# Speckle Noise Removal by Energy Models with New Regularization Setting

Authors: Bo Chen; Jinbin Zou; Weiqiang Zhang
Journal: Journal of Function Spaces
 (2020)

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## Abstract

In this paper, we introduce two novel total variation models to deal with speckle noise in ultrasound image in order to retain the fine details more effectively and to improve the speed of energy diffusion during the process. Firstly, two new convex functions are introduced as regularization term in the adaptive total variation model, and then, the diffusion performances of Hypersurface Total Variation (HYPTV) model and Logarithmic Total Variation (LOGTV) model are analyzed mathematically through the physical characteristics of local coordinates. We have shown that the larger positive parameter in the model is set, the greater energy diffusion speed appears to be, but it will cause the image to be too smooth that required adequate attention. Numerical experimental results show that our proposed LOGTV model for speckle noise removal is superior to traditional models, not only in visual effect but also in quantitative measures.


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## Body

## 1. Introduction

With the development of digital image technology, a large number of digital images are transmitted and compressed through various channels. However, image corruption usually is unavoidable during transmission and storage, and the resultant noise quite often seriously affects the visual effect of the image. Clearly, high-quality images are desirable in many areas, such as in medical imaging and pattern recognition. Thus, image denoising is a critical step in image processing and computer version, which plays an important role in various applied areas, especially in medical imaging, video processing, and remote sensing. On the other hand, it is also an important preprocessing process for other image processing that relies on subsequent processing.Today, image denoising becomes research of focus, and many image denoising methods have been proposed such as Lee filter [1], Kuan filter [2], locally adaptive statistic filters [3–5], PDE-based and curvature-based methods [6, 7], wavelet transform based thresholding methods [8], and total variational [9–11]. In addition, the method based on machine learning [12–15] has received wide attention in recent years, such as deep learning [12, 13], linear regression [14], and Bayesian learning [13, 15].In 1992, Rudin et al. [10] proposed a denoising model based on the total variation:
(1)Eλ1u=∫ΩDudx+λ12∫Ωu−u02dx,where ∫ΩDudx=sup∫Ωudivφ∣φ∈Cc1Ω,ℝn,φ∞≤1 represents the TV regularization term, u:Ω⟶ℝ is an original image that is without noise, u0=u+n is noisy image, and n represents the Gaussian random noise with mean zero and standard deviation σ. λ1>0 represents the regularization parameter which can be used to balance the fidelity terms and regularized terms in TV model, and Du represents the L1 norm of the image gradient.It is well known that medical ultrasonic images may have many noises of speckle, which will bring a significant problem in terms of the quality of ultrasonic images and cover up the lesions of certain important tissues. Further, it brings great difficulties to the actuate diagnosis and identification of certain specific diseases and can create the potential risk of missed diagnosis and misdiagnosis. Thus, it is very desirable to eliminate the speckle noise in ultrasound image and simultaneously retain the important features in practices. As mentioned in article [3], the speckle noise in medical ultrasonic images can be formulated in the following form:
(2)f=u+un,where f is a noisy image, and n represents the Gaussian random noise with zeros mean and standard deviation σ.In this paper, we focus on the image denoising form by using variation method, where the noisy image is ultrasound speckle noise. Based on the model (2) and the characteristics of the Gaussian distribution, Krissian et al. in the article [16] derived a date fidelity term:
(3)Fu,f=∫Ωf−u2udx.Within the variational framework, the data fidelity function is derived from the degradation model (2). One of the technical approaches to solve the variational model is the regularization technique, which minimizes the cost function to obtain stable and accurate solutions. In general, the image denoising variation method is to consider:
(4)minEuu∈Ω=TVu+λFu,f,where TVu is a regularization term that represents a prior information about the object to be restored, and Fu,f is a fidelity term to ensure that the restoration u is not far from the original observation f. λ>0 represents the regularization parameter which can balance the fidelity term and the regularization term.In [11], motivated by the classical ROF model [10], the authors proposed a convex variational model for removing the speckle noise in ultrasound image. The convex variational model involves the TV regularization term and convex fidelity term (see Equation (5)):
(5)minu∫ΩDudx+λ∫Ωf−u2udx,where ∫ΩDudx represents the TV regularization term, and Du represents the directional gradient of u. The existence and uniqueness of the solution of model (5) is proved in [11]. In this paper, we call the model in (5) the “JIN’s model.” In [17], the authors proposed a well-balanced speckle noise reduction (WBSN) model that can detect edges.Although TV regularization is effective for image denoising, it also leads to some staircase effects that is undesirable. In order to solve this problem, many methods based on improved TV regularization are proposed, such as high-order TV regularization [18–20], several hybrid TV regularization [21], the improved infimal convolution [22, 23], nonlocal TV model [24, 25], fractional order TV model [26], and anisotropic TV model [27, 28]. Fractional theory [29, 30], wavelet [31], and statistical information [32–34] are also employed to deal with intensity inhomogeneity or noise. Although these denoising methods can reduce the staircase effects in the restoration of additive noise images, however, there are more staircase effects that appeared in the restoration of speckle noise images. In this paper, we introduce HYPTV model and LOGTV model to reduce speckle noise and staircase effects in ultrasound images in an effective way.In numerical algorithms, most of the energy function minimization problems can be transformed to an Euler-Lagrange equation and then be solved by using the finite difference method. However, the choice of adequate regularization terms is critical in terms of solution accuracy. Moreover, when solving the Euler-Lagrange equation, the energy diffusion form of the noise image in different regions is required to handle differently based on the physical characteristics of local coordinates, since the diffusion velocity of different parameters to the energy function is different in the two models. The new proposed HYPTV model and the LOGTV model, as shown in this paper, not only can preserve the edges of the restored images well when restoring the ultrasonic image with speckle noise but also better reduce the staircase effect generated during the recovery process.The rest of this paper is as follows: in Section2, we review some background knowledge. In Section 3, we propose two new models based on variation; meanwhile, we not only analyze diffusion performance of the proposed models but also give the corresponding numerical algorithms. Section 4 shows five different experiments and results. The paper ends with concluding remarks in Section 5.


## 2. Preliminary

### 2.1. Some Theoretical Background

The Euler-Lagrange equation was developed in the 1750s by Euler and Lagrange in connection with their studies of the tautochrone problem.Theorem 1.
A multidimensional generalization comes from considering a function onm variables. If Ω is open, bounded Lipschitz domain in Rn, then
(6)Ah=∫ΩLx1,x2,⋯,xm,h,h1,h2,⋯,hmdx,is extremized only if h satisfies the partial differential equation:
(7)∂L∂h−∑i=1n∂∂xi∂L∂hi=0,
For model (5), the corresponding Euler-Lagrange equation is as follow:
(8)div∇u∇u+λf2u2−1=0,where ∇ and div, respectively, represent gradient operators and divergence operators. Using gradient descent method, we can get the model as follows:
(9)ut=div∇u∣∇u∣+λf2u2−1,t>0x,yinΩ,∂u∂n→=0ontheboundarofΩ,ut=0=u0inΩ¯,where n→ is the unit out normal vector of ∂Ω.Without losing generality, in the following, we consider the grayscale images asM×N matrices.Definition 2.
Letu∈U=C22Ω,R,g=g1,g2∈G=C22Ω,R2, the gradient operators on the space U and the divergence operators on the space G are defined as:
(10)∇:U⟶G,∇u=∂x+u,∂y+u,div:G⟶U,divg=∂x−g1+∂y−g2,where ∂x+,∂y+,∂x−,and∂y− are the first-order forward and backward discrete derivation operators in the x-direction and y-direction, respectively, which are defined as:
(11)∂x+u=ui,j+1−ui,j,if1≤j≤N−1,0,ifj=N,∂y+u=ui+1,j−ui,j,if1≤i≤M−1,0,ifi=M,∂x−u=ui,j−ui,j−1,if2≤j≤N,0,ifj=1,∂y−u=ui,j−ui−1,j,if2≤i≤M,0,ifi=1.
Applying the above gradient operators and divergence operators to model (9), we can obtain the equivalent minimization problem.Definition 3.
LetC be a convex subset of R, A function ϕ:C⟶R is called convex if
(12)ϕtx+1−ty≤tϕx+1−tϕy,∀x,y∈C,∀t∈0,1.The following facts are easily checked:Theorem 4.
If functionsϕ1 and ϕ2 are convex and have the same domain definition, then ϕ=ϕ1+ϕ2 is also convex.Proof.
Let functionsϕ1 and ϕ2 are convex. According to the Definition 3, for any x,y∈R and t∈0,1, we have
(13)ϕ1tx+1−ty≤tϕ1x+1−tϕ1y,ϕ2tx+1−ty≤tϕ2x+1−tϕ2y,ϕtx+1−ty=ϕ1tx+1−ty+ϕ2tx+1−ty≤tϕ1x+1−tϕ1y+tϕ2x+1−tϕ2y≤tϕ1x+ϕ2x+1−tϕ1y+ϕ2y=tϕx+1−tϕx.
Henceϕ=ϕ1+ϕ2 is convex.Theorem 5.
If a differentiable functionϕ:C⟶Rsatisfied
(14)ϕ″x≥0,∀x∈C,then ϕ is convex.


