# A Convex Adaptive Total Variation Model Based on the Gray Level Indicator for Multiplicative Noise Removal

Authors: Gang Dong; Zhichang Guo; Boying Wu
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

This paper focuses on the problem of multiplicative noise removal. Using a gray level indicator, we derive a
new functional which consists of the adaptive total variation term and the global convex fidelity term. We prove the existence, uniqueness, and comparison principle of the minimizer for the variational problem. The existence, uniqueness, and long-time behavior of the associated evolution equation are established. Finally, experimental results illustrate the effectiveness of the model in multiplicative noise reduction. Different from the other methods, the parameters in the proposed algorithms are found dynamically.


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## Body

## 1. Introduction

Multiplicative noise occurs while one deals with active imaging system, such as laser images, microscope images, and SAR images. Given a noisy imagef:Ω→R, where Ω is a bounded open subset of ℝ2, we assume
(1)f=uη,
where u is the true image and η is the noise. In what follows we will always assume that f>0 and u>0. Some prior information about the mean and variance of the multiplicative noise is as follows:
(2)1|Ω|∫Ω‍ηdx=1,(3)1|Ω|∫Ω‍(η-1)2dx=σ2,
where |Ω|=∫Ω‍1dx. Our purpose is to remove noise in images while preserving the maximum information about u.The goal of this paper is to propose a globally strictly convex functional well-adapted to removing multiplicative noise, which is as follows:(4)E(u)=∫Ω‍α(x)|Du|+∫Ω‍(u+flog1u)dx,
where f is noisy image and ∫Ωα(x)|Du| stands for the adaptive total variation of u. In the new model, we introduce a control factor, α(x), which controls the speed of diffusion at different region: at the low gray level (α(x)→0), the speed is slow; at the high gray level (α(x)→1), the speed is fast. The second term in the functional, that is, the fidelity term is global convex, which implies the constraint (2).Various adaptive filters for multiplicative noise removal have been proposed. In the beginning, variational methods for multiplicative noise reduction deal with the Gaussian multiplicative noise [1]. In actual life, the speckle noise [2] is more widespread, such as synthetic aperture radar (SAR) imagery [3, 4]. Then, Aubert and Aujol [5] propose a new variational method for Gamma multiplicative noise reduction. By the logarithm transform, a large variety of methods rely on the conversion of the multiplicative noise into additive noise.


### 1.1. Gaussian Noise

In the additive noise case, the most classical assumption is that the noise is a white Gaussian noise. So one case in which dealing with multiplicative noise is the white Gaussian noise. Using the framework in [6], Rudin et al. [1] consider the following optimization problem for Gaussian multiplicative noise reduction:
(5)u=argminu∈BV(Ω){J(u)+λH(u,f)},
where J(u)=∫Ω‍|∇u| stands for the total variation of u, and H(u,f) is a fidelity term which consists of two integrals with two Lagrange multipliers:
(6)H(u,f)=λ1∫Ω‍fudx+λ2∫Ω‍(fu-1)2dx.In order to make sure the two constraints (2) and (3) are always satisfied during the evolution, by the gradient projection method, the authors evolve the following evolution equation:
(7)∂u∂t=div(∇u|∇u|)+λ1fu2+λ2f2u3.If by the gradient projection method the values ofλ1 and λ2 are found dynamically, the method is not always convex; while if λ1,λ2>0 are fixed, the corresponding minimization problem will lead to a sequence of constant function u approaching +∞.


### 1.2. Gamma Noise

Generally, the speckle noise is treated as Gamma noise with mean equal to one. The probability distribution functiong(η) of the noise η takes the following form:
(8)g(η)=LLΓ(L)ηL-1exp(-Lη)1{η≥0}.Gamma noise is more complex than Gaussian noise [2]. Based on maximum a posteriori (MAP) on p(u∣f), Aubert and Aujol [5] assume that the noise η follows a Gamma probability distribution with mean equal to one and p(u) following a Gibbs prior and then derive a functional formulating the following minimization problem (called AA model):
(9)u=argminu∈S(Ω){∫Ω‍|Du|+λ∫Ω‍(logu+fu)dx},
where S(Ω)={u>0,u∈BV(Ω)}. The new fidelity term H(u,f)=∫Ω‍(logu+(f/u))dx is strictly convex for u∈(0,2f). The authors prove the existence of a minimizer u∈S(Ω) to the minimization problem and derive existence and uniqueness results of the solution to the associated evolution equation:
(10)∂u∂t=div(∇u|∇u|)+λf-uu2.


### 1.3. The Model Based on the Logarithm Transform

The simplest idea is to take the log of both sides of (1),
(11)logf=logu+logη=w+n,
which essentially converts the multiplicative problem into an additive one. If the distribution of the noise η takes the form (8), the expectation and the variance of the log-noise n are
(12)𝔼[n]=Ψ0(L)-Ψk(L),Var[n]=Ψ1(L),
where
(13)Ψk(z)=(ddz)k+1logΓ(z)
is the polygamma function [7]. In [8], Shi and Osher use relaxed inverse scale space (RISS) flows to deal with various noises and provide iterative TV regularization. First, using the log-data logf, they propose the following model:
(14)u=argminw∈BV(Ω){∫Ω‍|Dw|+λ2∥w-logf∥L2(Ω)2dx},
where w=logu. The corresponding RISS flow reads
(15)∂tw=div(∇w|∇w|)+λ(logf-w+v),∂tv=α(logf-w),
with v(0)=0, w(0)=c0, for c0=∫Ω‍logfdx/|Ω|. Then, they generalize several multiplicative noise models [5, 9, 10], and for convergence reasons, use iterative TV regularization on exp(w) to obtain the RISS flow. And they investigate the properties of the flow and study the dependence on flow parameters.


### 1.4. The Model Based on the Exponent Transform

Recently, Huang et al. [11] utilize an exponential transformation u→eu in the fidelity term of AA model to propose the following denoising model:
(16)minu,w{∫Ω‍(u+fe-u)dx+α1∫Ω‍|u-w|2dx----+α2∫Ω‍|Dw|},
where α1>0 and α2>0 are regularization parameters, and w is an auxiliary variable. Next, they further develop an alternating minimization algorithm for the model (16) by incorporating another way of modified TV regularization in [12]. However, the mathematical analysis to the variational problem (16) is not given in [11]. By the exponential transformation u→eu, Jin and Yang [13] change the fidelity term logu+fu of AA model (2) into u+feu and then study the following denoising model:
(17)minu{∫Ω‍|Du|+λ∫Ω‍(u+fe-u)dx}.Notice that the fidelity termH(u,f)=∫Ω‍(u+fe-u)dx is globally strictly convex. Based on this, they prove the uniqueness of the solutions to the variational problem (17) and show the existence and uniqueness of the weak solution for the following evolution equation corresponding to (17):
(18)∂u∂t=div(∇u|∇u|)+λ(fe-u-1).The paper is organized as follows. In Section2, inspired from [1, 5, 14], based on the gray level indicator α(x), we derive a convex adaptive total variation model (4) for multiplicative noise removal. In Section 3, we prove the existence and uniqueness of the solution of the minimization problem (4). In Section 4, we study the associated evolution problem to the minimization problem (4). Specifically, we define the weak solution of the evolution problem, derive estimates for the solution of an approximating problem, prove existence and uniqueness of the weak solution, and discuss the behavior of the weak solution as t→∞. In Section 5, we provide two numerical algorithms and experimental results to illustrate the effectiveness of our algorithms in image denoising. We also compare new algorithms with other ones.


## 1.1. Gaussian Noise

In the additive noise case, the most classical assumption is that the noise is a white Gaussian noise. So one case in which dealing with multiplicative noise is the white Gaussian noise. Using the framework in [6], Rudin et al. [1] consider the following optimization problem for Gaussian multiplicative noise reduction:
(5)u=argminu∈BV(Ω){J(u)+λH(u,f)},
where J(u)=∫Ω‍|∇u| stands for the total variation of u, and H(u,f) is a fidelity term which consists of two integrals with two Lagrange multipliers:
(6)H(u,f)=λ1∫Ω‍fudx+λ2∫Ω‍(fu-1)2dx.In order to make sure the two constraints (2) and (3) are always satisfied during the evolution, by the gradient projection method, the authors evolve the following evolution equation:
(7)∂u∂t=div(∇u|∇u|)+λ1fu2+λ2f2u3.If by the gradient projection method the values ofλ1 and λ2 are found dynamically, the method is not always convex; while if λ1,λ2>0 are fixed, the corresponding minimization problem will lead to a sequence of constant function u approaching +∞.


## 1.2. Gamma Noise

Generally, the speckle noise is treated as Gamma noise with mean equal to one. The probability distribution functiong(η) of the noise η takes the following form:
(8)g(η)=LLΓ(L)ηL-1exp(-Lη)1{η≥0}.Gamma noise is more complex than Gaussian noise [2]. Based on maximum a posteriori (MAP) on p(u∣f), Aubert and Aujol [5] assume that the noise η follows a Gamma probability distribution with mean equal to one and p(u) following a Gibbs prior and then derive a functional formulating the following minimization problem (called AA model):
(9)u=argminu∈S(Ω){∫Ω‍|Du|+λ∫Ω‍(logu+fu)dx},
where S(Ω)={u>0,u∈BV(Ω)}. The new fidelity term H(u,f)=∫Ω‍(logu+(f/u))dx is strictly convex for u∈(0,2f). The authors prove the existence of a minimizer u∈S(Ω) to the minimization problem and derive existence and uniqueness results of the solution to the associated evolution equation:
(10)∂u∂t=div(∇u|∇u|)+λf-uu2.


## 1.3. The Model Based on the Logarithm Transform

The simplest idea is to take the log of both sides of (1),
(11)logf=logu+logη=w+n,
which essentially converts the multiplicative problem into an additive one. If the distribution of the noise η takes the form (8), the expectation and the variance of the log-noise n are
(12)𝔼[n]=Ψ0(L)-Ψk(L),Var[n]=Ψ1(L),
where
(13)Ψk(z)=(ddz)k+1logΓ(z)
is the polygamma function [7]. In [8], Shi and Osher use relaxed inverse scale space (RISS) flows to deal with various noises and provide iterative TV regularization. First, using the log-data logf, they propose the following model:
(14)u=argminw∈BV(Ω){∫Ω‍|Dw|+λ2∥w-logf∥L2(Ω)2dx},
where w=logu. The corresponding RISS flow reads
(15)∂tw=div(∇w|∇w|)+λ(logf-w+v),∂tv=α(logf-w),
with v(0)=0, w(0)=c0, for c0=∫Ω‍logfdx/|Ω|. Then, they generalize several multiplicative noise models [5, 9, 10], and for convergence reasons, use iterative TV regularization on exp(w) to obtain the RISS flow. And they investigate the properties of the flow and study the dependence on flow parameters.


## 1.4. The Model Based on the Exponent Transform

Recently, Huang et al. [11] utilize an exponential transformation u→eu in the fidelity term of AA model to propose the following denoising model:
(16)minu,w{∫Ω‍(u+fe-u)dx+α1∫Ω‍|u-w|2dx----+α2∫Ω‍|Dw|},
where α1>0 and α2>0 are regularization parameters, and w is an auxiliary variable. Next, they further develop an alternating minimization algorithm for the model (16) by incorporating another way of modified TV regularization in [12]. However, the mathematical analysis to the variational problem (16) is not given in [11]. By the exponential transformation u→eu, Jin and Yang [13] change the fidelity term logu+fu of AA model (2) into u+feu and then study the following denoising model:
(17)minu{∫Ω‍|Du|+λ∫Ω‍(u+fe-u)dx}.Notice that the fidelity termH(u,f)=∫Ω‍(u+fe-u)dx is globally strictly convex. Based on this, they prove the uniqueness of the solutions to the variational problem (17) and show the existence and uniqueness of the weak solution for the following evolution equation corresponding to (17):
(18)∂u∂t=div(∇u|∇u|)+λ(fe-u-1).The paper is organized as follows. In Section2, inspired from [1, 5, 14], based on the gray level indicator α(x), we derive a convex adaptive total variation model (4) for multiplicative noise removal. In Section 3, we prove the existence and uniqueness of the solution of the minimization problem (4). In Section 4, we study the associated evolution problem to the minimization problem (4). Specifically, we define the weak solution of the evolution problem, derive estimates for the solution of an approximating problem, prove existence and uniqueness of the weak solution, and discuss the behavior of the weak solution as t→∞. In Section 5, we provide two numerical algorithms and experimental results to illustrate the effectiveness of our algorithms in image denoising. We also compare new algorithms with other ones.


## 2. A New Variational Model for Multiplicative Noise Removal

The goal of this section is to propose a new variational model for multiplicative noise removal. First, we proposed a new fidelity with global convexity, which can always satisfy the constraint (2) during the evolution. Then, by analyzing the properties of the noise, we propose the adaptive total variation based on a gray level indicator α(x).


