Inertial Forward–Backward–Forward Algorithm with Moving Point Projection for Monotone Inclusions and Image Restoration

Abstract

This paper introduces a novel inertial forward–backward–forward algorithm driven by a newly conceptualized moving point projection technique for solving monotone inclusion problems in real Hilbert spaces. By leveraging the properties of a Lipschitz continuous, monotone operator and a maximally monotone operator alongside this innovative projection strategy, we dynamically construct a sequence of nonempty, closed, and convex sets that contain the zeros of the sum of the two operators. This geometric construction ensures that the resulting sequence is well defined and guarantees its weak convergence to a solution. Furthermore, to validate the practical efficacy of the proposed theoretical framework, we evaluate our method on image restoration problems. Numerical experiments measuring the improvement in signal-to-noise ratio (ISNR) and the structural similarity index measure (SSIM) confirm that the proposed algorithm is highly efficient and significantly outperforms existing state-of-the-art methods.

1. Introduction

Throughout this work, we assume that $\mathcal{H}$ is a real Hilbert space equipped with the inner product $\langle ·,·\rangle$, and we denote by $\parallel s\parallel =\sqrt{\langle s,s\rangle }$ the norm induced by this inner product for all $s\in \mathcal{H}$. Let $\mathcal{P}:\mathcal{H}\to \mathcal{H}$ be a single-valued operator, and let $\mathcal{Q}:\mathcal{H}\to 2^{\mathcal{H}}$ be a multivalued operator.

We say that the operator $\mathcal{P}$ is:

• Lipschitz continuous if there exists a constant $\mathcal{l}\ge 0$ such that

  $$∥\mathcal{P}w-\mathcal{P}s∥\le \mathcal{l}∥w-s∥,\forall w,s\in \mathcal{H}.$$
  Whether $\mathcal{P}$ is contractive or nonexpansive depends on the value of the Lipschitz constant ℓ; specifically, $\mathcal{P}$ is contractive if $\mathcal{l}<1$ and nonexpansive if $\mathcal{l}\le 1$.
• Firmly nonexpansive if

  $${∥\mathcal{P}w-\mathcal{P}s∥}^2\le 〈\mathcal{P}w-\mathcal{P}s,w-s〉,\forall w,s\in \mathcal{H}.$$
  Equivalently,

  $${∥\mathcal{P}w-\mathcal{P}s∥}^2\le {∥w-s∥}^2-{∥\left(\mathcal{I}-\mathcal{P}\right)w-\left(\mathcal{I}-\mathcal{P}\right)s∥}^2,\forall w,s\in \mathcal{H}.$$

  (1)
• Quasi-nonexpansive if $\mathcal{F}\left(\mathcal{P}\right)\ne \varnothing$ and

  $$∥\mathcal{P}w-q∥\le ∥w-q∥,\forall w\in \mathcal{H},\forall q\in \mathcal{F}\left(\mathcal{P}\right).$$
• Monotone if

  $$〈w-s,\mathcal{P}w-\mathcal{P}s〉\ge 0,\forall w,s\in \mathcal{H}.$$
• $\sigma$-cocoercive if there exists a constant $\sigma >0$ such that

  $$〈w-s,\mathcal{P}w-\mathcal{P}s〉\ge \sigma {∥\mathcal{P}w-\mathcal{P}s∥}^2,\forall w,s\in \mathcal{H}.$$

Associated with the multivalued mapping $\mathcal{Q}$ are the sets $zer\left(\mathcal{Q}\right):=\left\{\tilde{q}\in \mathcal{H}\mid 0\in \mathcal{Q}\left(\tilde{q}\right)\right\}$ and $\mathcal{G}\left(\mathcal{Q}\right):=\left\{(w,\tilde{w})\in \mathcal{H}\times \mathcal{H}\mid \tilde{w}\in \mathcal{Q}\left(w\right)\right\}$, which denote the zero set and the graph of $\mathcal{Q}$, respectively. We say that $\mathcal{Q}$ is:

• Monotone if

  $$〈w-s,\tilde{w}-\tilde{s}〉\ge 0,\forall \left(w,\tilde{w}\right),\left(s,\tilde{s}\right)\in \mathcal{G}\left(\mathcal{Q}\right).$$
• $\sigma$-cocoercive if there is $\sigma >0$ such that

  $$〈w-s,\tilde{w}-\tilde{s}〉\ge \sigma {∥\tilde{w}-\tilde{s}∥}^2,\forall \left(w,\tilde{w}\right),\left(s,\tilde{s}\right)\in \mathcal{G}\left(\mathcal{Q}\right).$$
• Maximally monotone if $\mathcal{Q}$ is monotone and there exists no monotone operator whose graph properly contains the graph of $\mathcal{Q}$. More precisely, if $\widehat{\mathcal{Q}}:\mathcal{H}\to 2^{\mathcal{H}}$ is a monotone operator satisfying

  $$\mathcal{G}\left(\mathcal{Q}\right)\subseteq \mathcal{G}\left(\widehat{\mathcal{Q}}\right),$$

  then it necessarily follows that

  $$\mathcal{G}\left(\mathcal{Q}\right)=\mathcal{G}\left(\widehat{\mathcal{Q}}\right).$$

Let $\Gamma :\mathcal{H}\to \mathcal{H}$ be an operator. The classical fixed point problem consists of finding

$$s\in \mathcal{H}\mathrm{such}\mathrm{that}(\Gamma -\mathcal{I})s=0.$$

(2)

A related problem that frequently arises in nonlinear analysis and optimization is the monotone inclusion problem. Given two operators $\mathcal{P}:\mathcal{H}\to \mathcal{H}$ and $\mathcal{Q}:\mathcal{H}\to 2^{\mathcal{H}}$, the problem is formulated as finding

$$s\in \mathcal{H}\mathrm{such}\mathrm{that}0\in (\mathcal{P}+\mathcal{Q})s.$$

(3)

Here, $\mathcal{P}$ is a single-valued operator and $\mathcal{Q}$ is a multivalued operator. Recall that for $\mathcal{Q}:\mathcal{H}\to 2^{\mathcal{H}}$, the resolvent operator associated with $\mathcal{Q}$ is defined by

$${\mathcal{J}}_{\omega \mathcal{Q}}={(\mathcal{I}+\omega \mathcal{Q})}^{-1},$$

where $\omega >0$. It is well known that if $\mathcal{Q}$ is maximally monotone and $\omega >0$, then the domain of ${\mathcal{J}}_{\omega \mathcal{Q}}$ coincides with $\mathcal{H}$, i.e., $\mathrm{Dom}\left({\mathcal{J}}_{\omega \mathcal{Q}}\right)=\mathcal{H}$. Moreover, the mapping ${\mathcal{J}}_{\omega \mathcal{Q}}:\mathcal{H}\to \mathcal{H}$ is single-valued and firmly nonexpansive. For further details, see [1,2,3,4,5].