### 2.2. The Condition of TV Regularization Term

Although TV regularization is very effective in image restoration, it usually generates some staircase effects. Thus, it is suggested in literature to use general variational methods, i.e., to consider:(15)J1u=∫Ωφ∣∇u∣dx,where φs represents a potential function, and the case φs=s leads to the total variation regularization term. In the literature [35], the author Costanzino chooses φs=s2 that leads to the well-known harmonic model.In practice, we prefer good smoothing in some domain where the intensity of variations is relatively weak. This can be achieved by requiring a functionφs to satisfy the following conditions:
(16)φ′0=0,lims→0+φ″s=lims→0+φ′ss=c>0.Near the edge of the image, the intensity of variations is strong. If we would like to preserve the edge, then the functionφs should satisfy the following conditions:
(17)lims→∞φ″s=lims→∞φ′ss=0,lims→∞sφ″sφ′s=0.With the conditions (16) and (17), the function φs is convex and nondecreasing function, such as:
(18)proportionalfunction:φs=s,s≥0,exponentialfunction:φs=s2,s≥0,hypersurfacefunction:φs=1+α1s2,s≥0,α1>0,logarithmicfunction:φs=slnα2+s,s≥0,α2>0.There functions are convex and nondecreasing ons≥0, as shown in Figure 1.Figure 1
The plots of convex and nondecreasing functions.In this paper, we will use two new functionsφ1s=1+α1s2 and φ2s=slnα2+s, which appear to be quite effective for image processing, in particular, for ultrasound image denoising. Obviously, the functions φ1s and φ2s are convex and nondecreasing.


## 2.1. Some Theoretical Background

The Euler-Lagrange equation was developed in the 1750s by Euler and Lagrange in connection with their studies of the tautochrone problem.Theorem 1.
A multidimensional generalization comes from considering a function onm variables. If Ω is open, bounded Lipschitz domain in Rn, then
(6)Ah=∫ΩLx1,x2,⋯,xm,h,h1,h2,⋯,hmdx,is extremized only if h satisfies the partial differential equation:
(7)∂L∂h−∑i=1n∂∂xi∂L∂hi=0,
For model (5), the corresponding Euler-Lagrange equation is as follow:
(8)div∇u∇u+λf2u2−1=0,where ∇ and div, respectively, represent gradient operators and divergence operators. Using gradient descent method, we can get the model as follows:
(9)ut=div∇u∣∇u∣+λf2u2−1,t>0x,yinΩ,∂u∂n→=0ontheboundarofΩ,ut=0=u0inΩ¯,where n→ is the unit out normal vector of ∂Ω.Without losing generality, in the following, we consider the grayscale images asM×N matrices.Definition 2.
Letu∈U=C22Ω,R,g=g1,g2∈G=C22Ω,R2, the gradient operators on the space U and the divergence operators on the space G are defined as:
(10)∇:U⟶G,∇u=∂x+u,∂y+u,div:G⟶U,divg=∂x−g1+∂y−g2,where ∂x+,∂y+,∂x−,and∂y− are the first-order forward and backward discrete derivation operators in the x-direction and y-direction, respectively, which are defined as:
(11)∂x+u=ui,j+1−ui,j,if1≤j≤N−1,0,ifj=N,∂y+u=ui+1,j−ui,j,if1≤i≤M−1,0,ifi=M,∂x−u=ui,j−ui,j−1,if2≤j≤N,0,ifj=1,∂y−u=ui,j−ui−1,j,if2≤i≤M,0,ifi=1.
Applying the above gradient operators and divergence operators to model (9), we can obtain the equivalent minimization problem.Definition 3.
LetC be a convex subset of R, A function ϕ:C⟶R is called convex if
(12)ϕtx+1−ty≤tϕx+1−tϕy,∀x,y∈C,∀t∈0,1.The following facts are easily checked:Theorem 4.
If functionsϕ1 and ϕ2 are convex and have the same domain definition, then ϕ=ϕ1+ϕ2 is also convex.Proof.
Let functionsϕ1 and ϕ2 are convex. According to the Definition 3, for any x,y∈R and t∈0,1, we have
(13)ϕ1tx+1−ty≤tϕ1x+1−tϕ1y,ϕ2tx+1−ty≤tϕ2x+1−tϕ2y,ϕtx+1−ty=ϕ1tx+1−ty+ϕ2tx+1−ty≤tϕ1x+1−tϕ1y+tϕ2x+1−tϕ2y≤tϕ1x+ϕ2x+1−tϕ1y+ϕ2y=tϕx+1−tϕx.
Henceϕ=ϕ1+ϕ2 is convex.Theorem 5.
If a differentiable functionϕ:C⟶Rsatisfied
(14)ϕ″x≥0,∀x∈C,then ϕ is convex.


## 2.2. The Condition of TV Regularization Term

Although TV regularization is very effective in image restoration, it usually generates some staircase effects. Thus, it is suggested in literature to use general variational methods, i.e., to consider:(15)J1u=∫Ωφ∣∇u∣dx,where φs represents a potential function, and the case φs=s leads to the total variation regularization term. In the literature [35], the author Costanzino chooses φs=s2 that leads to the well-known harmonic model.In practice, we prefer good smoothing in some domain where the intensity of variations is relatively weak. This can be achieved by requiring a functionφs to satisfy the following conditions:
(16)φ′0=0,lims→0+φ″s=lims→0+φ′ss=c>0.Near the edge of the image, the intensity of variations is strong. If we would like to preserve the edge, then the functionφs should satisfy the following conditions:
(17)lims→∞φ″s=lims→∞φ′ss=0,lims→∞sφ″sφ′s=0.With the conditions (16) and (17), the function φs is convex and nondecreasing function, such as:
(18)proportionalfunction:φs=s,s≥0,exponentialfunction:φs=s2,s≥0,hypersurfacefunction:φs=1+α1s2,s≥0,α1>0,logarithmicfunction:φs=slnα2+s,s≥0,α2>0.There functions are convex and nondecreasing ons≥0, as shown in Figure 1.Figure 1
The plots of convex and nondecreasing functions.In this paper, we will use two new functionsφ1s=1+α1s2 and φ2s=slnα2+s, which appear to be quite effective for image processing, in particular, for ultrasound image denoising. Obviously, the functions φ1s and φ2s are convex and nondecreasing.


## 3. The Proposed Restoration Model

In this section, we propose adaptive total variation model for image restoration. We use the finite difference method to solve the Euler-Lagrange equation directly, and then find the minimum value of the energy function.