### 2.1. A Global Convex Fidelity Term

Based on the idea in [15], we can obtain the following fidelity:
(19)H(u,f)=u+flog1u.Note thatH′(u,f)=1-(f/u). Let us consider the following Euler-Lagrange equation for some functional-based problem:
(20)0=ℱ(∇u,∇2u)-λ(1-fu),inΩ,(21)∂u∂n→=0,on∂Ω,
where ∇u and ∇2u stand, respectively, for the gradient and the Hessian matrix of u with respect to the space variable x, and ℱ(∇u,∇2u) is divergence operator from the functional. By integrating (20) in space, using integration by parts and the boundary condition (21) in the sense of distributions, we have
(22)∫Ω‍(1-fu)dx=0,
which implies the constraint (2). Moreover, using the idea in [6], the parameter λ can be calculated as
(23)λ=1σ2|Ω|∫Ω‍ℱ(∇u,∇2u)(1-fu)dx.Remark 1.
(1) It is easy to check that the functionu→u+flog(1/u) reaches its minimum value f+flog(1/f) over ℝ+ for u=f.
(2) Let us denote by(24)h(u)=u+flog1u.
We haveh′(u)=1-(f/u)=(u-f)/u, and h′(u)=(f/u2)>0, which deduce that the new fidelity term H(u,f) is globally strictly convex.


### 2.2. The Adaptive Total Variation Model

Assumeu is a piecewise function, that is, u=∑i=1Ngi1Ωi, where Ωi∩Ωj=∅, for i≠j, ⋃i=1N‍Ωi=Ω, and gi is the gray level. Moreover, we assume that the samples of the noise on each pixel x are mutually independent and identically distributed (i.i.d.) with the probability density function η(x). For x∈Ω, f(x)=giη(x), and therefore, Var[f]=gi2Var[η]=gi2σ2, where Var[f] and Var[η] are the variance of the noise image f and the noise at the pixel x, respectively. It is noticed that the variance of the noise is the constant σ2 on each pixel, but the variance of the noise image f is influenced by the gray level. The higher the gray level is, the more remarkable the influence of the noise is. Especially, f=u when u=0, and therefore f is noise free in this case. The fact is illustrious in Figure 1. It can be seen that in despite of the independent identically distribution of noise (see Figure 1(b)), the noise image shows different features on the pixel, where the gray levels are different (see Figure 1(d)).The relation between the influence of the noise and the gray levels. (a) 1D signalf; (b) 1D speckled noise with mean equal to 1 and L=5; (c) f degraded by the speckled noise; (d) the compare between f and the noise signal f.
(a)
Noise-free signal
(b)
Speckled noise
(c)
Noise signal
(d)
Original/noise signalIn [14], Chan and Strong propose the following adaptive total variation model:
(25)min∫Ω‍g(x)|∇u|dx,
where the weight function g(x) controls the speed of diffusion at different region. Utilizing this idea, we proposed a gray level indicator α(x). The indicator α(x) has such properties as follows: α(s) is monotonically increasing, g(0)=0, α(s)≥0, and α(s)→1, as s→supx∈Ωu. Therefore, we propose the following gray indicators α(x):
(26)α(x)=(1-11+k|Gσ*f|2)1+kM2kM2,
or
(27)α(x)=Gσ*fM,
where M=supx∈Ω(Gσ*f)(x), Gσ(x)=(1/4πσ)exp{-|x|2/4σ2}, σ>0, and k>0 are parameters. With this choice, α(x) is a positive-valued continuous function; α(x) is much smaller value at low gray levels (α(x)→0) than at high gray levels (α(x)→1) so that some small features at low gray levels are much less smoothed and therefore are preserved. As previously stated, at high gray levels, the region of the noise image is degraded more, while the region at low gray levels is degraded less (see Figure 1(d)). Then, from (26)/(27), at the high gray levels, α(x)→1, the new model is more smooth at the region; at low gray levels, α(x)→0, the new model is less smooth at the region.The previous analysis leads us to propose a convex adaptive total variation model for multiplicative noise removal,(28)minu∈BV(Ω)∫Ω‍α(x)|Du|+λ∫Ω‍(u+flog1u)dx.The evolution of the Euler-Lagrange equation for (28) is as follows:
(29)∂u∂t=div(α(x)∇u|∇u|)-λ(1-fu),(x,t)∈Ω×(0,T),(30)∂u∂n→=0,(x,t)∈∂Ω×[0,T],(31)u(0,x)=f,x∈Ω.


## 2.1. A Global Convex Fidelity Term

Based on the idea in [15], we can obtain the following fidelity:
(19)H(u,f)=u+flog1u.Note thatH′(u,f)=1-(f/u). Let us consider the following Euler-Lagrange equation for some functional-based problem:
(20)0=ℱ(∇u,∇2u)-λ(1-fu),inΩ,(21)∂u∂n→=0,on∂Ω,
where ∇u and ∇2u stand, respectively, for the gradient and the Hessian matrix of u with respect to the space variable x, and ℱ(∇u,∇2u) is divergence operator from the functional. By integrating (20) in space, using integration by parts and the boundary condition (21) in the sense of distributions, we have
(22)∫Ω‍(1-fu)dx=0,
which implies the constraint (2). Moreover, using the idea in [6], the parameter λ can be calculated as
(23)λ=1σ2|Ω|∫Ω‍ℱ(∇u,∇2u)(1-fu)dx.Remark 1.
(1) It is easy to check that the functionu→u+flog(1/u) reaches its minimum value f+flog(1/f) over ℝ+ for u=f.
(2) Let us denote by(24)h(u)=u+flog1u.
We haveh′(u)=1-(f/u)=(u-f)/u, and h′(u)=(f/u2)>0, which deduce that the new fidelity term H(u,f) is globally strictly convex.


## 2.2. The Adaptive Total Variation Model

Assumeu is a piecewise function, that is, u=∑i=1Ngi1Ωi, where Ωi∩Ωj=∅, for i≠j, ⋃i=1N‍Ωi=Ω, and gi is the gray level. Moreover, we assume that the samples of the noise on each pixel x are mutually independent and identically distributed (i.i.d.) with the probability density function η(x). For x∈Ω, f(x)=giη(x), and therefore, Var[f]=gi2Var[η]=gi2σ2, where Var[f] and Var[η] are the variance of the noise image f and the noise at the pixel x, respectively. It is noticed that the variance of the noise is the constant σ2 on each pixel, but the variance of the noise image f is influenced by the gray level. The higher the gray level is, the more remarkable the influence of the noise is. Especially, f=u when u=0, and therefore f is noise free in this case. The fact is illustrious in Figure 1. It can be seen that in despite of the independent identically distribution of noise (see Figure 1(b)), the noise image shows different features on the pixel, where the gray levels are different (see Figure 1(d)).The relation between the influence of the noise and the gray levels. (a) 1D signalf; (b) 1D speckled noise with mean equal to 1 and L=5; (c) f degraded by the speckled noise; (d) the compare between f and the noise signal f.
(a)
Noise-free signal
(b)
Speckled noise
(c)
Noise signal
(d)
Original/noise signalIn [14], Chan and Strong propose the following adaptive total variation model:
(25)min∫Ω‍g(x)|∇u|dx,
where the weight function g(x) controls the speed of diffusion at different region. Utilizing this idea, we proposed a gray level indicator α(x). The indicator α(x) has such properties as follows: α(s) is monotonically increasing, g(0)=0, α(s)≥0, and α(s)→1, as s→supx∈Ωu. Therefore, we propose the following gray indicators α(x):
(26)α(x)=(1-11+k|Gσ*f|2)1+kM2kM2,
or
(27)α(x)=Gσ*fM,
where M=supx∈Ω(Gσ*f)(x), Gσ(x)=(1/4πσ)exp{-|x|2/4σ2}, σ>0, and k>0 are parameters. With this choice, α(x) is a positive-valued continuous function; α(x) is much smaller value at low gray levels (α(x)→0) than at high gray levels (α(x)→1) so that some small features at low gray levels are much less smoothed and therefore are preserved. As previously stated, at high gray levels, the region of the noise image is degraded more, while the region at low gray levels is degraded less (see Figure 1(d)). Then, from (26)/(27), at the high gray levels, α(x)→1, the new model is more smooth at the region; at low gray levels, α(x)→0, the new model is less smooth at the region.The previous analysis leads us to propose a convex adaptive total variation model for multiplicative noise removal,(28)minu∈BV(Ω)∫Ω‍α(x)|Du|+λ∫Ω‍(u+flog1u)dx.The evolution of the Euler-Lagrange equation for (28) is as follows:
(29)∂u∂t=div(α(x)∇u|∇u|)-λ(1-fu),(x,t)∈Ω×(0,T),(30)∂u∂n→=0,(x,t)∈∂Ω×[0,T],(31)u(0,x)=f,x∈Ω.


## 3. The Minimization Problem (28)

In this section, we study the existence and uniqueness of the solution to the minimization problem (28), and then we consider the comparison principle for the problem (28).


### 3.1. Preliminaries

Iff∈L∞(Ω) with 0<infx∈Ωf≤f≤supx∈Ωf, then
(32)0<α(infx∈Ωf)≤α(x)≤1,x∈Ω.We always assume thatΩ is a bounded open subset of ℝn with Lipschitz boundary. As in [16], we introduce the following definition of α-total variation.Definition 2.
A functionu∈L1(Ω) has bounded α-total variation in Ω, if
(33)supϕ∈Φα∫Ω‍fdivϕdx<∞,
where
(34)Φα≕{ϕ∈C01(Ω,ℝn),|ϕ|≤α}.Remark 3 (see [16]).
(1) Ifu∈L1(Ω) has bounded α-total variation in Ω, there is a Radon vector measure Du on Ω such that
(35)∫Ω‍α|Du|≕supϕ∈Φα∫Ω‍fdivϕdx.
(2) From (32), u∈L1(Ω) having bounded α-total variation in Ω implies that u∈BV(Ω).
Now, we directly show some lemmas onα-total variation from [16].Lemma 4.
Assume that{uk}k=1∞⊂BV(Ω) and uk→u in L1(Ω). Then, u∈BV(Ω), and
(36)∫Ω‍α|Duk|≤liminfk→∞∫Ω‍α|Duk|.Lemma 5.
Assume thatu∈BV(Ω). Then, there exists a sequence {uj}j=1∞∈H1(Ω) such that
(37)uj⟶uinL1(Ω),∫Ω‍α|Duj|⟶∫Ω‍α|Du|.By a minor modification of the proof of Lemma 1 in Section 4.3 of [17], we can have the following lemma.Lemma 6.
Letu∈BV(Ω), and let φa,b be the cut-off function defined by
(38)φa,b(x)={a,ifx≤a,x,ifa≤x≤b,b,ifx≥b.
Then, Dφa,b(u)∈BV(Ω), and
(39)∫Ω‍α|Dφa,b(u)|≤∫Ω‍α|∇u|.


### 3.2. Existence and Uniqueness of the Problem (28)

In this subsection, we show that problem (28) has at least one solution in BV(Ω).Theorem 7.
Letf∈L∞(Ω) with infx∈Ωf>0. Then, the problem (28) admits a unique solution u∈BV(Ω) such that
(40)0<infx∈Ωf≤u≤supx∈Ωf.Proof.
Let us rewrite(41)E(u)=∫Ω‍α(x)|Du|+λ∫Ω‍(u+flog1u)dx.
From Remark 1(1), we have
(42)E(u)≥λ∫Ω‍(f+flog1f)dx,
for u∈BV(Ω) with u>0. This implies that E(u) has a lower bound for all u∈BV(Ω) with u>0. Hence, there exists a minimizing sequence {un}⊂BV(Ω) for the problem (28).
Step
1. Let us denote a=infx∈Ωf and b=supx∈Ωf. We first claim that 0<a≤un≤b.
In fact, we remark thath(s) is decreasing for s∈(0,f) and increasing for s∈(f,+∞). Therefore, if M≥f, one always has
(43)log(min(s,M))+fmin(s,M)≤logs+fs.
Hence, if M=b=supx∈Ωf, we have
(44)∫Ω‍(log(inf(u,M))+finf(u,M))dx≤∫Ω‍(logu+fu)dx.
Moreover, utilizing Lemma 6, we can have
(45)∫Ω‍α|D(inf(u,b))|≤∫Ω‍α|Du|.
Combining (44) and (45), we deduce that
(46)E(inf(u,b))≤E(u).
On the other hand, in the same way, we get that E(sup(u,a))≤E(u). Therefore, we can assume that a≤un≤b without restriction.
Step
2. Let us prove that there exists u∈BV(Ω) such that
(47)E(u)=minv∈BV(Ω)E(v).
The proof in the Step 1 implies in particular that un is bounded in L1(Ω). Since a≤un≤b and h(s)∈C[a,b], h(un) is bounded. Moreover, by the definition of {un}, there exists a constant C such that
(48)∫Ω‍α|Dun|+∫Ω‍h(un)dx≤C.
Then,
(49)∫Ω‍α|Dun|≤C,
which implies that the sequence {un}n=1∞ is bounded in BV(Ω). Consequently, there exists a function u∈BV(Ω) and a subsequence {un}n=1∞, denoted by itself, such that as n→∞,
(50)un⇀uinweak*inBV(Ω),un⟶ustronglyinL1(Ω),---------a≤u≤b.
Utilizing Lemma 4 and Fatou’s lemma, we get that u is a solution of the problem (28).
Finally, from Remark1(2), h is strictly convex as f>0, and then the uniqueness of the minimizer follows from the strict convexity of the energy functional in (28).