One of the classical approaches for solving monotone inclusion problems is the forward–backward splitting method (FBSM), introduced by Lions and Mercier [6] and Passty [7]. The method generates a sequence $\left\{v_n\right\}$ starting from an arbitrary initial point $v_0\in \mathcal{H}$ according to the following iterative scheme:

$$v_{n+1}={\mathcal{J}}_{{\omega }_n\mathcal{Q}}\left(\mathcal{I}-{\omega }_n\mathcal{P}\right)v_n,$$

(4)

where ${\omega }_n\in (0,+\infty )$ for all $n\in \mathbb{N}$, ${\mathcal{J}}_{{\omega }_n\mathcal{Q}}={(\mathcal{I}+{\omega }_n\mathcal{Q})}^{-1}$ denotes the resolvent of $\mathcal{Q}$, refs. [1,2] and the operators ${\mathcal{J}}_{{\omega }_n\mathcal{Q}}$ and $(\mathcal{I}-{\omega }_n\mathcal{P})$ are commonly referred to as the backward and forward operators, respectively. The operators $\mathcal{P}:\mathcal{H}\to \mathcal{H}$ and $\mathcal{Q}:\mathcal{H}\to 2^{\mathcal{H}}$ are assumed to be monotone. Since its introduction, the forward–backward splitting method has been widely studied and further developed in the literature.

Subsequently, Tseng [8] proposed a modification of (4) by incorporating an additional correction step. This modification allows the algorithm to guarantee convergence under less restrictive conditions compared with those required by the original scheme (4). The resulting iterative process, known as Tseng’s algorithm, is given by

$$\left\{\begin{array}{c}{u}_n={\mathcal{J}}_{{\omega }_n\mathcal{Q}}(\mathcal{I}-{\omega }_n\mathcal{P})\left(v_n\right),\hfill \\ v_{n+1}={proj}_{\mathcal{C}}(u_n-{\omega }_n(\mathcal{P}{u}_n-\mathcal{P}{v}_n)),\hfill \end{array}\right.$$

(5)

where $\mathcal{C}\subseteq \mathcal{H}$ is a nonempty closed and convex set satisfying $\mathcal{C}\cap zer(\mathcal{P}+\mathcal{Q})\ne ⌀$. The mapping ${proj}_{\mathcal{C}}:\mathcal{H}\to \mathcal{C}$ denotes the metric projection.

Furthermore, when $\mathcal{C}=\mathcal{H}$, the scheme (5) simplifies to

$$\left\{\begin{array}{c}{u}_n={\mathcal{J}}_{{\omega }_n\mathcal{Q}}(\mathcal{I}-{\omega }_n\mathcal{P})\left(v_n\right),\hfill \\ v_{n+1}=u_n-{\omega }_n(\mathcal{P}{u}_n-\mathcal{P}{v}_n),\hfill \end{array}\right.$$

(6)

The parameters are chosen as follows. Let $\gamma >1$ and $\tau ,\vartheta \in (0,1)$. The stepsize ${\omega }_n$ is selected as the largest value $\omega \in \left\{\gamma ,\gamma \tau ,\gamma {\tau }^2,\dots \right\}$ such that

$$\omega ∥\mathcal{P}{u}_n-\mathcal{P}{v}_n∥\le \vartheta ∥u_n-v_n∥.$$

This algorithm is also referred to as the forward–backward–forward (FBF) algorithm.

In 1964, Polyak [9] introduced the idea of accelerating the convergence of iterative methods by incorporating the term ${\omega }_n(v_n-v_{n-1})$, which is now commonly referred to as the inertial extrapolation term. The inertial technique was later refined by Nesterov [10] for convex optimization. Since then, inertial extrapolation has attracted considerable interest and has been widely developed in various contexts, including monotone inclusion problems—see, e.g., refs. [11,12,13,14,15]—and equilibrium problems and variational inequalities—see, e.g., refs. [16,17,18,19,20].

In 2021, Padcharoen et al. [21] incorporated an inertial extrapolation step into Tseng’s splitting algorithm and reported improved performance compared with several existing methods. Their iterative process is defined as

$$\left(\mathbf{PKKK}\mathbf{2021}\right)\left\{\begin{array}{c}\mathrm{Let}{v}_0,v_1\in \mathcal{H},\hfill \\ u_n=v_n+{\theta }_n(v_n-v_{n-1}),\hfill \\ w_n={\mathcal{J}}_{{\omega }_n\mathcal{Q}}(\mathcal{I}-{\omega }_n\mathcal{P})u_n,\hfill \\ v_{n+1}=w_n-{\omega }_n(\mathcal{P}{w}_n-\mathcal{P}{u}_n),\hfill \end{array}\right.$$

(7)

where $\mathcal{P}:\mathcal{H}\to \mathcal{H}$ is a monotone and Lipschitz continuous operator, while $\mathcal{Q}:\mathcal{H}\to 2^{\mathcal{H}}$ is maximally monotone. They established that the sequence $\left\{v_n\right\}$ generated by (7) converges weakly to a solution of the problem (3), that is, an element of $zer(\mathcal{P}+\mathcal{Q})$.

On the other hand, Nakajo and Takahashi [22] introduced the hybrid projection method for solving (2). Let $\mathcal{C}$ be a nonempty closed and convex subset of $\mathcal{H}$ and define the following iterative process:

$$\left\{\begin{array}{c}{u}_n={\theta }_n{v}_n+(1-{\theta }_n)\Gamma v_n,\hfill \\ {\Omega }_n=\left\{q\in \mathcal{C}\mid \parallel u_n-q\parallel \le \parallel v_n-q\parallel \right\},\hfill \\ {\Phi }_n=\left\{q\in \mathcal{C}\mid 〈v_n-q,v_0-v_n〉\ge 0\right\},\hfill \\ v_{n+1}={proj}_{{\Omega }_n\cap {\Phi }_n}{v}_0,n\in \mathbb{N},\hfill \end{array}\right.$$

(8)

In this setting, ${\theta }_n\in \left[0,1\right)$, the operator $\Gamma :\mathcal{C}\to \mathcal{C}$ is nonexpansive, and $\mathbf{pro}{\mathbf{j}}_{{\Omega }_n\cap {\Phi }_n}$ denotes the metric projection from $\mathcal{C}$ onto ${\Omega }_n\cap {\Phi }_n$. Furthermore, they proved that the sequence $\left\{v_n\right\}$ converges strongly to ${proj}_{\mathcal{F}\left(\Gamma \right)}{v}_0$.