### 3.1. The Adaptive Total Variation Model

Apply the selected function to model (15), we propose adaptive total variation model,
(19)minuEu=∫Ωφi∇udx+λ∫Ωf−u2udx,where ∫Ωφi∣∇u∣dx is a regularization term, and i=1,2; ∫Ωf−u2/udx is a fidelity term; λ represents the regularization parameter which can balance fidelity term and regularization term.Firstly, the energy functionalEu is convex, which guarantees the existence of the minimal solution of the model (19).Theorem 6.
The energy functionalEu is convex. That is to say, for any u1,u2∈Ω, and t∈0,1, we have:
(20)Etu1+1−tu2≤tEu1+1−tEu2,where Eu=∫Ωφi∣∇u∣dx+λ∫Ωf−u2/udx.Proof.
Firstly, the functionφis is convex; according to Definition 3, for any u1,u2∈Ω, and t∈0,1, we have
(21)∫Ωφi∇tu1+1−tu2dx≤t∫Ωφi∇u1dx+1−t∫Ωφi∇u2dx.Meanwhile, we have (The proof step of the inequality is in the appendix.)
(22)∫Ωf−tu1+1−tu22tu1+1−tu2dx=∫Ωf2tu1+1−tu2−2f+tu1+1−tu2dx≤∫Ωf2tu2+1−tu1u1u2−2f+tu1+1−tu2dx=∫Ωtf2u1−2tf+tu1+1−tf2u2−21−tf+1−tu2dx=∫Ωtf−u12u1+1−tf−u22u2dx.
Therefore,(23)Etu1+1−tu2≤tEu1+1−tEu2.
This proof is established.
Secondly, the uniqueness of the minimum solution of the model (19) can also be proved.Theorem 7.
Ifu1 and u2 are two minimize solutions of model (19), then we have u1=u2.Proof.
According to Theorem6, we have
(24)Eu1+u22=∫Ωφi∇12u1+12u2dx+λ∫Ωf−u1/2+u2/22u1+u2/2dx≤∫Ω12φi∇u1+12φi∇u2+λ∫Ωf21/4u1+u22−1/4u1−u221/2u1+1/2u2u1u2−2f+12u1+12u2dx≤12∫Ωφi∇u1dx+12∫Ωφi∇u2dx+λ∫Ωf−u122u1+f−u222u2−f2u1−u222u1+u2u1u2dx=12Eu1+12Eu2−λ∫Ωf2u1−u222u1+u2u1u2dx=Eu1−λ∫Ωf2u1−u222u1+u2u1u2dx.
Ifu1≠u2, then the above assumption gives a contradiction that u1 is not a minimize solution.


### 3.2. Diffusion Performance

In this subsection, we mainly analyze the diffusion performance and diffusion speed of the energy function of HYPTV model and LOGTV model.


#### 3.2.1. Diffusion Performance of HYPTV Model

Firstly, we use the finite difference method to solve the HYPTV model, associated with the potential functionφ1s=1+α1s2.From Definition2, we can obtain the corresponding Euler-Lagrange equation HYPTV model that as follows:
(25)divα11+α1∇u2∇u+λf2u2−1=0.Using gradient descent method, Equation (25) can be transformed to:
(26)ut=divα11+α1∇u2∇u+λf2u2−1inΩ,∂u∂n→=0ontheboundarofΩ,ut=0=u0inΩ¯,where n→ is the unit out normal vector of ∂Ω.In order to analyze the diffusion performance, local image coordinate systemξ‐η is established. As shown in Figure 2, the η-axis represents the direction parallel to the image gradient at the pixel level, and the ξ-axis is the corresponding vertical direction.Figure 2
Global and local coordinate schematic diagram.According to Figure2, we can know:
(27)ξ=1∣∇u∣−uy,ux,η=1∣∇u∣ux,uy.So, Equation (26) can be rewritten as:
(28)ut=ψ11∣∇u∣uξξ+ψ21∣∇u∣uηη+λf2u2−1,where
(29)ψ11∣∇u∣=α11+α1∇u2,ψ21∣∇u∣=α11+α1∇u23/2,uξξ=uy2uxx−2uxuyuxy+ux2uyy∇u2,uηη=ux2uxx+2uxuyuxy+uy2uyy∇u2.Theψ11∣∇u∣ and ψ21∣∇u∣ are control functions of the diffusion along the ξ-direction and η-direction, respectively. Now, we consider the diffusion of image restoration. Some test images are shown in Figure 3.Figure 3
Test images: (a) image 1 (256×256); (b) image 2 (256×256); (c) house (256×256); (d) pirate (512×512); (e) peppers (256×256); (f) boat (512×512); (g) bird (256×256).
(a)(b)(c)(d)(e)(f)(g)(1) Smooth area. When ∣∇u∣⟶0, lim∣∇u∣→0ψ11∣∇u∣=lim∣∇u∣→0ψ21∣∇u∣=α1. This shows that the diffusion form of the energy Equation (19) is isotropic. In other words, the energy diffusion rate along direction ξ and direction η is very close in the process of image restoration in the smooth region. And the rate of energy diffusion is obviously positively correlated with the parameter α1.(2) Sharp area. When ∣∇u∣⟶∞, we obtain lim∣∇u∣→∞ψ21∣∇u∣/ψ11(∣∇u∣)=0. This shows that the diffusion form of the energy Equation (19) is anisotropic. In other words, the energy diffusion rate in ξ-direction in Equation (28) is much larger than that in the η-direction in the sharp region. But the gradient ∣∇u∣ does not exceed 255, so lim∣∇u∣→255ψ21∣∇u∣/ψ11(∣∇u∣)=lim∣∇u∣→2551/1+α1∇u2=1/1+α1×2552. One can see that the larger the parameter α1 is set, the smaller the limit becomes. And the rate of energy diffusion is obviously positively correlated with the parameter α1.


#### 3.2.2. Diffusion Performance of LOGTV Model

Secondly, we use the finite difference method to solve the LOGTV model, which the potential function isφ2s=slnα2+s.From the Definition2, we can obtain the corresponding Euler-Lagrange equation LOGTV model that as follows:
(30)divlnα2+∣∇u∣∣∇u∣+1α2+∣∇u∣∇u+λf2u2−1=0.Using gradient descent method, Equation (30) can be transformed to:
(31)ut=divlnα2+∣∇u∣∣∇u∣+1α2+∣∇u∣∇u+αf2u2−1,where n→ is the unit out normal vector of ∂Ω.Hence, Equation (31) can be rewritten as:
(32)ut=ψ12∣∇u∣uξξ+ψ22∣∇u∣uηη+λf2u2−1,where
(33)ψ12∣∇u∣=lnα2+∣∇u∣∣∇u∣+1α2+∣∇u∣,ψ22∣∇u∣=1α2+∣∇u∣+α2α2+∣∇u∣2.ψ12∣∇u∣ and ψ22∣∇u∣ are control functions of the diffusion along the ξ-direction and η-direction, respectively. Now, we consider the diffusion of image restoration.(1) Smooth area. When α2=1, and ∣∇u∣⟶0, we can obtain lim∣∇u∣→0ψ12∣∇u∣=2 and lim∣∇u∣→0ψ22∣∇u∣=2. This shows that the diffusion form of the energy Equation (19) is isotropic. In other words, the energy diffusion rate along direction ξ and direction η is very close in the process of image restoration in the smooth region. When α2≠1, and ∣∇u∣⟶0, we can obtain lim∣∇u∣→0ψ1∣∇u∣=∞ and lim∣∇u∣→0ψ2∣∇u∣=2/α2. This shows that the diffusion form of the energy Equation (19) is anisotropic. However, the gradient of noise image is relatively large, so in the smooth region, whatever the value of α2, it has little effect on the model.(2) Sharp area. When ∣∇u∣⟶∞, we can obtain lim∣∇u∣→∞ψ22∣∇u∣/ψ12(∣∇u∣)=0. This shows that the diffusion form of the energy Equation (19) is anisotropic. In other words, the energy diffusion rate in ξ-direction in Equation (28) is much larger than that in the η-direction in the sharp region. But the gradient ∣∇u∣ does not exceed 255, so lim∣∇u∣→255ψ2∣∇u∣/ψ1(∣∇u∣)=510α2+2552/225α2+2552+α2+2552∗lnα2+255. One can see that the larger the parameter α2 is set, the smaller the limit becomes. And the rate of energy diffusion is clearly positively correlated with the parameter α2.


