IMAGE RESTORATION USING AN INERTIAL VISCOSITY FIXED POINT

ALGORITHM

1 EBRU ALTIPARMAK AND 2 IBRAHIM KARAHAN

Abstract. The image restoration problem is one of the popular topics in image processing studied by many

authors on account of its applications in various areas. The aim of this paper is to present a new algorithm byusing viscosity approximation with inertial effect for finding a common fixed point of an infinite family of non-

expansive mappings in a Hilbert space and obtaining more quality images from degenerate images. Some strong

convergence theorems are proved under mild conditions. The obtained results are applied to solve monotoneinclusion problems, convex minimization problems, variational inequality problems and generalized equilibrium

problems. It is shown that the proposed algorithm performs better than some other algorithms. Also, the effects

of inertial and viscosity terms in the algorithm on image restoration have been investigated.

1. Introduction

Image processing is a method to facilitate the perception of images by computers and interpretation byhumans. Images may be distorted for some reason such as camera or object movement, electricity, heat, sharpand sudden disturbances in the image signal and sensor illumination levels. Image processing methods performsome operations on an image to obtain an enhanced image or to extract some useful information from it. Inrecent years, these methods have been used in almost every field such as military industry, Forensic sciences,underwater imaging, astronomy, physics, art, biomedical remote sensing applications, geographic sciences, imageand data storage, medical imaging, astronomical imaging and industrial automation. One of the areas that havean important place in image processing is image restoration. Image restoration is the process of obtaining arelatively clear image from the distorted or noisy image. So, the goal of image restoration techniques is toincrease the quality of the images. After reconstruction, the quality of the images can be measured with thevalues of signal to noise ratio (SNR), improvement in signal to noise ratio (ISNR) and peak signal to noise ratio(PSNR).The mathematical model for the image restoration problem is formulated by

υ = Ax+ b

where x is the original image, A is the blurring matrix, b is the additive noise and v is the observed image. Theaim of the image restoration problem is to minimize additive noise b by using the observed image v. The mainapproach for this problem is to solve the regularized least squares minimization problem given by:

x∗ = argminx

1

2‖Ax− v‖22 + λK(x)

, (1.1)

where λ > 0 is a regularization parameter, K(x) is a regularizer function that should be convex and ‖Ax− v‖22is a least squares term that measures the distance between h and Ax. In 1977, Tikhonov and Arsenin [27]introduced the following Tikhonov regularization problem by taking Tikhonov matrix L as a special case of K:

x∗ = argminx

1

2‖Ax− v‖22 + λ ‖Lx‖22

.

By taking the W wavelet transform matrix instead of the function K in problem (1.1), we get the wavelet-basedregularization given by:

x∗ = argminx

1

2‖Ax− v‖22 + λ ‖Wx‖1

.

On the other hand, another successful regularization problem is known as l1 regularization which is given asfollows:

x∗ = argminx

1

2‖Ax− v‖22 + λ ‖x‖1

.

Key words and phrases. image restoration problem, viscosity, inertial, nonexpansive mapping, monotone operator, Hilbert space2010 Mathematics Subject Classification: 47H20, 49M20, 49M25, 49M27, 47J25, 47H05.

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IMAGE RESTORATION USING AN INERTIAL VISCOSITY FIXED POINT ALGORITHM 2

All these problems can be reformulated by in the following general way:

x∗ = argminxh (x) + g (x) . (1.2)

So, it is clear that the image restoration problem is a classical minimization problem for the sum of two specialfunctions. It is known that the problem (1.2) is equivalent to the following variational inclusion problem:

0 ∈ ∇h (x∗) + ∂g (x∗) , (1.3)

where ∂g is the subdifferential of function g defined by ∂g (x) := u ∈ H : g (x) + 〈y − x, u〉 ≤ g (y) ,∀y ∈ Hand ∇h is the gradient operator of function h and to the following fixed point problem:

x∗ = J∂gλ (I − λ∇f)x∗, (1.4)

where J∂gλ is the resolvent operator defined by J∂gλ = (I + λ∂g)−1. Hence, fixed point theory plays a very

important role in solving image restoration problems and so fixed point algorithms can be used to find thesolutions of the considered problems, see, for instance [2, 5, 6, 8, 9, 10, 13, 17, 25]. Lions and Mercier [18]introduced the following classical forward backward splitting (FBS) algorithm which is one of the most importantalgorithm:

xn+1 = J∂gλ (I − λ∇f)xn, (1.5)

where λ > 0 and I is the identity operator. Lorenz and Pock [19] presented the inertial forward backwardalgorithm (IFBS) for monotone operators in the following form:

yn = xn + θn (xn − xn−1)

xn+1 = J∂gλ (I − λ∇f)yn, ∀n ≥ 1(1.6)

where θn is the inertial parameter which controls the momentum xn−xn−1. In 2016, Shehu and Cai [7] presentedthe following algorithm by combining the algorithm (1.5) with viscosity approximation method:

xn+1 = αnf (xn) + (1− αn) JBλn(I − λnA)xn, ∀n ≥ 1 (1.7)

where f is a contraction mapping, A is a ν-inverse strongly accretive mapping and B is an m-accretive mapping.They proved the convergence of the sequence generated by the algorithm in uniformly smooth Banach spaces. In2019, Kithuan et al. [15] proved some strong convergence theorems for the following inertial viscosity forward-backward splitting algorithm in a Hilbert space H :

yn = xn + θn (xn − xn−1)xn+1 = δn∇h (xn) + (1− δn) JBλn(I − λnA)yn,∀n ≥ 1

(1.8)

where A : H → H is a ν-inverse strongly monotone mapping and B : H → 2H is a maximal monotoneoperator. In 2017, Verma and Shukla [29] studied the new accelerated proximal gradient algorithm (NAGA) fora nonexpansive mapping in the following way :

yn = xn + θn (xn − xn−1)xn+1 = JBλn(I − λnA)

((1− βn) yn + βnJ

Bλn

(I − λnA)yn),∀n ≥ 1.

(1.9)

Padcharoen and Kumam [22] introduced a modified MM algorithm (FBMMMA) for finding a common fixedpoint of a countable family of nonexpansive operators in the following manner:

yn = xn + θn (xn − xn−1)zn = (1− ρn) yn + ρnJ

Bλn

(I − λnA)ynwn = (1− δn − ρn) zn + δnJ

Bλn

(I − λnA)zn + φnJBλn

(I − λnA)ynxn+1 = JBλn(I − λnA)wn.

(1.10)

They obtained the weak convergence of the sequence generated by the algorithm in Hilbert space. Puangpeeand Suantai [23] presented a new accelerated fixed point algorithm (AV FB) as follows :

yn = xn + θn (xn − xn−1)zn = (1− σn) yn + σnJ

Bλn

(I − λnA)ynxn+1 = ψnf (xn) + %nJ

Bλn

(I − λnA)yn + ϕnJBλn

(I − λnA)zn.(1.11)

They showed the strong convergence of the proposed algorithm for an infinite family of nonexpansive mappingsin Hilbert space. All these authors gave some applications for image restoration problems and intended to obtainmore quality images.