In 2020, Tan et al. [23] proposed an inertial forward–backward–forward algorithm combined with a hybrid projection technique for solving monotone inclusion problems. This method, commonly referred to as the TZQ2020 algorithm, integrates an inertial extrapolation step with Tseng’s forward–backward–forward framework in order to accelerate the iterative process while maintaining strong convergence properties. The iterative scheme is given by

$$\left(\mathbf{TZQ}\mathbf{2020}\right)\left\{\begin{array}{c}\mathrm{Let}{v}_0,v_1\in \mathcal{H},\hfill \\ u_n=v_n+{\theta }_n(v_n-v_{n-1}),\hfill \\ w_n={\mathcal{J}}_{{\omega }_n\mathcal{Q}}(\mathcal{I}-{\omega }_n\mathcal{P})u_n,\hfill \\ y_n=w_n-{\omega }_n(\mathcal{P}{w}_n-\mathcal{P}{u}_n),\hfill \\ {\Omega }_n=\left\{q\in \mathcal{H}\left|{∥y_n-q∥}^2\le {∥u_n-q∥}^2-\left(1-{\mu }^2{\displaystyle \frac{{\omega }_n^2}{{\omega }_{n+1}^2}}\right){∥u_n-w_n∥}^2\right.\right\},\hfill \\ {\Phi }_n=\left\{q\in \mathcal{H}\mid \langle v_n-q,v_n-v_0\rangle \le 0\right\},\hfill \\ v_{n+1}={proj}_{{\Omega }_n\cap {\Phi }_n}\left(v_0\right),n\in \mathbb{N}.\hfill \end{array}\right.$$

(9)

Although the TZQ2020 algorithm guarantees strong convergence, its projection step involves the intersection of two sets, which may increase the computational cost in practical implementations.

In 2026, Ungchittrakool et al. [24] proposed an inertial forward–backward algorithm with a moving point projection method, which is defined as follows:

$$\left\{\begin{array}{c}{u}_n=v_n+{\theta }_n\left(v_n-v_{n-1}\right),\hfill \\ w_n={\alpha }_n{v}_n+\left(1-{\alpha }_n\right){\mathcal{J}}_{{\omega }_n\mathcal{Q}}\left(\mathcal{I}-{\omega }_n\mathcal{P}\right)u_n,\hfill \\ {\Omega }_{n+1}=\left\{q\in \mathcal{H}\left|\begin{array}{c}{∥w_n-q∥}^2\le {∥v_n-q∥}^2+\left(1-{\alpha }_n\right){\theta }_n^2{∥v_n-v_{n-1}∥}^2\hfill \\ +2{\theta }_n\left(1-{\alpha }_n\right)〈v_n-q,v_n-v_{n-1}〉\hfill \end{array}\right.\right\},\hfill \\ v_{n+1}={proj}_{{\Omega }_{n+1}}{v}_n,n\in \mathbb{N},\hfill \end{array}\right.$$

(10)

where the constraint set ${\Omega }_{n+1}$ is dynamically constructed at each iteration using Fejér-type inequalities. Unlike classical hybrid projection methods that project onto intersections of fixed reference sets, the projection in this scheme is performed onto a moving convex set determined by the current iterate. This algorithm simplifies the computational structure and ensures weak convergence under suitable control conditions on the parameters.

Motivated by recent advances in the literature, this paper introduces a novel inertial forward–backward–forward algorithm equipped with a moving point projection technique for solving monotone inclusion problems and image restoration in real Hilbert spaces. To clarify the precise novelty and contributions of our work with respect to existing methods, we highlight the following key aspects:

• Algorithmic Novelty: The genuinely new component of our proposed method (see Algorithm 1 in Section 3) is the core novelty lies in the integration of the moving point projection (MPP) strategy into the inertial forward–backward–forward (I-FBF) framework. Specifically, we dynamically construct a single half-space ${\Omega }_{n+1}$ at each iteration and project a convex combination point $z_n$ onto it.
• Distinction from State-of-the-Art: Our method fundamentally differs from the most closely related algorithms in its projection architecture. Unlike the PKKK2021 algorithm [21], which relies on a standard FBF step without a projection mechanism, and the TZQ2020 algorithm [23], which requires calculating the metric projection onto the intersection of two reference sets (${\Omega }_n\cap {\Phi }_n$), our algorithm calculates the projection onto a single, dynamically updated moving set. Finally, while the recent work by Ungchittrakool et al. [24] applied the MPP technique to the standard forward–backward method, the proposed algorithm extends this concept to the FBF framework, which accommodates a broader class of monotone operators by relaxing the cocoercivity assumption to mere Lipschitz continuity.
• Concrete Computational Advantage: By avoiding the intersection of multiple sets, our moving point projection design significantly simplifies the computational structure and reduces the computational cost per iteration. This yields a highly efficient algorithm that theoretically guarantees weak convergence under suitable control conditions, and empirically outperforms state-of-the-art methods in high-dimensional computational tasks.

In addition to the rigorous theoretical convergence analysis, the algorithm is specifically designed for practical implementation in image processing. Numerical experiments evaluating the improvement in signal-to-noise ratio (ISNR) and the structural similarity index measure (SSIM) confirm that our proposed method is highly efficient and achieves superior reconstruction quality compared to current state-of-the-art algorithms.

2. Preliminaries

In this section, we compile key theoretical tools that will play a crucial role in the convergence analysis presented in the following section. Let $\mathcal{C}$ be a non-empty, closed, and convex subset of a real Hilbert space $\mathcal{H}$. Throughout this work, the symbols “⇀” and “→” will be used to denote weak and strong convergence, respectively. For a given sequence $\left\{v_n\right\}\subseteq \mathcal{H}$, we denote by ${\omega }_w\left(v_n\right):=\left\{w\in \mathcal{H}|\exists \left\{v_{n_k}\right\}\subseteq \left\{v_n\right\},v_{n_k}\rightharpoonup w\right\}$ the set of all weak sequential cluster points of $\left\{v_n\right\}$. Let $\Gamma :\mathcal{H}\to \mathcal{H}$ be a mapping. The set of all fixed points of $\Gamma$ is denoted by $\mathcal{F}\left(\Gamma \right):=\left\{s\in \mathcal{H}|\left(\Gamma -\mathcal{I}\right)s=\mathbf{0}\right\}$ where $\mathcal{I}$ denotes the identity mapping.

Lemma 1. 

Let $\mathcal{Q}:\mathcal{H}\to 2^{\mathcal{H}}$ be a maximally monotone operator and $\mathcal{P}:\mathcal{H}\to \mathcal{H}$ be an operator on $\mathcal{H}$. Define $T_{\omega }:={\mathcal{J}}_{\omega \mathcal{Q}}\left(\mathcal{I}-\omega \mathcal{P}\right)$, $\omega >0$. Then we have

$$\mathcal{F}\left(T_{\omega }\right)=zer(\mathcal{P}+\mathcal{Q}),\forall \omega >0.$$

Proof. 

