# Solving Adaptive Image Restoration Problems via a Modified Projection Algorithm

Authors: Hao Yang; Xueping Luo; Leiting Chen
Journal: Mathematical Problems in Engineering
 (2016)

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## Abstract

We introduce a new general TV regularizer, namely, generalized TV regularization, to study image denoising and nonblind image deblurring problems. In order to discuss the generalized TV image restoration with solution-driven adaptivity, we consider the existence and uniqueness of the solution for mixed quasi-variational inequality. Moreover, the convergence of a modified projection algorithm for solving mixed quasi-variational inequalities is also shown. The corresponding experimental results support our theoretical findings.


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## Body

## 1. Introduction

Digital image restoration plays an important role in many applications of sciences and engineering such as medical and astronomical imaging, film restoration, and image and video coding. Recovering an image from a degraded image is usually an ill-posed inverse problem and it should be dealt with through selecting a suitable regularizer. Since the work of Rudin, Osher, and Fatemi (ROF) in [1], the regularization methods based on total variation (TV) have known a success, mostly due to their ability to preserve edges in the image. In recent years, a number of research works have been proposed in the field of TV regularization approaches, which are used for the task of image denoising and nonblind image deblurring. Aujol et al. [2] replaced L2 norm of the data fidelity term by L1 norm to modify the ROF functional model. The TV regularization approaches in [3, 4] can be described by means of locally dependent constraint sets; that is, the functional is adaptive to the input data. Another class of approaches are the nonlocal methods [5, 6] including nonlocal variants of TV regularization. [7–9] extended TV regularization to second- or higher-order cases. These works mentioned above considered TV regularization approaches for solving the image denoising problems. In addition, these approaches can also be utilized for the image deblurring; see, for example, [10–12]. Chambolle [10] proposed algorithm for minimizing the TV model and applied the algorithm to image zooming. A TV deblurring approach with adaptive choice of the regularization parameter was presented in [12]. In all these literatures, the image restoration problem is always regarded as optimization problem using discrete TV regularization. How to solve such optimization problem with a TV regularization, which is fundamental and crucial, is the core problem in our discussion.It is well known that the theory of variational inequality has been developed as a class of important tools for the study of minimization problems; see, for example, [13]. Among such variational inequalities, inverse variational inequalities, mixed variational inequalities, and quasi-variational inequalities are very significant generalizations, which have been applied to a wide range of problems, such as mechanics, economics, finance, optimal control, and transportation. References [14, 15] proposed Tikhonov regularization method and a general regularization method for solving inverse variational inequality problems. Luo and Yang [16, 17] further extended the results of [14, 15] to the inverse mixed variational inequality problems. The generalized quasi-variational inequality problem was introduced in [18]. However, to our knowledge, a few works implemented quasi-variational inequality to deal with image restoration problem. Recently, Lenzen et al. [19–21] firstly considered a class of quasi-variational inequalities for studying adaptive image restoration, where adaptivity is solution-driven adaptivity. Moreover, they showed that a lot of experimental results support their theoretical findings.Inspired and motivated by the works of [19, 20], in this paper, we introduce a general TV regularization which includes TV regularization of the classical ROF model [1] as its special case. For solving the minimization problem with generalized TV regularization, we discuss its dual problem, which is like the following formulation: (1)minp∈D⁡Fp+Φp,where D is a convex constraint set. For generalizing the regularization approach to solution-driven adaptivity, we find a fixed point of the following mapping: (2)p0⟼p¯≔argminp∈Dp0⁡Fp+Φp.The above fixed point problem is equivalent to solving a mixed quasi-variational inequality [22]. We provide the existence and uniqueness of a fixed point for the mixed quasi-variational inequality for adaptive image restoration. Thus, our theoretical results generalize the research works of [19]. Meanwhile, we propose a modified projection algorithm for solving mixed quasi-variational inequality and prove its convergence. Finally, we give improved experimental results compared to the experiments presented in [19]. Moreover, our experimental results show that the solution-driven adaptive generalized TV model produces excellent restoration effects for different test images.The rest of this paper is organized as follows. In Section2, we recall some notations concerned with generalized Φ-projection operator. In Section 3, we introduce the generalized TV regularization which covers other TV regularizers given in literature. Our model of solution-driven adaptivity described by means of mixed quasi-variational inequalities is shown in Section 4. We consider the theoretical results in Section 5, where we prove the existence and uniqueness of the solution for mixed quasi-variational inequality. In Section 6, we present a modified projection algorithm and its convergence. We give a lot of numerical experiments supporting our theoretical results and showing our better improvement in Section 7. Finally, we conclude this paper in Section 8.


## 2. Preliminaries

In this section, we recall the concept of the generalizedΦ-projection operator, together with its properties.LetΩ be a nonempty closed convex subset of Rn. Let G:Rn×Ω→R∪{+∞} be a function defined as follows: (3)Gx,ξ=x2-2x,ξ+ξ2+2ρΦξ,where ξ∈Ω,x∈Rn,ρ is a positive number, and Φ:Ω→R∪{+∞} is a proper, convex, and lower semicontinuous function.Definition 1.
The generalizedΦ-projection operator ΠΩΦ:Rn⇉Ω is defined as (4)ΠΩΦx=u∈Ω:Gx,u=infξ∈Ω⁡Gx,ξ,∀x∈Rn.From Lemmas3.1 and 3.2 of [25], we know that ΠΩΦ is a single valued and nonexpansive mapping and x∗=ΠΩΦ(x) if and only if(5)x∗-x,y-x∗+ρΦy-ρΦx∗≥0,∀y∈Ω.