IMAGE RESTORATION USING AN INERTIAL VISCOSITY FIXED POINT ALGORITHM 3

In this paper, motivated and inspired by the given algorithms and K- iteration algorithm introduced byHussain et al. [14], we introduced a new algorithm which is more effective than the algorithms exist in theliterature. We proved the strong convergence of generated sequence and gave some application of the proposedalgorithm to the different problems especially to the image restoration problem to get better quality images.This research is organized as follows. In section 2, Preliminaries, we give some definitions and lemmas whichwe need to prove the main result. In section 3, Main results, we prove the strong convergence of the proposedalgorithm. In the next section, Applications, we apply our main result to solving inclusion problems, variationalinequality problems, generalized equilibrium problems and image restoration problems.

2. Preliminaries

Throughout this paper, let H be a real Hilbert space with the inner product 〈., .〉 and the induced norm ‖.‖.Also, let C be a nonempty closed and convex subset of a real Hilbert space H and T a mapping on C. Then,the set of all fixed points of T is denoted by F (T ) := x ∈ C : x = Tx .

The metric projection PC : H → C is defined as follows: by given x ∈ H there exist a unique point in C suchthat

‖x− PCx‖ ≤ infy∈C‖x− y‖ .

It is well-known that PC is a nonexpansive mapping and it can be characterized by

〈x− PCx, y − PCx〉 ≤ 0 (2.1)

for all y ∈ C, see [12] for more details.

Definition 2.1. [3] Let T : C → H be a nonlinear operator. Then T is said to be:

(1) L-Lipschitz continuous, if there exists a constant L > 0 such that:

‖Tx− Ty‖ ≤ L ‖x− y‖ ,∀x, y ∈ C.

If L = 1, then T is said to be nonexpansive mapping and if L < 1 then T is said to be contractionmapping.

(2) υ-strongly monotone operator if there exists υ > 0 such that:

〈Tx− Ty, x− y〉 ≥ υ ‖x− y‖2 , ∀x, y ∈ C

(3) η-inverse strongly monotone operator if there exists η > 0 such that:

〈Tx− Ty, x− y〉 ≥ η ‖Tx− Ty‖2 , ∀x, y ∈ C.

We can see that if T is η-inverse strongly monotone, then it is 1η -Lipschitz continuous.

Let A : H → 2H be a set-valued operator. A is called monotone if 〈z − w, x− y〉 ≥ 0, for all z ∈ Ax andw ∈ Ay. If the graph of a monotone operator is not properly contained in the graph of any other monotoneoperators, then it is called the maximal monotone operator. The following lemmas give some useful informationsregarding maximal monotonicity.

Lemma 2.2. [3] Let A : H → 2H be a monotone operator. Then A is maximal monotone if and only ifR (I +A) = H.

Lemma 2.3. [3] Let Γ0 (H) denotes the class of all lower semi-continuous convex functions defined from H to(−∞,∞] . If g ∈ Γ0 (H) , then ∂g is maximal monotone.

Let A : H → 2H be a maximal monotone operator. Then the resolvent operator JAλ : H → H associated with

A is defined by JAλ = (I + λA)−1

for all x ∈ H and for some λ > 0. It is well-known that JAλ is a nonexpansivemapping and F

(JAλ)

= A−10 where A−10 = x ∈ H : 0 ∈ Ax which is called the set of all zero points of A.

Definition 2.4. [3] Let g ∈ Γ0 (H) and λ > 0. The proximal operator of parameter λ of g at x is defined by

proxλg (x) = argmin

y∈H

g (y) +

1

2λ‖y − x‖2

.

IMAGE RESTORATION USING AN INERTIAL VISCOSITY FIXED POINT ALGORITHM 4

It is known that if g ∈ Γ0 (H) , then J∂gλ = proxλg. By taking l1-norm instead of g in the definition of proximaloperator, the proximal operator is called the soft thresholding operator and it can be given by the following way:

proxλ‖.‖1 (x) = sign(x) max ‖x‖1 − λ, 0 .

We recall the following properties which is useful to prove our main result, please see for details [21, 24].

Let Tn and Λ be two families of nonexpansive mappings of H into itself such that ∅ 6= F (Λ) ⊂ ∩∞n=1F (Tn)where F (Λ) is the set of all common fixed points of mappings belongs to Λ.Then we say that Tn satisfiesNST-condition (I) with Λ if for every bounded sequence xn,

limn→∞

‖xn − Tnxn‖ = 0 implies limn→∞

‖xn − Txn‖ = 0, for all T ∈ Λ.

Especially, if Λ consist of one mapping, that is, Λ = T , then Tn is said to satisfy the NST-condition (I)with T.

Lemma 2.5. [4] For a real Hilbert space H, let A : H → H be a L-Lipschitz operator and let B : H → 2Hbe amaximal monotone operator. If Tn is a forward backward operator, i.e. Tn = JBλn(I − λnA) where λn ∈ (0, 2/L)

for all n ≥ 1 such that λn converges to λ, then Tn satisfies NST-condition (I) with T , where T = JBλ (I −λA)is a forward backward operator with λ ∈ (0, 2/L) .

The following lemmas are important for proving our main theorem.

Lemma 2.6. [11] Let T be a nonexpansive operator from H into itself with F (T ) 6= ∅. Then, the mapping I−Tis demiclosed at zero, that is, for any sequences xn ∈ H such that xn x ∈ H and ‖xn − Txn‖ → 0 asn→∞, then it implies x ∈ F (T ) .

Lemma 2.7. [3] Let H be a real Hilbert space. Then for all x, y ∈ H, and λ ∈ [0, 1] , the following propertieshold:

(1) ‖x± y‖2 = ‖x‖2 ± 2 〈x, y〉+ ‖y‖2 ,(2) ‖x+ y‖2 ≤ ‖x‖2 + 2 〈y, x+ y〉 ,(3) ‖λx+ (1− λ) y‖2 = λ ‖x‖2 + (1− λ) ‖y‖2 − λ (1− λ) ‖x− y‖2 .

Lemma 2.8. [1, 28] Let sn and εn be sequences of nonnegative real numbers such that

sn+1 ≤ (1− δn) sn + δnrn + εn,

where δn is a sequence in [0, 1] and rn is a real sequence. If the following conditions hold, then limn→∞ sn =0 :

(1)∞∑n=1

δn =∞,

(2)∞∑n=1

εn <∞,

(3) lim supn→∞ rn ≤ 0.