See, for example, ([21] Lemma 1).   □

Lemma 2 

([3,4]). Let $\mathcal{H}$ be a real Hilbert space. Then the following holds for all $w,s\in \mathcal{H}$ and $\kappa \in \mathbb{R}$:

$${∥\kappa w+(1-\kappa )s∥}^2=\kappa {∥w∥}^2+(1-\kappa ){∥s∥}^2-\kappa (1-\kappa ){∥w-s∥}^2.$$

(11)

Lemma 3 

([12] Lemma 2.3). Let $\left\{{\Gamma }_n\right\}$ and $\left\{{\delta }_n\right\}$ be sequences of nonnegative real numbers such that

$${\Gamma }_{n+1}\le {\Gamma }_n+{\theta }_n({\Gamma }_n-{\Gamma }_{n-1})+{\delta }_n,$$

where $\left\{{\theta }_n\right\}\subseteq [0,1)$ satisfies $0\le {\theta }_n\le \theta <1$ and ${\sum }_{n=1}^{\infty }{\delta }_n<\infty$. Then the following statements hold:

1.

${\displaystyle \sum _{n=1}^{\infty }{[{\Gamma }_n-{\Gamma }_{n-1}]}_{+}<\infty }$, where ${\left[\sigma \right]}_{+}:=max\{\sigma ,0\}$.

2.

There exists ${\gamma }^{*}\ge 0$ such that ${\displaystyle \underset{n\to \infty }{lim}{\Gamma }_n={\gamma }^{*}}$.

Lemma 4 

([25]). Let $\mathcal{Q}:\mathcal{H}\to 2^{\mathcal{H}}$ be a maximally monotone operator and $\mathcal{P}:\mathcal{H}\to \mathcal{H}$ be a Lipschitz continuous and monotone operator. Then the operator $\mathcal{P}+\mathcal{Q}$ is a maximally monotone operator.

Lemma 5 

([26]). Let $\mathcal{C}$ be a nonempty set of $\mathcal{H}$ and $\left\{v_n\right\}$ be a sequence in $\mathcal{H}$ such that the following two conditions hold:

1.

There exists $l\in \mathbb{R}$ such that ${lim}_{n\to \infty }∥v_n-v∥=l,\mathrm{for}\mathrm{all}v\in \mathcal{C}$,

2.

For every subsequence $\left\{v_{n_k}\right\}$ of $\left\{v_n\right\},v_{n_k}\rightharpoonup v\in \mathcal{H}$, we have $v\in \mathcal{C}$.

Then $\left\{v_n\right\}$ converges weakly to a point in $\mathcal{C}$.

Lemma 6 

([27] Lemma 2.6). Let $\mathcal{P}:\mathcal{H}\to \mathcal{H}$ be an operator and $\mathcal{Q}:\mathcal{H}\to 2^{\mathcal{H}}$ be a maximally monotone operator. Then, for any $y,z\in \mathcal{H}$ and $r>0$, the following statements are equivalent:

1.

$z={\mathcal{J}}_{r\mathcal{Q}}(\mathcal{I}-r\mathcal{P})y$,

2.

$\mathit{There}\mathit{exists}w\in \mathcal{Q}z\mathit{satisfying}w={\displaystyle \frac{1}{r}}\left(y-z-r\mathcal{P}y\right)$.

Lemma 7 

([27] Lemma 2.7). Let $\mathcal{P}:\mathcal{H}\to \mathcal{H}$ be a Lipschitz continuous and monotone operator, and let $\mathcal{Q}:\mathcal{H}\to 2^{\mathcal{H}}$ be a maximally monotone operator, where $\mathcal{H}$ is a real Hilbert space. For any $y,z\in \mathcal{H}$ and any $r>0$ satisfying

$$z={\mathcal{J}}_{r\mathcal{Q}}(\mathcal{I}-r\mathcal{P})y$$

and for any $x^{*}\in zer(\mathcal{P}+\mathcal{Q})$, the following inequality holds:

$$〈y-z-r\left(\mathcal{P}y-\mathcal{P}z\right),z-x^{*}〉\ge 0.$$

3. Main Results

This section establishes the convergence properties of the proposed inertial forward-backward-forward algorithm with moving point projection (I-FBF-MPP). The subsequent theoretical analysis relies on the following standard assumptions:

(H1)

The monotone inclusion problem admits at least one solution, i.e., $\exists \tilde{v}\in {\rm Y}:=\left\{v\in \mathcal{H}|0\in \left(\mathcal{P}+\mathcal{Q}\right)v\right\}$

(H2)

Let $\mathcal{Q}:\mathcal{H}\to 2^{\mathcal{H}}$ and $\mathcal{P}:\mathcal{H}\to \mathcal{H}$ be operators such that $\mathcal{Q}$ is maximally monotone, while $\mathcal{P}$ is monotone and ℓ-Lipschitz continuous.

By integrating a Tseng-type evaluation with inertial acceleration, we introduce the I-FBF-MPP method presented in Algorithm 1.

Algorithm 1 I-FBF-MPP Algorithm
Require:	(i) $\overline{\theta }\in [0,1)$,
	(ii) $\left\{{\epsilon }_n\right\}\subseteq (0,\infty )$ with ${\sum }_{n=1}^{\infty }{\epsilon }_n<+\infty$,
	(iii)$\left\{{\omega }_n\right\}\subseteq [a,b]\subseteq (0,1/\mathcal{l})$,
	(iv) $\left\{{\phi }_n\right\}\subseteq (0,1)$ with ${\displaystyle \underset{n\to \infty }{lim\; inf}{\phi }_n>0}$.
1: $v_0,v_1\in \mathcal{H}$ chosen arbitrarily 2: for  $n=1,2,\dots$  do 3:         Compute inertial parameter: $${\theta }_n=\left\{\begin{array}{cc}min\left\{\overline{\theta },\frac{{\epsilon }_n}{\parallel v_n-v_{n-1}{\parallel }^2}\right\},\hfill & \mathrm{if}{v}_n\ne v_{n-1},\hfill \\ \overline{\theta },\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$ 4:         $u_n=(1+{\theta }_n)v_n-{\theta }_n{v}_{n-1}$ 5:         $w_n={\mathcal{J}}_{{\omega }_n\mathcal{Q}}(\mathcal{I}-{\omega }_n\mathcal{P})u_n$ 6:         $y_n=w_n-{\omega }_n\left(\mathcal{P}{w}_n-\mathcal{P}{u}_n\right)$ 7:         Define half-space: $${\Omega }_{n+1}=\left\{q\in \mathcal{H}|\parallel y_n{-q\parallel }^2\le \parallel u_n{-q\parallel }^2-(1-{\omega }_n^2{\mathcal{l}}^2){\parallel u_n-w_n\parallel }^2\right\}$$ 8:         $z_n={\phi }_n{y}_n+(1-{\phi }_n)u_n$ 9:         $v_{n+1}={proj}_{{\Omega }_{n+1}}\left(z_n\right)$ 10: end for

Convergence Analysis

Lemma 8. 

The sequence $\left\{v_n\right\}$ generated by Algorithm 1 is well defined and ${\rm Y}:={(\mathcal{P}+\mathcal{Q})}^{-1}\left(0\right)\subseteq {\Omega }_n$ for every $n\in \mathbb{N}$. Furthermore, the following inequality holds for each $n\in \mathbb{N}$:

$$\parallel y_n-g^{*}{\parallel }^2 \le \parallel u_n-g^{*}{\parallel }^2-\left(1-{\omega }_n^2{\mathcal{l}}^2\right){\parallel u_n-w_n\parallel }^2,\forall g^{*}\in {\rm Y}.$$

(12)

Proof. 