## 3. Generalized TV Regularization

In this section, we introduce a new variational approach for image denoising and nonblind image deblurring that is based on total variation regularization. Our general approach covers various adaptive and anisotropic types of TV regularization approaches.The image deblurring problem formulation is as follows. Letf∈Rn be a degraded noisy image, which is obtained from a noise-free image u∈Rn by convolution with a blurring kernel M, followed by an addition of Gaussian noise; that is, (6)f=Mu+δ,where M is n×n invertible matrix and δ is a Gaussian random variable with zero mean. The above problem is a typical inverse problem. In order to recover u from f, assuming that u↦M∗u is a mapping from Rn→Rn, we aim at considering the following optimization problem:(7)argminu∈Rn⁡Eu=12Mu-fL22+αTVu,α>0.In particular, if M≔Id, the minimization problem (7) reduces to image denoising problem, where Id denotes an identity mapping.Now we denote byL:Rnd→Rn the discretization of the divergence operator div and denote by ·2L2 norm. Let us define the following generalized total variation regularizer: (8)αTVu≔supp∈D⁡Lp,uL2-M-⊤Lp2,where (9)D=p∈Rnd,pi∈Bα0,i=1,…,nfor p=(p1,p2,…,pn)⊤ with pi∈Rd and Bα(0) is d-dimensional closed ball with radius α centered at 0. Then the optimization problem is given as(10)argminu∈Rn⁡Eu=12Mu-f22+supp∈D⁡Lp,uL2-M-⊤Lp2.Problem (10) includes a large variety of problems as its special cases: (i)
IfM-⊤Lp2=0, then (10) reduces to problem (2.12) of [19].(ii)
IfM-⊤Lp2=0 and M≔Id, then (10) reduces to the classical ROF model of [1].We derive the corresponding dual problem of (10) as follows. The optimality condition for u reads(11)M⊤Mu-f+Lp=0.It follows from (11) that(12)u=M-1f-M-⊤Lp,(13)Mu=f-M-⊤Lp,where M-⊤≔(M⊤)-1. Using the abbreviation A≔M-⊤L, from (11) and (12) we obtain the dual problem (14)E∗p=12Mu-f22+Lp,uL2-M-⊤Lp2=12M-⊤Lp22+f⊤M-⊤Lp-M-1M-⊤Lp⊤Lp-M-⊤Lp2=12Ap22+fAp-Ap⊤Ap-Ap2=-12Ap22-2fAp-Ap2=-12Ap-f,Ap-fL2-f22-Ap2=-12Ap-f22+12f22-Ap2.When maximizing E∗(p) over D, the constant term 1/2f22 can be omitted without changing the optimum. Moreover, the maximization of E∗ equals the minimization of G(p)≔-E∗(p), and we can formulate the dual problem of (10) as(15)argminp∈D⁡Gp=Fp+Φp=12Ap-f22+Ap2with D≔Dloc⁡1×Dloc⁡2×⋯×Dloc⁡n, where each local constraint set Dloci is a d-dimensional closed ball.From a solutionp¯ of the dual problem, we can retrieve the solution u¯ of the primal problem by u¯=M-1(f-Ap¯). Therefore, the key issue in our discussion of generalized TV image restoration problem is to solve the above minimization problem (15).


## 4. Solution-Driven Adaptivity

In [20], Lenzen et al. proposed a kind of adaptivity, where the constraint set D depends on the unknown solution p of the problem. Naturally, the adaptivity is determined by the noise-free image u, which can be obtained by u=M-1(f-Ap). Moreover, the experimental results of [19] showed that the adaptivity was improved by solution-driven model. In the following, in order to study generalized TV image restoration with solution-driven adaptivity, we generalize problem (15) by introducing a dependency of D on the dual variable: find a fixed point p∗ of the mapping(16)p0⟼p¯≔argminp∈Dp0⁡Gp=Fp+Φp.Having found a fixed point p∗, the corresponding constraint set is Dp∗; that is, the adaptivity is solution-driven. Solving problem (16) is equivalent to considering the following mixed quasi-variational inequality (MQVI):(17)findp∗∈Dp∗such that∇Fp∗,p-p∗+Φp-Φp∗≥0,∀p∈Dp∗,where F(p)=1/2Ap-f22 and Φ(p)=Ap2.From (5), it is easy to see that the MQVI (17) is equivalent to the following projection equation:(18)p∗=ΠDp∗Φp∗-ρ∇Fp∗,∀ρ>0.


## 5. Theory for MQVI

In this section, we provide the existence and uniqueness results for the MQVI (17).