### 3.3. Numerical Implementation

We will describe the corresponding numerical algorithm in this section. Firstly, the HYPTV model can be solved by discretization as follows:(34)ui,jk+1=ui,jk+ΔtdivT1∇uk∇uki,j+λkf2uk2−1i,j,where T1∇uk=α1/1+α1∇uk2, and Δt represents time step. Furthermore, the iterative formula can approximate as:
(35)ui,jk+1=ui,jk+ΔtA1∇uki,j+λkf2uk2−1i,j,for i=1,⋯,M; j=1,⋯,N, and M×N represent the size of the image. Here:
(36)A1∇uki,j=∂x−T1∇xuk∂x+uki,j+∂y−T1∇yuk∂y+uki,j,∇xui,j=∂x+ui,j2+m∂y+ui,j,∂y−ui,j2+δ,∇yui,j=∂y+ui,j2+m∂x+ui,j,∂x−ui,j2+δ,where ma,b=signa+signb/2⋅mina,b, and δ>0 is a positive parameter that is close to zero. With boundary conditions:
(37)u0,jk=u1,jk;uN+1,jk=uN,jk,ui,0k=ui,1k;ui,N+1k=ui,Nk.Now, note Equation (25), the two sides are multiplied by f−uu/f+u, and then, the integral on the domain Ω can be obtained:
(38)λ∫Ωf−u2u=∫Ωdivα11+α1∇u2∇uu−fuu+f,Because the Gaussian noisen have mean 0 and variance σ2, we can obtain:
(39)λk=1σ2Ω∑i,jA1∇ukuk−fukuk+f,In the process of iteration, we always use the previous solution to calculate the next solution. The optimization algorithm for HYPTV model is given in the following (Algorithm1).Algorithm 1
1:Initializeλ0=0, u0=f, k=02:Repeat3:While NDR>limit do4:  Updateuk by (35)5:  Updateλk by (39)6:  Computer NDR7:  Setk=k+18:End While9:Final Input:uSecondly, the LOGTV model can be solve by discretization as follows:(40)ui,jk+1=ui,jk+ΔtdivT2∇uk∇uki,j+λkf2uk2−1i,j,where T2∇uk=lnα2+∇uk/∇uk+1/α2+∇uk. Furthermore, the iterative formula can approximate as:
(41)ui,jk+1=ui,jk+ΔtA2∇uki,j+λkf2uk2−1i,j,(42)A2∇uki,j=∂x−T2∇x+uk∂x+uki,j+∂y−T2∇y+uk∂y+uki,j,(43)λk=1σ2Ω∑i,jA2∇ukuk−fuk+f.Similar to the HYPTV model, the optimization algorithm for LOGTV model is given in the following (Algorithm2).Algorithm 2
1:Initializeλ0=0, u0=f, k=02:Repeat3:While NDR>limit do4:  Updateuk by (41)5:  Updateλk by (43)6:  Computer NDR7:  Setk=k+18:End While9:Final Input:u


## 3.1. The Adaptive Total Variation Model

Apply the selected function to model (15), we propose adaptive total variation model,
(19)minuEu=∫Ωφi∇udx+λ∫Ωf−u2udx,where ∫Ωφi∣∇u∣dx is a regularization term, and i=1,2; ∫Ωf−u2/udx is a fidelity term; λ represents the regularization parameter which can balance fidelity term and regularization term.Firstly, the energy functionalEu is convex, which guarantees the existence of the minimal solution of the model (19).Theorem 6.
The energy functionalEu is convex. That is to say, for any u1,u2∈Ω, and t∈0,1, we have:
(20)Etu1+1−tu2≤tEu1+1−tEu2,where Eu=∫Ωφi∣∇u∣dx+λ∫Ωf−u2/udx.Proof.
Firstly, the functionφis is convex; according to Definition 3, for any u1,u2∈Ω, and t∈0,1, we have
(21)∫Ωφi∇tu1+1−tu2dx≤t∫Ωφi∇u1dx+1−t∫Ωφi∇u2dx.Meanwhile, we have (The proof step of the inequality is in the appendix.)
(22)∫Ωf−tu1+1−tu22tu1+1−tu2dx=∫Ωf2tu1+1−tu2−2f+tu1+1−tu2dx≤∫Ωf2tu2+1−tu1u1u2−2f+tu1+1−tu2dx=∫Ωtf2u1−2tf+tu1+1−tf2u2−21−tf+1−tu2dx=∫Ωtf−u12u1+1−tf−u22u2dx.
Therefore,(23)Etu1+1−tu2≤tEu1+1−tEu2.
This proof is established.
Secondly, the uniqueness of the minimum solution of the model (19) can also be proved.Theorem 7.
Ifu1 and u2 are two minimize solutions of model (19), then we have u1=u2.Proof.
According to Theorem6, we have
(24)Eu1+u22=∫Ωφi∇12u1+12u2dx+λ∫Ωf−u1/2+u2/22u1+u2/2dx≤∫Ω12φi∇u1+12φi∇u2+λ∫Ωf21/4u1+u22−1/4u1−u221/2u1+1/2u2u1u2−2f+12u1+12u2dx≤12∫Ωφi∇u1dx+12∫Ωφi∇u2dx+λ∫Ωf−u122u1+f−u222u2−f2u1−u222u1+u2u1u2dx=12Eu1+12Eu2−λ∫Ωf2u1−u222u1+u2u1u2dx=Eu1−λ∫Ωf2u1−u222u1+u2u1u2dx.
Ifu1≠u2, then the above assumption gives a contradiction that u1 is not a minimize solution.


## 3.2. Diffusion Performance

In this subsection, we mainly analyze the diffusion performance and diffusion speed of the energy function of HYPTV model and LOGTV model.


### 3.2.1. Diffusion Performance of HYPTV Model

Firstly, we use the finite difference method to solve the HYPTV model, associated with the potential functionφ1s=1+α1s2.From Definition2, we can obtain the corresponding Euler-Lagrange equation HYPTV model that as follows:
(25)divα11+α1∇u2∇u+λf2u2−1=0.Using gradient descent method, Equation (25) can be transformed to:
(26)ut=divα11+α1∇u2∇u+λf2u2−1inΩ,∂u∂n→=0ontheboundarofΩ,ut=0=u0inΩ¯,where n→ is the unit out normal vector of ∂Ω.In order to analyze the diffusion performance, local image coordinate systemξ‐η is established. As shown in Figure 2, the η-axis represents the direction parallel to the image gradient at the pixel level, and the ξ-axis is the corresponding vertical direction.Figure 2
Global and local coordinate schematic diagram.According to Figure2, we can know:
(27)ξ=1∣∇u∣−uy,ux,η=1∣∇u∣ux,uy.So, Equation (26) can be rewritten as:
(28)ut=ψ11∣∇u∣uξξ+ψ21∣∇u∣uηη+λf2u2−1,where
(29)ψ11∣∇u∣=α11+α1∇u2,ψ21∣∇u∣=α11+α1∇u23/2,uξξ=uy2uxx−2uxuyuxy+ux2uyy∇u2,uηη=ux2uxx+2uxuyuxy+uy2uyy∇u2.Theψ11∣∇u∣ and ψ21∣∇u∣ are control functions of the diffusion along the ξ-direction and η-direction, respectively. Now, we consider the diffusion of image restoration. Some test images are shown in Figure 3.Figure 3
Test images: (a) image 1 (256×256); (b) image 2 (256×256); (c) house (256×256); (d) pirate (512×512); (e) peppers (256×256); (f) boat (512×512); (g) bird (256×256).
(a)(b)(c)(d)(e)(f)(g)(1) Smooth area. When ∣∇u∣⟶0, lim∣∇u∣→0ψ11∣∇u∣=lim∣∇u∣→0ψ21∣∇u∣=α1. This shows that the diffusion form of the energy Equation (19) is isotropic. In other words, the energy diffusion rate along direction ξ and direction η is very close in the process of image restoration in the smooth region. And the rate of energy diffusion is obviously positively correlated with the parameter α1.(2) Sharp area. When ∣∇u∣⟶∞, we obtain lim∣∇u∣→∞ψ21∣∇u∣/ψ11(∣∇u∣)=0. This shows that the diffusion form of the energy Equation (19) is anisotropic. In other words, the energy diffusion rate in ξ-direction in Equation (28) is much larger than that in the η-direction in the sharp region. But the gradient ∣∇u∣ does not exceed 255, so lim∣∇u∣→255ψ21∣∇u∣/ψ11(∣∇u∣)=lim∣∇u∣→2551/1+α1∇u2=1/1+α1×2552. One can see that the larger the parameter α1 is set, the smaller the limit becomes. And the rate of energy diffusion is obviously positively correlated with the parameter α1.