Lemma 2.9. [20] Let Φn be a sequence of real numbers that does not decrease at infinity such that there existsa subsequence Φni of Φn which satisfies Φni < Φni+1

for all i ∈ N. Let τ (n)n≥n0be a sequence of integer,

defined as follows:

τ (n) := max l ≤ n : Φl < Φl+1 .

Then the followings are satisfied:

(1) τ (n0) ≤ τ (n0 + 1) ≤ ... and τ (n)→∞,(2) Φτ(n) ≤ Φτ(n)+1 and Φn ≤ Φτ(n)+1 , for all n ≥ n0.

3. Main Results

In this section, we present a new algorithm for finding a common fixed point of an infinite family of nonex-pansive mappings in real Hilbert space and prove its strong convergence under some mild assumptions.

IMAGE RESTORATION USING AN INERTIAL VISCOSITY FIXED POINT ALGORITHM 5

Theorem 3.1. Let Tn be a family of nonexpansive mappings on H into itself which satisfies the NST-condition (I) with a nonexpansive mapping T : H → H. Let x0, x1 ∈ H, f : H → H be a k-contraction mappingand xn be a sequence generated by

zn = xn + θn (xn − xn−1)yn = (1− βn) zn + βnTnznωn = Tn ((1− αn)Tnxn + αnTnyn)xn+1 = (1− γn)Tnωn + γnf (ωn)

(3.1)

where,

θn :=

min

θ, ηnγn‖xn−xn−1‖

if xn 6= xn−1

θ, otherwise,

for θ ≥ 0, ηn ∈ (0,∞) and αn , βn , γn ∈ (0, 1) be sequences which satisfy the conditions:

(1) 0 < a ≤ αn < a′< 1,

(2) limn→∞ ηn = 0,

(3) limn→∞ γn = 0,∞∑n=1

γn =∞,

for some positive real numbers a and a′. Then the sequence xn converges strongly to a point x∗of F (T ) ,

where x∗ = PF (T )f (x∗) .

Proof. First, we prove that xn is bounded. Let x∗ ∈ F (T ) such that x∗ = PF (T )f (x∗). By Algorithm 3.1, wecan write

‖zn − x∗‖ = ‖xn + θn (xn − xn−1)− x∗‖≤ ‖xn − x∗‖+ θn ‖xn − xn−1‖ , (3.2)

and, since Tn is a nonexpansive mapping, we have

‖yn − x∗‖ = ‖(1− βn) zn + βnTnzn − x∗‖≤ (1− βn) ‖zn − x∗‖+ βn ‖Tnzn − x∗‖= (1− βn) ‖zn − x∗‖+ βn ‖Tnzn − Tnx∗‖≤ ‖zn − x∗‖ , (3.3)

and also,

‖ωn − x∗‖ = ‖Tn ((1− αn)Tnxn + αnTnyn)− x∗‖= ‖Tn ((1− αn)Tnxn + αnTnyn)− Tnx∗‖≤ ‖(1− αn)Tnxn + αnTnyn − x∗‖= ‖(1− αn) (Tnxn − x∗) + αn (Tnyn − x∗)‖= ‖(1− αn) (Tnxn − Tnx∗) + αn (Tnyn − Tnx∗)‖≤ (1− αn) ‖xn − x∗‖+ αn ‖yn − x∗‖ . (3.4)

Combining (3.2), (3.3) and (3.4), we obtain that

‖xn+1 − x∗‖ = ‖(1− γn)Tnωn + γnf (ωn)− x∗‖= ‖(1− γn) (Tnωn − x∗) + γn (f (ωn)− f (x∗)) + γn (f (x∗)− x∗)‖≤ (1− γn) ‖Tnωn − x∗‖+ γn ‖f (ωn)− f (x∗)‖+ γn ‖f (x∗)− x∗‖= (1− γn) ‖Tnωn − Tnx∗‖+ γn ‖f (ωn)− f (x∗)‖+ γn ‖f (x∗)− x∗‖≤ (1− γn) ‖ωn − x∗‖+ γnk ‖ωn − x∗‖+ γn ‖f (x∗)− x∗‖≤ (1− γn (1− k)) [(1− αn) ‖xn − x∗‖+ αn ‖yn − x∗‖] + γn ‖f (x∗)− x∗‖≤ (1− γn (1− k)) [(1− αn) ‖xn − x∗‖+ αn ‖xn − x∗‖

+αnθn ‖xn − xn−1‖] + γn ‖f (x∗)− x∗‖

≤ (1− γn (1− k)) ‖xn − x∗‖+ a′γn.

θnγn‖xn − xn−1‖+ γn ‖f (x∗)− x∗‖ . (3.5)

IMAGE RESTORATION USING AN INERTIAL VISCOSITY FIXED POINT ALGORITHM 6

By using the definition of θn and condition (2), it is clear that

θnγn‖xn − xn−1‖ → 0 as n→∞.

Hence, there exists a positive constant M1 > 0 such that, for all n ≥ 1,

θnγn‖xn − xn−1‖ ≤M1.

It follows from (3.5) that,

‖xn+1 − x∗‖ ≤ (1− γn (1− k)) ‖xn − x∗‖+ γn

(a′M1 + ‖f (x∗)− x∗‖

)= (1− γn (1− k)) ‖xn − x∗‖+ γn (1− k)

[a′M1 + ‖f (x∗)− x∗‖

(1− k)

]

≤ max

‖xn − x∗‖ ,

a′M1 + ‖f (x∗)− x∗‖

(1− k)

...

≤ max

‖x1 − x∗‖ ,

a′M1 + ‖f (x∗)− x∗‖

(1− k)

,

for all n ≥ 1. So, we obtain that xn is bounded and hence zn , yn and ωn are also bounded.

Secondly, we want to prove that xn → x∗ = PF (T )f (x∗) . Indeed, we have the followings for all

‖zn − x∗‖2 = ‖xn + θn (xn − xn−1)− x∗‖2

≤ ‖xn − x∗‖2 + 2θn ‖xn − x∗‖ ‖xn − xn−1‖+ θ2n ‖xn − xn−1‖2 , (3.6)

and

‖ωn − x∗‖2 = ‖Tn ((1− αn)Tnxn + αnTnyn)− x∗‖2

≤ ‖(1− αn)Tnxn + αnTnyn − x∗‖2

= ‖(1− αn) (Tnxn − Tnx∗) + αn (Tnyn − Tnx∗)‖2 .