We will show that the sequence $\left\{v_n\right\}$ is well defined for each $n\in \mathbb{N}$. First, we prove that the following inequality holds:

$$\parallel y_1-g^{*}{\parallel }^2 \le \parallel u_1-g^{*}{\parallel }^2-\left(1-{\omega }_1^2{\mathcal{l}}^2\right){\parallel u_1-w_1\parallel }^2,\forall g^{*}\in {\rm Y}.$$

(13)

For $n=1$, let $g^{*}\in {\rm Y}$. We now consider the following inequality

$$\begin{array}{cc}\hfill {∥y_1-g^{*}∥}^2& ={∥w_1-{\omega }_1(\mathcal{P}{w}_1-\mathcal{P}{u}_1)-g^{*}∥}^2\hfill \\ \hfill & ={∥w_1-g^{*}∥}^2+{\omega }_1^2{∥\mathcal{P}{w}_1-\mathcal{P}{u}_1∥}^2-2{\omega }_1〈w_1-g^{*},\mathcal{P}{w}_1-\mathcal{P}{u}_1〉\hfill \\ \hfill & ={∥w_1-u_1∥}^2+{∥u_1-g^{*}∥}^2+2〈w_1-u_1,u_1-g^{*}〉\hfill \\ \hfill & +{\omega }_1^2{∥\mathcal{P}{w}_1-\mathcal{P}{u}_1∥}^2-2{\omega }_1〈w_1-g^{*},\mathcal{P}{w}_1-\mathcal{P}{u}_1〉\hfill \\ \hfill & ={∥u_1-g^{*}∥}^2-{∥w_1-u_1∥}^2+{\omega }_1^2{∥\mathcal{P}{w}_1-\mathcal{P}{u}_1∥}^2\hfill \\ \hfill & -2〈u_1-w_1-{\omega }_1\left(\mathcal{P}{u}_1-\mathcal{P}{w}_1\right),w_1-g^{*}〉\hfill \\ \hfill & \le {∥u_1-g^{*}∥}^2-\left(1-{\omega }_1^2{\mathcal{l}}^2\right){∥w_1-u_1∥}^2\hfill \\ \hfill & -2〈u_1-w_1-{\omega }_1\left(\mathcal{P}{u}_1-\mathcal{P}{w}_1\right),w_1-g^{*}〉.\hfill \end{array}$$

(14)

By Lemma 7, (14) and ${\omega }_1\in \left(0,{\displaystyle \frac{1}{\mathcal{l}}}\right)$, we obtain (13), and then $g^{*}\in {\Omega }_2$. Therefore ${\rm Y}\subseteq {\Omega }_2\ne ⌀$. By rewriting (13) to be the form of inequality with inner product and the linearity of the inner product, we can verify that ${\Omega }_2$ is closed and convex, so there exists a unique $v_2\in {\Omega }_2$ such that $v_2={proj}_{{\Omega }_2}{z}_1$, where $z_1={\phi }_1{y}_1+(1-{\phi }_1)u_1$.

Let $j>1$, and suppose that ${\rm Y}\subseteq {\Omega }_j\ne ⌀$. By the same process as above, we can rewrite the inequality defining ${\Omega }_j$ into the form of an inequality involving inner products, which proves that ${\Omega }_j$ is closed and convex. And consequently there exists a unique $v_j\in {\Omega }_j$ such that $v_j={proj}_{{\Omega }_j}{z}_{j-1}$. Let $g^{*}\in {\rm Y}$, and consider the following inequality

$$\begin{array}{cc}\hfill {∥y_j-g^{*}∥}^2& ={∥w_j-{\omega }_j(\mathcal{P}{w}_j-\mathcal{P}{u}_j)-g^{*}∥}^2\hfill \\ \hfill & ={∥w_j-g^{*}∥}^2+{\omega }_j^2{∥\mathcal{P}{w}_j-\mathcal{P}{u}_j∥}^2-2{\omega }_j〈w_j-g^{*},\mathcal{P}{w}_j-\mathcal{P}{u}_j〉\hfill \\ \hfill & ={∥w_j-u_j∥}^2+{∥u_j-g^{*}∥}^2+2〈w_j-u_j,u_j-g^{*}〉\hfill \\ \hfill & +{\omega }_j^2{∥\mathcal{P}{w}_j-\mathcal{P}{u}_j∥}^2-2{\omega }_j〈w_j-g^{*},\mathcal{P}{w}_j-\mathcal{P}{u}_j〉\hfill \\ \hfill & ={∥u_j-g^{*}∥}^2-{∥w_j-u_j∥}^2+{\omega }_j^2{∥\mathcal{P}{w}_j-\mathcal{P}{u}_j∥}^2\hfill \\ \hfill & -2〈u_j-w_j-{\omega }_j\left(\mathcal{P}{u}_j-\mathcal{P}{w}_j\right),w_j-g^{*}〉\hfill \\ \hfill & \le {∥u_j-g^{*}∥}^2-\left(1-{\omega }_j^2{\mathcal{l}}^2\right){∥w_j-u_j∥}^2\hfill \\ \hfill & -2〈u_j-w_j-{\omega }_j\left(\mathcal{P}{u}_j-\mathcal{P}{w}_j\right),w_j-g^{*}〉.\hfill \end{array}$$

(15)

By Lemma 7, (15) and ${\omega }_j\in \left(0,{\displaystyle \frac{1}{\mathcal{l}}}\right)$, we obtain

$$\parallel y_j-g^{*}{\parallel }^2\le \parallel u_j-g^{*}{\parallel }^2-\left(1-{\omega }_j^2{\mathcal{l}}^2\right){\parallel u_j-w_j\parallel }^2.$$

(16)

Then $g^{*}\in {\Omega }_{j+1}$, therefore ${\rm Y}\subseteq {\Omega }_{j+1}\ne ⌀$. By rewriting (16) to be the form of inequality with inner product and the linearity of the inner product, we can guarantee that ${\Omega }_{j+1}$ is closed and convex. Then there exists a unique $v_{j+1}\in {\Omega }_{j+1}$ such that $v_{j+1}={proj}_{{\Omega }_{j+1}}{z}_j$. By mathematical induction, we can conclude that $v_{n+1}={proj}_{{\Omega }_{n+1}}{z}_n$ for all $n\in \mathbb{N}$ which means $\left\{v_n\right\}$ is well defined, and ${\rm Y}\subseteq {\Omega }_n$ for each $n\in \mathbb{N}$. This completes the proof. □

Lemma 9 (Existence of weak cluster point in ${\rm Y}$).