### 3.2.2. Diffusion Performance of LOGTV Model

Secondly, we use the finite difference method to solve the LOGTV model, which the potential function isφ2s=slnα2+s.From the Definition2, we can obtain the corresponding Euler-Lagrange equation LOGTV model that as follows:
(30)divlnα2+∣∇u∣∣∇u∣+1α2+∣∇u∣∇u+λf2u2−1=0.Using gradient descent method, Equation (30) can be transformed to:
(31)ut=divlnα2+∣∇u∣∣∇u∣+1α2+∣∇u∣∇u+αf2u2−1,where n→ is the unit out normal vector of ∂Ω.Hence, Equation (31) can be rewritten as:
(32)ut=ψ12∣∇u∣uξξ+ψ22∣∇u∣uηη+λf2u2−1,where
(33)ψ12∣∇u∣=lnα2+∣∇u∣∣∇u∣+1α2+∣∇u∣,ψ22∣∇u∣=1α2+∣∇u∣+α2α2+∣∇u∣2.ψ12∣∇u∣ and ψ22∣∇u∣ are control functions of the diffusion along the ξ-direction and η-direction, respectively. Now, we consider the diffusion of image restoration.(1) Smooth area. When α2=1, and ∣∇u∣⟶0, we can obtain lim∣∇u∣→0ψ12∣∇u∣=2 and lim∣∇u∣→0ψ22∣∇u∣=2. This shows that the diffusion form of the energy Equation (19) is isotropic. In other words, the energy diffusion rate along direction ξ and direction η is very close in the process of image restoration in the smooth region. When α2≠1, and ∣∇u∣⟶0, we can obtain lim∣∇u∣→0ψ1∣∇u∣=∞ and lim∣∇u∣→0ψ2∣∇u∣=2/α2. This shows that the diffusion form of the energy Equation (19) is anisotropic. However, the gradient of noise image is relatively large, so in the smooth region, whatever the value of α2, it has little effect on the model.(2) Sharp area. When ∣∇u∣⟶∞, we can obtain lim∣∇u∣→∞ψ22∣∇u∣/ψ12(∣∇u∣)=0. This shows that the diffusion form of the energy Equation (19) is anisotropic. In other words, the energy diffusion rate in ξ-direction in Equation (28) is much larger than that in the η-direction in the sharp region. But the gradient ∣∇u∣ does not exceed 255, so lim∣∇u∣→255ψ2∣∇u∣/ψ1(∣∇u∣)=510α2+2552/225α2+2552+α2+2552∗lnα2+255. One can see that the larger the parameter α2 is set, the smaller the limit becomes. And the rate of energy diffusion is clearly positively correlated with the parameter α2.


## 3.2.1. Diffusion Performance of HYPTV Model

Firstly, we use the finite difference method to solve the HYPTV model, associated with the potential functionφ1s=1+α1s2.From Definition2, we can obtain the corresponding Euler-Lagrange equation HYPTV model that as follows:
(25)divα11+α1∇u2∇u+λf2u2−1=0.Using gradient descent method, Equation (25) can be transformed to:
(26)ut=divα11+α1∇u2∇u+λf2u2−1inΩ,∂u∂n→=0ontheboundarofΩ,ut=0=u0inΩ¯,where n→ is the unit out normal vector of ∂Ω.In order to analyze the diffusion performance, local image coordinate systemξ‐η is established. As shown in Figure 2, the η-axis represents the direction parallel to the image gradient at the pixel level, and the ξ-axis is the corresponding vertical direction.Figure 2
Global and local coordinate schematic diagram.According to Figure2, we can know:
(27)ξ=1∣∇u∣−uy,ux,η=1∣∇u∣ux,uy.So, Equation (26) can be rewritten as:
(28)ut=ψ11∣∇u∣uξξ+ψ21∣∇u∣uηη+λf2u2−1,where
(29)ψ11∣∇u∣=α11+α1∇u2,ψ21∣∇u∣=α11+α1∇u23/2,uξξ=uy2uxx−2uxuyuxy+ux2uyy∇u2,uηη=ux2uxx+2uxuyuxy+uy2uyy∇u2.Theψ11∣∇u∣ and ψ21∣∇u∣ are control functions of the diffusion along the ξ-direction and η-direction, respectively. Now, we consider the diffusion of image restoration. Some test images are shown in Figure 3.Figure 3
Test images: (a) image 1 (256×256); (b) image 2 (256×256); (c) house (256×256); (d) pirate (512×512); (e) peppers (256×256); (f) boat (512×512); (g) bird (256×256).
(a)(b)(c)(d)(e)(f)(g)(1) Smooth area. When ∣∇u∣⟶0, lim∣∇u∣→0ψ11∣∇u∣=lim∣∇u∣→0ψ21∣∇u∣=α1. This shows that the diffusion form of the energy Equation (19) is isotropic. In other words, the energy diffusion rate along direction ξ and direction η is very close in the process of image restoration in the smooth region. And the rate of energy diffusion is obviously positively correlated with the parameter α1.(2) Sharp area. When ∣∇u∣⟶∞, we obtain lim∣∇u∣→∞ψ21∣∇u∣/ψ11(∣∇u∣)=0. This shows that the diffusion form of the energy Equation (19) is anisotropic. In other words, the energy diffusion rate in ξ-direction in Equation (28) is much larger than that in the η-direction in the sharp region. But the gradient ∣∇u∣ does not exceed 255, so lim∣∇u∣→255ψ21∣∇u∣/ψ11(∣∇u∣)=lim∣∇u∣→2551/1+α1∇u2=1/1+α1×2552. One can see that the larger the parameter α1 is set, the smaller the limit becomes. And the rate of energy diffusion is obviously positively correlated with the parameter α1.


## 3.2.2. Diffusion Performance of LOGTV Model

Secondly, we use the finite difference method to solve the LOGTV model, which the potential function isφ2s=slnα2+s.From the Definition2, we can obtain the corresponding Euler-Lagrange equation LOGTV model that as follows:
(30)divlnα2+∣∇u∣∣∇u∣+1α2+∣∇u∣∇u+λf2u2−1=0.Using gradient descent method, Equation (30) can be transformed to:
(31)ut=divlnα2+∣∇u∣∣∇u∣+1α2+∣∇u∣∇u+αf2u2−1,where n→ is the unit out normal vector of ∂Ω.Hence, Equation (31) can be rewritten as:
(32)ut=ψ12∣∇u∣uξξ+ψ22∣∇u∣uηη+λf2u2−1,where
(33)ψ12∣∇u∣=lnα2+∣∇u∣∣∇u∣+1α2+∣∇u∣,ψ22∣∇u∣=1α2+∣∇u∣+α2α2+∣∇u∣2.ψ12∣∇u∣ and ψ22∣∇u∣ are control functions of the diffusion along the ξ-direction and η-direction, respectively. Now, we consider the diffusion of image restoration.(1) Smooth area. When α2=1, and ∣∇u∣⟶0, we can obtain lim∣∇u∣→0ψ12∣∇u∣=2 and lim∣∇u∣→0ψ22∣∇u∣=2. This shows that the diffusion form of the energy Equation (19) is isotropic. In other words, the energy diffusion rate along direction ξ and direction η is very close in the process of image restoration in the smooth region. When α2≠1, and ∣∇u∣⟶0, we can obtain lim∣∇u∣→0ψ1∣∇u∣=∞ and lim∣∇u∣→0ψ2∣∇u∣=2/α2. This shows that the diffusion form of the energy Equation (19) is anisotropic. However, the gradient of noise image is relatively large, so in the smooth region, whatever the value of α2, it has little effect on the model.(2) Sharp area. When ∣∇u∣⟶∞, we can obtain lim∣∇u∣→∞ψ22∣∇u∣/ψ12(∣∇u∣)=0. This shows that the diffusion form of the energy Equation (19) is anisotropic. In other words, the energy diffusion rate in ξ-direction in Equation (28) is much larger than that in the η-direction in the sharp region. But the gradient ∣∇u∣ does not exceed 255, so lim∣∇u∣→255ψ2∣∇u∣/ψ1(∣∇u∣)=510α2+2552/225α2+2552+α2+2552∗lnα2+255. One can see that the larger the parameter α2 is set, the smaller the limit becomes. And the rate of energy diffusion is clearly positively correlated with the parameter α2.