From (3.3), (3.6) and the property (3) of Lemma (2.7), we get

‖ωn − x∗‖2 ≤ αn ‖Tnyn − Tnx∗‖2 + (1− αn) ‖Tnxn − Tnx∗‖2

−αn (1− αn) ‖Tnyn − Tnxn‖2 (3.7)

≤ αn ‖Tnyn − Tnx∗‖2 + (1− αn) ‖Tnxn − Tnx∗‖2

≤ αn ‖yn − x∗‖2 + (1− αn) ‖xn − x∗‖2

≤ αn ‖zn − x∗‖2 + (1− αn) ‖xn − x∗‖2

≤ αn

[‖xn − x∗‖2 + 2θn ‖xn − x∗‖ ‖xn − xn−1‖

+θ2n ‖xn − xn−1‖2

]+ (1− αn) ‖xn − x∗‖2

= ‖xn − x∗‖2 + 2αnθn ‖xn − x∗‖ ‖xn − xn−1‖ (3.8)

+αnθ2n ‖xn − xn−1‖2 .

IMAGE RESTORATION USING AN INERTIAL VISCOSITY FIXED POINT ALGORITHM 7

Also again by using the properties (2) and (3) of Lemma (2.7), we get

‖xn+1 − x∗‖2 = ‖(1− γn)Tnωn + γnf (ωn)− x∗‖2

≤ ‖(1− γn) (Tnωn − x∗) + γn (f (ωn)− f (x∗)) + γn (f (x∗)− x∗)‖2

≤ ‖(1− γn) (Tnωn − x∗) + γn (f (ωn)− f (x∗))‖2

+2 〈γn (f (x∗)− x∗) , xn+1 − x∗〉≤ (1− γn) ‖ωn − x∗‖2 + γn ‖f (ωn)− f (x∗)‖2

−γn (1− γn) ‖f (ωn)− f (x∗)− ωn − x∗‖+2 〈γn (f (x∗)− x∗) , xn+1 − x∗〉

≤ (1− γn) ‖ωn − x∗‖2 + γnk2 ‖ωn − x∗‖2

+2γn 〈f (x∗)− x∗, xn+1 − x∗〉 .

So, It follows from (3.8) that,

‖xn+1 − x∗‖2 ≤ (1− γn (1− k)) ‖ωn − x∗‖2 + 2γn 〈f (x∗)− x∗, xn+1 − x∗〉

= (1− γn (1− k))[‖xn − x∗‖2 + 2αnθn ‖xn − x∗‖ ‖xn − xn−1‖

+αnθ2n ‖xn − xn−1‖2

]+ 2γn 〈f (x∗)− x∗, xn+1 − x∗〉

≤ (1− γn (1− k)) ‖xn − x∗‖2 + αnθn ‖xn − xn−1‖ [2 ‖xn − x∗‖+ θn ‖xn − xn−1‖]+2γn 〈f (x∗)− x∗, xn+1 − x∗〉 . (3.9)

Since

θn ‖xn − xn−1‖ = γn.θnγn‖xn − xn−1‖ → 0 as n→∞, (3.10)

there exists a positive constant M2 > 0 such that

θn ‖xn − xn−1‖ ≤M2,

for all n ≥ 1.From (3.9), we can write

‖xn+1 − x∗‖2 ≤ (1− γn (1− k)) ‖xn − x∗‖2 + 3M3a′θn ‖xn − xn−1‖

+2γn 〈f (x∗)− x∗, xn+1 − x∗〉

≤ (1− γn (1− k)) ‖xn − x∗‖2 + γn (1− k)

[3M3a

′

(1− k).θnγn‖xn − xn−1‖

2

(1− k)2γn 〈f (x∗)− x∗, xn+1 − x∗〉

],

where M3 = supn≥1 ‖xn − x∗‖ ,M2 . In the above inequality, if we set

sn = ‖xn − x∗‖2 , δn = γn (1− k)

and

rn =3M3a

′

(1− k)

θnγn‖xn − xn−1‖+

2

(1− k)2γn 〈f (x∗)− x∗, xn+1 − x∗〉 ,

then we obtainsn+1 ≤ (1− δn) sn + δnrn,∀n ≥ 1. (3.11)

Now we need to prove lim supn→∞ rn ≤ 0 in order to complete the proof. So, we consider the following twocases.

In the first case, we assume that there exists n0 ∈ N such that the sequence ‖xn − x∗‖n≥n0is nonincreas-

ing. Since the sequence xn is bounded, it follows that ‖xn − x∗‖ is a convergent sequence. By using the

assumption (3) of theorem, we get∞∑n=1

δn =∞. Next, we claim that

lim supn→∞

〈f (x∗)− x∗, xn+1 − x∗〉 ≤ 0.

IMAGE RESTORATION USING AN INERTIAL VISCOSITY FIXED POINT ALGORITHM 8

Since

‖yn − x∗‖2 = ‖(1− βn) zn + βnTnzn − x∗‖2

= ‖(1− βn) (zn − x∗) + βn (Tnzn − x∗)‖2

≤ βn ‖Tnzn − x∗‖2 + (1− βn) ‖zn − x∗‖2 − βn (1− βn) ‖Tnzn − zn‖2

≤ ‖zn − x∗‖2 − βn (1− βn) ‖Tnzn − zn‖2 , (3.12)

it follows from (3.6) and the property (4) of Lemma (2.7) that

‖ωn − x∗‖2 ≤ αn ‖Tnyn − Tnx∗‖2 + (1− αn) ‖Tnxn − Tnx∗‖2

−αn (1− αn) ‖Tnyn − Tnxn‖2

≤ αn ‖zn − x∗‖2 − αnβn (1− βn) ‖Tnzn − zn‖2 + (1− αn) ‖xn − x∗‖2

≤ αn ‖xn − x∗‖2 + 2αnθn ‖xn − x∗‖ ‖xn − xn−1‖+ αnθ2n ‖xn − xn−1‖2

−αnβn (1− βn) ‖Tnzn − zn‖2 + (1− αn) ‖xn − x∗‖2

= ‖xn − x∗‖2 + 2αnθn ‖xn − x∗‖ ‖xn − xn−1‖+αnθ

2n ‖xn − xn−1‖2 − αnβn (1− βn) ‖Tnzn − zn‖2 . (3.13)

Also, from (3.13) and the property (3) of Lemma (2.7), we get

‖xn+1 − x∗‖2 = ‖(1− γn) (Tnωn − x∗) + γn (f (ωn)− x∗)‖2

≤ γn ‖f (ωn)− x∗‖2 + (1− γn) ‖Tnωn − Tnx∗‖2

−γn (1− γn) ‖f (ωn)− Tnωn‖2

≤ γn ‖f (ωn)− x∗‖2 + (1− γn) ‖ωn − x∗‖2

≤ γn ‖f (ωn)− x∗‖2 + (1− γn)[‖xn − x∗‖2 + 2αnθn ‖xn − x∗‖ ‖xn − xn−1‖

+αnθ2n ‖xn − xn−1‖2 − αnβn (1− βn) ‖Tnzn − zn‖2

]= γn ‖f (ωn)− x∗‖2 + (1− γn) ‖xn − x∗‖2 + (1− γn)αnθn ‖xn − xn−1‖ [2 ‖xn − x∗‖

θn ‖xn − xn−1‖]− (1− γn)αnβn (1− βn) ‖Tnzn − zn‖2 .