Assume that hypotheses (H1) and (H2) hold. Let $\left\{u_n\right\}$, $\left\{w_n\right\}$, $\left\{v_n\right\}$ be the sequences generated by Algorithm 1. If

$$\underset{n\to \infty }{lim}\parallel u_n-v_n\parallel =0=\underset{n\to \infty }{lim}\parallel u_n-w_n\parallel$$

and $\left\{v_{n_k}\right\}$ converges weakly to $\widehat{l}\in \mathcal{H}$, then $\widehat{l}\in {\rm Y}$.

Proof. 

Suppose that ${lim}_{n\to \infty }\parallel u_n-v_n\parallel =0$ and ${lim}_{n\to \infty }\parallel u_n-w_n\parallel =0$. Then, from the definition of $\left\{w_n\right\}$ in Algorithm 1 it follows that $w_{n_k}={(\mathcal{I}+{\omega }_{n_k}\mathcal{Q})}^{-1}(\mathcal{I}-{\omega }_{n_k}\mathcal{P})u_{n_k}$, and it is easy to verify the following

$${\displaystyle \frac{1}{{\omega }_{n_k}}}\left(u_{n_k}-w_{n_k}-{\omega }_{n_k}\mathcal{P}{u}_{n_k}\right)\in \mathcal{Q}{w}_{n_k}.$$

Let $(r,q)\in \mathcal{G}(\mathcal{P}+\mathcal{Q})$. Then, $q\in (\mathcal{P}+\mathcal{Q})r$, that is, $q-\mathcal{P}r\in \mathcal{Q}r$. By the monotonicity of $\mathcal{Q}$, we have

$$〈r-w_{n_k},q-\mathcal{P}r-{\displaystyle \frac{1}{{\omega }_{n_k}}}\left(u_{n_k}-w_{n_k}-{\omega }_{n_k}\mathcal{P}{u}_{n_k}\right)〉\ge 0.$$

(17)

By rearranging the inequality (17), we then obtain

$$\begin{array}{cc}\hfill \langle r-w_{n_k},q\rangle & \ge \langle r-w_{n_k},\mathcal{P}r+{\displaystyle \frac{1}{{\omega }_{n_k}}}(u_{n_k}-w_{n_k}-{\omega }_{n_k}\mathcal{P}{u}_{n_k})\rangle \hfill \\ \hfill & =\langle r-w_{n_k},\mathcal{P}r-\mathcal{P}{u}_{n_k}\rangle +\langle r-w_{n_k},{\displaystyle \frac{1}{{\omega }_{n_k}}}(u_{n_k}-w_{n_k})\rangle \hfill \\ \hfill & =\langle r-w_{n_k},\mathcal{P}r-\mathcal{P}{w}_{n_k}\rangle +\langle r-w_{n_k},\mathcal{P}{w}_{n_k}-\mathcal{P}{u}_{n_k}\rangle \hfill \\ \hfill & +\langle r-w_{n_k},{\displaystyle \frac{1}{{\omega }_{n_k}}}(u_{n_k}-w_{n_k})\rangle \hfill \\ \hfill & \ge \langle r-w_{n_k},\mathcal{P}{w}_{n_k}-\mathcal{P}{u}_{n_k}\rangle +\langle r-w_{n_k},{\displaystyle \frac{1}{{\omega }_{n_k}}}(u_{n_k}-w_{n_k})\rangle .\hfill \end{array}$$

By using the fact that $\parallel u_n-w_n\parallel \to 0$ and the Lipschitz continuity of $\mathcal{P}$, we obtain $\parallel \mathcal{P}{u}_{n_k}-\mathcal{P}{w}_{n_k}\parallel \to 0$. Next, since $\parallel u_n-v_n\parallel \to 0$ and $v_{n_k}\rightharpoonup \widehat{l}$, it follows that $u_{n_k}\rightharpoonup \widehat{l}$, which in turn implies $w_{n_k}\rightharpoonup \widehat{l}$. Together with ${\omega }_{n_k}\in \left[a,b\right]$, we obtain

$$\langle r-\widehat{l},q-0\rangle =\langle r-\widehat{l},q\rangle =\underset{k\to \infty }{lim}\langle r-w_{n_k},q\rangle \ge 0.$$

From Lemma 4, we deduce that $0\in \left(\mathcal{P}+\mathcal{Q}\right)\widehat{l}$, and thus $\widehat{l}\in {\rm Y}$. □

Lemma 10. 

Let $\left\{v_n\right\}$ be the sequence generated by Algorithm 1. Then, for any $g^{*}\in {\rm Y}$, $\underset{n\to \infty }{lim}∥v_n-g^{*}∥$ exists.

Proof. 

Let $g^{*}\in {\rm Y}$. From the definition of $v_{n+1}$ in Algorithm 1, we have $v_{n+1}={proj}_{{\Omega }_{n+1}}\left(z_n\right)$. Since $g^{*}\in {\rm Y}$, and Lemma 8 guarantees that ${\rm Y}\subseteq {\Omega }_{n+1}$ for all $n\in \mathbb{N}$, it follows that $g^{*}\in {\Omega }_{n+1}$. Recall that the metric projection onto a closed and convex set is a firmly nonexpansive mapping. Thus, by applying (1) and the fact that ${proj}_{{\Omega }_{n+1}}\left(g^{*}\right)=g^{*}$, we obtain the following inequality

$$\begin{array}{cc}\hfill {∥v_{n+1}-g^{*}∥}^2& ={∥{proj}_{{\Omega }_{n+1}}\left(z_n\right)-{proj}_{{\Omega }_{n+1}}\left(g^{*}\right)∥}^2\hfill \\ \hfill & \le {∥z_n-g^{*}∥}^2-{∥\left(\mathcal{I}-{proj}_{{\Omega }_{n+1}}\right)\left(z_n\right)-\left(\mathcal{I}-{proj}_{{\Omega }_{n+1}}\right)\left(g^{*}\right)∥}^2\hfill \\ \hfill & ={∥z_n-g^{*}∥}^2-{∥v_{n+1}-z_n∥}^2.\hfill \end{array}$$

(18)

By substituting the definition of $z_n={\phi }_n{y}_n+(1-{\phi }_n)u_n$ into $\parallel z_n-g^{*}{\parallel }^2$ and applying (11), we obtain

$$\begin{array}{cc}\hfill \parallel z_n-g^{*}{\parallel }^2& =\parallel {\phi }_n(y_n-g^{*})+(1-{\phi }_n)(u_n-g^{*}){\parallel }^2.\hfill \\ \hfill & ={\phi }_n\parallel y_n-g^{*}{\parallel }^2+(1-{\phi }_n)\parallel u_n-g^{*}{\parallel }^2-{\phi }_n(1-{\phi }_n){\parallel y_n-u_n\parallel }^2.\hfill \end{array}$$

(19)