## 3.3. Numerical Implementation

We will describe the corresponding numerical algorithm in this section. Firstly, the HYPTV model can be solved by discretization as follows:(34)ui,jk+1=ui,jk+ΔtdivT1∇uk∇uki,j+λkf2uk2−1i,j,where T1∇uk=α1/1+α1∇uk2, and Δt represents time step. Furthermore, the iterative formula can approximate as:
(35)ui,jk+1=ui,jk+ΔtA1∇uki,j+λkf2uk2−1i,j,for i=1,⋯,M; j=1,⋯,N, and M×N represent the size of the image. Here:
(36)A1∇uki,j=∂x−T1∇xuk∂x+uki,j+∂y−T1∇yuk∂y+uki,j,∇xui,j=∂x+ui,j2+m∂y+ui,j,∂y−ui,j2+δ,∇yui,j=∂y+ui,j2+m∂x+ui,j,∂x−ui,j2+δ,where ma,b=signa+signb/2⋅mina,b, and δ>0 is a positive parameter that is close to zero. With boundary conditions:
(37)u0,jk=u1,jk;uN+1,jk=uN,jk,ui,0k=ui,1k;ui,N+1k=ui,Nk.Now, note Equation (25), the two sides are multiplied by f−uu/f+u, and then, the integral on the domain Ω can be obtained:
(38)λ∫Ωf−u2u=∫Ωdivα11+α1∇u2∇uu−fuu+f,Because the Gaussian noisen have mean 0 and variance σ2, we can obtain:
(39)λk=1σ2Ω∑i,jA1∇ukuk−fukuk+f,In the process of iteration, we always use the previous solution to calculate the next solution. The optimization algorithm for HYPTV model is given in the following (Algorithm1).Algorithm 1
1:Initializeλ0=0, u0=f, k=02:Repeat3:While NDR>limit do4:  Updateuk by (35)5:  Updateλk by (39)6:  Computer NDR7:  Setk=k+18:End While9:Final Input:uSecondly, the LOGTV model can be solve by discretization as follows:(40)ui,jk+1=ui,jk+ΔtdivT2∇uk∇uki,j+λkf2uk2−1i,j,where T2∇uk=lnα2+∇uk/∇uk+1/α2+∇uk. Furthermore, the iterative formula can approximate as:
(41)ui,jk+1=ui,jk+ΔtA2∇uki,j+λkf2uk2−1i,j,(42)A2∇uki,j=∂x−T2∇x+uk∂x+uki,j+∂y−T2∇y+uk∂y+uki,j,(43)λk=1σ2Ω∑i,jA2∇ukuk−fuk+f.Similar to the HYPTV model, the optimization algorithm for LOGTV model is given in the following (Algorithm2).Algorithm 2
1:Initializeλ0=0, u0=f, k=02:Repeat3:While NDR>limit do4:  Updateuk by (41)5:  Updateλk by (43)6:  Computer NDR7:  Setk=k+18:End While9:Final Input:u


## 4. Experimental Results

In this section, we present numerical results to demonstrate the effectiveness of the HYPTV and LOGTV model in image restoration. Firstly, to evaluate the quality of restored images, we use the peak signal-to-noise ratio (PSNR) value and the structure similarity (SSIM) index, which are defined as follows:(44)PSNRu,u¯=10log102552mnu−u¯22,SSIMu,u¯=2μu¯μu+c1σu¯u+c2μu¯2+μu2+c1σu¯2+σu2+c2,where u∈ℝm×n is the clean image, and u¯∈ℝm×n is the restored image. μa is the average of a, σa is the standard deviation of a, and c1 and c2 are some constants for stability. Secondly, we calculated the noise deviation reduction (NDR) at each iteration as a convergence condition;
(45)NDR=meanf−uk2uk,where uk represents the results of the kth iterations, respectively. Finally, the stopping condition (NDR) for the HYPTV and LOGTV models is as follows:
(46)NDR−σ≤10−1,In the numerical experiment, we will use the noise image as the initial value, that is,f=u0. Moreover, the gray values of all original images are in range [0, 255].


### 4.1. Denoising Effect of Different Parameters of HYPTV Model and LOGTV Model

In this example, we use different parameterαii=1,2 values in the algorithm to test the effect of HYPTV and LOGTV models. The test images is shown in Figure 3(a), and the noise levels are σ=2 and σ=3. Figure 4 shows the different PSNR values and iteration numbers when different β values are used in the HYPTV model algorithm. Figure 5 shows the different PSNR values and iteration numbers when different β values are used in the LOGTV model algorithm. Firstly, we can also see that PSNR is the largest in αii=1,2=15. Secondly, with the increase of parameters, the speed of image restoration is faster. Based on the above analysis, when αii=1,2=15, the denoising performance of HPYTV and LOGTV models is close to the best. Therefore, in the following experiment, we choose αii=1,2=15 in HPYTV and LOGTV models.Figure 4
The PSNR values and iteration times of different parameterα2 (HYPTV model).Figure 5
The PSNR values and iteration times of different parameterα2 (LOGTV model).


### 4.2. Denoising Effect of the HYPTV and LOGTV Model

In this subsection, we use the algorithm to test the effect of the HYPTV and LOGTV model on image denoising. In Figures6 and 7, we show the original images, the noise images, and the restored images by HYPTV and LOGTV models. Figures 6(a) and 7(a) are original images, Figures 6(b) and 7(b) are noisy images, which the noise levels are σ=2 and σ=3, respectively. Figures 6(c) and 7(c) are corresponding restored images by HYPTV model. Figures 6(d) and 7(d) are corresponding restored images by LOGTV model. Table 1 shows that the PSNR values, SSIM values, and the numbers of iteration for the different test images can be got by using the HYPTV model and the LOGTV model. “Noise image PSNR” is the peak signal-to-noise ratio of noisy images and original images. “Denoising image PSNR” is the peak signal-to-noise ratio of restored images and original images. “Iter” is the number of iterations of the algorithm. From the results, it is obvious that the HYPTV model and the LOGTV model are fairly effective in reducing the speckle noise and edge-preserving.Figure 6
Numerical result of the “image 1” image with noise standard deviationσ=2. (a) Original image (image 1); (b) noisy image; (c) restored image by HYPTV model α1=15; (d) restored image by LOGTV model α2=15.
(a)(b)(c)(d)Figure 7
Numerical result of the “image 2” image with noise standard deviationσ=3. (a) Original image (image 2); (b) noisy image; (c) restored image by HYPTV model α1=15; (d) restored image by LOGTV model α2=15.
(a)(b)(c)(d)Table 1
Numerical result of the “image 1” and “image 2” images by the HYPTV and LOGTV model.
ImageσNoise image PSNRHYPTV (PSNR/SSIM)LOGTV (PSNR/SSIM)Image 1223.7632.31/0.778632.98/0.8025Image 2223.2634.78/0.836035.31/0.8562Image 1320.2431.10/0.780631.57/0.7948Image 2319.7633.47/0.830034.07/0.8568Image 1417.7729.97/0.792030.45/0.8113Image 2417.5132.51/0.864932.63/0.8667


### 4.3. Reduction of Staircase

In this subsection, we test the reduction of the staircase effect in HYPTV model and LOGTV model by image in Figure8. Figure 8(a) shows original images (“house”). Figure 8(b) shows detailed images of Figure 8(a), respectively. Figure 8(c) shows noise images which noise standard deviation σ=3. Figure 8(d) shows detailed images of Figure 8(c). The original images contain a lot of details information, such as textures edges and in homogeneous regions.Figure 8
Numerical result of the “lena” and “house” images with noise standard deviationσ=3. (a) Original images (house); (b) about the detailed image of (a); (c) noisy images; (d) about the detailed image of (c).
(a)(b)(c)(d)Figure9 displays the restoration results of the noisy “house” image, respectively. Figures 9(a)–9(e) was restored ROF model [10], ATV model [36], JIN’s model [11], HYPTV model, and LOGTV model, respectively. Figures 9(f)–9(j) is corresponding details images.Figure 9
Numerical result of the “house” images with noise standard deviationσ=3. (a) Restored image by the ROF model; (b) restored image by the ATV model; (c) restored image by the JIN’s model; (d) restored image by HYPTV model; (e) restored image by LOGTV model; (f), (g), (h), (i), and (j) about the detailed image of (a), (b), (c), (d), and (e), respectively.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)According to the results, we can see that JIN’s model, HYPTV model, and LOGTV model had a good effect in removing image noise and preserving image edges. In addition, we can clearly see staircase effect in the detailed images obtained by ROF model, ATV model, and JIN’s model. At the same time, the staircase effect of HYPTV model and LOGTV model restored images is reduced. Although both the HYPTV model and the LOGTV model can reduce the ladder effect, the LOGTV model is better than the HYPTV model.


