reduced. Conversely, more details are preserved by Algorithm 1 and SRAD method. Secondly, the PPB method in [27] can remove the multiplicative noise effectively because it is a patch-based method, which is a different framework from TV-based methods (e.g., AA-like method and Algorithm 1). The images obtained from the PPB method provide smoother regions and better shape preservation (e.g., “Remote1” shown in Figure 4 and the backgrounds in “Lena” shown in Figure 3). However, we still notice some artifacts (e.g., the edge of the homogeneous regions in “Lena” and “Remote1”), and the PPB method seems to attenuate the image sharpness (e.g., the hat of “Lena” and trees in “Remote1”) shown in Figures 5–10. Indeed, those spurious artifacts can be removed by the TV regularization in our method, but Algorithm 1 may suffer from staircasing effects (e.g., the skin of “Lena”). Finally, we observe that Algorithm 2, which combines Algorithm 1 and PPB method, can resolve the problems of PPB method and Algorithm 1 and performs better than the PPB-AA method in attenuating the unpleasant artifacts.Figure 3
Denoising results of different methods on Lena image withL=5: (a) PSNR = 18.73dB; (b) PSNR = 27.75dB; (c) PSNR = 29.02dB; (d) PSNR = 28.12dB; (e) PSNR = 28.91dB; (f) PSNR = 30.37dB; (g) PSNR = 30.18dB; (h) PSNR = 30.82dB.
(a)(b)(c)(d)(e)(f)(g)(h)Figure 4
Denoising results of different methods on Remote1 image withL=5: (a) PSNR = 20.44dB; (b) PSNR = 24.46dB; (c) PSNR = 26.60dB; (d) PSNR = 25.91dB; (e) PSNR = 26.69dB; (f) PSNR = 26.07dB; (g) PSNR = 26.39dB; (h) PSNR = 26.86dB.
(a)(b)(c)(d)(e)(f)(g)(h)Figure 5
Partial enlarged detail of the denoising results on Lena image withL=10: (a) PSNR = 21.73dB; (b) PSNR = 29.52dB; (c) PSNR = 29.75dB; (d) PSNR = 30.01dB; (e) PSNR = 30.99dB; (f) PSNR = 31.82dB; (g) PSNR = 32.05dB; (h) PSNR = 32.28dB.
(a)(b)(c)(d)(e)(f)(g)(h)Figure 6
Partial enlarged detail of the denoising results on Lena image withL=7: (a) PSNR = 20.16dB; (b) PSNR = 28.58dB; (c) PSNR = 29.42dB; (d) PSNR = 29.01dB; (e) PSNR = 29.95dB; (f) PSNR = 31.08dB; (g) PSNR = 31.09dB; (h) PSNR = 31.29dB.
(a)(b)(c)(d)(e)(f)(g)(h)Figure 7
Partial enlarged detail of the denoising results on Lena image withL=5: (a) PSNR = 18.73dB; (b) PSNR = 27.75dB; (c) PSNR = 29.02dB; (d) PSNR = 28.12dB; (e) PSNR = 28.91dB; (f) PSNR = 30.37dB; (g) PSNR = 30.18dB; (h) PSNR = 30.82dB.
(a)(b)(c)(d)(e)(f)(g)(h)Figure 8
Partial enlarged detail of the denoising results on Remote1 image withL=10: (a) PSNR = 23.45dB; (b) PSNR = 26.12dB; (c) PSNR = 27.46dB; (d) PSNR = 27.98dB; (e) PSNR = 28.67dB; (f) PSNR = 27.54dB; (g) PSNR = 28.03dB; (h) PSNR = 28.79dB.
(a)(b)(c)(d)(e)(f)(g)(h)Figure 9
Partial enlarged detail of the denoising results on Remote1 image withL=7: (a) PSNR = 21.90dB; (b) PSNR = 25.23dB; (c) PSNR = 27.13dB; (d) PSNR = 26.91dB; (e) PSNR = 27.52dB; (f) PSNR = 26.81dB; (g) PSNR = 27.22dB; (h) PSNR = 27.90dB.
(a)(b)(c)(d)(e)(f)(g)(h)Figure 10
Partial enlarged detail of the denoising results on Remote1 image withL=5: (a) PSNR = 20.44dB; (b) PSNR = 24.46dB; (c) PSNR = 26.60dB; (d) PSNR = 25.91dB; (e) PSNR = 26.69dB; (f) PSNR = 26.07dB; (g) PSNR = 26.39dB; (h) PSNR = 26.86dB.
(a)(b)(c)(d)(e)(f)(g)(h)