### 3.3. Comparison Principle

In this subsection, we state a comparison principle for problem (28).Proposition 8.
Letf1,  f2∈L∞(Ω) with infx∈Ωf1,  infx∈Ωf2>0, and f1<f2. Assume that u1 (resp., u2) is a solution of the problem (28) for f=f1 (resp., f=f2). Then, u1≤u2 a.e. in Ω.Proof.
Let us denoteu∨v=sup(u,v), and u∧v=inf(u,v).
From Theorem7, It is noticed that there exist the solutions u1 and u2 for f1 and f2, respectively. Since ui is a minimizer with data fi, for i=1,2,
(51)∫Ω‍α|D(u1∧u2)|+λ∫Ω‍(u1∧u2+f1log1u1∧u2)≥∫Ω‍α(x)|Du1|+λ∫Ω‍(u1+f1log1u1)dx,∫Ω‍α|D(u1∨u2)|+λ∫Ω‍(u1∨u2+f1log1u1∨u2)≥∫Ω‍α(x)|Du2|+λ∫Ω‍(u2+f2log1u2)dx.
Adding these two inequalities and by a minor modification of the facts in [18, 19], which is as follows:
(52)∫Ω‍α|D(u1∧u2)|+∫Ω‍α|D(u1∨u2)|≤∫Ω‍α|Du1|+∫Ω‍α|Du2|,
then we can deduce that
(53)∫Ω‍(u1∧u2-u1)dx+∫Ω‍f1(log1u1∧u2-log1u1)dx+∫Ω‍(u1∨u2-u2)dx+∫Ω‍f2(log1u1∨u2-log1u2)dx≥0.
Writing Ω={u1>u2}∪{u1≤u2}, we easily deduce that
(54)∫{u1>u2}‍(logu1-logu2)(f1-f2)dx≥0.
Since f1<f2, the set {u1>u2} is a zero Lebesgue measure, that is, u1≤u2 a.e. in Ω.


## 3.1. Preliminaries

Iff∈L∞(Ω) with 0<infx∈Ωf≤f≤supx∈Ωf, then
(32)0<α(infx∈Ωf)≤α(x)≤1,x∈Ω.We always assume thatΩ is a bounded open subset of ℝn with Lipschitz boundary. As in [16], we introduce the following definition of α-total variation.Definition 2.
A functionu∈L1(Ω) has bounded α-total variation in Ω, if
(33)supϕ∈Φα∫Ω‍fdivϕdx<∞,
where
(34)Φα≕{ϕ∈C01(Ω,ℝn),|ϕ|≤α}.Remark 3 (see [16]).
(1) Ifu∈L1(Ω) has bounded α-total variation in Ω, there is a Radon vector measure Du on Ω such that
(35)∫Ω‍α|Du|≕supϕ∈Φα∫Ω‍fdivϕdx.
(2) From (32), u∈L1(Ω) having bounded α-total variation in Ω implies that u∈BV(Ω).
Now, we directly show some lemmas onα-total variation from [16].Lemma 4.
Assume that{uk}k=1∞⊂BV(Ω) and uk→u in L1(Ω). Then, u∈BV(Ω), and
(36)∫Ω‍α|Duk|≤liminfk→∞∫Ω‍α|Duk|.Lemma 5.
Assume thatu∈BV(Ω). Then, there exists a sequence {uj}j=1∞∈H1(Ω) such that
(37)uj⟶uinL1(Ω),∫Ω‍α|Duj|⟶∫Ω‍α|Du|.By a minor modification of the proof of Lemma 1 in Section 4.3 of [17], we can have the following lemma.Lemma 6.
Letu∈BV(Ω), and let φa,b be the cut-off function defined by
(38)φa,b(x)={a,ifx≤a,x,ifa≤x≤b,b,ifx≥b.
Then, Dφa,b(u)∈BV(Ω), and
(39)∫Ω‍α|Dφa,b(u)|≤∫Ω‍α|∇u|.


## 3.2. Existence and Uniqueness of the Problem (28)

In this subsection, we show that problem (28) has at least one solution in BV(Ω).Theorem 7.
Letf∈L∞(Ω) with infx∈Ωf>0. Then, the problem (28) admits a unique solution u∈BV(Ω) such that
(40)0<infx∈Ωf≤u≤supx∈Ωf.Proof.
Let us rewrite(41)E(u)=∫Ω‍α(x)|Du|+λ∫Ω‍(u+flog1u)dx.
From Remark 1(1), we have
(42)E(u)≥λ∫Ω‍(f+flog1f)dx,
for u∈BV(Ω) with u>0. This implies that E(u) has a lower bound for all u∈BV(Ω) with u>0. Hence, there exists a minimizing sequence {un}⊂BV(Ω) for the problem (28).
Step
1. Let us denote a=infx∈Ωf and b=supx∈Ωf. We first claim that 0<a≤un≤b.
In fact, we remark thath(s) is decreasing for s∈(0,f) and increasing for s∈(f,+∞). Therefore, if M≥f, one always has
(43)log(min(s,M))+fmin(s,M)≤logs+fs.
Hence, if M=b=supx∈Ωf, we have
(44)∫Ω‍(log(inf(u,M))+finf(u,M))dx≤∫Ω‍(logu+fu)dx.
Moreover, utilizing Lemma 6, we can have
(45)∫Ω‍α|D(inf(u,b))|≤∫Ω‍α|Du|.
Combining (44) and (45), we deduce that
(46)E(inf(u,b))≤E(u).
On the other hand, in the same way, we get that E(sup(u,a))≤E(u). Therefore, we can assume that a≤un≤b without restriction.
Step
2. Let us prove that there exists u∈BV(Ω) such that
(47)E(u)=minv∈BV(Ω)E(v).
The proof in the Step 1 implies in particular that un is bounded in L1(Ω). Since a≤un≤b and h(s)∈C[a,b], h(un) is bounded. Moreover, by the definition of {un}, there exists a constant C such that
(48)∫Ω‍α|Dun|+∫Ω‍h(un)dx≤C.
Then,
(49)∫Ω‍α|Dun|≤C,
which implies that the sequence {un}n=1∞ is bounded in BV(Ω). Consequently, there exists a function u∈BV(Ω) and a subsequence {un}n=1∞, denoted by itself, such that as n→∞,
(50)un⇀uinweak*inBV(Ω),un⟶ustronglyinL1(Ω),---------a≤u≤b.
Utilizing Lemma 4 and Fatou’s lemma, we get that u is a solution of the problem (28).
Finally, from Remark1(2), h is strictly convex as f>0, and then the uniqueness of the minimizer follows from the strict convexity of the energy functional in (28).


## 3.3. Comparison Principle

In this subsection, we state a comparison principle for problem (28).Proposition 8.
Letf1,  f2∈L∞(Ω) with infx∈Ωf1,  infx∈Ωf2>0, and f1<f2. Assume that u1 (resp., u2) is a solution of the problem (28) for f=f1 (resp., f=f2). Then, u1≤u2 a.e. in Ω.Proof.
Let us denoteu∨v=sup(u,v), and u∧v=inf(u,v).
From Theorem7, It is noticed that there exist the solutions u1 and u2 for f1 and f2, respectively. Since ui is a minimizer with data fi, for i=1,2,
(51)∫Ω‍α|D(u1∧u2)|+λ∫Ω‍(u1∧u2+f1log1u1∧u2)≥∫Ω‍α(x)|Du1|+λ∫Ω‍(u1+f1log1u1)dx,∫Ω‍α|D(u1∨u2)|+λ∫Ω‍(u1∨u2+f1log1u1∨u2)≥∫Ω‍α(x)|Du2|+λ∫Ω‍(u2+f2log1u2)dx.
Adding these two inequalities and by a minor modification of the facts in [18, 19], which is as follows:
(52)∫Ω‍α|D(u1∧u2)|+∫Ω‍α|D(u1∨u2)|≤∫Ω‍α|Du1|+∫Ω‍α|Du2|,
then we can deduce that
(53)∫Ω‍(u1∧u2-u1)dx+∫Ω‍f1(log1u1∧u2-log1u1)dx+∫Ω‍(u1∨u2-u2)dx+∫Ω‍f2(log1u1∨u2-log1u2)dx≥0.
Writing Ω={u1>u2}∪{u1≤u2}, we easily deduce that
(54)∫{u1>u2}‍(logu1-logu2)(f1-f2)dx≥0.
Since f1<f2, the set {u1>u2} is a zero Lebesgue measure, that is, u1≤u2 a.e. in Ω.


## 4. The Associated Evolution Equations (29)–(31)

In this section, by an approach from the theory used in both [16, 20], we define the weak solution of the evolution problems (29)–(31), derive estimates for the solution of an approximating problem, prove existence and uniqueness of the weak solution, and discuss the behavior of the weak solution as t→∞.


### 4.1. Definition of a Pseudosolution to the Problems (29)–(31)

DenoteQT=Ω×[0,T), 0<T≤∞. Suppose that v∈L2(0,T;H1(Ω))∩L∞(QT) with v>0, ∂v/∂n→=0 and u is a classical solution of (29)–(31) with u>0. Multiplying (29) by (v-u), integrating over Ω, we have
(55)∫Ω‍∂tu(v-u)dx+∫Ω‍α(x)∇u|∇u|∇(v-u)dx+λ∫Ω‍u-fu(v-u)dx=0.
Since
(56)∫Ω‍∇u|∇u|∇(v-u)dx=∫Ω‍α(x)∇u|∇u|∇vdx-∫Ω‍α(x)|∇u|dx≤∫Ω‍α(x)|∇v|dx-∫Ω‍α(x)|∇u|dx,
and utilizing the Lagrange mean value theorem,
(57)∫Ω‍(v+flog1v)dx-∫Ω‍(u+flog1u)dx=∫Ω‍(v-u)dx-∫Ω‍fξ(v-u)dx≥∫Ω‍(v-u)dx-∫Ω‍fu(v-u)dx,
for either v≥ξ≥u>0 or u≥ξ≥v>0, we have
(58)∫Ω‍∂t(v-u)dx+E(v)≥E(u).
Integrating over [0,t] for any t∈[0,T] then yields
(59)∫0t‍∫Ω‍∂su(v-u)dxds+∫0t‍E(v)ds≥∫0t‍E(u)ds.On the other hand, letv=u+ϵϕ in (59) with ϕ∈C∞(Ω¯), ∂ϕ/∂n→=0, and ϕ>0. Then left-hand side of the inequality (59) has a minimum at ϵ=0. Hence, if u satisfies (59) and u∈L2(0,T;BV∩L2(Ω))∩L∞(QT) with ut∈L2(QT), u>0, and ∂u/∂n→=0, u is also a solution of the problems (29)–(31) in the sense of distribution. Based on this fact which is similar to the ideas in [13, 16, 21], we give the following definition.Definition 9.
A functionu∈L2(0,T;BV(Ω)∩L2(Ω))∩L∞(QT) is called a pseudosolution of (29)–(31), if ∂u/∂t∈L2(QT), ∂u/∂n→=0, u>0, and u satisfies (59) for all t∈[0,T], v∈L2(0,T;BV(Ω)∩L2(Ω))∩L∞(QT) with v>0 and ∂v/∂n→=0. Moreover,
(60)limt→0+∫Ω‍|u(x,t)-f(x)|2dx=0.