### 4.4. Comparison with ROF Model, ATV Model, and JIN’s Model

In this subsection, we compare the effect of the ROF model, ATV model, and JIN’s model with HYPTV and LOGTV models for some images. The test original images are shown in Figure3(“house,” “peppers,” “boat,” “pirate,” and “bird”), with two sizes of 512×512 and three sizes of 256×256. Figure 10 shows some noise images with different standard deviation.Figure 10
Noisy images. (a) Boat (σ=3); (b) pirate (σ=4); (c) peppers (σ=2); (d) house (σ=3); (e) bird (σ=4).
(a)(b)(c)(d)(e)Figures11–15 display the restoration results for images (“peppers,” “boat,” “house,” “pirate,” and “bird”) through ROF model, ATV model, JIN’s model, HYPTV model, and LOGTV model. The noise versions of “peppers”, “boat” and “house”, and “pirate” and “bird” are obtained by model (2) with standard deviations 2, 3, and 4, respectively. In addition, the detailed images of the restored images are also displayed. Table 2 shows the PSNR and SSIM values for different test images by using the ROF model, ATV model, JIN’s model, HYPTV model, and LOGTV model. From the results of Figures 11–15 and Table 2, the noise standard deviation σ=2. We can observe that although the four models can effectively remove the noises while preserving the edges and details, the restored images by HYPTV model and LOGTV model have better visual effect with less staircase effects than by the ROF model, ATV model, and JIN’s model. The noise standard deviation σ=3,4. The visual effect of restored images by the ROF model and ATV model is particularly poor, but JIN’s model, HYPTV model, and LOGTV model can effectively remove the noises. Finally, Table 2 shows that LOGTV model has higher PSNR and SSIM values than other four models. This means that our proposed LOGTV model is available in reducing the speckle noise in some images.Figure 11
Numerical result of the “peppers” image with noise standard deviationσ=2; (a) restored image by the ROF model; (b) restored image by the ATV model; (c) restored image by the JIN’s model; (d) restored image by HYPTV model; (e) restored image by LOGTV model; (f), (g), (h), (i), and (j) about the detailed image of (a), (b), (c), (d) and (e), respectively.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)Figure 12
Numerical result of the “boat” image with noise standard deviationσ=3. (a) Restored image by the ROF model; (b) restored image by the ATV model; (c) restored image by the JIN’s model; (d) restored image by HYPTV model; (e) restored image by LOGTV model; (f), (g), (h), (i), and (j) about the detailed image of (a), (b), (c), (d), and (e), respectively.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)Figure 13
Numerical result of the “house” image with noise standard deviationσ=3. (a) Restored image by the ROF model; (b) restored image by the ATV model; (c) restored image by the JIN’s model; (d) restored image by HYPTV model; (e) restored image by LOGTV model; (f), (g), (h), (i), and (j) about the detailed image of (a), (b), (c), (d), and (e), respectively.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)Figure 14
Numerical result of the “pirate” image with noise standard deviationσ=4. (a) Restored image by the ROF model; (b) restored image by the ATV model; (c) restored image by the JIN’s model; (d) restored image by HYPTV model; (e) restored image by LOGTV model; (f), (g), (h), (i), and (j) about the detailed image of (a), (b), (c), (d), and (e), respectively.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)Figure 15
Numerical result of the “bird” image with noise standard deviationσ=4. (a) Restored image by the ROF model; (b) restored image by the ATV model; (c) restored image by the JIN’s model; (d) restored image by HYPTV model; (e) restored image by LOGTV model; (f), (g), (h), (i), and (j) about the detailed image of (a), (b), (c), (d), and (e), respectively.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)Table 2
The PSNR of the restored images by the different model.
ImageσROF (PSNR/SSIM)ATV (PSNR/SSIM)JIN’s (PSNR/SSIM)HYPTV (PSNR/SSIM)LOGTV (PSNR/SSIM)Peppers227.60/0.718328.66/0.837729.46/0.819529.43/0.825029.53/0.8274Boat227.26/0.824027.90/0.857428.35/0.862228.49/0.865928.64/0.8661House227.45/0.618529.06/0.810128.77/0.687629.55/0.737429.74/0.7447Pirate228.33/0.897527.22/0.853028.51/0.904228.55/0.904728.74/0.9060Bird228.99/0.690730.28/0.784930.36/0.777430.88/0.813731.07/0.8173Peppers325.93/0.675826.70/0.740927.13/0.744627.19/0.749227.37/0.7516Boat325.79/0.748126.47/0.798626.68/0.802926.64/0.798526.82/0.7989House326.08/0.575227.33/0.661127.24/0.655127.62/0.686627.82/0.6896Pirate326.35/0.839226.38/0.833226.81/0.857826.85/0.858026.98/0.8583Bird327.36/0.645127.92/0.693028.48/0.730728.92/0.768629.13/0.7701Peppers424.49/0.615325.59/0.715325.63/0.680025.65/0.698125.83/0.7011Boat424.47/0.680724.95/0.733825.24/0.737025.31/0.738225.51/0.7388House424.63/0.513025.30/0.576026.18/0.639926.19/0.641926.43/0.6437Pirate424.03/0.765224.33/0.720524.75/0.778725.64/0.811825.77/0.8159Bird425.66/0.567926.49/0.658427.25/0.726327.25/0.727027.51/0.7282Best denoising performance are given in bold.


### 4.5. Denoising Results of Real Ultrasound Images

In this subsection, we test some real ultrasound images. Figure16 shows the experimental results of real ultrasound images by applying JIN’s model, HYPTV model, and LOGTV model. Table 3 shows the different iteration for the different test images by using the JIN’s model, HYPTV model, and LOGTV model. We find that LOGTV model is much effective than JIN’s model and HYPTV model in obtaining the satisfactory restored images.Figure 16
Numerical result of the real ultrasound image (the real ultrasound image from [18]). (a, e, i) Noisy image; (b, f, j) restored image by the JIN’s model; (c, g, k) restored image by HYPTV model; (d, h, l) restored image by LOGTV model.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)Table 3
Iteration times and time of different models.
ImageJIN’s (iter/time)HYPTV (iter/time)LOGTV (iter/time)Ultra 181/0.46 s26/0.31 s22/0.24 sUltra 276/0.49 s22/0.27 s20/0.23 sUltra 388/0.45 s27/0.27 s24/0.23 sBest performance are given in bold.


## 4.1. Denoising Effect of Different Parameters of HYPTV Model and LOGTV Model

In this example, we use different parameterαii=1,2 values in the algorithm to test the effect of HYPTV and LOGTV models. The test images is shown in Figure 3(a), and the noise levels are σ=2 and σ=3. Figure 4 shows the different PSNR values and iteration numbers when different β values are used in the HYPTV model algorithm. Figure 5 shows the different PSNR values and iteration numbers when different β values are used in the LOGTV model algorithm. Firstly, we can also see that PSNR is the largest in αii=1,2=15. Secondly, with the increase of parameters, the speed of image restoration is faster. Based on the above analysis, when αii=1,2=15, the denoising performance of HPYTV and LOGTV models is close to the best. Therefore, in the following experiment, we choose αii=1,2=15 in HPYTV and LOGTV models.Figure 4
The PSNR values and iteration times of different parameterα2 (HYPTV model).Figure 5
The PSNR values and iteration times of different parameterα2 (LOGTV model).


## 4.2. Denoising Effect of the HYPTV and LOGTV Model

In this subsection, we use the algorithm to test the effect of the HYPTV and LOGTV model on image denoising. In Figures6 and 7, we show the original images, the noise images, and the restored images by HYPTV and LOGTV models. Figures 6(a) and 7(a) are original images, Figures 6(b) and 7(b) are noisy images, which the noise levels are σ=2 and σ=3, respectively. Figures 6(c) and 7(c) are corresponding restored images by HYPTV model. Figures 6(d) and 7(d) are corresponding restored images by LOGTV model. Table 1 shows that the PSNR values, SSIM values, and the numbers of iteration for the different test images can be got by using the HYPTV model and the LOGTV model. “Noise image PSNR” is the peak signal-to-noise ratio of noisy images and original images. “Denoising image PSNR” is the peak signal-to-noise ratio of restored images and original images. “Iter” is the number of iterations of the algorithm. From the results, it is obvious that the HYPTV model and the LOGTV model are fairly effective in reducing the speckle noise and edge-preserving.Figure 6
Numerical result of the “image 1” image with noise standard deviationσ=2. (a) Original image (image 1); (b) noisy image; (c) restored image by HYPTV model α1=15; (d) restored image by LOGTV model α2=15.
(a)(b)(c)(d)Figure 7
Numerical result of the “image 2” image with noise standard deviationσ=3. (a) Original image (image 2); (b) noisy image; (c) restored image by HYPTV model α1=15; (d) restored image by LOGTV model α2=15.
(a)(b)(c)(d)Table 1
Numerical result of the “image 1” and “image 2” images by the HYPTV and LOGTV model.
ImageσNoise image PSNRHYPTV (PSNR/SSIM)LOGTV (PSNR/SSIM)Image 1223.7632.31/0.778632.98/0.8025Image 2223.2634.78/0.836035.31/0.8562Image 1320.2431.10/0.780631.57/0.7948Image 2319.7633.47/0.830034.07/0.8568Image 1417.7729.97/0.792030.45/0.8113Image 2417.5132.51/0.864932.63/0.8667