So the following is true for all n > 1 :

(1− γn)αnβn (1− βn) ‖Tnzn − zn‖2 ≤ γn

(‖f (ωn)− x∗‖2 − ‖xn − x∗‖2

)+ ‖xn − x∗‖2 − ‖xn+1 − x∗‖2

(1− γn)αnθn ‖xn − xn−1‖ [2θn ‖xn − x∗‖+ θn ‖xn − xn−1‖] .

Hence, we conclude from the assumptions (1) and (2) of theorem and the convergence of the sequences‖xn − x∗‖ and of θn ‖xn − xn−1‖ that

‖Tnzn − zn‖ → 0 as n→∞. (3.14)

Since Tn satisfies the NST-condition (I) with T, we obtain that

‖Tzn − zn‖ → 0 as n→∞ . (3.15)

On the other hand, since

‖zn − xn‖ = ‖xn + θn (xn − xn−1)− xn‖ = θn ‖xn − xn−1‖ → 0 as n→∞, (3.16)

by using (3.14) and (3.16), we have

‖yn − xn‖ = ‖(1− βn) zn + βnTnzn − xn‖≤ ‖zn − xn‖+ βn ‖Tzn − zn‖ ,

and so it is provided that

‖yn − xn‖ → 0 as n→∞. (3.17)

IMAGE RESTORATION USING AN INERTIAL VISCOSITY FIXED POINT ALGORITHM 9

Also, from (3.14), (3.16) and (3.17), we get

‖ωn − zn‖ = ‖ωn − zn + Tnzn − Tnzn‖≤ ‖ωn − Tnzn‖+ ‖Tnzn − zn‖= ‖Tn ((1− αn)Tnxn + αnTnyn)− Tnzn‖+ ‖Tnzn − zn‖≤ ‖(1− αn)Tnxn + αnTnyn − zn‖+ ‖Tnzn − zn‖≤ αn ‖Tnyn − Tnxn‖+ ‖Tnxn − Tnzn‖+ ‖Tnzn − zn‖+ ‖Tnzn − zn‖≤ αn ‖yn − xn‖+ ‖xn − zn‖+ 2 ‖Tnzn − zn‖ ,

and so

‖ωn − zn‖ → 0 as n→∞. (3.18)

By using (3.14) and (3.18), we obtain

‖xn+1 − zn‖ ≤ ‖xn+1 − Tnzn‖+ ‖zn − Tnzn‖= ‖(1− γn)Tnωn + γnf (ωn)− Tnzn‖+ ‖Tnzn − zn‖≤ γn ‖f (ωn)− Tnωn‖+ ‖Tnωn − Tnzn‖+ ‖Tnzn − zn‖≤ γn ‖f (ωn)− Tnωn‖+ ‖ωn − zn‖+ ‖Tnzn − zn‖

which implies

‖xn+1 − zn‖ → 0 as n→∞. (3.19)

Hence, we get

‖xn+1 − xn‖ ≤ ‖xn+1 − zn‖+ ‖zn − xn‖ → 0 as n→∞.

Now, we are in a position to prove that xn → x∗. Let

υ = lim supn→∞

〈f (x∗)− x∗, xn+1 − x∗〉 .

Since xn is bounded, there exists a subsequence xni of xn such that xni t and

υ = limi→∞

〈f (x∗)− x∗, xni+1 − x∗〉 .

By using (3.15) and (3.16), we obtain that

‖xn − Txn‖ = ‖xn − zn + zn + Tzn − Tzn − Txn‖≤ ‖zn − xn‖+ ‖Tzn − zn‖+ ‖Tzn − Txn‖≤ 2 ‖zn − xn‖+ ‖Tzn − zn‖

and so, we conclude that

‖xn − Txn‖ → 0 as n→∞.

Hence, it is obvious from Lemma 2.6 that t ∈ F (T ) . On the other hand, since ‖xn+1 − xn‖ → 0 as n → ∞and xni t this implies that xni+1 → t. Furthermore, by using x∗ = PF (T )f (x∗) and the property of the metricprojection operators (2.1), we can write

υ = limi→∞

〈f (x∗)− x∗, xni+1 − x∗〉 = 〈f (x∗)− x∗, t− x∗〉 ≤ 0. (3.20)

Then, we have

lim supn→∞

〈f (x∗)− x∗, xn+1 − x∗〉 ≤ 0. (3.21)

It follows from (3.10) and (3.21) that lim supn→∞ rn ≤ 0. Finally, we deduce that xn → x∗.

In the second case, we assume that there exists a n0 ∈ N such that the sequence ‖xn − x∗‖n≥n0is not

monotonically decreasing. Let Φn = ‖xn − x∗‖2 for all n ∈ N. So, there exists a subsequence

Φnj

of Φnsuch that Φnj < Φnj+1

for all j ∈ N. Then, we define τ : n : n ≥ n0 → N as follows:

τ (n) := max l ∈ N : l ≤ n,Φl < Φl+1 .

IMAGE RESTORATION USING AN INERTIAL VISCOSITY FIXED POINT ALGORITHM 10

Obviously, τ is a nondecreasing sequence. Then, by using Lemma 2.9, we have Φτ(n) ≤ Φτ(n)+1, i.e.,∥∥xτ(n) − x∗∥∥ ≤ ∥∥xτ(n)+1 − x∗

∥∥ for all n ≥ n0. Similarly to the first case, we can obtain the everything provedin the first case by taking τ (n) instead of n. So we have

lim supn→∞

∥∥xτ(n) − x∗∥∥2 ≤ 0.

Therefore, we get ∥∥xτ(n) − x∗∥∥2 → 0 and

∥∥xτ(n)+1 − x∗∥∥→ 0 as n→∞. (3.22)

So, it follows from (3.22) and Lemma 2.9 that

‖xn − x∗‖ ≤∥∥xτ(n)+1 − x∗

∥∥→ 0 as n→∞.

Therefore, xn converges strongly to x∗ and this completes the proof.

4. Applications

In this section, we will give some applications of Algorithm 3.1 to the convex minimization, variationalinequality, generalized equilibrium, monotone inclusion and image restoration problems.

4.1. Applicaton to monotone inclusion problems. Finding the zero sum of two monotone operators is oneof the most important problems in monotone operator theory. We study the following inclusion problem: findingx ∈ H such that

0 ∈ (A+B)x (4.1)

where A : H → H is an operator and B : H → 2H is a set-valued operator. This problem includes, as specialcases, convex minimization problems, variational inequalities, and equilibrium problems. Also, some tangibleproblems in statistical regression, machine learning, image processing, signal processing and the linear inverseproblem can be formulated mathematically in the form (4.1). It is well-known that the problem (4.1) is equivalentto the problem of finding x which satisfies the following equation:

JBλ (I − λA)x = x,

where A : H → H a η-inverse strongly monotone operator, B : H → 2H a maximal monotone operator andλ ∈ [0, 2η] . Also, it can be seen that JBλ (I − λA) is a nonexpansive mapping, see for details [3].