### 4.2. The Approximating Problem for Problems (29)–(31)

In this subsection, we consider the approximating problem(61)∂up,δ∂t-div(α|∇up,δ|p-2∇up,δ)+λup,δ-fδup,δ=0,---------------in(x,t)∈Ω×ℝ+,(62)∂up,δ∂n→=0,on(x,t)∈∂Ω×ℝ+,(63)up,δ(x,0)=fδ,onx∈Ω,
where 1<p≤2, fδ∈H1(Ω)∩L∞(Ω) such that as δ→0,
(64)fδ⟶finL2(Ω),∫Ω‍α|Dfδ|⟶∫Ω‍α|Df|,0<infx∈Ωf≤fδ≤supx∈Ωf.
From Lemma 5, we can have the existence of fδ.Let us denote(65)Ep(u)=1p∫Ω‍α(x)|∇u|p+λ∫Ω‍(u+flog1u)dx,
which is associate with the problems (61)–(63). Combining the proof of Theorem  3.4 in [16] with the proof of Theorem 7, we also have the following lemma.Lemma 10.
Letfδ∈L∞(Ω) with infx∈Ωfδ>0. Then, there is a unique solution to the problem
(66)minu∈W+1,p(Ω)∩L∞(Ω)Ep(u),
which is satisfying
(67)0<infx∈Ωfδ≤u≤supx∈Ωfδ,
where W+1,p(Ω)={u>0,u∈W1,p(Ω)}.Based on this fact, we have the following existence and uniqueness result for the problems (61)–(63).Theorem 11.
Letfδ∈L∞(Ω) with infx∈Ωfδ>0 and supx∈Ωf≤1. Then, the problems (61)–(63) admit a unique pseudosolution up,δ∈L∞(0,∞;W1,p(Ω)∩L2(Ω)) with ∂tup,δ∈L2(Q∞), that is,
(68)∫0t‍∫Ω‍∂su(v-up,δ)dxds+∫0t‍Ep(v)ds≥∫0t‍Ep(up,δ)ds,
for any t∈[0,T] and v∈L2(0,T;H1(Ω)) with v>0 and ∂v/∂n→=0. Moreover,
(69)0<infx∈Ωfδ≤u≤supx∈Ωfδ,
and for any T>0,
(70)∫0T‍∫Ω‍|∂tup,δ|2dxdt--+supt∈[0,T]{∫Ω‍(up,δ+fδlog1up,δ)dx1p∫Ω‍α|∇up,δ|pdx-------+λ∫Ω‍(up,δ+fδlog1up,δ)dx}-≤1p∫Ω‍α|∇fδ|pdx+λ∫Ω‍(fδ+fδlog1fδ)dx.Proof.
Let us fixk=infx∈Ωfδ>0 and introduce the following functions:
(71)[u]k=max{u,k}≕{u,ifu≥k,k,ifu<k.
We consider the auxiliary problem as follows:
(72)∂up,δ∂t-div(α|∇up,δ|p-2∇up,δ)+λup,δ-fδ[u]k=0,--------------in(x,t)∈Ω×ℝ+,(73)∂up,δ∂n→=0,on(x,t)∈∂Ω×ℝ+,(74)up,δ(x,0)=fδ,onx∈Ω.
Since p-Laplacian operator is a maximal monotone operator, using Galerkin method and Lebesgue convergence theorem, from standard results for parabolic equations [22], we get that the problems (72)–(74) admit a unique weak solution up,δ such that
(75)up,δ∈L∞(0,∞;W1,p(Ω)∩L2(Ω)),∂tup,δ∈L2(Q∞).
Next, let us verify that the truncated function[·]k in the problems (72)–(74) can be omitted. Multiply (72) by (up,δ-k)-, where
(76)(up,δ-k)-={up,δ-k,ifup,δ-k≤0,0,otherwise,
and integrate over Ω to get
(77)12ddt∫Ω‍(up,δ-k)-2dx+∫Ω‍α|∇(up,δ-k)-|pdx+∫Ω‍(up,δ-fδ)(up,δ-k)-[u]kdx=0.
Then,
(78)12ddt∫Ω‍(up,δ-k)-2dx≤0.
Therefore, (1/2)(d/dt)∫Ω‍(up,δ-k)-2dx is decreasing in t and since
(79)12∫Ω‍(up,δ-k)-2dx≥0,12∫Ω‍(up,δ-k)-2dx|t=0=0,
we have that
(80)12∫Ω‍(up,δ-k)-2dx=0,
for all t∈[0,T], and so
(81)u(t)≥k=infx∈Ωfδ>0ℒ-a.e.onΩ∀t>0.
Then, up,δ is also the solution to the problems (61)–(63). Let us fix K=supx∈Ωfδ. Multiplying (61) by (u-K)+, where
(82)(up,δ-K)+={up,δ-K,ifup,δ-K≥0,0,otherwise,
a similar argument yields that u(t)≤K=supx∈Ωfδ for all t, and then (69) follows.
Moreover, multiplying (61) by ∂tup,δ and integrating it over Ωt yield
(83)∫0t‍∫Ω‍|∂sup,δ|2dxds+1p∫0t‍∂s(∫Ω‍α|∇up,δ|pdx)ds+λ∫0t‍∂s(up,δ+flog1up,δdx)ds=0,
for t∈[0,T]. Then (70) follows.
Finally, we will verify that the above weak solution will be the pseudosolution for the problems (61)–(63). Suppose that v∈L2(0,T;H1(Ω)) with v>0 and ∂v/∂n→=0. Multiplying (61) by v-up,δ, then integrating by parts over ΩT, we obtain
(84)∫0T‍∫Ω‍∂tup,δ(v-up,δ)dxdt+∫0T‍∫Ω‍α|∇up,δ|p-2(∇v-∇up,δ)dxdt+∫0T‍∫Ω‍(up,δ-fδ)(v-up,δ)up,δdxdt=0.
Utilizing Young’s inequality with ϵ=|∇up,δ|2-p yields for the second term in the left-hand side of the above inequality (84), we obtain
(85)|∇up,δ|p-2∇up,δ∇v-|∇up,δ|p≤|∇v|pp+|∇up,δ|pp.
Utilizing the Lagrange mean value theorem yields
(86)∫Ω‍(v+fδlog1v)dx-∫Ω‍(up,δ+fδlog1up,δ)dx=∫Ω‍(v-up,δ)dx-∫Ω‍fδξ(v-up,δ)dx≥∫Ω‍(v-up,δ)dx-∫Ω‍fδup,δ(v-up,δ)dx,
for either v≥ξ≥up,δ>0 or up,δ≥ξ≥v>0. Returning to (84), we therefore conclude
(87)∫0T‍∫Ω‍∂tup,δ(v-up,δ)dxdt+1p∫0T‍∫Ω‍α|∇v|pdx+λ∫0T‍∫Ω‍(v+fδlog1v)dxdt≥1p∫0T‍∫Ω‍α|∇up,δ|pdx+λ∫0T‍∫Ω‍(up,δ+fδlog1up,δ)dxdt,
that is,
(88)∫0t‍∫Ω‍∂su(v-up,δ)dxds+∫0t‍Ep(v)ds≥∫0t‍Ep(up,δ)ds,
for any t∈[0,T]. This fact implies that up,δ is a pseudosolution for the problems (61)–(63). The uniqueness follows by the uniqueness of the weak solution for the problems (61)–(63).


### 4.3. Existence and Uniqueness of the Problems (29)–(31)

In this subsection, we will prove the main theorem for the existence and uniqueness for the solution to the problems (29)–(31).Theorem 12.
Supposef∈BV(Ω)∩L∞(Ω) with infx∈Ωf>0 and supx∈Ωf≤1. Then, there exists a unique pseudosolution u∈L∞(0,∞,BV(Ω)∩L∞(Ω)) to the problems (29)–(31). Moverover,
(89)0<infx∈Ωf≤u≤supx∈Ωf,(90)∫0∞‍∫Ω‍|∂tu|2dxdt+supt∈[0,∞){∫Ω‍α|Du|+λ∫Ω‍(u+flog1u)dx}≤∫Ω‍α|Df|+λ∫Ω‍(f+flog1f)dx.Proof
Step
1. First we fix δ>0 and pass to the limit p→1.
Letup,δ be the pseudosolution solution of (61)–(63). From (69)-(70), we know that, for fixed δ>0, up,δ is uniformly bounded in L∞(0,∞,BV(Ω)∩L∞(Ω)) and ∂tup,δ is uniformly bounded in L2(Q∞), that is,
(91)∥up,δ∥L∞(Q∞)+∥up,δ∥L∞(0,∞,BV(Ω))≤C,∥∂tup,δ∥L2(Q∞)≤C,
where C>0 is constant. Now we claim that there exists a sequence of functions upj,δ and a function uδ∈L∞(0,∞,BV(Ω)∩L∞(Ω)) such that, as pj→1,
(92)∂tupj,δ⟶∂tuδweaklyinL2(Q∞),(93)upj,δ⟶uδweakly*inL∞(Q∞)-with0<infx∈Ωfδ≤uδ≤supx∈Ωfδ,(94)upj,δ⟶uδstronglyinL1(Ω)foreacht∈[0,∞),(95)upj,δ⟶uδinL2(Ω)uniformlyint,(96)limt→0+∫Ω‍|upj,δ(x,t)-fδ(x)|2dx=0.
In fact, from (91), there is a sequence upj,δ and a function uδ∈L∞(Q∞) with ∂tuδ∈L2(Q∞) such that (92) and (93) hold.
Note that, for anyψ∈L2(Ω), as j→∞,
(97)∫Ω‍(upj,δ(x,t)-fδ(x))ψ(x)dx=∫Qt‍∂supj,δ(x,s)1[0,t](s)ψ(x)dxds⟶∫Qt‍∂suδ(x,s)1[0,t](s)ψ(x)dxds=∫Ω‍(uδ(x,t)-fδ(x))ψ(x)dx,
which shows that, for each t,
(98)upj,δ⟶uδweaklyinL2(Ω).
By (69) and (70), for each t∈[0,∞), upj(·,t) is a bounded sequence in W1,1(Ω). Hence, combining this with (98), we get that, for each t as pj→1,
(99)upj,δ⟶uδinL1(Ω),
which implies (94).
Since(100)∥upj,δ(·,t)-upj,δ(·,t′)∥L2(Ω)2≤|t-t′|∫QT‍(∂tupj,δ)2dxdt,
(96) follows.
To see (95), noting that from (100), t→upj,δ(·,t)∈L2(Ω) is equicontinuous, and from (93) and (99), we have, for each t∈[0,∞),
(101)upj,δ⟶uδstronglyinL2(Ω).
Then, a standard argument yields (95).
From (70) and (92), we also obtain that uδ∈L∞(0,∞,BV(Ω)∩L∞(Ω)) with ∂tuδ∈L2(Q∞). The claim is proved.
Next we show that, for allv∈L2(0,T;H1(Ω)) with v>0 and ∂v/∂n→=0, and for each t∈[0,∞),
(102)∫0t‍∫Ω‍∂suδ(v-uδ)dxds+∫0t‍∫Ω‍α|∇v|dxds+λ∫0t‍∫Ω‍(v+fδlog1v)dxds≥∫0t‍∫Ω‍α|Duδ|ds+λ∫0t‍∫Ω‍(uδ+fδlog1uδ)dxds.
By Theorem 11, we obtain that
(103)∫0T‍∫Ω‍∂tupj,δ(v-upj,δ)dxdt+1pj∫0T‍∫Ω‍α|∇v|pjdxdt+λ∫0T‍∫Ω‍(v+fδlog1v)dxdt≥1pj∫0T‍∫Ω‍α|∇upj,δ|pjdxdt+λ∫0T‍∫Ω‍(upj,δ+fδlog1upj,δ)dxdt,
Notice that, using Lemma 4, we have
(104)∫Ω‍α|Duδ|≤liminfj→∞∫Ω‍α|Dupj,δ|≤liminfj→∞(∫Ω‍α|∇upj,δ|pjdx)1/pj(∫Ω‍α(x)dx)1-1/pj=liminfj→∞1pj(∫Ω‍α|∇upj,δ|pjdx)1/pj.
Letting j→∞, pj→1 in (103) and using (92), (94), and (104), we obtain (102) for all v∈L2(0,T;H1(Ω)) with v>0 and ∂v/∂n→=0.
Step
2. Now it only remains to pass to the limit as δ→0 in (102) to complete the existence of the solution to (29)–(31).
Replacingup by upj in (70), letting j→∞ (pj→1), and using (92)–(95), (104), and (64), we obtain
(105)∫0∞‍∫Ω‍|∂tuδ|2dxdt+supt∈[0,∞){∫Ω‍α|Duδ|+λ∫Ω‍(uδ+fδlog1uδ)dx}≤∫Ω‍α|∇fδ|dx+λ∫Ω‍(fδ+fδlog1fδ)dx.
Combining this with (69), we have
(106)uδisuniformlyboundedinL∞(0,∞,BV(Ω)∩L∞(Ω)),∂tuδisuniformlyboundedinL2(Q∞).
Then, by a processing similar to the one for getting (92)–(96), we can obtain a sequence of functions uδj and a function u∈L∞(0,∞;BV(Ω)∩L∞(Ω) such that, as j→∞, δj→0,
(107)∂tuδj⟶∂tuweaklyinL2(Q∞),(108)uδj⟶uweakly*inL∞(Q∞)with0<infx∈Ωf≤u≤supx∈Ωf,(109)uδj⟶ustronglyinL1(Ω)foreacht∈[0,∞),(110)uδj⟶uinL2(Ω)uniformlyint,(111)limt→0+∫Ω‍|uδj(x,t)-f|2dx=0.
Replacinguδ by uδj in (102), letting j→∞, δj→0, and using Lemma 4, we can have from (107)–(110)
(112)∫0t‍∫Ω‍∂su(v-u)dxds+∫0t‍∫Ω‍α|∇v|dxds+λ∫0t‍∫Ω‍(v+flog1v)dxds≥∫0t‍∫Ω‍α|Du|ds+λ∫0t‍∫Ω‍(u+flog1u)dxds,
for all v∈L2(0,T;H1(Ω)) with v>0 and ∂v/∂n→=0. Then, the existence of the pseudosolution to (29)–(31) is completed.
Moreover, replacinguδ by uδj, letting j→∞ in (105), and using (107)–(110) and (64), we have
(113)∫0∞‍∫Ω‍|∂tu|2dxdt+supt∈[0,∞){∫Ω‍α|Du|+λ∫Ω‍(u+flog1u)dx}≤∫Ω‍α|Df|dx+λ∫Ω‍(f+flog1f)dx.
Finally, the uniqueness of pseudosolutions to the problems (29)–(31) follows as in [21, 23]. Let u1, u2 be two pseudosolutions to (29)–(31) with u1(x,0)=u2(x,0)=f. Then, we have
(114)∫0t‍∫Ω‍∂su1(u2-u1)dxds+∫0t‍∫Ω‍α|∇u2|dxds+λ∫0t‍∫Ω‍(u2+flog1u2)dxds≥∫0t‍∫Ω‍α|Du1|ds+λ∫0t‍∫Ω‍(u1+flog1u1)dxds,∫0t‍∫Ω‍∂su2(u1-u2)dxds+∫0t‍∫Ω‍α|∇u1|dxds+λ∫0t‍∫Ω‍(u1+flog1u1)dxds≥∫0t‍∫Ω‍α|Du2|ds+λ∫0t‍∫Ω‍(u2+flog1u2)dxds.
Adding the above two inequalities, we get
(115)∫0t‍∫Ω‍∂s(u1-u2)2dxds≤0,
for all t>0. This implies that u1=u2 a.e. in (x,t)∈Q∞.