## 3.2. Deblurring Experiment

In the second experiment, we compare our proposed method with AA-like method for the simultaneous deblurring and denoising case. The two different blurring operators are the motion blur with length 5 and angle 30 and Gaussian blur with a window size7×7 and a standard deviation of 2. And after the test images are blurred, they are corrupted by multiplicative Nakagami noise with L=10, L=7, and L=5, respectively. We list the parameter values in Table 3, and the empirical choice of β is the same as denoising case. The numerical results are presented in Table 4. We can see that the PSNR and SSIM values of the restored images by Algorithm 1 are higher than that of the AA-like method. Moreover, we show the degraded images and the restored results of both methods for the images “Cameraman” in Figures 11–13. One can see Algorithm 1 preserves more details (e.g., the tripod in “Cameraman”). Due to oversmoothing, the AA-like method tends to lose details and attenuate the image contrast. Note that both methods cost more time than that in the denoising case because of the existence of the blurring operator.Table 3
The parameter values used in deblurring experiment.
LImageMethodMotion blurGaussian blurλαβλαβ10LenaAA-like0.03——0.04——Algorithm 10.013.510.017.11BoatAA-like0.04——0.04——Algorithm 10.015.510.0111.51CameramanAA-like0.03——0.03——Algorithm 10.01710.011217LenaAA-like0.03——0.04——Algorithm 10.011.910.0141BoatAA-like0.04——0.04——Algorithm 10.013.210.016.21CameramanAA-like0.03——0.03——Algorithm 10.013.610.01715LenaAA-like0.03——0.04——Algorithm 10.0111.10.0121.1BoatAA-like0.03——0.04——Algorithm 10.011.81.10.013.41.1CameramanAA-like0.03——0.03——Algorithm 10.011.81.10.013.71.1Table 4
The comparison of different deblurring and denoising methods.
LImageMethodMotion blurGaussian blurPSNR (dB)SSIMTime (s)PSNR (dB)SSIMTime (s)10LenaAA-like27.730.83252226.800.828554Algorithm 129.400.90394128.290.874869BoatAA-like25.500.80458724.470.751518Algorithm 126.480.840230225.540.7962349CameramanAA-like23.580.6748822.210.61497Algorithm 125.210.75048623.570.7246387LenaAA-like27.030.80660326.470.815713Algorithm 128.760.890104227.840.8631012BoatAA-like25.030.77777824.170.731648Algorithm 125.960.818221025.060.7692215CameramanAA-like23.310.64912022.090.594120Algorithm 124.660.74846023.370.7164915LenaAA-like26.340.77873825.960.805991Algorithm 127.990.874116427.150.8471131BoatAA-like24.510.74170823.800.710922Algorithm 125.430.798219824.560.7482319CameramanAA-like22.990.61914221.810.575143Algorithm 124.250.73652822.970.709562Figure 11
Deblurring results of different methods on Cameraman image withL=10: (a) degraded (motion); (b) AA-like (23.58 dB); (c) Algorithm 1 (25.21 dB); (d) degraded (Gaussian); (e) AA-like (22.21 dB); (f) Algorithm 1 (23.57 dB).
(a)(b)(c)(d)(e)(f)Figure 12
Deblurring results of different methods on Cameraman image withL=7: (a) degraded (motion); (b) AA-like (23.31 dB); (c) Algorithm 1 (24.66 dB); (d) degraded (Gaussian); (e) AA-like (22.09 dB); (f) Algorithm 1 (23.37 dB).
(a)(b)(c)(d)(e)(f)Figure 13
Deblurring results of different methods on Cameraman image withL=5: (a) degraded (motion); (b) AA-like (22.99 dB); (c) Algorithm 1 (24.25 dB); (d) degraded (Gaussian); (e) AA-like (21.81 dB); (f) Algorithm 1 (22.97 dB).
(a)(b)(c)(d)(e)(f)


## 3.3. Real SAR Images Restoration Experiment

Furthermore, we evaluate the performance of a newly proposed model for despeckling based on three real SAR images. The stopping criterion is the same as that of the denoising experiment. Since there are no original noise-free images to be compared, PSNR and SSIM cannot be used to evaluate the quality of recovered images. AssumingL=5, we tune the parameters until the recovered images show the best results. The parameters settings are reported in Table 5. Figures 14–16 present the denoising results on real SAR images. It is obvious that all of their objective function values are monotonically decreasing. The less speckle is eliminated by Lee filter, SRAD method, and AA-like method. Lee filter does not enhance edges. There are staircasing effects existing in the results of AA-like method. Algorithm 1 can remove noise more effectively than AA-like method. PPB method performs well in restoring images, such as smoother regions and better shape preservation. But there are some spurious artifacts in the edge of the images. Comparing the experimental results, we can find that Algorithm 2 reduces the artifacts more effectively than PPB-AA method. At the same time, the staircasing effects also decrease in these two methods.Table 5
The parameter values used in SAR images restoration experiment.
ImagesMethodsλαβSAR image 1AA-like0.12——Algorithm 10.021/121.1PPB-AA0.12——Algorithm 20.020.11.1SAR image 2AA-like0.15——Algorithm 10.021/121.1PPB-AA0.17——Algorithm 20.011/121.1SAR image 3AA-like0.15——Algorithm 10.021/121.1PPB-AA0.15——Algorithm 20.021/121.1Figure 14
Denoising results on SAR image 1: (a) degraded image; (b) Lee; (c) SRAD; (d) PPB; (e) AA-like; (f) Algorithm 1; (g) PPB-AA; (h) Algorithm 2.
(a)(b)(c)(d)(e)(f)(g)(h)Figure 15
Denoising results on SAR image 2: (a) degraded image; (b) Lee; (c) SRAD; (d) PPB; (e) AA-like; (f) Algorithm 1; (g) PPB-AA; (h) Algorithm 2.
(a)(b)(c)(d)(e)(f)(g)(h)Figure 16
Denoising results on SAR image 3: (a) degraded image; (b) Lee; (c) SRAD; (d) PPB; (e) AA-like; (f) Algorithm 1; (g) PPB-AA; (h) Algorithm 2.
(a)(b)(c)(d)(e)(f)(g)(h)