Now, as a corollary of Theorem 3.1, we give the following to approximate a solution of the inclusion problem(4.1) by swaping Tn and T with JBλn (I − λnA) and JBλ (I − λA) , respectively.

Theorem 4.1. Let f be a k-contraction mapping on H, A : H → H a η-inverse strongly monotone operator andB : H → 2H a maximal monotone operator such that Ω = (A+B)

−1(0) 6= ∅. Let λn ∈ (0, 2η) be a sequence

such that λn → λ where λ is a constant belongs to (0, 2η) . Let x0, x1 ∈ H, θ ≥ 0 and xn be a sequencegenerated by

zn = xn + θn (xn − xn−1)yn = (1− βn) zn + βnJ

Bλn

(I − λnA) znωn = JBλn (I − λnA)

((1− αn) JBλn (I − λnA)xn + αnJ

Bλn

(I − λnA) yn)

xn+1 = (1− γn) JBλn (I − λnA)ωn + γnf (ωn) .

(4.2)

where all the parameters satisfy the same conditions as in Theorem 3.1 . Then xn converges strongly to apoint x∗ of Ω, where x∗ = PΩf (x∗) .

Proof. The proof is clear from Lemma 2.5.

We next give a numerical example to show the convergence of the sequence generated by Algorithm 4.2 tothe solution of variational inclusion problem.

Example 4.2. Let A : l2 → l2 and B : l2 → l2 be two operators defined by Ax = 3x + (1, 2, 3, 0, ...) andBx = 8x, where x = (x1, x2, x3,...) ∈ l2. We can easily see that A is a 1/3-inverse strongly monotone and B is amaximal monotone operator. Indeed, for x, y ∈ l2, we have

〈x− y,Ax−Ay〉 = 〈x− y, 3x− 3y〉 = 3 ‖x− y‖2l2 ≥1

3‖Ax−Ay‖2l2

IMAGE RESTORATION USING AN INERTIAL VISCOSITY FIXED POINT ALGORITHM 11

and〈x− y,Bx−By〉 = 〈x− y, 8x− 8y〉 = 8 ‖x− y‖2l2 ≥ 0.

So B is a monotone operator. On the other hand, since R (I + λnB) = l2, we obtain from Lemma 2.2 that Bis a maximal monotone operator. By a direct calculation, we obtain

JBλn (x− λnAx) = (I + λA)−1

(x− λnAx)

=1− 3λn1 + 8λn

x− λn1 + 8λn

(1, 2, 3, 0, ...) ,

and (A+B)−1

(0) = (−0.0909,−0.1818,−0.2727, 0, ...). In Algorithm 4.2, we choose x1 = (−3,−5,−1, 0, ...) , αn =1

108 , βn = 1n+1 , γn = 1

(n+1)6, ηn = 10

n and f (x) = 0.1x. If θ = 0.99, then we get

θn :=

min

0.99, 10

n(n+1)‖xn−xn−1‖

if xn 6= xn−1,

0.99, otherwise..

In Table 1, we give the iteration steps of Algorithm 4.2.

No.Iteration xn ‖xn+1 − xn‖1 (−3.0000,−5.0000,−1.0000, 0, ...) .2 (−0.2587,−0.4593,−0.3112, 0, ...) 5.3485083 (−0.1007,−0.1979,−0.2746, 0, ...) 0.3076974 (−0.0915,−0.1827,−0.2728, 0, ...) 0.0178425 (−0.0909,−0.1819,−0.2727, 0, ...) 0.0010136 (−0.0909,−0.1819,−0.2727, 0, ...) 0.0000527 (−0.0909,−0.1819,−0.2727, 0, ...) 0.0000048 (−0.0909,−0.1819,−0.2727, 0, ...) 0.0000029 (−0.0909,−0.1819,−0.2727, 0, ...) 0.00000110 (−0.0909,−0.1819,−0.2727, 0, ...) 0.000000...

......

20 (−0.0909,−0.1818,−0.2727, 0, ...) 0.00000021 (−0.0909,−0.1818,−0.2727, 0, ...) 0.000000

Table 1. Iteration steps and error terms of Algorithm 4.2

In Figure 1, we compare the performance of Algorithm 4.2 and the FBS algorithm.

Figure 1. Comparison of Algorithm 4.2 and FBS algorithm

IMAGE RESTORATION USING AN INERTIAL VISCOSITY FIXED POINT ALGORITHM 12

4.2. Applicaton to convex minimization problems. Let h : H → R be a differentiable and convex functionsuch that ∇h is a η-inverse strongly monotone and g : H → R be a proper convex and lower semi-continuousfunction. We consider the following convex minimization problem: finding x∗ ∈ H such that

h (x∗) + g (x∗) = minx∈Hh (x) + g (x) . (4.3)

The solution set of convex minimization problem is denoted by Π. As mentioned in the introduction part, it isknown that the convex minimization problem (4.3) is equivalent to the fixed point problem (1.4). Also, since

∇h is an inverse strongly monotone and ∂g is a maximal monotone operators, we know that J∂gλn (I − λn∇h) is

a nonexpansive mapping for λ ∈ [0, 2η]. So, we obtain the following theorem.

Theorem 4.3. Let f be a k-contraction mapping on H, h : H → R be a differentiable and convex function suchthat ∇h is a η-inverse strongly monotone and g : H → R be a proper convex and lower semi-continuous functionsuch that Π 6= ∅. Let x0, x1 ∈ H and xn be a sequence generated by

zn = xn + θn (xn − xn−1)

yn = (1− βn) zn + βnJ∂gλn

(I − λn∇h) zn

ωn = J∂gλn (I − λn∇h)(

(1− αn) J∂gλn (I − λn∇h)xn + αnJ∂gλn

(I − λn∇h) yn

)xn+1 = (1− γn) J∂gλn (I − λn∇h)ωn + γnf (ωn)

(4.4)

where,

θn :=

min

θ, ηnγn‖xn−xn−1‖

if xn 6= xn−1

θ, otherwise,

for θ ≥ 0. Let αn , βn , γn , θn and ηn be sequences which satisfy the same conditions as in Theorem3.1 such that λn → λ, for λn, λ ∈ (0, 2η) . Then xn converges strongly to a x∗ solution of convex minimizationproblem, where x∗ = PΠf (x∗) .