### 5.1. Existence of Solutions

Theorem 2.
LetFp≔1/2Ap-f22 and Φ(p)≔Ap2, where A:Rmn→Rn is a linear operator. Let D(p) be defined as follows: (19)D:p¯⇉Dp¯≔p∈Rmn:pi∈Dlocip¯⊂Rmn,i=1,…,n,where each Dloci:Rmn⇉Rmn, i=1,…,n, has the following properties: (i)
For fixedp the set Dloci(p) is a closed convex subset of Rmn.(ii)
There existsC>0 such that, for all i,p, one has Dlocip⊂BC0.(iii)
There existsc>0 such that, for every p and every i, one has Bc(0)⊂Dloci(p). In particular, Dloci(p) is nonempty.(iv)
The generalizedΦ-projection operator ΠDlocipΦq of q onto Dlocip for a fixed q is continuous with respect to p. Then mixed quasi-variational inequality (17) has a solution.Proof.
Firstly, we know from the definition ofDp that (20)Dp≔Dloc1p×Dloc2p×⋯×Dlocnp.Thus, from assumption (ii), we immediately derive that (21)Dp⊂BC0n⊂BnC0,p∈Rmn.Since BnC(0) is a bounded closed convex ball, BnC(0) is compact convex.
Assumptions (i) and (iii) imply thatD is a nonempty closed convex valued mapping on BnC(0). Moreover, Φ(p)=Ap2 is a proper, convex, and lower semicontinuous function. Hence the generalized Φ-projection operator ΠD(p)Φ is well defined. By ΠD(p)Φ(q)=(ΠDloc⁡1(p)Φ(q1),…,ΠDloc⁡n(p)Φ(qn))⊤ and (iv), we obtain that ΠD(p)Φ is continuous with respect to p.
Now we define the mappingS:BnC(0)→BnC(0) by (22)Sp≔ΠDpΦp-ρ∇Fp,∀ρ>0.It follows from the continuity of ∇F(p)=A⊤(Ap-f) and ΠD(p)Φ that S is continuous. Hence by (18) and the Brouwer fixed point theorem, we have that problem (17) has a solution.In the case thatΦ≡0, a similar result of Theorem 2 was obtained (see Proposition 4.2 of [19]). Therefore, Theorem 2 can also be considered as a generalization of Proposition 4.2 of [19].


### 5.2. Uniqueness Result of the Proposed Approach

In this subsection, let us consider the uniqueness results for MQVI (17). In [19], Lenzen et al. discussed the uniqueness of solution of quasi-variational inequality on only a subspace of Rmn, because ∇F is not strongly monotone on the null space N(A) of A (N(A)≔{x∈Rn:Ax=0}). On the other hand, our main aim is to find u=M-1(f-Ap), which does not depend on the component of p in N(A). In view of these reasons mentioned, [19] restricted the problem of quasi-variational inequality to the complement N⊥(A) of N(A). Now we utilize similar method to study problem (17). Firstly, we give the following search model that is restricted to N⊥(A):(23)Findp∗∈ΠN⊥ADp∗(24)such that∇Fp∗,p-p∗+Φp-Φp∗≥0,∀p∈ΠN⊥ADp∗.Theorem 3.
Assume that the setD(p) depends only on pres≔ΠN⊥(A)(p).(i)
Letpres∗ be a solution to the restricted problem (23), and then any p∗∈ΠN⊥(A)-1(pres∗)∩D(pres∗) is a solution to the original problem (17).(ii)
Letp∗ be a solution to the unrestricted problem (17), and then any pres∗∈ΠN⊥(A)-1(p∗) is a solution to the restricted problem (23).Proof.
Letpres∗∈ΠN⊥(A)(D(pres∗)) be a solution to the restricted problem (23); that is,(25)∇Fpres∗,p-pres∗+Φp-Φpres∗≥0,∀p∈ΠN⊥ADpres∗.For any p∗∈D(pres∗) such that pres∗=ΠN⊥(A)p∗=p∗-ΠN(A)p∗, it holds that p∗∈D(p∗)=D(pres∗). Now let p∈D(pres∗)=D(p∗) be arbitrary. We decompose p into p=pres+pN, where pres≔ΠN⊥(A)p and pN≔ΠN(A)p. Then it follows from (25) and Ap=Apres and Ap∗=Apres∗ that (26)∇Fp∗,p-p∗+Φp-Φp∗=A⊤Ap∗-f,p-p∗+Ap2-Ap∗2=Ap∗-f,Ap-p∗+Ap2-Ap∗2=Apres∗-f,Apres-pres∗+Apres2-Apres∗2=A⊤Apres∗-f,pres-pres∗+Apres2-Apres∗2≥0.Thus, p∗ is a solution of (17).
Letp∗ be a solution of problem (17). In particular, p∗∈D(p∗). We consider the decomposition p∗=pres∗+pN∗, where pres∗≔ΠN⊥(A)p∗ and pN∗≔ΠN(A)p∗. Then (27)pres∗∈ΠN⊥ADp∗=ΠN⊥ADpres∗.Let pres∈ΠN⊥(A)(D(pres∗)) be arbitrary. There exists p∈D(pres∗) such that(28)pres=ΠN⊥Ap=p-ΠNAp.It follows from (28) that (29)∇Fpres∗,pres-pres∗+Φpres-Φpres∗=Apres∗-f,Apres-pres∗+Apres2-Apres∗2=Ap∗-f,Ap-p∗+Ap2-Ap∗2=A⊤Ap∗-f,p-p∗+Ap2-Ap∗2≥0,where the last inequality holds since p∈D(p∗) due to D(p∗)=D(pres∗) and p∗ solves (17). Thus pres∗ is a solution of (23).From Section3, we can see that the final purpose of finding the existence and uniqueness of p∗ is to solve the optimal problem (10), because u∗=M-1(f-Ap∗) which does not depend on pN∗. We therefore focus on v∗=Ap∗, which depends only on the component pres∗ of p∗ in N⊥A; that is, we only need to consider the restricted problem (23). Based on Theorem 3, the restricted problem has a solution if and only if the original problem has a solution. Here we specify v∗2=p∗B≔(p∗)⊤A⊤Ap∗. In the following discussion, D~(v) denotes a nonempty, closed, convex set such that D~(v)=D(p)=D~(Ap).Before showing the uniqueness, let us defineT:Rn⇉Rmn such that p¯∈T(v) if and only if p¯∈D~(v) and p¯ is a solution to the following MQVI: (30)∇Fp¯,p-p¯+Φp-Φp¯≥0,∀p∈D~v.Theorem 4.
Under all the assumption conditions of Theorem2 and the assumptions (i)
∇F is Lipschitz continuous with Lipschitz constant μ2≔A⊤A2>0, that is, (31)∇Fx-∇Fy2≤μ2x-y2,∀x,y∈Rmn,(ii)
the generalizedΦ-projection operator ΠD~(v)Φq is Lipschitz continuous with respect to v with the variation rate η~>0, that is, (32)ΠD~vΦq-ΠD~v′Φq2≤η~v-v′2,∀q∈Rmn,(iii)
η~<1/μ2, if p∗ is a solution of mixed quasi-variational inequality (17), then v∗=Ap∗ is unique.Proof.
Fixv1,v2∈Im⁡(A). Let Di≔D~vi,pi∈Tvi:(i)
IfA(p1-p2)=0, we obtain (33)ATv1-ATv22=Ap1-Ap22=0<v1-v22.(ii)
IfA(p1-p2)≠0, since pi, i=1,2, solve argminp∈D~(vi)1/2Ap-f22+Ap2, the following MQVI(34)∇Fpi,q-pi+Φq-Φpi≥0,∀q∈Di,holds; that is, for any ρ≥0,(35)pi=ΠDiΦpi-ρ∇Fpi.In particular, q≔ΠD2Φ(p1-ρ∇F(p1)), and it follows from (35) and (ii) that(36)p1-q2=ΠD1Φp1-ρ∇Fp1-ΠD2Φp1-ρ∇Fp12≤η~v1-v22.On the other hand, p2∈D2 implies that(37)q-p1-ρ∇Fp1,p2-q+Φp2-Φq≥0.Therefore, from (34) and (37), we have (38)q-p1,p2-q≥ρ∇Fp1,q-p2-Φp2+Φq=ρ∇Fp1,q-p1+ρ∇Fp2,p1-q+ρ∇Fp2,q-p2+Φq-Φp2+ρ∇Fp1-∇Fp2,p1-p2≥ρ∇Fp1-∇Fp2,q-p1+ρ∇Fp1-∇Fp2,p1-p2=ρA⊤Ap1-p2,q-p1+ρAp1-Ap222;that is,(39)ρAp1-Ap222≤q-p1,p2-q+ρA⊤Ap1-p2,p1-q≤q-p12·q-p22+ρAp1-Ap22·Ap1-Aq2and, by dividing by ρAp1-Ap22>0, because Ap1-Ap2≠0,(40)Ap1-Ap22≤q-p12·q-p22ρAp1-Ap22+Ap1-Aq2.Since ρ is arbitrarily large, we find(41)Ap1-Ap22≤Ap1-Aq2.By (i), we have A2=μ2. Thus, it follows from (36) that(42)Ap1-Aq2≤A2p1-q2≤μ2η~v1-v22;that is, (43)ATv1-ATv22≤μ2η~v1-v22.Therefore, we can see from (iii) that A∘T:Rn→Rn is a contractive mapping. Moreover, the Banach fixed point theorem implies that there exists a unique fixed point v∗=Ap∗ of A∘T in Rn.We already mentioned that, in the considered applications for image restoration, we are actually interested in the variableu≔M-1(f-Ap). Obviously, it follows from Theorem 4 that u is unique.