## 3.4. The Influence of Parameters and Mean

Here, we study the sensitivities of parameters and the effect of mean of our restoration model. According to the analysis in Section2.1, the value of parameter β varies with the level of the noise. Experimentally, β is chosen as 1 when L>5. When L=5, β is set as 1.1. In the proposed model, parameter α needs to satisfy condition α≥1/12. Parameter λ is set as λ2<1/72 based on the description in Section 2.3, which is changing from 0.01 to 0.11 with an interval 0.01 in experiment. In the denoising case, we take the noisy images “Lena” and “Cameraman” as examples. For each image, we vary the values of α with three levels of the noise, L=10,7,5. The PSNR values of restored images using Algorithm 1 are presented in Figure 17. We observe that there is an upmost point on each curve. With the increasing α, the performance of “Lena” with L=5 and “Cameraman” with L=10 decreases rapidly, which means that the denoising results are sensitive to value of α for each L. In the simultaneous deblurring and denoising case, we take the noisy image of “Cameraman” blurred by the motion blur with length 5 and angle 30 and then corrupted by multiplicative Nakagami noise with L=10 as an example. The parameter settings are λ=0.01, α=7, and β=1 in this case known from Table 3. At first, for the case of λ=0.01, we study the sensitivity of the parameter α in terms of solution quality and computation time. The PSNR (dB), SSIM, and CPU time (s) used for various values of α are listed in Table 6. We observe that if α is less than 7, the PSNR value of the restored image by Algorithm 1 increases with the increase in α. However, the PSNR and SSIM values decrease as the value of α increases when α is more than 7. The numerical results indicate that there is a certain value of α to make the model get the best quality of recovered image for a fixed value of λ. Then, for the case of α=7, we further analyze the sensitivity of the parameter λ. The PSNR (dB), SSIM, and CPU time (s) used for various values of λ are listed in Table 7. From Table 7, we observe that λ=0.01 is the best choice for obtaining the highest PSNR value of denoised image.Figure 17
The PSNR values against the values ofα under three noise levels: (a) Lena; (b) Cameraman.
(a)(b)Table 6
Numerical results of Algorithm 1 with parameterλ=0.01 on Cameraman image.
α0.13789PSNR (dB)24.1524.9925.2125.2025.17SSIM0.7620.7760.7500.7380.726Time (s)274395486528529Table 7
Numerical results of Algorithm 1 with parameterα=7 on Cameraman image.
λ0.010.020.050.080.11PSNR (dB)25.2124.8523.7723.1022.62SSIM0.7500.7740.7510.7340.725Time (s)486367246231229Finally, in order to discuss the influence of preserving the mean of the original image, there are four assumptions, including recovered images obtained from model (9) and model (9) with preserving 1 times, 1.2 times, 0.8 times the mean of the original image, respectively. Table 8 reports the PSNR (dB) and SSIM values on Cameraman image with different mean. We find the fact the PSNR (dB) and SSIM values obtained from model (9) with preserving mean of the original image are almost higher than the others. As shown in Figure 18, when the mean of recovered image is larger than that of the original image, the restored image by Algorithm 1 looks lighter. In contrast, the restored image is darker. PSNR and SSIM values of both decrease significantly. Although there is not much difference between the recovered images obtained from preserving the mean and not preserving the mean, the PSNR value is higher when the mean is maintained as the original. These indicate that bias correction technique [36] is helpful to enhance the performance of the model (9).Table 8
Numerical results on Cameraman image with different mean.
MeanPreservedNot preservedIncreasedDecreasedPSNR (dB)25.2124.3419.1118.22SSIM0.7500.7500.7260.732Figure 18
Results on Cameraman image with different mean: (a) preserved (25.21 dB); (b) not preserved (24.34 dB); (c) increased (19.11 dB); (d) decreased (18.22 dB).
(a)(b)(c)(d)


## 4. Conclusion

This paper proposes a new variational model for solving the SAR image restoration problem by investigating the statistical property of multiplicative noise from Nakagami distribution. Under a mild condition, we show that the new model is strictly convex. We further analyze some mathematical properties of the model in detail. We adopt the primal-dual algorithm to solve the convex optimization problem. Two kinds of simulated experiments and real SAR images experiments are conducted. The experimental results demonstrate the excellent performance of the proposed method over some state-of-the-art methods in terms of the noise removal capability and detail preservation capability.



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*Source: 1952782-2020-06-10.xml*