4.3. Application to the variational inequality problems (VIP). The variational inequality problem isdefined as the problem of finding a point x∗ ∈ C such that

〈Ax∗, y − x∗〉 , ∀y ∈ C (4.5)

where A : C → H is a nonlinear monotone operator. We denote the solution set of (4.5) by V I(C,A). It is known

that the variational inequality problem (4.5) is equivalent to finding a point x∗ such that x∗ = J∂iCλn(I − λnA)x∗

where ∂iC is the subdifferential of the indicator function iC : H → (−∞,∞] of C defined by

iC (x) =

0, if x ∈ C∞, if x /∈ C .

It is well-known that the indicator function iC is a proper, lower semi-continuous and convex function on H. So,the subdifferential ∂iC is a maximal monotone operator. Based on these facts, we can easily see that,

y = J∂iCλ x ⇔ x ∈ (y + λ∂iCy)⇔ x− y ∈ λ∂iCy⇔ y = PCx.

So, the variational inequality problem (4.5) is equivalent to the fixed point problem x∗ = PC (I − λnA)x∗.Since PC (I − λnA) is a nonexpansive mapping when A is an inverse strongly monotone operator, the followingtheorem can be obtained from Theorem 3.1.

Theorem 4.4. Let A : H → H be an η-inverse strongly monotone operator such that V I(C,A) 6= ∅. Letx0, x1 ∈ H and let xn be a sequence generated by

zn = xn + θn (xn − xn−1)yn = (1− βn) zn + βnPC (I − λnA) znωn = PC (I − λnA) ((1− αn)PC (I − λnA)xn + αnPC (I − λnA) yn)xn+1 = (1− γn)PC (I − λnA)ωn + γnf (ωn)

where,

θn :=

min

θ, ηnγn‖xn−xn−1‖

if xn 6= xn−1

θ, otherwise,

and for θ ≥ 0. Then under the same conditions as in Theorem 3.1, the sequence xn converges strongly to asolution x∗ of variational inequality problem (4.5), where x∗ = PV I(C,A)f (x∗) .

IMAGE RESTORATION USING AN INERTIAL VISCOSITY FIXED POINT ALGORITHM 13

4.4. Application to generalized equilibrium problems (GEPs). Let F : C ×C → R be a bifunction andA : H → H be a monotone operator. The generalized equilibrium problem is formulated by finding x ∈ C suchthat

F (x, y) + 〈Ax, y − x〉 ≥ 0 (4.6)

for all y ∈ C. In order to solve the generalized equilibrium problem (4.6), we assume that F satisfies thefollowing:

(1) F (x, x) = 0, for all x, y ∈ C,(2) F is monotone, that is, F (x, y) + F (y, x) ≤ 0, for all x, y ∈ C,(3) for each x, y, z ∈ C limt→0 F (tz + (1− t)x, y) ≤ F (x, y) ,(4) for each x, y, z ∈ C, y 7→ F (x, y) is convex and lower semicontinuous.

We denote the set of solutions of (4.6) by GEP (F,A). We need the following lemmas to give an applicationof our main theorem to the generalized equilibrium problem.

Lemma 4.5. [26] Let F be a bifunction from C×C to R which satisfies (1)-(4). Let AF be a set-valued mappingfrom H into itself defined by

AFx =

z ∈ H : F (x, y) ≥ 〈y − x, z〉 , x ∈ C∅, x /∈ C. (4.7)

Then AF is a maximal monotone operator with the domain D (AF ) ⊂ C and EP (F ) = A−1F 0.

Lemma 4.6. [16] Let F : C × C → R be a bifunction satisfying (1)-(2) and A : H → H is continuous andmonotone on H, hence maximal monotone. For r > 0 and x ∈ H, let Wr : H → C be a mapping defined asfollows:

Wrx =

z ∈ C : F (z, y) +

1

r〈z − x+ rAx, y − z〉 ≥ 0, ∀y ∈ C

.

Then the following hold:

(1) Wr is single-valued,(2) F (Wr) = GEP (F,A) ,(3) Wr is a nonexpansive mapping,(4) GEP (F,A) is closed and convex.

Kitkuan et al. [16] showed in the proof of the lemma that GEP (F,A) = (A+AF )−1

(0). On the other hand,

it is know that JAFλn (I − λnA)x = x⇔ x ∈ (A+AF )−1

(0). So, by taking B = AF in Theorem 4.1 and by usingLemma 4.5, we obtain the following theorem.

Theorem 4.7. Let A : H → H be an η-inverse strongly monotone operator for η > 0, F a bifunction fromC × C to R which satisfies (1)-(4) , and AF : H → 2H a maximal monotone operator defined by (4.7) such that

Γ = (A+AF )−1

(0) 6= ∅. Let x0, x1 ∈ H and let xn be a sequence generated byzn = xn + θn (xn − xn−1)

yn = (1− βn) zn + βnJAFλn

(I − λnA) zn

ωn = JAFλn (I − λnA)(

(1− αn) JAFλn (I − λnA)xn + αnJAFλn

(I − λnA) yn

)xn+1 = (1− γn) JAFλn (I − λnA)ωn + γnf (ωn) .

where,

θn :=

min

θ, ηnγn‖xn−xn−1‖

if xn 6= xn−1,

θ, otherwise.

and for θ ≥ 0.Then under the same conditions as in Theorem 3.1, the sequence xn converges strongly to apoint x∗of generalized equilibrium problem, where x∗ = PΓf (x∗) .

4.5. Application to image restoration problems. In this section, we apply Algorithm 4.4 to solve theimage restoration problem and also compare the efficiency of Algorithm 4.4 with the Algorithm 1.8 FBS, AVFB,FBMMMA and NAGA. All algorithms were written in Matlab 2020b and an Asus Intel Core i7 laptop.

Recall that the image restoration problem can be formulated as the following linear inverse problem:

υ = Ax+ b, (4.8)

IMAGE RESTORATION USING AN INERTIAL VISCOSITY FIXED POINT ALGORITHM 14

where x ∈ Rn×1 is the original image, A ∈ Rm×n is a blurring matrix, b ∈ Rm×1 is the additive noise and v isthe observed image. Also, it is well-known that solving (4.8) is equivalent to solving the convex minimizationproblem

x∗ = argminx∈Rn

1

2‖Ax− v‖22 + τ ‖x‖1

. (4.9)

To solve the problem (4.9), we use Theorem 4.3. We set h (x) = 12 ‖Ax− v‖

22 and g (x) = τ ‖x‖1. Then, it is

easy to see that the gradient of h is ∇h (x) = AT (Ax− v), where AT is a transpose of A. The Lipschitz constantL of ∇h is computed by the maximum eigenvalues of the matrix ATA. In all comparisons, we use the test imagesLena, Pepper and Cameraman. At this point, the degenerate image is obtained by adding motion blur andrandom noise to the test images. In order to add the blur, we use the Matlab function fspecial(’motion’, 15,60).We will try to obtain an image close to the original image using Algorithm 4.4. The quality of the restoredimage is measured by the signal to noise ratio (SNR) which is defined by

SNR = 20 log‖x‖2

‖x− xn‖2,

where x and xn is the original image and the estimated image at iteration n, respectively.