## 5.1. Existence of Solutions

Theorem 2.
LetFp≔1/2Ap-f22 and Φ(p)≔Ap2, where A:Rmn→Rn is a linear operator. Let D(p) be defined as follows: (19)D:p¯⇉Dp¯≔p∈Rmn:pi∈Dlocip¯⊂Rmn,i=1,…,n,where each Dloci:Rmn⇉Rmn, i=1,…,n, has the following properties: (i)
For fixedp the set Dloci(p) is a closed convex subset of Rmn.(ii)
There existsC>0 such that, for all i,p, one has Dlocip⊂BC0.(iii)
There existsc>0 such that, for every p and every i, one has Bc(0)⊂Dloci(p). In particular, Dloci(p) is nonempty.(iv)
The generalizedΦ-projection operator ΠDlocipΦq of q onto Dlocip for a fixed q is continuous with respect to p. Then mixed quasi-variational inequality (17) has a solution.Proof.
Firstly, we know from the definition ofDp that (20)Dp≔Dloc1p×Dloc2p×⋯×Dlocnp.Thus, from assumption (ii), we immediately derive that (21)Dp⊂BC0n⊂BnC0,p∈Rmn.Since BnC(0) is a bounded closed convex ball, BnC(0) is compact convex.
Assumptions (i) and (iii) imply thatD is a nonempty closed convex valued mapping on BnC(0). Moreover, Φ(p)=Ap2 is a proper, convex, and lower semicontinuous function. Hence the generalized Φ-projection operator ΠD(p)Φ is well defined. By ΠD(p)Φ(q)=(ΠDloc⁡1(p)Φ(q1),…,ΠDloc⁡n(p)Φ(qn))⊤ and (iv), we obtain that ΠD(p)Φ is continuous with respect to p.
Now we define the mappingS:BnC(0)→BnC(0) by (22)Sp≔ΠDpΦp-ρ∇Fp,∀ρ>0.It follows from the continuity of ∇F(p)=A⊤(Ap-f) and ΠD(p)Φ that S is continuous. Hence by (18) and the Brouwer fixed point theorem, we have that problem (17) has a solution.In the case thatΦ≡0, a similar result of Theorem 2 was obtained (see Proposition 4.2 of [19]). Therefore, Theorem 2 can also be considered as a generalization of Proposition 4.2 of [19].