By (12), we have

$$\begin{array}{cc}\hfill \parallel z_n-g^{*}{\parallel }^2& \le {\phi }_n\left(\parallel u_n-g^{*}{\parallel }^2-(1-{\omega }_n^2{\mathcal{l}}^2){\parallel u_n-w_n\parallel }^2\right)\hfill \\ \hfill & +(1-{\phi }_n)\parallel u_n-g^{*}{\parallel }^2-{\phi }_n(1-{\phi }_n){\parallel y_n-u_n\parallel }^2\hfill \\ \hfill & =\parallel u_n-g^{*}{\parallel }^2-{\phi }_n(1-{\omega }_n^2{\mathcal{l}}^2)\parallel u_n-w_n{\parallel }^2-{\phi }_n(1-{\phi }_n){\parallel y_n-u_n\parallel }^2.\hfill \end{array}$$

(20)

By applying the identity (11) to expand $\parallel u_n-g^{*}{\parallel }^2$, we obtain

$$\begin{array}{cc}\hfill \parallel u_n-g^{*}{\parallel }^2& =\parallel (1+{\theta }_n)(v_n-g^{*})-{\theta }_n(v_{n-1}-g^{*}){\parallel }^2\hfill \\ \hfill & =(1+{\theta }_n)\parallel v_n-g^{*}{\parallel }^2-{\theta }_n\parallel v_{n-1}-g^{*}{\parallel }^2+{\theta }_n(1+{\theta }_n){\parallel v_n-v_{n-1}\parallel }^2.\hfill \end{array}$$

(21)

By combining (18), (20) and (21), we have

$$\begin{array}{cc}\hfill \parallel v_{n+1}-g^{*}{\parallel }^2& \le (1+{\theta }_n)\parallel v_n-g^{*}{\parallel }^2-{\theta }_n\parallel v_{n-1}-g^{*}{\parallel }^2+{\theta }_n(1+{\theta }_n)\parallel v_n-v_{n-1}{\parallel }^2\hfill \\ \hfill & -{\phi }_n(1-{\omega }_n^2{\mathcal{l}}^2)\parallel u_n-w_n{\parallel }^2-{\phi }_n(1-{\phi }_n)\parallel y_n-u_n{\parallel }^2\hfill \\ \hfill & -\parallel v_{n+1}-z_n{\parallel }^2.\hfill \end{array}$$

(22)

According to Algorithm 1, the parameter ${\theta }_n$ is updated such that ${\theta }_n=min\left\{\overline{\theta },{\displaystyle \frac{{\epsilon }_n}{\parallel v_n-v_{n-1}{\parallel }^2}}\right\}$ when $v_n\ne v_{n-1}$ and ${\theta }_n=\overline{\theta }$ when $v_n=v_{n-1}$. These enforce the condition ${\theta }_n{\parallel v_n-v_{n-1}\parallel }^2\le {\epsilon }_n$, where ${\sum }_{n=1}^{\infty }{\epsilon }_n<+\infty$. Since $0\le {\theta }_n\le \overline{\theta }<1$, we can deduce that

$${\theta }_n(1+{\theta }_n){\parallel v_n-v_{n-1}\parallel }^2\le (1+\overline{\theta }){\epsilon }_n.$$

Define ${\delta }_n:={\theta }_n(1+{\theta }_n){\parallel v_n-v_{n-1}\parallel }^2$. Since ${\sum }_{n=1}^{\infty }{\epsilon }_n<\infty$, it follows that ${\sum }_{n=1}^{\infty }{\delta }_n<\infty$. Setting ${\Gamma }_n:={\parallel v_n-g^{*}\parallel }^2$ and discarding the negative terms in (22), we obtain

$${\Gamma }_{n+1}\le {\Gamma }_n+{\theta }_n({\Gamma }_n-{\Gamma }_{n-1})+{\delta }_n.$$

(23)

By applying Lemma 3 to (23), we can conclude that ${lim}_{n\to \infty }\parallel v_n-g^{*}\parallel$ exists. This completes the proof. □

Theorem 1. 

Assume that hypotheses (H1) and (H2) hold. Let $\left\{v_n\right\}$ be the sequence generated by Algorithm 1. Then the sequence $\left\{v_n\right\}$ converges weakly to a point $q^{*}\in {\rm Y}$, that is, $v_n\rightharpoonup q^{*}\in {\rm Y}$.

Proof. 

Let $g^{*}\in {\rm Y}$. By (22), and ${\Gamma }_n:={\parallel v_n-g^{*}\parallel }^2$, we have

$$\begin{array}{c}\hfill {\phi }_n(1-{\omega }_n^2{\mathcal{l}}^2){\parallel u_n-w_n\parallel }^2\le {\Gamma }_n-{\Gamma }_{n+1}+{\theta }_n({\Gamma }_n-{\Gamma }_{n-1})+{\delta }_n.\end{array}$$

(24)

We know that ${\omega }_n\in [a,b]\subseteq (0,1/\mathcal{l})$, which guarantees that $(1-{\omega }_n^2{\mathcal{l}}^2)\ge (1-b^2{\mathcal{l}}^2)>0$. From Lemma 10, we know that $\underset{n\to \infty }{lim}{\Gamma }_n$ exists, then by letting $n\to \infty$ in (24) and using condition () in Algorithm 1, it yields that

$$\underset{n\to \infty }{lim}\parallel u_n-w_n\parallel =0.$$

(25)

On the other hand, it can be observed that

$${∥u_n-v_n∥}^2={\theta }_n^2{∥v_n-v_{n-1}∥}^2\le {\theta }_n(1+{\theta }_n){∥v_n-v_{n-1}∥}^2\le (1+\overline{\theta }){\epsilon }_n.$$

(26)

Since ${\sum }_{n=1}^{\infty }{\epsilon }_n<+\infty$, it implies by (26) that

$$\underset{n\to \infty }{lim}\parallel u_n-v_n\parallel =0.$$

(27)

By employing the triangle inequality, $\parallel v_n-w_n\parallel \le \parallel v_n-u_n\parallel + \parallel u_n-w_n\parallel$. Let $n\to \infty$ and the consequences of (25) and (27), we have

$$\underset{n\to \infty }{lim}\parallel v_n-w_n\parallel =0.$$

(28)

Next, we will show that ${\omega }_w\left(v_n\right)\subseteq {\rm Y}$. By Lemma 10, we have the sequence $\left\{v_n\right\}$ is bounded, and we can verify that ${\omega }_w\left(v_n\right)\ne ⌀$. Let $\widehat{q}\in {\omega }_w\left(v_n\right)$, then there exists a subsequence $\left\{v_{n_k}\right\}\subseteq \left\{v_n\right\}$ such that $v_{n_k}\rightharpoonup \widehat{q}$. By Lemma 9, (25) and (27), we can conclude that $\widehat{q}\in {\rm Y}$, that is ${\omega }_w\left(v_n\right)\subseteq {\rm Y}$. Now, we have shown that

1.

For each $g^{*}\in {\rm Y}$, $\underset{n\to \infty }{lim}∥v_n-g^{*}∥$ exists,

2.

${\omega }_w\left(v_n\right)\subseteq {\rm Y}$.