In Theorem 4.3, we take λn = nL(n+1) , βn = γn = 1

n , αn = 12 , ηn = 1020

n and f (x) = 0.99x. If θ = 0.99, then

we obtain

θn :=

min

0.99, 1020

n2‖xn−xn−1‖

if xn 6= xn−1,

0.99, otherwise.

Firstly, we investigate the effectiveness of inertial and viscosity terms in Algorithm 4.4. The obtained results aregiven in Figure 2, Figure 3 and Table 2.

Figure 2. (a) Pepper image (b) Blurred image (c) exclusion of viscosity and inertial termsfrom Algorithm 4.4 (d) exclusion of viscosity term from Algorithm 4.4 (e) exclusion of inertialterm from Algorithm 4.4 (f) Algorithm 4.4

The experimental results show that the viscosity and inertial terms enable us to obtain more quality resultsin image restoration.

IMAGE RESTORATION USING AN INERTIAL VISCOSITY FIXED POINT ALGORITHM 15

Figure 3. Graphic of SNR values for Pepper image

SNR ValuesNo.Iterations Algorithm 4.4 Exclusion of inertial Exclusion of viscosity Exclusion of inertial and viscosity

1 25.292058 22.368991 0.000000 0.0000005 43.403404 42.632692 29.530821 29.34544910 48.643235 47.275684 39.653579 39.29498220 52.490425 50.871112 46.844475 46.24720630 54.397355 52.757836 49.890838 49.19486440 55.626238 54.027441 51.688664 50.95062950 56.494179 54.971150 52.927414 52.17418370 57.613901 56.302456 54.587938 53.84523080 57.977812 56.788446 55.183919 54.45848290 57.977812 57.190757 55.679869 54.976793100 58.458689 57.526028 56.098940 55.421989

Table 2. SNR values for Pepper image

In what follows, we compare Algorithm 4.4 with AVFB, FBMMMA, and NAGA for the infinite family ofnonexpansive mappings. We take the parameters as λn = n

L(n+1) , βn = γn = ρn = σn = 1n , αn = δn = ϕn = 1

2 ,

%n = 1 − 12 −

1n , ηn = 1020

n , f (x) = 0.7x. In this case, the experimental results have been given in Figure 4Figure 5 and Table 3.

Algorithm 4.4 provides image restoration with higher SNR, so the performance of image restoration of Algo-rithm 4.4 is better than FBMMMA, AVFB, and NAGA.

Finally, we compare Algorithm 4.4 with Algorithm (1.8) and FBS algorithm. If we choose λn = 1L , βn =

γn = δn = ψn = φn = 1n , ηn = 1020

n , θ = 0.99, αn = 12 , and ∇h (x) = f (x) = 0.6x then the experimental results

are given in Figure 6, Figure 7 and Table 4 .

We deduce that Algorithm 4.4 have higher SNR than Algorithm 1.8 and FBS algorithm. That is, Algorithm4.4 has more effective in image restoration than the other algorithms.

References

[1] Aoyama K., Yasunori Y., Takahashi W., Toyoda M., On a strongly nonexpansive sequence in a Hilbert space, J. Nonlinear

Convex Anal., 8, (2007), 471-489.[2] Artsawang N., Ungchittrakool K., Inertial Mann-Type Algorithm for a Nonexpansive Mapping to solve Monotone Inclusion

and Image Restoraton Problem, Symmetry, 12, (5) (2020) 750.

IMAGE RESTORATION USING AN INERTIAL VISCOSITY FIXED POINT ALGORITHM 16

Figure 4. (a) Cameraman image (b) Blurred image (c) NAGA (d)FBMMMA (e) AVFBA (f)Algorithm 4.4

Figure 5. Graphic of SNR values for Cameraman image

[3] Bauschke H.H., Combettes P. L., Convex Analysis and Monotone Operator Theory in Hilbert Space, CMS Books in Mathematics

Springer, New York, 2011.[4] Bussaban L., Suantai S., Kaewkhao A., A parallel inerrtial S-iteration forward-bacward algorithm for regression and classifica-

tion problems, Carpathian. J., 36, (2020), 35-44.[5] Byrne C., Aunified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse Probl., 20,

(2004), 103-120.[6] Cholamjiak, P.; Shehu, Y. Inertial forward-backward splitting method in Banach spaces with application to compressed sensing,

Appl. Math., 64, (2019), 409-435.

[7] Cholamjiak P, Suantai S. Viscosity approximation methods for a nonexpansive semigroup in Banach spaces with gauge functions,J. Glob. Optim.,54, (2012), 185-197.

IMAGE RESTORATION USING AN INERTIAL VISCOSITY FIXED POINT ALGORITHM 17

SNR ValuesNo.Iterations Algorithm 4.4 AVFBA FBMMMA NAGA

1 23.367583 15.283348 30.855847 24.0051465 36.649323 33.006604 36.84145 35.54021710 40.897499 38.751068 41.140872 36.34563820 44.460392 43.223013 42.465793 36.84394030 46.393900 44.970841 42.365621 37.09798940 47.709275 46.126490 42.353539 37.26960950 48.684256 46.944487 42.355574 37.39894770 50.040212 48.358723 41.786430 37.58869780 50.525856 48.881213 41.589025 37.66242190 50.924759 49.328276 41.462924 37.726771100 51.256042 49.738586 41.258755 37.783805

Table 3. SNR values for Cameraman image

Figure 6. (a) Lena image (b) Blurred image (c) FBS (d) Algorithm 1.8 (e) Algorithm 4.4

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IMAGE RESTORATION USING AN INERTIAL VISCOSITY FIXED POINT ALGORITHM 18

Figure 7. Graphic of SNR values for Lena image

SNR ValuesNo.Iteration Algorithm 4.4 Algorithm 1.8 FBS

1 17.748951 17.372599 44.9225495 43.22688 39, .82411 47.59390810 50.193068 47.884830 49.43194520 54.994313 53.046121 51.90083030 57.174776 54.929475 53.59715940 58.497187 56.231519 54.84997450 59.389414 57.219987 55.82022570 60.479787 58.659111 57.24681180 60.814045 59.112947 57.79384190 61.056274 59.487271 58.264392100 61.228103 59.805114 58.673967

Table 4. SNR values for Lena image

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IMAGE RESTORATION USING AN INERTIAL VISCOSITY FIXED POINT ALGORITHM 19

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1 Department of Mathematics, Faculty of Science, Erzurum Technical University, Erzurum, 25700, Turkey,

2 Department of Mathematics, Faculty of Science, Erzurum Technical University, Erzurum, 25700, Turkey,

Email address: 1ebru.altiparmak@erzurum.edu.tr,

Email address: 2ibrahimkarahan@erzurum.edu.tr,