## 5.2. Uniqueness Result of the Proposed Approach

In this subsection, let us consider the uniqueness results for MQVI (17). In [19], Lenzen et al. discussed the uniqueness of solution of quasi-variational inequality on only a subspace of Rmn, because ∇F is not strongly monotone on the null space N(A) of A (N(A)≔{x∈Rn:Ax=0}). On the other hand, our main aim is to find u=M-1(f-Ap), which does not depend on the component of p in N(A). In view of these reasons mentioned, [19] restricted the problem of quasi-variational inequality to the complement N⊥(A) of N(A). Now we utilize similar method to study problem (17). Firstly, we give the following search model that is restricted to N⊥(A):(23)Findp∗∈ΠN⊥ADp∗(24)such that∇Fp∗,p-p∗+Φp-Φp∗≥0,∀p∈ΠN⊥ADp∗.Theorem 3.
Assume that the setD(p) depends only on pres≔ΠN⊥(A)(p).(i)
Letpres∗ be a solution to the restricted problem (23), and then any p∗∈ΠN⊥(A)-1(pres∗)∩D(pres∗) is a solution to the original problem (17).(ii)
Letp∗ be a solution to the unrestricted problem (17), and then any pres∗∈ΠN⊥(A)-1(p∗) is a solution to the restricted problem (23).Proof.
Letpres∗∈ΠN⊥(A)(D(pres∗)) be a solution to the restricted problem (23); that is,(25)∇Fpres∗,p-pres∗+Φp-Φpres∗≥0,∀p∈ΠN⊥ADpres∗.For any p∗∈D(pres∗) such that pres∗=ΠN⊥(A)p∗=p∗-ΠN(A)p∗, it holds that p∗∈D(p∗)=D(pres∗). Now let p∈D(pres∗)=D(p∗) be arbitrary. We decompose p into p=pres+pN, where pres≔ΠN⊥(A)p and pN≔ΠN(A)p. Then it follows from (25) and Ap=Apres and Ap∗=Apres∗ that (26)∇Fp∗,p-p∗+Φp-Φp∗=A⊤Ap∗-f,p-p∗+Ap2-Ap∗2=Ap∗-f,Ap-p∗+Ap2-Ap∗2=Apres∗-f,Apres-pres∗+Apres2-Apres∗2=A⊤Apres∗-f,pres-pres∗+Apres2-Apres∗2≥0.Thus, p∗ is a solution of (17).
Letp∗ be a solution of problem (17). In particular, p∗∈D(p∗). We consider the decomposition p∗=pres∗+pN∗, where pres∗≔ΠN⊥(A)p∗ and pN∗≔ΠN(A)p∗. Then (27)pres∗∈ΠN⊥ADp∗=ΠN⊥ADpres∗.Let pres∈ΠN⊥(A)(D(pres∗)) be arbitrary. There exists p∈D(pres∗) such that(28)pres=ΠN⊥Ap=p-ΠNAp.It follows from (28) that (29)∇Fpres∗,pres-pres∗+Φpres-Φpres∗=Apres∗-f,Apres-pres∗+Apres2-Apres∗2=Ap∗-f,Ap-p∗+Ap2-Ap∗2=A⊤Ap∗-f,p-p∗+Ap2-Ap∗2≥0,where the last inequality holds since p∈D(p∗) due to D(p∗)=D(pres∗) and p∗ solves (17). Thus pres∗ is a solution of (23).From Section3, we can see that the final purpose of finding the existence and uniqueness of p∗ is to solve the optimal problem (10), because u∗=M-1(f-Ap∗) which does not depend on pN∗. We therefore focus on v∗=Ap∗, which depends only on the component pres∗ of p∗ in N⊥A; that is, we only need to consider the restricted problem (23). Based on Theorem 3, the restricted problem has a solution if and only if the original problem has a solution. Here we specify v∗2=p∗B≔(p∗)⊤A⊤Ap∗. In the following discussion, D~(v) denotes a nonempty, closed, convex set such that D~(v)=D(p)=D~(Ap).Before showing the uniqueness, let us defineT:Rn⇉Rmn such that p¯∈T(v) if and only if p¯∈D~(v) and p¯ is a solution to the following MQVI: (30)∇Fp¯,p-p¯+Φp-Φp¯≥0,∀p∈D~v.Theorem 4.
Under all the assumption conditions of Theorem2 and the assumptions (i)
∇F is Lipschitz continuous with Lipschitz constant μ2≔A⊤A2>0, that is, (31)∇Fx-∇Fy2≤μ2x-y2,∀x,y∈Rmn,(ii)
the generalizedΦ-projection operator ΠD~(v)Φq is Lipschitz continuous with respect to v with the variation rate η~>0, that is, (32)ΠD~vΦq-ΠD~v′Φq2≤η~v-v′2,∀q∈Rmn,(iii)
η~<1/μ2, if p∗ is a solution of mixed quasi-variational inequality (17), then v∗=Ap∗ is unique.Proof.
Fixv1,v2∈Im⁡(A). Let Di≔D~vi,pi∈Tvi:(i)
IfA(p1-p2)=0, we obtain (33)ATv1-ATv22=Ap1-Ap22=0<v1-v22.(ii)
IfA(p1-p2)≠0, since pi, i=1,2, solve argminp∈D~(vi)1/2Ap-f22+Ap2, the following MQVI(34)∇Fpi,q-pi+Φq-Φpi≥0,∀q∈Di,holds; that is, for any ρ≥0,(35)pi=ΠDiΦpi-ρ∇Fpi.In particular, q≔ΠD2Φ(p1-ρ∇F(p1)), and it follows from (35) and (ii) that(36)p1-q2=ΠD1Φp1-ρ∇Fp1-ΠD2Φp1-ρ∇Fp12≤η~v1-v22.On the other hand, p2∈D2 implies that(37)q-p1-ρ∇Fp1,p2-q+Φp2-Φq≥0.Therefore, from (34) and (37), we have (38)q-p1,p2-q≥ρ∇Fp1,q-p2-Φp2+Φq=ρ∇Fp1,q-p1+ρ∇Fp2,p1-q+ρ∇Fp2,q-p2+Φq-Φp2+ρ∇Fp1-∇Fp2,p1-p2≥ρ∇Fp1-∇Fp2,q-p1+ρ∇Fp1-∇Fp2,p1-p2=ρA⊤Ap1-p2,q-p1+ρAp1-Ap222;that is,(39)ρAp1-Ap222≤q-p1,p2-q+ρA⊤Ap1-p2,p1-q≤q-p12·q-p22+ρAp1-Ap22·Ap1-Aq2and, by dividing by ρAp1-Ap22>0, because Ap1-Ap2≠0,(40)Ap1-Ap22≤q-p12·q-p22ρAp1-Ap22+Ap1-Aq2.Since ρ is arbitrarily large, we find(41)Ap1-Ap22≤Ap1-Aq2.By (i), we have A2=μ2. Thus, it follows from (36) that(42)Ap1-Aq2≤A2p1-q2≤μ2η~v1-v22;that is, (43)ATv1-ATv22≤μ2η~v1-v22.Therefore, we can see from (iii) that A∘T:Rn→Rn is a contractive mapping. Moreover, the Banach fixed point theorem implies that there exists a unique fixed point v∗=Ap∗ of A∘T in Rn.We already mentioned that, in the considered applications for image restoration, we are actually interested in the variableu≔M-1(f-Ap). Obviously, it follows from Theorem 4 that u is unique.