### 4.4. Long-Time Behavior

At last, we will show the asymptotic limit of the solutionu(·,t) as t→∞.Theorem 13.
Ast→∞, the pseudosolutions to the problems (29)–(31), u(x,t), converges strongly in L2(Ω) to the minimizer u~ of the functional E(u), that is, the solution of the problem (28).Proof.
Take a functionv∈BV(Ω)∩L∞(Ω) with v>0 in (59), then
(116)∫Ω‍(u(x,t)-f(x))v(x)dx-12∫Ω‍(u2(x,t)-f2(x))dx+t∫Ω‍α|∇v|dxds+λt∫Ω‍(v+flog1v)dxds≥∫0t‍∫Ω‍α|Du|ds+λ∫0t‍∫Ω‍(u+flog1u)dxds.
As in [16], let
(117)w(x,t)=1t∫0t‍u(x,s)ds.
Since u∈L∞(0,∞;BV(Ω)∩L∞(Ω)) with u>0, for each t>0, we have that w(·,t)∈BV(Ω)∩L∞(Ω) with {w(·,t)} uniformly bounded in BV(Ω) and L∞(Ω). Then, there exists a subsequence {w(·,ti)} of {w(·,t)} and a function u~∈BV(Ω)∩L∞(Ω), such that as ti→∞,
(118)w(·,ti)⟶u~stronglyinL1(Ω),w(·,ti)⟶u~weakly*inBV(Ω).
Since {w(·,t)} is uniformly bounded in L∞(Ω), we have
(119){w(·,ti)}⟶u~stronglyinL2(Ω).
By dividingt in (116) and then taking the limit along si→∞, we get that, for any v∈BV(Ω)∩L∞ with v>0,
(120)∫Ω‍α|Dv|dx+λ∫Ω‍(v+flog1v)dx≥∫Ω‍α|Du~|ds+λ∫Ω‍(u~+flog1u~)dx,
which implies that u~ is the minimizer of the problem (28).


## 4.1. Definition of a Pseudosolution to the Problems (29)–(31)

DenoteQT=Ω×[0,T), 0<T≤∞. Suppose that v∈L2(0,T;H1(Ω))∩L∞(QT) with v>0, ∂v/∂n→=0 and u is a classical solution of (29)–(31) with u>0. Multiplying (29) by (v-u), integrating over Ω, we have
(55)∫Ω‍∂tu(v-u)dx+∫Ω‍α(x)∇u|∇u|∇(v-u)dx+λ∫Ω‍u-fu(v-u)dx=0.
Since
(56)∫Ω‍∇u|∇u|∇(v-u)dx=∫Ω‍α(x)∇u|∇u|∇vdx-∫Ω‍α(x)|∇u|dx≤∫Ω‍α(x)|∇v|dx-∫Ω‍α(x)|∇u|dx,
and utilizing the Lagrange mean value theorem,
(57)∫Ω‍(v+flog1v)dx-∫Ω‍(u+flog1u)dx=∫Ω‍(v-u)dx-∫Ω‍fξ(v-u)dx≥∫Ω‍(v-u)dx-∫Ω‍fu(v-u)dx,
for either v≥ξ≥u>0 or u≥ξ≥v>0, we have
(58)∫Ω‍∂t(v-u)dx+E(v)≥E(u).
Integrating over [0,t] for any t∈[0,T] then yields
(59)∫0t‍∫Ω‍∂su(v-u)dxds+∫0t‍E(v)ds≥∫0t‍E(u)ds.On the other hand, letv=u+ϵϕ in (59) with ϕ∈C∞(Ω¯), ∂ϕ/∂n→=0, and ϕ>0. Then left-hand side of the inequality (59) has a minimum at ϵ=0. Hence, if u satisfies (59) and u∈L2(0,T;BV∩L2(Ω))∩L∞(QT) with ut∈L2(QT), u>0, and ∂u/∂n→=0, u is also a solution of the problems (29)–(31) in the sense of distribution. Based on this fact which is similar to the ideas in [13, 16, 21], we give the following definition.Definition 9.
A functionu∈L2(0,T;BV(Ω)∩L2(Ω))∩L∞(QT) is called a pseudosolution of (29)–(31), if ∂u/∂t∈L2(QT), ∂u/∂n→=0, u>0, and u satisfies (59) for all t∈[0,T], v∈L2(0,T;BV(Ω)∩L2(Ω))∩L∞(QT) with v>0 and ∂v/∂n→=0. Moreover,
(60)limt→0+∫Ω‍|u(x,t)-f(x)|2dx=0.


## 4.2. The Approximating Problem for Problems (29)–(31)

In this subsection, we consider the approximating problem(61)∂up,δ∂t-div(α|∇up,δ|p-2∇up,δ)+λup,δ-fδup,δ=0,---------------in(x,t)∈Ω×ℝ+,(62)∂up,δ∂n→=0,on(x,t)∈∂Ω×ℝ+,(63)up,δ(x,0)=fδ,onx∈Ω,
where 1<p≤2, fδ∈H1(Ω)∩L∞(Ω) such that as δ→0,
(64)fδ⟶finL2(Ω),∫Ω‍α|Dfδ|⟶∫Ω‍α|Df|,0<infx∈Ωf≤fδ≤supx∈Ωf.
From Lemma 5, we can have the existence of fδ.Let us denote(65)Ep(u)=1p∫Ω‍α(x)|∇u|p+λ∫Ω‍(u+flog1u)dx,
which is associate with the problems (61)–(63). Combining the proof of Theorem  3.4 in [16] with the proof of Theorem 7, we also have the following lemma.Lemma 10.
Letfδ∈L∞(Ω) with infx∈Ωfδ>0. Then, there is a unique solution to the problem
(66)minu∈W+1,p(Ω)∩L∞(Ω)Ep(u),
which is satisfying
(67)0<infx∈Ωfδ≤u≤supx∈Ωfδ,
where W+1,p(Ω)={u>0,u∈W1,p(Ω)}.Based on this fact, we have the following existence and uniqueness result for the problems (61)–(63).Theorem 11.
Letfδ∈L∞(Ω) with infx∈Ωfδ>0 and supx∈Ωf≤1. Then, the problems (61)–(63) admit a unique pseudosolution up,δ∈L∞(0,∞;W1,p(Ω)∩L2(Ω)) with ∂tup,δ∈L2(Q∞), that is,
(68)∫0t‍∫Ω‍∂su(v-up,δ)dxds+∫0t‍Ep(v)ds≥∫0t‍Ep(up,δ)ds,
for any t∈[0,T] and v∈L2(0,T;H1(Ω)) with v>0 and ∂v/∂n→=0. Moreover,
(69)0<infx∈Ωfδ≤u≤supx∈Ωfδ,
and for any T>0,
(70)∫0T‍∫Ω‍|∂tup,δ|2dxdt--+supt∈[0,T]{∫Ω‍(up,δ+fδlog1up,δ)dx1p∫Ω‍α|∇up,δ|pdx-------+λ∫Ω‍(up,δ+fδlog1up,δ)dx}-≤1p∫Ω‍α|∇fδ|pdx+λ∫Ω‍(fδ+fδlog1fδ)dx.Proof.
Let us fixk=infx∈Ωfδ>0 and introduce the following functions:
(71)[u]k=max{u,k}≕{u,ifu≥k,k,ifu<k.
We consider the auxiliary problem as follows:
(72)∂up,δ∂t-div(α|∇up,δ|p-2∇up,δ)+λup,δ-fδ[u]k=0,--------------in(x,t)∈Ω×ℝ+,(73)∂up,δ∂n→=0,on(x,t)∈∂Ω×ℝ+,(74)up,δ(x,0)=fδ,onx∈Ω.
Since p-Laplacian operator is a maximal monotone operator, using Galerkin method and Lebesgue convergence theorem, from standard results for parabolic equations [22], we get that the problems (72)–(74) admit a unique weak solution up,δ such that
(75)up,δ∈L∞(0,∞;W1,p(Ω)∩L2(Ω)),∂tup,δ∈L2(Q∞).
Next, let us verify that the truncated function[·]k in the problems (72)–(74) can be omitted. Multiply (72) by (up,δ-k)-, where
(76)(up,δ-k)-={up,δ-k,ifup,δ-k≤0,0,otherwise,
and integrate over Ω to get
(77)12ddt∫Ω‍(up,δ-k)-2dx+∫Ω‍α|∇(up,δ-k)-|pdx+∫Ω‍(up,δ-fδ)(up,δ-k)-[u]kdx=0.
Then,
(78)12ddt∫Ω‍(up,δ-k)-2dx≤0.
Therefore, (1/2)(d/dt)∫Ω‍(up,δ-k)-2dx is decreasing in t and since
(79)12∫Ω‍(up,δ-k)-2dx≥0,12∫Ω‍(up,δ-k)-2dx|t=0=0,
we have that
(80)12∫Ω‍(up,δ-k)-2dx=0,
for all t∈[0,T], and so
(81)u(t)≥k=infx∈Ωfδ>0ℒ-a.e.onΩ∀t>0.
Then, up,δ is also the solution to the problems (61)–(63). Let us fix K=supx∈Ωfδ. Multiplying (61) by (u-K)+, where
(82)(up,δ-K)+={up,δ-K,ifup,δ-K≥0,0,otherwise,
a similar argument yields that u(t)≤K=supx∈Ωfδ for all t, and then (69) follows.
Moreover, multiplying (61) by ∂tup,δ and integrating it over Ωt yield
(83)∫0t‍∫Ω‍|∂sup,δ|2dxds+1p∫0t‍∂s(∫Ω‍α|∇up,δ|pdx)ds+λ∫0t‍∂s(up,δ+flog1up,δdx)ds=0,
for t∈[0,T]. Then (70) follows.
Finally, we will verify that the above weak solution will be the pseudosolution for the problems (61)–(63). Suppose that v∈L2(0,T;H1(Ω)) with v>0 and ∂v/∂n→=0. Multiplying (61) by v-up,δ, then integrating by parts over ΩT, we obtain
(84)∫0T‍∫Ω‍∂tup,δ(v-up,δ)dxdt+∫0T‍∫Ω‍α|∇up,δ|p-2(∇v-∇up,δ)dxdt+∫0T‍∫Ω‍(up,δ-fδ)(v-up,δ)up,δdxdt=0.
Utilizing Young’s inequality with ϵ=|∇up,δ|2-p yields for the second term in the left-hand side of the above inequality (84), we obtain
(85)|∇up,δ|p-2∇up,δ∇v-|∇up,δ|p≤|∇v|pp+|∇up,δ|pp.
Utilizing the Lagrange mean value theorem yields
(86)∫Ω‍(v+fδlog1v)dx-∫Ω‍(up,δ+fδlog1up,δ)dx=∫Ω‍(v-up,δ)dx-∫Ω‍fδξ(v-up,δ)dx≥∫Ω‍(v-up,δ)dx-∫Ω‍fδup,δ(v-up,δ)dx,
for either v≥ξ≥up,δ>0 or up,δ≥ξ≥v>0. Returning to (84), we therefore conclude
(87)∫0T‍∫Ω‍∂tup,δ(v-up,δ)dxdt+1p∫0T‍∫Ω‍α|∇v|pdx+λ∫0T‍∫Ω‍(v+fδlog1v)dxdt≥1p∫0T‍∫Ω‍α|∇up,δ|pdx+λ∫0T‍∫Ω‍(up,δ+fδlog1up,δ)dxdt,
that is,
(88)∫0t‍∫Ω‍∂su(v-up,δ)dxds+∫0t‍Ep(v)ds≥∫0t‍Ep(up,δ)ds,
for any t∈[0,T]. This fact implies that up,δ is a pseudosolution for the problems (61)–(63). The uniqueness follows by the uniqueness of the weak solution for the problems (61)–(63).