## 4.3. Reduction of Staircase

In this subsection, we test the reduction of the staircase effect in HYPTV model and LOGTV model by image in Figure8. Figure 8(a) shows original images (“house”). Figure 8(b) shows detailed images of Figure 8(a), respectively. Figure 8(c) shows noise images which noise standard deviation σ=3. Figure 8(d) shows detailed images of Figure 8(c). The original images contain a lot of details information, such as textures edges and in homogeneous regions.Figure 8
Numerical result of the “lena” and “house” images with noise standard deviationσ=3. (a) Original images (house); (b) about the detailed image of (a); (c) noisy images; (d) about the detailed image of (c).
(a)(b)(c)(d)Figure9 displays the restoration results of the noisy “house” image, respectively. Figures 9(a)–9(e) was restored ROF model [10], ATV model [36], JIN’s model [11], HYPTV model, and LOGTV model, respectively. Figures 9(f)–9(j) is corresponding details images.Figure 9
Numerical result of the “house” images with noise standard deviationσ=3. (a) Restored image by the ROF model; (b) restored image by the ATV model; (c) restored image by the JIN’s model; (d) restored image by HYPTV model; (e) restored image by LOGTV model; (f), (g), (h), (i), and (j) about the detailed image of (a), (b), (c), (d), and (e), respectively.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)According to the results, we can see that JIN’s model, HYPTV model, and LOGTV model had a good effect in removing image noise and preserving image edges. In addition, we can clearly see staircase effect in the detailed images obtained by ROF model, ATV model, and JIN’s model. At the same time, the staircase effect of HYPTV model and LOGTV model restored images is reduced. Although both the HYPTV model and the LOGTV model can reduce the ladder effect, the LOGTV model is better than the HYPTV model.


## 4.4. Comparison with ROF Model, ATV Model, and JIN’s Model

In this subsection, we compare the effect of the ROF model, ATV model, and JIN’s model with HYPTV and LOGTV models for some images. The test original images are shown in Figure3(“house,” “peppers,” “boat,” “pirate,” and “bird”), with two sizes of 512×512 and three sizes of 256×256. Figure 10 shows some noise images with different standard deviation.Figure 10
Noisy images. (a) Boat (σ=3); (b) pirate (σ=4); (c) peppers (σ=2); (d) house (σ=3); (e) bird (σ=4).
(a)(b)(c)(d)(e)Figures11–15 display the restoration results for images (“peppers,” “boat,” “house,” “pirate,” and “bird”) through ROF model, ATV model, JIN’s model, HYPTV model, and LOGTV model. The noise versions of “peppers”, “boat” and “house”, and “pirate” and “bird” are obtained by model (2) with standard deviations 2, 3, and 4, respectively. In addition, the detailed images of the restored images are also displayed. Table 2 shows the PSNR and SSIM values for different test images by using the ROF model, ATV model, JIN’s model, HYPTV model, and LOGTV model. From the results of Figures 11–15 and Table 2, the noise standard deviation σ=2. We can observe that although the four models can effectively remove the noises while preserving the edges and details, the restored images by HYPTV model and LOGTV model have better visual effect with less staircase effects than by the ROF model, ATV model, and JIN’s model. The noise standard deviation σ=3,4. The visual effect of restored images by the ROF model and ATV model is particularly poor, but JIN’s model, HYPTV model, and LOGTV model can effectively remove the noises. Finally, Table 2 shows that LOGTV model has higher PSNR and SSIM values than other four models. This means that our proposed LOGTV model is available in reducing the speckle noise in some images.Figure 11
Numerical result of the “peppers” image with noise standard deviationσ=2; (a) restored image by the ROF model; (b) restored image by the ATV model; (c) restored image by the JIN’s model; (d) restored image by HYPTV model; (e) restored image by LOGTV model; (f), (g), (h), (i), and (j) about the detailed image of (a), (b), (c), (d) and (e), respectively.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)Figure 12
Numerical result of the “boat” image with noise standard deviationσ=3. (a) Restored image by the ROF model; (b) restored image by the ATV model; (c) restored image by the JIN’s model; (d) restored image by HYPTV model; (e) restored image by LOGTV model; (f), (g), (h), (i), and (j) about the detailed image of (a), (b), (c), (d), and (e), respectively.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)Figure 13
Numerical result of the “house” image with noise standard deviationσ=3. (a) Restored image by the ROF model; (b) restored image by the ATV model; (c) restored image by the JIN’s model; (d) restored image by HYPTV model; (e) restored image by LOGTV model; (f), (g), (h), (i), and (j) about the detailed image of (a), (b), (c), (d), and (e), respectively.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)Figure 14
Numerical result of the “pirate” image with noise standard deviationσ=4. (a) Restored image by the ROF model; (b) restored image by the ATV model; (c) restored image by the JIN’s model; (d) restored image by HYPTV model; (e) restored image by LOGTV model; (f), (g), (h), (i), and (j) about the detailed image of (a), (b), (c), (d), and (e), respectively.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)Figure 15
Numerical result of the “bird” image with noise standard deviationσ=4. (a) Restored image by the ROF model; (b) restored image by the ATV model; (c) restored image by the JIN’s model; (d) restored image by HYPTV model; (e) restored image by LOGTV model; (f), (g), (h), (i), and (j) about the detailed image of (a), (b), (c), (d), and (e), respectively.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)Table 2
The PSNR of the restored images by the different model.
ImageσROF (PSNR/SSIM)ATV (PSNR/SSIM)JIN’s (PSNR/SSIM)HYPTV (PSNR/SSIM)LOGTV (PSNR/SSIM)Peppers227.60/0.718328.66/0.837729.46/0.819529.43/0.825029.53/0.8274Boat227.26/0.824027.90/0.857428.35/0.862228.49/0.865928.64/0.8661House227.45/0.618529.06/0.810128.77/0.687629.55/0.737429.74/0.7447Pirate228.33/0.897527.22/0.853028.51/0.904228.55/0.904728.74/0.9060Bird228.99/0.690730.28/0.784930.36/0.777430.88/0.813731.07/0.8173Peppers325.93/0.675826.70/0.740927.13/0.744627.19/0.749227.37/0.7516Boat325.79/0.748126.47/0.798626.68/0.802926.64/0.798526.82/0.7989House326.08/0.575227.33/0.661127.24/0.655127.62/0.686627.82/0.6896Pirate326.35/0.839226.38/0.833226.81/0.857826.85/0.858026.98/0.8583Bird327.36/0.645127.92/0.693028.48/0.730728.92/0.768629.13/0.7701Peppers424.49/0.615325.59/0.715325.63/0.680025.65/0.698125.83/0.7011Boat424.47/0.680724.95/0.733825.24/0.737025.31/0.738225.51/0.7388House424.63/0.513025.30/0.576026.18/0.639926.19/0.641926.43/0.6437Pirate424.03/0.765224.33/0.720524.75/0.778725.64/0.811825.77/0.8159Bird425.66/0.567926.49/0.658427.25/0.726327.25/0.727027.51/0.7282Best denoising performance are given in bold.


## 4.5. Denoising Results of Real Ultrasound Images

In this subsection, we test some real ultrasound images. Figure16 shows the experimental results of real ultrasound images by applying JIN’s model, HYPTV model, and LOGTV model. Table 3 shows the different iteration for the different test images by using the JIN’s model, HYPTV model, and LOGTV model. We find that LOGTV model is much effective than JIN’s model and HYPTV model in obtaining the satisfactory restored images.Figure 16
Numerical result of the real ultrasound image (the real ultrasound image from [18]). (a, e, i) Noisy image; (b, f, j) restored image by the JIN’s model; (c, g, k) restored image by HYPTV model; (d, h, l) restored image by LOGTV model.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)Table 3
Iteration times and time of different models.
ImageJIN’s (iter/time)HYPTV (iter/time)LOGTV (iter/time)Ultra 181/0.46 s26/0.31 s22/0.24 sUltra 276/0.49 s22/0.27 s20/0.23 sUltra 388/0.45 s27/0.27 s24/0.23 sBest performance are given in bold.


## 5. Concluding Remarks

In this paper, we propose a new speckle noise restoration model based on adaptive TV method. Two new convex functions are introduced as the TV regularization term. By analyzing the diffusion performance of the proposed two models, one can see that the LOGTV model has faster diffusion speed than the HYPTV model. Moreover, we introduced two iterative numerical algorithms to solve the proposed models. The experiment results show the affectivity of our proposed model and the similarity between JIN’s model and HYPTV model. In addition, we compared the effect of the ROF model, ATV model, and JIN’s model with the LOGTV model, and the experiment results show high efficiency of LOGTV model in image restoration.



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*Source: 3936975-2020-07-20.xml*