## 6. Numerics

Throughout this section, assume that all the assumption conditions of Theorems2 and 4 are satisfied. Next, we propose an iterative algorithm to solve the MQVI (17) and show convergence of the proposed algorithm.


### 6.1. Proposed Iterative Algorithm

See Algorithm1.Algorithm 1:Modified projection method for solving the MQVI (17).
Input: The maximal number of iterations N; the constraint set D(p0); the starting point
p
0
∈
D
(
p
0
)
⊂
B
n
C
(
0
); the parameter number ρ and such that 0<ρ<1/μ2.
Output: pN.
begin
for
k
=
0 to N-1 do
Step  1.
p
-
k
-
1
≔
Π
D
p
k
-
1
Φ
p
k
-
1
-
ρ
∇
F
p
k
-
1;
Step  2.
p
k
≔
Π
D
p
-
k
-
1
Φ
p
k
-
1
-
ρ
∇
F
p
-
k
-
1;
Step  3.If |pk-pk-1|≤ε, for all ε>0, then, stop; otherwise, k=k+1 and go the Step 1.
end
end


### 6.2. Convergence of Algorithm1

Theorem 5.
Let all the assumptions of Theorems2 and 4 be satisfied. Moreover, assume that ∇F is monotone; that is, (44)∇Fx-∇Fy,x-y≥0,∀x,y∈Rmn.Then the sequence {pk} generated by Algorithm 1 converges to the unique solution p∗ of MQVI (17).Proof.
For the proof, see Theorem2.5 in [26].


## 6.1. Proposed Iterative Algorithm

See Algorithm1.Algorithm 1:Modified projection method for solving the MQVI (17).
Input: The maximal number of iterations N; the constraint set D(p0); the starting point
p
0
∈
D
(
p
0
)
⊂
B
n
C
(
0
); the parameter number ρ and such that 0<ρ<1/μ2.
Output: pN.
begin
for
k
=
0 to N-1 do
Step  1.
p
-
k
-
1
≔
Π
D
p
k
-
1
Φ
p
k
-
1
-
ρ
∇
F
p
k
-
1;
Step  2.
p
k
≔
Π
D
p
-
k
-
1
Φ
p
k
-
1
-
ρ
∇
F
p
-
k
-
1;
Step  3.If |pk-pk-1|≤ε, for all ε>0, then, stop; otherwise, k=k+1 and go the Step 1.
end
end


## 6.2. Convergence of Algorithm1

Theorem 5.
Let all the assumptions of Theorems2 and 4 be satisfied. Moreover, assume that ∇F is monotone; that is, (44)∇Fx-∇Fy,x-y≥0,∀x,y∈Rmn.Then the sequence {pk} generated by Algorithm 1 converges to the unique solution p∗ of MQVI (17).Proof.
For the proof, see Theorem2.5 in [26].


## 7. Experiment Results

### 7.1. Improvement of Solution-Driven Adaptive TV Regularization

In this section, we show that the adaptivity is improved by switching from a solution-driven adaptive TV regularization to a solution-driven adaptive generalized TV model. To this end, we consider the image Cameraman. We generate test data for the denoising problem by adding Gaussian noise with zero mean and standard deviation 0.1 and for the deblurring problem by applying a blurring operator and adding Gaussian noise with zero mean and standard deviation 0.01. For comparison, we make use of a mean SSIM (MSSIM) index [27] to evaluate the restored image quality in Figures 1 and 2. The definitions of the similarity measures SSIM and MSSIM are given as follows: (45)SSIMx,y=2μxμy+c12σxσy+c2μx2+μy2+c1σx2+σy2+c2,where μx, μy are the average values of two signals x and y, σx2 and σy2 are the variances of x and y, and c1, c2 are variables to stabilize the division with weak denominator (46)MSSIMX,Y=1M∑j=1MSSIMxj,yj,where X and Y are the reference and the distorted images, respectively; xj and yj are the image contents at the jth local window; and M is the number of local windows of the image.Figure 1
(a) Noisy input data, MSSIM = 0.327. (b) Solution-driven adaptive TV, MSSIM = 0.812. (c) Solution-driven adaptive generalized TV, MSSIM = 0.822.
(a)
(b)
(c)Figure 2
(a) Blurred input data, MSSIM = 0.032. (b) Solution-driven adaptive TV, MSSIM = 0.866. (c) Solution-driven adaptive generalized TV, MSSIM = 0.868.
(a)
(b)
(c)In Figure1, we give the close-up of the results of denoising the image Cameraman with TV regularization and generalized TV regularization approaches. The values for similarity given show that generalized TV regularization approach enhances the reconstruction compared to the TV regularization.In Figure2, close-up of the results of deblurring the image Cameraman with TV regularization and generalized TV regularization approaches is shown. In terms of similarity to the original data, we can see that generalized TV regularization approach improves the reconstruction compared to the TV regularization.