## 4.3. Existence and Uniqueness of the Problems (29)–(31)

In this subsection, we will prove the main theorem for the existence and uniqueness for the solution to the problems (29)–(31).Theorem 12.
Supposef∈BV(Ω)∩L∞(Ω) with infx∈Ωf>0 and supx∈Ωf≤1. Then, there exists a unique pseudosolution u∈L∞(0,∞,BV(Ω)∩L∞(Ω)) to the problems (29)–(31). Moverover,
(89)0<infx∈Ωf≤u≤supx∈Ωf,(90)∫0∞‍∫Ω‍|∂tu|2dxdt+supt∈[0,∞){∫Ω‍α|Du|+λ∫Ω‍(u+flog1u)dx}≤∫Ω‍α|Df|+λ∫Ω‍(f+flog1f)dx.Proof
Step
1. First we fix δ>0 and pass to the limit p→1.
Letup,δ be the pseudosolution solution of (61)–(63). From (69)-(70), we know that, for fixed δ>0, up,δ is uniformly bounded in L∞(0,∞,BV(Ω)∩L∞(Ω)) and ∂tup,δ is uniformly bounded in L2(Q∞), that is,
(91)∥up,δ∥L∞(Q∞)+∥up,δ∥L∞(0,∞,BV(Ω))≤C,∥∂tup,δ∥L2(Q∞)≤C,
where C>0 is constant. Now we claim that there exists a sequence of functions upj,δ and a function uδ∈L∞(0,∞,BV(Ω)∩L∞(Ω)) such that, as pj→1,
(92)∂tupj,δ⟶∂tuδweaklyinL2(Q∞),(93)upj,δ⟶uδweakly*inL∞(Q∞)-with0<infx∈Ωfδ≤uδ≤supx∈Ωfδ,(94)upj,δ⟶uδstronglyinL1(Ω)foreacht∈[0,∞),(95)upj,δ⟶uδinL2(Ω)uniformlyint,(96)limt→0+∫Ω‍|upj,δ(x,t)-fδ(x)|2dx=0.
In fact, from (91), there is a sequence upj,δ and a function uδ∈L∞(Q∞) with ∂tuδ∈L2(Q∞) such that (92) and (93) hold.
Note that, for anyψ∈L2(Ω), as j→∞,
(97)∫Ω‍(upj,δ(x,t)-fδ(x))ψ(x)dx=∫Qt‍∂supj,δ(x,s)1[0,t](s)ψ(x)dxds⟶∫Qt‍∂suδ(x,s)1[0,t](s)ψ(x)dxds=∫Ω‍(uδ(x,t)-fδ(x))ψ(x)dx,
which shows that, for each t,
(98)upj,δ⟶uδweaklyinL2(Ω).
By (69) and (70), for each t∈[0,∞), upj(·,t) is a bounded sequence in W1,1(Ω). Hence, combining this with (98), we get that, for each t as pj→1,
(99)upj,δ⟶uδinL1(Ω),
which implies (94).
Since(100)∥upj,δ(·,t)-upj,δ(·,t′)∥L2(Ω)2≤|t-t′|∫QT‍(∂tupj,δ)2dxdt,
(96) follows.
To see (95), noting that from (100), t→upj,δ(·,t)∈L2(Ω) is equicontinuous, and from (93) and (99), we have, for each t∈[0,∞),
(101)upj,δ⟶uδstronglyinL2(Ω).
Then, a standard argument yields (95).
From (70) and (92), we also obtain that uδ∈L∞(0,∞,BV(Ω)∩L∞(Ω)) with ∂tuδ∈L2(Q∞). The claim is proved.
Next we show that, for allv∈L2(0,T;H1(Ω)) with v>0 and ∂v/∂n→=0, and for each t∈[0,∞),
(102)∫0t‍∫Ω‍∂suδ(v-uδ)dxds+∫0t‍∫Ω‍α|∇v|dxds+λ∫0t‍∫Ω‍(v+fδlog1v)dxds≥∫0t‍∫Ω‍α|Duδ|ds+λ∫0t‍∫Ω‍(uδ+fδlog1uδ)dxds.
By Theorem 11, we obtain that
(103)∫0T‍∫Ω‍∂tupj,δ(v-upj,δ)dxdt+1pj∫0T‍∫Ω‍α|∇v|pjdxdt+λ∫0T‍∫Ω‍(v+fδlog1v)dxdt≥1pj∫0T‍∫Ω‍α|∇upj,δ|pjdxdt+λ∫0T‍∫Ω‍(upj,δ+fδlog1upj,δ)dxdt,
Notice that, using Lemma 4, we have
(104)∫Ω‍α|Duδ|≤liminfj→∞∫Ω‍α|Dupj,δ|≤liminfj→∞(∫Ω‍α|∇upj,δ|pjdx)1/pj(∫Ω‍α(x)dx)1-1/pj=liminfj→∞1pj(∫Ω‍α|∇upj,δ|pjdx)1/pj.
Letting j→∞, pj→1 in (103) and using (92), (94), and (104), we obtain (102) for all v∈L2(0,T;H1(Ω)) with v>0 and ∂v/∂n→=0.
Step
2. Now it only remains to pass to the limit as δ→0 in (102) to complete the existence of the solution to (29)–(31).
Replacingup by upj in (70), letting j→∞ (pj→1), and using (92)–(95), (104), and (64), we obtain
(105)∫0∞‍∫Ω‍|∂tuδ|2dxdt+supt∈[0,∞){∫Ω‍α|Duδ|+λ∫Ω‍(uδ+fδlog1uδ)dx}≤∫Ω‍α|∇fδ|dx+λ∫Ω‍(fδ+fδlog1fδ)dx.
Combining this with (69), we have
(106)uδisuniformlyboundedinL∞(0,∞,BV(Ω)∩L∞(Ω)),∂tuδisuniformlyboundedinL2(Q∞).
Then, by a processing similar to the one for getting (92)–(96), we can obtain a sequence of functions uδj and a function u∈L∞(0,∞;BV(Ω)∩L∞(Ω) such that, as j→∞, δj→0,
(107)∂tuδj⟶∂tuweaklyinL2(Q∞),(108)uδj⟶uweakly*inL∞(Q∞)with0<infx∈Ωf≤u≤supx∈Ωf,(109)uδj⟶ustronglyinL1(Ω)foreacht∈[0,∞),(110)uδj⟶uinL2(Ω)uniformlyint,(111)limt→0+∫Ω‍|uδj(x,t)-f|2dx=0.
Replacinguδ by uδj in (102), letting j→∞, δj→0, and using Lemma 4, we can have from (107)–(110)
(112)∫0t‍∫Ω‍∂su(v-u)dxds+∫0t‍∫Ω‍α|∇v|dxds+λ∫0t‍∫Ω‍(v+flog1v)dxds≥∫0t‍∫Ω‍α|Du|ds+λ∫0t‍∫Ω‍(u+flog1u)dxds,
for all v∈L2(0,T;H1(Ω)) with v>0 and ∂v/∂n→=0. Then, the existence of the pseudosolution to (29)–(31) is completed.
Moreover, replacinguδ by uδj, letting j→∞ in (105), and using (107)–(110) and (64), we have
(113)∫0∞‍∫Ω‍|∂tu|2dxdt+supt∈[0,∞){∫Ω‍α|Du|+λ∫Ω‍(u+flog1u)dx}≤∫Ω‍α|Df|dx+λ∫Ω‍(f+flog1f)dx.
Finally, the uniqueness of pseudosolutions to the problems (29)–(31) follows as in [21, 23]. Let u1, u2 be two pseudosolutions to (29)–(31) with u1(x,0)=u2(x,0)=f. Then, we have
(114)∫0t‍∫Ω‍∂su1(u2-u1)dxds+∫0t‍∫Ω‍α|∇u2|dxds+λ∫0t‍∫Ω‍(u2+flog1u2)dxds≥∫0t‍∫Ω‍α|Du1|ds+λ∫0t‍∫Ω‍(u1+flog1u1)dxds,∫0t‍∫Ω‍∂su2(u1-u2)dxds+∫0t‍∫Ω‍α|∇u1|dxds+λ∫0t‍∫Ω‍(u1+flog1u1)dxds≥∫0t‍∫Ω‍α|Du2|ds+λ∫0t‍∫Ω‍(u2+flog1u2)dxds.
Adding the above two inequalities, we get
(115)∫0t‍∫Ω‍∂s(u1-u2)2dxds≤0,
for all t>0. This implies that u1=u2 a.e. in (x,t)∈Q∞.


## 4.4. Long-Time Behavior

At last, we will show the asymptotic limit of the solutionu(·,t) as t→∞.Theorem 13.
Ast→∞, the pseudosolutions to the problems (29)–(31), u(x,t), converges strongly in L2(Ω) to the minimizer u~ of the functional E(u), that is, the solution of the problem (28).Proof.
Take a functionv∈BV(Ω)∩L∞(Ω) with v>0 in (59), then
(116)∫Ω‍(u(x,t)-f(x))v(x)dx-12∫Ω‍(u2(x,t)-f2(x))dx+t∫Ω‍α|∇v|dxds+λt∫Ω‍(v+flog1v)dxds≥∫0t‍∫Ω‍α|Du|ds+λ∫0t‍∫Ω‍(u+flog1u)dxds.
As in [16], let
(117)w(x,t)=1t∫0t‍u(x,s)ds.
Since u∈L∞(0,∞;BV(Ω)∩L∞(Ω)) with u>0, for each t>0, we have that w(·,t)∈BV(Ω)∩L∞(Ω) with {w(·,t)} uniformly bounded in BV(Ω) and L∞(Ω). Then, there exists a subsequence {w(·,ti)} of {w(·,t)} and a function u~∈BV(Ω)∩L∞(Ω), such that as ti→∞,
(118)w(·,ti)⟶u~stronglyinL1(Ω),w(·,ti)⟶u~weakly*inBV(Ω).
Since {w(·,t)} is uniformly bounded in L∞(Ω), we have
(119){w(·,ti)}⟶u~stronglyinL2(Ω).
By dividingt in (116) and then taking the limit along si→∞, we get that, for any v∈BV(Ω)∩L∞ with v>0,
(120)∫Ω‍α|Dv|dx+λ∫Ω‍(v+flog1v)dx≥∫Ω‍α|Du~|ds+λ∫Ω‍(u~+flog1u~)dx,
which implies that u~ is the minimizer of the problem (28).


## 5. Numerical Methods and Experimental Results

We present in this section some numerical examples illustrating the capability of our model. We also compare it with the known model (AA). In the next two subsections, two numerical discrete schemes, theα-total variation (α-TV) scheme and the p-Laplace approximate (p-LA) scheme, will be proposed.


### 5.1.α-Total Variation Scheme

Numerically we get a solution to the problem (28) by computing the associated equation (29) to a steady state. To discretize (29), the finite difference scheme in [6] is used. Denote the space step by h=1 and the time step by τ. Thus, we have
(121)Dx±(ui,j)=±[ui±1,j-ui,j],Dy±(ui,j)=±[ui,j±1-ui,j],|Dx(ui,j)|=(Dx+(ui,j))2+(m[Dy+(ui,j),Dy-(ui,j)])2+ϵ,|Dx(ui,j)|=(Dy+(ui,j))2+(m[Dx+(ui,j),Dx-(ui,j)])2+ϵ,
where m[a,b]=(signa+signb)min(|a|,|b|)/2 and ϵ>0 is the regularized parameter chosen near 0.The numerical algorithms for the problems (29)–(31) are given as follows:
(122)αi,j=(1-11+k|Gσ*f|i,j2)1+kM2kM2,(123)Zi,jn=Dx-(αi,jDx+ui,jn|Dx+ui,jn|)+Dy-(αi,jDy+ui,jn|Dy+ui,jn|),(124)λn=∑i,j(Zi,jn(1-(f/(u+ϵ)))L)IJ,(125)ui,jn+1=ui,jn+τZi,jn-τλn(1-fu+ϵ),(126)ui,j0=fi,j=f(ih,jh),0≤i≤I,0≤j≤J,(127)ui,0n=ui,1n,u0,jn=u1,jnuI,in=uI-1,in,ui,Jn=ui,J-1n.Here the MATLAB function “conv2" is used to represent the two-dimensional discrete convolution transform of the matrixui,j, that is, G*u. Through the above lines, we can obtain ui,j1 by ui,j0. The program will stop when it achieves our desire.


### 5.2.p-Laplace Approximate Scheme

From the proof of Theorem12, we can know that the term ∫Ω‍α|Du| can be approximated by the term (1/p)∫Ω‍α|∇u|pdx. Based on this, we can use the numerical algorithms for the problems (61)–(63) to get a solution to the problem (28), as p→1. As in [24], the numerical discrete scheme for the problems (61)–(63) is given as follows:
(128)αi,j=(1-11+k|Gσ*f|i,j2)1+kM2kM2,Z1i,jn=αi,j((Dx+ui,jn)2+(Dy+ui,jn)2)(pn-2)/2Dx+ui,jn,Z2i,jn=αi,j((Dx+ui,jn)2+(Dy+ui,jn)2)(pn-2)/2Dy+ui,jn,λn=∑i,j((Dx-Z1i,jn+Dy-Z2i,jn)(1-f/(u+ϵ))L)IJ,ui,jn+1=ui,jn+τ(Dx-Z1i,jn+Dy-Z2i,jn)-τλn(1-fu+ϵ),pn+1=pn+n(pL-pU)N,p0=pU,ui,j0=fi,j=f(ih,jh),0≤i≤I,0≤j≤J,ui,0n=ui,1n,u0,jn=u1,jn,uI,in=uI-1,in,ui,Jn=ui,J-1n,
where pL=1, 1<pU≤2 is upper limit of the exponent p, n=1,2,…,N, and N is the iteration time.


### 5.3. Comparison with Other Methods

In this section, we used the similar way for numerical experiments as [25]. For comparison purposes, some very recent multiplicative noise removal algorithms from the literature are considered, such as the SO Algorithm [8] (see (16)-(17)) and the AA Algorithm [5] (see (10)). As recommended in [8], the stopping rule about SO Algorithm is to reach K such that K=max{k∈ℕ:Var[wk-wO]≥Var[η]=Ψ1(L)}, where wO is the underlying log-image, and η is the relevant noise, and the variance is defined by (16).The denoising algorithms were tested on three images: a synthetic image (300×300 pixels), an aerial image (256×256 pixels), and a cameraman image (256×256 pixels). In all numerical experiments by our algorithms, the images do not need to be normalized in the range [1,256]. This is different from the other algorithms. For each image, a noisy observation is generated by multiplying the original image by the speckle noise according to the model in (8) with the choice L∈{1,4,10}.For a noise-free imageuO and its denoised version by any algorithm u, the denoising performance is measured in terms of peak signal to noise ratio (PSNR) [25],
(129)PSNR=10log10MN|maxuO-minuO|2∥u-uO∥L22dB,
and mean absolute-deviation error (MAE),
(130)MAE=∥u-uO∥L1MN,
where |maxuO-minuO| gives the gray-scale range of the original image, M,N is the size of the image.For fair comparison, the parameters of SO and AA were tweaked manually to reach their best performance level. Their values are summarized in Table1. In the α-total variation (α-TV) scheme, there are four parameters: the influencing factor k, the scale of convolution σ, the time step τ, and the turning factor λ. However, the parameter λ is found dynamically by (124). To ensure stability as well as optimal results, we choose τ=0.05. In p-Laplace approximate (p-LA) scheme, besides the same four parameters with α-TV scheme, there is a new parameter pU with 1<pU≤2. We need not the the exact value of pU but an approximate estimate (e.g., pU=1.2). Notice that the parameters of our method are very stable with respect to the image.Table 1
Parameters used in the comparison study.
Algorithm
Parameters
L
=
1
L
=
4
L
=
10
The synthetic image (300×300)
α-TV
σ
=
2
,
k
=
0.05
σ
=
2
,
k
=
0.05
σ
=
2
,
k
=
0.05
p-LA
p
U
=
1.3
p
U
=
1.3
p
U
=
1.2
SO
λ
=
0.1
,
α
=
0.25
λ
=
0.7
,
α
=
0.25
λ
=
0.2
,
α
=
0.25
AA
λ
=
20
λ
=
150
λ
=
240
The aerial image (512×512)
α-TV
σ
=
2
,
k
=
0.03
σ
=
2
,
k
=
0.015
σ
=
2
,
k
=
0.015
p-LA
p
U
=
1.4
p
U
=
1.4
p
U
=
1.4
SO
λ
=
0.1
,
α
=
0.25
λ
=
0.3
,
α
=
0.25
λ
=
1.2
,
α
=
0.25
AA
λ
=
25
λ
=
120
λ
=
130
The cameraman image (256×256)
α-TV
σ
=
2
,
k
=
0.005
σ
=
2
,
k
=
0.005
σ
=
2
,
k
=
0.005
p-LA
p
U
=
1.3
p
U
=
1.2
p
U
=
1.2
SO
λ
=
0.04
,
α
=
0.25
λ
=
0.2
,
α
=
0.25
λ
=
1
,
α
=
0.25
AA
λ
=
125
λ
=
240
λ
=
390The results are depicted in Figures2, 3, and 4 for the synthetic image, Figures 5, 6, and 7 for the aerial image, and Figures 8, 9, and 10 for the cameraman image. Our methods do a good job at restoring faint geometrical structures of the images even for low values of L, see for instance the results on the aerial image for L=1 and L=4. Our algorithm performs among the best and even outperforms its competitors most of the time both visually and quantitatively as revealed by the PSNR and MAE values. For SO method, the number of iterations necessary to satisfy the stopping rule rapidly increases when L decreases [25]. For AA method, the appropriate parameter λ is necessary.Synthetic image (300×300). (a) Noisy image corrupted by speckle noise for L=1 in (8). (b) Original image. (c) Our algorithm by α-TV, σ=2, k=0.05, and τ=0.05. (d) Our algorithm by p-LA, pU=1.3. (e) SO algorithm, λ=0.1, α=0.25. (f) AA algorithm, λ=50.
(a)
Noisy:L=1, PSNR = −3.46
(b)
Original
(c)
α-TV: PSNR = 19.67, MAE = 2.45
(d)
p-LA: PSNR = 19.76, MAE = 2.51
(e)
SO: PSNR = 4.14, MAE = 23.94
(f)
AA: PSNR = 16.33, MAE = 4.50Synthetic image (300×300) (a) Noisy image corrupted by speckle noise for L=4 in (8). (b) Original image. (c) Our algorithm by α-TV, σ=2, k=0.05, τ=0.05. (d) Our algorithm by p-LA, pU=1.3. (e) SO algorithm, λ=0.7, α=0.25. (f) AA algorithm, λ=150.
(a)
Noisy:L=4, PSNR = 2.52
(b)
Original
(c)
α-TV: PSNR = 23.86, MAE = 1.27
(d)
p-LA: PSNR = 23.14, MAE = 1.43
(e)
SO: PSNR = 15.18, MAE = 6.15
(f)
AA: PSNR = 20.14, MAE = 2.94Synthetic image (300×300) (a) Noisy image corrupted by speckle noise for L=10 in (8). (b) Original image. (c) Our algorithm by α-TV, σ=2, k=0.05, τ=0.05. (d) Our algorithm by p-LA, pU=1.2. (e) SO algorithm, λ=0.7, α=0.25. (f) AA algorithm, λ=240.
(a)
Noisy:L=10, PSNR = 6.52
(b)
Original
(c)
α-TV: PSNR = 26.37, MAE = 0.92
(d)
p-LA: PSNR = 25.68, MAE = 0.96
(e)
SO: PSNR = 21.00, MAE = 2.84
(f)
AA: PSNR = 23.94, MAE = 1.49Aerial image (512×512). (a) Noisy image corrupted by speckle noise for L=1 in (8). (b) Original image. (c) Our algorithm by α-TV, σ=2, k=0.03, and τ=0.05. (d) Our algorithm by p-LA, pU=1.4. (e) SO algorithm, λ=0.1, α=0.25. (f) AA algorithm, λ=25.
(a)
Noisy:L=1, PSNR = 12.39
(b)
Original
(c)
α-TV: PSNR = 23.25, MAE = 12.00
(d)
p-LA: PSNR = 23.47, MAE = 11.64
(e)
SO: PSNR = 17.82, MAE = 25.44
(f)
AA: PSNR = 22.34, MAE = 13.35Aerial image (512×512). (a) Noisy image corrupted by speckle noise for L=4 in (8). (b) Original image. (c) Our algorithm by α-TV, σ=2, k=0.015, and τ=0.05. (d) Our algorithm by p-LA, pU=1.4. (e) SO algorithm, λ=0.3, α=0.25. (f) AA algorithm, λ=120.
(a)
Noisy:L=4, PSNR = 18.39
(b)
Original
(c)
α-TV: PSNR = 25.82, MAE = 8.95
(d)
p-LA: PSNR = 26.16, MAE = 8.81
(e)
SO: PSNR = 24.00, MAE = 11.05
(f)
AA: PSNR = 24.20, MAE = 10.30Aerial image (512×512). (a) Noisy image corrupted by speckle noise for L=10 in (8). (b) Original image. (c) Our algorithm by α-TV, σ=2, k=0.015, and τ=0.05. (d) Our algorithm by p-LA, pU=1.4. (e) SO algorithm, λ=1.2, α=0.25. (f) AA algorithm, λ=130.
(a)
Noisy:L=10, PSNR = 22.45
(b)
Original
(c)
α-TV: PSNR = 27.92, MAE = 7.12
(d)
p-LA: PSNR = 28.18, MAE = 6.98
(e)
SO: PSNR = 27.27, MAE = 7.42
(f)
AA: PSNR = 26.04, MAE = 8.13Cameraman image (512×512). (a) Noisy image corrupted by speckle noise for L=1 in (8). (b) Original image. (c) Our algorithm by α-TV, σ=2, k=0.01, and τ=0.05. (d) Our algorithm by p-LA, pU=1.3. (e) SO algorithm, λ=0.04, α=0.25. (f) AA algorithm, λ=125.
(a)
Noisy:L=1, PSNR = 5.26
(b)
Original
(c)
α-TV: PSNR = 20.81, MAE = 14.65
(d)
p-LA: PSNR = 21.03, MAE = 13.94
(e)
SO: PSNR = 14.15, MAE = 38.92
(f)
AA: PSNR = 18.81, MAE = 22.14Cameraman image (512×512). (a) Noisy image corrupted by speckle noise for L=4 in (8). (b) Original image. (c) Our algorithm by α-TV, σ=2, k=0.005, and τ=0.05. (d) Our algorithm by p-LA, pU=1.2. (e) SO algorithm, λ=0.2, α=0.25. (f) AA algorithm, λ=240.
(a)
Noisy:L=4, PSNR = 11.23
(b)
Original
(c)
α-TV: PSNR = 24.22, MAE = 8.88
(d)
p-LA: PSNR = 24.10, MAE = 9.18
(e)
SO: PSNR = 21.61, MAE = 14.64
(f)
AA: PSNR = 21.91, MAE = 14.25Cameraman image (512×512). (a) Noisy image corrupted by speckle noise for L=10 in (8). (b) Original image. (c) Our algorithm by α-TV, σ=8/6, k=0.005, and τ=0.05. (d) Our algorithm by p-LA, pU=1.2. (e) SO algorithm, λ=1, α=0.25. (f) AA algorithm, λ=390.
(a)
Noisy:L=10, PSNR = 15.27
(b)
Original
(c)
α-TV: PSNR = 26.29, MAE = 7.14
(d)
p-LA: PSNR = 26.03, MAE = 7.43
(e)
SO: PSNR = 24.96, MAE = 8.36
(f)
AA: PSNR = 24.25, MAE = 9.97In the numerical experiments, we can see that for the nontexture image, our methods and AA method work well (see Figure4); for the texture image, our methods and SO method work well (see Figure 7). The denoising performance results are tabulated in Table 2, where the best PSNR and MAE values are shown in boldface. The PSNR improvement brought by our approach can be quite high particularly for L=1 (see e.g., Figures 2–4), and the visual resolution is quite respectable. This is an important achievement since in practice L has a small value, usually 1, rarely above 4. This improvement becomes less salient as L increases which is intuitively expected. But even for L=10, the PSNR of our algorithm can be higher than AA and SO methods (see Table 2).Table 2
PSNR and MAE.
PSNR
MAE
L
1
4
10
L
1
4
10
The synthetic image (300 × 300)
α-TV
19.67
23.86
26.37
α-TV
2.45
1.27
0.92
p-LA
19.76
23.14
25.68
p-LA
2.51
1.43
0.96
SO
4.14
15.81
21.00
SO
23.94
6.15
2.84
AA
16.33
20.14
23.94
AA
4.50
2.94
1.49
The aerial image (512×512)
α-TV
23.25
25.82
27.92
α-TV
12.00
8.95
7.12
p-LA
23.47
26.16
28.18
p-LA
11.64
8.81
6.98
SO
17.82
24.00
27.27
SO
25.44
11.05
7.42
AA
22.34
24.20
26.04
AA
13.35
10.30
8.13
The cameraman image (256×256)
α-TV
20.81
24.22
26.29
α-TV
14.65
8.88
7.14
p-LA
24.10
23.14
26.03
p-LA
13.94
9.18
7.43
SO
14.15
21.61
24.96
SO
38.92
14.64
8.36
AA
18.81
21.91
24.25
AA
22.14
14.25
9.97


## 5.1.α-Total Variation Scheme

Numerically we get a solution to the problem (28) by computing the associated equation (29) to a steady state. To discretize (29), the finite difference scheme in [6] is used. Denote the space step by h=1 and the time step by τ. Thus, we have
(121)Dx±(ui,j)=±[ui±1,j-ui,j],Dy±(ui,j)=±[ui,j±1-ui,j],|Dx(ui,j)|=(Dx+(ui,j))2+(m[Dy+(ui,j),Dy-(ui,j)])2+ϵ,|Dx(ui,j)|=(Dy+(ui,j))2+(m[Dx+(ui,j),Dx-(ui,j)])2+ϵ,
where m[a,b]=(signa+signb)min(|a|,|b|)/2 and ϵ>0 is the regularized parameter chosen near 0.The numerical algorithms for the problems (29)–(31) are given as follows:
(122)αi,j=(1-11+k|Gσ*f|i,j2)1+kM2kM2,(123)Zi,jn=Dx-(αi,jDx+ui,jn|Dx+ui,jn|)+Dy-(αi,jDy+ui,jn|Dy+ui,jn|),(124)λn=∑i,j(Zi,jn(1-(f/(u+ϵ)))L)IJ,(125)ui,jn+1=ui,jn+τZi,jn-τλn(1-fu+ϵ),(126)ui,j0=fi,j=f(ih,jh),0≤i≤I,0≤j≤J,(127)ui,0n=ui,1n,u0,jn=u1,jnuI,in=uI-1,in,ui,Jn=ui,J-1n.Here the MATLAB function “conv2" is used to represent the two-dimensional discrete convolution transform of the matrixui,j, that is, G*u. Through the above lines, we can obtain ui,j1 by ui,j0. The program will stop when it achieves our desire.