### 7.2. Comparison of Image Restoration for Different Test Images

In this subsection, we report that our method can remove Gaussian noise efficiently while preserving details very well. For noise removal, we compare our method with some recently developed models, namely, the local TV [28], the NLTV [29], the anisotropic TV [30], and the isotropic TV [30], when the Split Bregman method is considered. The quality of our restoration is measured by the peak signal-to-noise ratio (PSNR) in decibels (dB): (47)PSNR=10log10⁡M×N∑i=1M∑j=1Nui,j-u~i,j2,where M×N indicates the image size and u and u~ denote the original image and the restored one, respectively. Generally, the higher PSNR values indicate better quality of the restored images.The test images are shown in Figure3.Figure 3
Test images. (a) Cameraman. (b) Lena. (c) Pepper. (d) Building.
(a)
(b)
(c)
(d)In Figure4, we make use of the local TV model, the NLTV model, the anisotropic TV model, the isotropic TV model, and the solution-driven adaptive generalized TV model to display the restoration results for the test image (a) corrupted by 8% Gaussian noise. Here, our proposed model performs significantly better and can suppress the Gaussian noise successfully.Figure 4
(a) Noisy image (PSNR = 8.748 dB). (b) Local TV, Split Bregman (PSNR = 28.18 dB). (c) Nonlocal TV, Split Bregman (PSNR = 30.35 dB). (d) Anisotropic TV, Split Bregman (PSNR = 27.687 dB). (e) Isotropic TV, Split Bregman (PSNR = 28.594 dB). (f) Solution-driven adaptive generalized TV, modified projection (PSNR = 30.56 dB).
(a)
(b)
(c)
(d)
(e)
(f)Table1 lists the restoration results in the PSNR of different methods for test images Cameraman, Lena, Pepper, and Building corrupted by Gaussian noise.Table 1
Comparison of restoration results in PSNR (dB) for images corrupted by Gaussian noise.
Methods
Cameraman
Lena
Pepper
Building
Local TV
28.18
27.68
26.68
27.55
NLTV
30.35
29.87
30.86
29.23
Anisotropic TV
27.687
26.1
26.591
29.3
Isotropic TV
28.594
27.902
27.661
26.603
Proposed
30.56
30.21
30.98
29.5From Table1, it is apparent that our proposed method generates the best restoration results for all the test images. Actually, in Table 1, the proposed method obtains the highest PSNR values for all the test images. This demonstrated that our method is more robust for images corrupted by Gaussian noise.


### 7.3. Expanded Experiments

Remote sensing very often deals with inverting a forward model. To this aim, one has to produce an accurate and robust model able to predict physical, chemical, geological, or atmospheric parameters from spectra, such as surface temperature, water vapour, and ozone; see, for example, [31]. Denoising images could be part of the accurate and robust model. In particular, our image restoration method could be used to remove noises in the class of satellite images.In Figure5, we give the close-up of the results of denoising the satellite image adding Gaussian noise with the standard derivation σ=0.02. Though the fourth-order PDE model removes noise more thoroughly, texture detail also had been changed. Comparing rivers and streets with those in the original image, some information was lost. Our method not only suppresses the noise very well but also keeps a lot of detail and texture information of the original image.Figure 5
(a) Original satellite image. (b) Gaussian noise (σ=0.02) image. (c) Fourth-order PDE model. (d) Proposed method.
(a)
(b)
(c)
(d)From Table2, we can see that our method has better denoising effects than pure anisotropic diffusion and fourth-order PDE.Table 2
Comparison of restoration results in PSNR (dB) for satellite images.
Methods
σ
=
0.01
σ
=
0.02
σ
=
0.05
σ
=
1
Pure anisotropic diffusion [23]
30.16
23.68
15.36
6.10
Fourth-order PDE [24]
31.64
25.40
17.40
11.36
Proposed
34.32
27.68
19.54
13.28


## 7.1. Improvement of Solution-Driven Adaptive TV Regularization

In this section, we show that the adaptivity is improved by switching from a solution-driven adaptive TV regularization to a solution-driven adaptive generalized TV model. To this end, we consider the image Cameraman. We generate test data for the denoising problem by adding Gaussian noise with zero mean and standard deviation 0.1 and for the deblurring problem by applying a blurring operator and adding Gaussian noise with zero mean and standard deviation 0.01. For comparison, we make use of a mean SSIM (MSSIM) index [27] to evaluate the restored image quality in Figures 1 and 2. The definitions of the similarity measures SSIM and MSSIM are given as follows: (45)SSIMx,y=2μxμy+c12σxσy+c2μx2+μy2+c1σx2+σy2+c2,where μx, μy are the average values of two signals x and y, σx2 and σy2 are the variances of x and y, and c1, c2 are variables to stabilize the division with weak denominator (46)MSSIMX,Y=1M∑j=1MSSIMxj,yj,where X and Y are the reference and the distorted images, respectively; xj and yj are the image contents at the jth local window; and M is the number of local windows of the image.Figure 1
(a) Noisy input data, MSSIM = 0.327. (b) Solution-driven adaptive TV, MSSIM = 0.812. (c) Solution-driven adaptive generalized TV, MSSIM = 0.822.
(a)
(b)
(c)Figure 2
(a) Blurred input data, MSSIM = 0.032. (b) Solution-driven adaptive TV, MSSIM = 0.866. (c) Solution-driven adaptive generalized TV, MSSIM = 0.868.
(a)
(b)
(c)In Figure1, we give the close-up of the results of denoising the image Cameraman with TV regularization and generalized TV regularization approaches. The values for similarity given show that generalized TV regularization approach enhances the reconstruction compared to the TV regularization.In Figure2, close-up of the results of deblurring the image Cameraman with TV regularization and generalized TV regularization approaches is shown. In terms of similarity to the original data, we can see that generalized TV regularization approach improves the reconstruction compared to the TV regularization.


## 7.2. Comparison of Image Restoration for Different Test Images

In this subsection, we report that our method can remove Gaussian noise efficiently while preserving details very well. For noise removal, we compare our method with some recently developed models, namely, the local TV [28], the NLTV [29], the anisotropic TV [30], and the isotropic TV [30], when the Split Bregman method is considered. The quality of our restoration is measured by the peak signal-to-noise ratio (PSNR) in decibels (dB): (47)PSNR=10log10⁡M×N∑i=1M∑j=1Nui,j-u~i,j2,where M×N indicates the image size and u and u~ denote the original image and the restored one, respectively. Generally, the higher PSNR values indicate better quality of the restored images.The test images are shown in Figure3.Figure 3
Test images. (a) Cameraman. (b) Lena. (c) Pepper. (d) Building.
(a)
(b)
(c)
(d)In Figure4, we make use of the local TV model, the NLTV model, the anisotropic TV model, the isotropic TV model, and the solution-driven adaptive generalized TV model to display the restoration results for the test image (a) corrupted by 8% Gaussian noise. Here, our proposed model performs significantly better and can suppress the Gaussian noise successfully.Figure 4
(a) Noisy image (PSNR = 8.748 dB). (b) Local TV, Split Bregman (PSNR = 28.18 dB). (c) Nonlocal TV, Split Bregman (PSNR = 30.35 dB). (d) Anisotropic TV, Split Bregman (PSNR = 27.687 dB). (e) Isotropic TV, Split Bregman (PSNR = 28.594 dB). (f) Solution-driven adaptive generalized TV, modified projection (PSNR = 30.56 dB).
(a)
(b)
(c)
(d)
(e)
(f)Table1 lists the restoration results in the PSNR of different methods for test images Cameraman, Lena, Pepper, and Building corrupted by Gaussian noise.Table 1
Comparison of restoration results in PSNR (dB) for images corrupted by Gaussian noise.
Methods
Cameraman
Lena
Pepper
Building
Local TV
28.18
27.68
26.68
27.55
NLTV
30.35
29.87
30.86
29.23
Anisotropic TV
27.687
26.1
26.591
29.3
Isotropic TV
28.594
27.902
27.661
26.603
Proposed
30.56
30.21
30.98
29.5From Table1, it is apparent that our proposed method generates the best restoration results for all the test images. Actually, in Table 1, the proposed method obtains the highest PSNR values for all the test images. This demonstrated that our method is more robust for images corrupted by Gaussian noise.


## 7.3. Expanded Experiments

Remote sensing very often deals with inverting a forward model. To this aim, one has to produce an accurate and robust model able to predict physical, chemical, geological, or atmospheric parameters from spectra, such as surface temperature, water vapour, and ozone; see, for example, [31]. Denoising images could be part of the accurate and robust model. In particular, our image restoration method could be used to remove noises in the class of satellite images.In Figure5, we give the close-up of the results of denoising the satellite image adding Gaussian noise with the standard derivation σ=0.02. Though the fourth-order PDE model removes noise more thoroughly, texture detail also had been changed. Comparing rivers and streets with those in the original image, some information was lost. Our method not only suppresses the noise very well but also keeps a lot of detail and texture information of the original image.Figure 5
(a) Original satellite image. (b) Gaussian noise (σ=0.02) image. (c) Fourth-order PDE model. (d) Proposed method.
(a)
(b)
(c)
(d)From Table2, we can see that our method has better denoising effects than pure anisotropic diffusion and fourth-order PDE.Table 2
Comparison of restoration results in PSNR (dB) for satellite images.
Methods
σ
=
0.01
σ
=
0.02
σ
=
0.05
σ
=
1
Pure anisotropic diffusion [23]
30.16
23.68
15.36
6.10
Fourth-order PDE [24]
31.64
25.40
17.40
11.36
Proposed
34.32
27.68
19.54
13.28


## 8. Conclusion

In this paper, our main contribution is to introduce the generalized TV regularization which includes various types of TV regularization as special cases. We studied the existence and uniqueness of the solution for mixed quasi-variational inequality for solution-driven adaptive image denoising and image deblurring with generalized TV regularization. The convergence of proposed algorithm for MQVI was proposed. Moreover, our experimental results showed that we improve the image restoration quality and apply the proposed model to deal with the images in many different fields. Our further work will consider utilizing the theory of other variational inequalities for solving optimization problems with other kinds of generalized TV regularizations.



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*Source: 6132356-2016-04-04.xml*