## 5.2.p-Laplace Approximate Scheme

From the proof of Theorem12, we can know that the term ∫Ω‍α|Du| can be approximated by the term (1/p)∫Ω‍α|∇u|pdx. Based on this, we can use the numerical algorithms for the problems (61)–(63) to get a solution to the problem (28), as p→1. As in [24], the numerical discrete scheme for the problems (61)–(63) is given as follows:
(128)αi,j=(1-11+k|Gσ*f|i,j2)1+kM2kM2,Z1i,jn=αi,j((Dx+ui,jn)2+(Dy+ui,jn)2)(pn-2)/2Dx+ui,jn,Z2i,jn=αi,j((Dx+ui,jn)2+(Dy+ui,jn)2)(pn-2)/2Dy+ui,jn,λn=∑i,j((Dx-Z1i,jn+Dy-Z2i,jn)(1-f/(u+ϵ))L)IJ,ui,jn+1=ui,jn+τ(Dx-Z1i,jn+Dy-Z2i,jn)-τλn(1-fu+ϵ),pn+1=pn+n(pL-pU)N,p0=pU,ui,j0=fi,j=f(ih,jh),0≤i≤I,0≤j≤J,ui,0n=ui,1n,u0,jn=u1,jn,uI,in=uI-1,in,ui,Jn=ui,J-1n,
where pL=1, 1<pU≤2 is upper limit of the exponent p, n=1,2,…,N, and N is the iteration time.


## 5.3. Comparison with Other Methods

In this section, we used the similar way for numerical experiments as [25]. For comparison purposes, some very recent multiplicative noise removal algorithms from the literature are considered, such as the SO Algorithm [8] (see (16)-(17)) and the AA Algorithm [5] (see (10)). As recommended in [8], the stopping rule about SO Algorithm is to reach K such that K=max{k∈ℕ:Var[wk-wO]≥Var[η]=Ψ1(L)}, where wO is the underlying log-image, and η is the relevant noise, and the variance is defined by (16).The denoising algorithms were tested on three images: a synthetic image (300×300 pixels), an aerial image (256×256 pixels), and a cameraman image (256×256 pixels). In all numerical experiments by our algorithms, the images do not need to be normalized in the range [1,256]. This is different from the other algorithms. For each image, a noisy observation is generated by multiplying the original image by the speckle noise according to the model in (8) with the choice L∈{1,4,10}.For a noise-free imageuO and its denoised version by any algorithm u, the denoising performance is measured in terms of peak signal to noise ratio (PSNR) [25],
(129)PSNR=10log10MN|maxuO-minuO|2∥u-uO∥L22dB,
and mean absolute-deviation error (MAE),
(130)MAE=∥u-uO∥L1MN,
where |maxuO-minuO| gives the gray-scale range of the original image, M,N is the size of the image.For fair comparison, the parameters of SO and AA were tweaked manually to reach their best performance level. Their values are summarized in Table1. In the α-total variation (α-TV) scheme, there are four parameters: the influencing factor k, the scale of convolution σ, the time step τ, and the turning factor λ. However, the parameter λ is found dynamically by (124). To ensure stability as well as optimal results, we choose τ=0.05. In p-Laplace approximate (p-LA) scheme, besides the same four parameters with α-TV scheme, there is a new parameter pU with 1<pU≤2. We need not the the exact value of pU but an approximate estimate (e.g., pU=1.2). Notice that the parameters of our method are very stable with respect to the image.Table 1
Parameters used in the comparison study.
Algorithm
Parameters
L
=
1
L
=
4
L
=
10
The synthetic image (300×300)
α-TV
σ
=
2
,
k
=
0.05
σ
=
2
,
k
=
0.05
σ
=
2
,
k
=
0.05
p-LA
p
U
=
1.3
p
U
=
1.3
p
U
=
1.2
SO
λ
=
0.1
,
α
=
0.25
λ
=
0.7
,
α
=
0.25
λ
=
0.2
,
α
=
0.25
AA
λ
=
20
λ
=
150
λ
=
240
The aerial image (512×512)
α-TV
σ
=
2
,
k
=
0.03
σ
=
2
,
k
=
0.015
σ
=
2
,
k
=
0.015
p-LA
p
U
=
1.4
p
U
=
1.4
p
U
=
1.4
SO
λ
=
0.1
,
α
=
0.25
λ
=
0.3
,
α
=
0.25
λ
=
1.2
,
α
=
0.25
AA
λ
=
25
λ
=
120
λ
=
130
The cameraman image (256×256)
α-TV
σ
=
2
,
k
=
0.005
σ
=
2
,
k
=
0.005
σ
=
2
,
k
=
0.005
p-LA
p
U
=
1.3
p
U
=
1.2
p
U
=
1.2
SO
λ
=
0.04
,
α
=
0.25
λ
=
0.2
,
α
=
0.25
λ
=
1
,
α
=
0.25
AA
λ
=
125
λ
=
240
λ
=
390The results are depicted in Figures2, 3, and 4 for the synthetic image, Figures 5, 6, and 7 for the aerial image, and Figures 8, 9, and 10 for the cameraman image. Our methods do a good job at restoring faint geometrical structures of the images even for low values of L, see for instance the results on the aerial image for L=1 and L=4. Our algorithm performs among the best and even outperforms its competitors most of the time both visually and quantitatively as revealed by the PSNR and MAE values. For SO method, the number of iterations necessary to satisfy the stopping rule rapidly increases when L decreases [25]. For AA method, the appropriate parameter λ is necessary.Synthetic image (300×300). (a) Noisy image corrupted by speckle noise for L=1 in (8). (b) Original image. (c) Our algorithm by α-TV, σ=2, k=0.05, and τ=0.05. (d) Our algorithm by p-LA, pU=1.3. (e) SO algorithm, λ=0.1, α=0.25. (f) AA algorithm, λ=50.
(a)
Noisy:L=1, PSNR = −3.46
(b)
Original
(c)
α-TV: PSNR = 19.67, MAE = 2.45
(d)
p-LA: PSNR = 19.76, MAE = 2.51
(e)
SO: PSNR = 4.14, MAE = 23.94
(f)
AA: PSNR = 16.33, MAE = 4.50Synthetic image (300×300) (a) Noisy image corrupted by speckle noise for L=4 in (8). (b) Original image. (c) Our algorithm by α-TV, σ=2, k=0.05, τ=0.05. (d) Our algorithm by p-LA, pU=1.3. (e) SO algorithm, λ=0.7, α=0.25. (f) AA algorithm, λ=150.
(a)
Noisy:L=4, PSNR = 2.52
(b)
Original
(c)
α-TV: PSNR = 23.86, MAE = 1.27
(d)
p-LA: PSNR = 23.14, MAE = 1.43
(e)
SO: PSNR = 15.18, MAE = 6.15
(f)
AA: PSNR = 20.14, MAE = 2.94Synthetic image (300×300) (a) Noisy image corrupted by speckle noise for L=10 in (8). (b) Original image. (c) Our algorithm by α-TV, σ=2, k=0.05, τ=0.05. (d) Our algorithm by p-LA, pU=1.2. (e) SO algorithm, λ=0.7, α=0.25. (f) AA algorithm, λ=240.
(a)
Noisy:L=10, PSNR = 6.52
(b)
Original
(c)
α-TV: PSNR = 26.37, MAE = 0.92
(d)
p-LA: PSNR = 25.68, MAE = 0.96
(e)
SO: PSNR = 21.00, MAE = 2.84
(f)
AA: PSNR = 23.94, MAE = 1.49Aerial image (512×512). (a) Noisy image corrupted by speckle noise for L=1 in (8). (b) Original image. (c) Our algorithm by α-TV, σ=2, k=0.03, and τ=0.05. (d) Our algorithm by p-LA, pU=1.4. (e) SO algorithm, λ=0.1, α=0.25. (f) AA algorithm, λ=25.
(a)
Noisy:L=1, PSNR = 12.39
(b)
Original
(c)
α-TV: PSNR = 23.25, MAE = 12.00
(d)
p-LA: PSNR = 23.47, MAE = 11.64
(e)
SO: PSNR = 17.82, MAE = 25.44
(f)
AA: PSNR = 22.34, MAE = 13.35Aerial image (512×512). (a) Noisy image corrupted by speckle noise for L=4 in (8). (b) Original image. (c) Our algorithm by α-TV, σ=2, k=0.015, and τ=0.05. (d) Our algorithm by p-LA, pU=1.4. (e) SO algorithm, λ=0.3, α=0.25. (f) AA algorithm, λ=120.
(a)
Noisy:L=4, PSNR = 18.39
(b)
Original
(c)
α-TV: PSNR = 25.82, MAE = 8.95
(d)
p-LA: PSNR = 26.16, MAE = 8.81
(e)
SO: PSNR = 24.00, MAE = 11.05
(f)
AA: PSNR = 24.20, MAE = 10.30Aerial image (512×512). (a) Noisy image corrupted by speckle noise for L=10 in (8). (b) Original image. (c) Our algorithm by α-TV, σ=2, k=0.015, and τ=0.05. (d) Our algorithm by p-LA, pU=1.4. (e) SO algorithm, λ=1.2, α=0.25. (f) AA algorithm, λ=130.
(a)
Noisy:L=10, PSNR = 22.45
(b)
Original
(c)
α-TV: PSNR = 27.92, MAE = 7.12
(d)
p-LA: PSNR = 28.18, MAE = 6.98
(e)
SO: PSNR = 27.27, MAE = 7.42
(f)
AA: PSNR = 26.04, MAE = 8.13Cameraman image (512×512). (a) Noisy image corrupted by speckle noise for L=1 in (8). (b) Original image. (c) Our algorithm by α-TV, σ=2, k=0.01, and τ=0.05. (d) Our algorithm by p-LA, pU=1.3. (e) SO algorithm, λ=0.04, α=0.25. (f) AA algorithm, λ=125.
(a)
Noisy:L=1, PSNR = 5.26
(b)
Original
(c)
α-TV: PSNR = 20.81, MAE = 14.65
(d)
p-LA: PSNR = 21.03, MAE = 13.94
(e)
SO: PSNR = 14.15, MAE = 38.92
(f)
AA: PSNR = 18.81, MAE = 22.14Cameraman image (512×512). (a) Noisy image corrupted by speckle noise for L=4 in (8). (b) Original image. (c) Our algorithm by α-TV, σ=2, k=0.005, and τ=0.05. (d) Our algorithm by p-LA, pU=1.2. (e) SO algorithm, λ=0.2, α=0.25. (f) AA algorithm, λ=240.
(a)
Noisy:L=4, PSNR = 11.23
(b)
Original
(c)
α-TV: PSNR = 24.22, MAE = 8.88
(d)
p-LA: PSNR = 24.10, MAE = 9.18
(e)
SO: PSNR = 21.61, MAE = 14.64
(f)
AA: PSNR = 21.91, MAE = 14.25Cameraman image (512×512). (a) Noisy image corrupted by speckle noise for L=10 in (8). (b) Original image. (c) Our algorithm by α-TV, σ=8/6, k=0.005, and τ=0.05. (d) Our algorithm by p-LA, pU=1.2. (e) SO algorithm, λ=1, α=0.25. (f) AA algorithm, λ=390.
(a)
Noisy:L=10, PSNR = 15.27
(b)
Original
(c)
α-TV: PSNR = 26.29, MAE = 7.14
(d)
p-LA: PSNR = 26.03, MAE = 7.43
(e)
SO: PSNR = 24.96, MAE = 8.36
(f)
AA: PSNR = 24.25, MAE = 9.97In the numerical experiments, we can see that for the nontexture image, our methods and AA method work well (see Figure4); for the texture image, our methods and SO method work well (see Figure 7). The denoising performance results are tabulated in Table 2, where the best PSNR and MAE values are shown in boldface. The PSNR improvement brought by our approach can be quite high particularly for L=1 (see e.g., Figures 2–4), and the visual resolution is quite respectable. This is an important achievement since in practice L has a small value, usually 1, rarely above 4. This improvement becomes less salient as L increases which is intuitively expected. But even for L=10, the PSNR of our algorithm can be higher than AA and SO methods (see Table 2).Table 2
PSNR and MAE.
PSNR
MAE
L
1
4
10
L
1
4
10
The synthetic image (300 × 300)
α-TV
19.67
23.86
26.37
α-TV
2.45
1.27
0.92
p-LA
19.76
23.14
25.68
p-LA
2.51
1.43
0.96
SO
4.14
15.81
21.00
SO
23.94
6.15
2.84
AA
16.33
20.14
23.94
AA
4.50
2.94
1.49
The aerial image (512×512)
α-TV
23.25
25.82
27.92
α-TV
12.00
8.95
7.12
p-LA
23.47
26.16
28.18
p-LA
11.64
8.81
6.98
SO
17.82
24.00
27.27
SO
25.44
11.05
7.42
AA
22.34
24.20
26.04
AA
13.35
10.30
8.13
The cameraman image (256×256)
α-TV
20.81
24.22
26.29
α-TV
14.65
8.88
7.14
p-LA
24.10
23.14
26.03
p-LA
13.94
9.18
7.43
SO
14.15
21.61
24.96
SO
38.92
14.64
8.36
AA
18.81
21.91
24.25
AA
22.14
14.25
9.97



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*Source: 912373-2013-06-06.xml*