By Lemma 5, we can conclude that $v_n\rightharpoonup q^{*}\in {\rm Y}$. This completes the proof. □

4. Numerical Experiments

For the numerical experiments, two test images (a flower image and a cat image) captured by the authors using a personal digital camera were used. These images were selected to demonstrate the performance of the proposed algorithm for image restoration under different types of blur and noise. The degraded images were generated by applying blur operators and additive noise, and the restoration performance was evaluated using standard image quality metrics such as ISNR and SSIM. All the experiments were implemented in MATLAB R2019a running on a personal computer with an Intel(R) Core(TM) i7-7700HQ CPU @ 2.80 GHz processor and 20 GB of RAM, running a 64-bit operating system.

Image restoration aims to remove blur artifacts and enhance the quality of degraded images, and has been extensively studied in the literature [28,29,30]. Mathematically, the objective of image restoration is to recover the original image v from a degraded observation u. A commonly used mathematical model describing the relationship between $v\in {\mathbb{R}}^{n\times 1}$ and $u\in {\mathbb{R}}^{m\times 1}$ is given by

$$u=\Pi v+\epsilon ,$$

where $\Pi \in {\mathbb{R}}^{m\times n}$ represents the blur operator, and $\epsilon \in {\mathbb{R}}^{m\times 1}$ denotes the additive noise. The reconstructed image can be obtained by solving the following LASSO (Least Absolute Shrinkage and Selection Operator) problem:

$$\mathrm{find}v\in \underset{v\in {\mathbb{R}}^{n\times 1}}{argmin}\left\{{\displaystyle \frac{1}{2}}{∥\Pi v-u∥}_2^2+\lambda {∥v∥}_1\right\}.$$

(29)

where $\lambda >0$ is a penalty parameter, ${∥·∥}_1$ denotes the ${\mathcal{l}}_1$-norm, and ${∥·∥}_2$ denotes the standard Euclidean norm.

Let $h_1:{\mathbb{R}}^{n\times 1}\to \mathbb{R}$ be defined by $h_1\left(v\right)={\displaystyle \frac{1}{2}}{∥\Pi v-u∥}_2^2,$ and let $h_2:{\mathbb{R}}^{n\times 1}\to \mathbb{R}$ be defined by $h_2\left(v\right)=\lambda {\parallel v\parallel }_1.$ Solving Equation (29) is equivalent to solving a monotone inclusion problem. In particular, the problem corresponds to an image restoration model when we set $\mathcal{P}$ as the gradient of $h_1$, that is, $\mathcal{P}:=\nabla h_1$, and $\mathcal{Q}$ as the subdifferential of $h_2$, that is, $\mathcal{Q}:=\partial h_2$. Hence, the problem can be written as follows:

$$\mathrm{find}v\in {\mathbb{R}}^{n\times 1}\mathrm{such}\mathrm{that}0\in (\mathcal{P}+\mathcal{Q})v,$$

(30)

where

$$\mathcal{P}=\nabla h_1=\nabla \left({\displaystyle \frac{1}{2}}{∥\Pi (·)-u∥}_2^2\right)={\Pi }^T(\Pi (·)-u),$$

where ${\Pi }^{\top }$ is the transpose of $\Pi$ and

$$\mathcal{Q}=\partial h_2\left(v\right)=\left\{p\in \mathcal{H}\left|h_2\left(a\right)\ge h_2\left(v\right)+\langle p,a-v\rangle ,\forall a\in \mathcal{H}\right.\right\}.$$

To quantify the quality of the reconstructed image, we gauge it using the improvement in signal-to-noise ratio (ISNR) for images, which is defined as follows:

$$\mathrm{ISNR}\left(n\right)=10{log}_{10}{\displaystyle \frac{{\parallel v-u\parallel }_2^2}{\parallel v-v_n{\parallel }_2^2}},$$

where v, u and $v_n$ represent the original image, degraded image and the restored image at iteration n, respectively. In addition, we also employ the structural similarity index measure (SSIM) [31] to assess the perceptual quality of the reconstructed images. The test images are original photographs taken by the authors and do not belong to any publicly available dataset. The following table summarizes the parameter configurations for each algorithm.

Table 1. Parameter settings of the algorithms.

Algorithm	Parameters
	$\overline{\mathit{\theta }}$	${\mathit{\epsilon }}_{\mathit{n}}$	${\mathit{\omega }}_{\mathit{n}}$	${\mathit{\phi }}_{\mathit{n}}$
TZQ2020	$0.9000$	$\frac{1000}{n^2}$	$0.5-\frac{150n}{1000n+100}$	–
PKKK2021	$0.9000$	$\frac{1000}{n^2}$	$0.5-\frac{150n}{1000n+100}$	–
Proposed (Algorithm 1)	$0.9000$	$\frac{1000}{n^2}$	$0.5-\frac{150n}{1000n+100}$	$0.999-\left(0.899\right)2^{-n}$

Table 1. Parameter settings of the algorithms.

Algorithm	Parameters
	$\overline{\mathit{\theta }}$	${\mathit{\epsilon }}_{\mathit{n}}$	${\mathit{\omega }}_{\mathit{n}}$	${\mathit{\phi }}_{\mathit{n}}$
TZQ2020	$0.9000$	$\frac{1000}{n^2}$	$0.5-\frac{150n}{1000n+100}$	–
PKKK2021	$0.9000$	$\frac{1000}{n^2}$	$0.5-\frac{150n}{1000n+100}$	–
Proposed (Algorithm 1)	$0.9000$	$\frac{1000}{n^2}$	$0.5-\frac{150n}{1000n+100}$	$0.999-\left(0.899\right)2^{-n}$

Table 2. Experimental settings for image restoration tests.

Image/Setting	Image Size	Type of Blur	Noise Level	Regularization Parameter
Flower	$512\times 512$	Motion $(l=9,\theta =40)$	$N\sim \mathcal{U}{(0,\sigma )}^{512\times 512}$	$\lambda =0.001$
Cat	$512\times 512$	Average $(15\times 15)$	$N\sim \mathcal{U}{(0,\sigma )}^{512\times 512}$	$\lambda =0.001$

Table 2. Experimental settings for image restoration tests.

Image/Setting	Image Size	Type of Blur	Noise Level	Regularization Parameter
Flower	$512\times 512$	Motion $(l=9,\theta =40)$	$N\sim \mathcal{U}{(0,\sigma )}^{512\times 512}$	$\lambda =0.001$
Cat	$512\times 512$	Average $(15\times 15)$	$N\sim \mathcal{U}{(0,\sigma )}^{512\times 512}$	$\lambda =0.001$

Here, $\theta$ denotes the motion direction in degrees measured counter-clockwise. The noise is modeled as $N\sim \mathcal{U}{(0,\sigma )}^{m\times n}$, where $\sigma$ controls the noise level. In this study, $\sigma =0.001$. The notation $\mathcal{U}(0,\sigma )$ represents the uniform distribution on the interval $(0,\sigma )$, meaning that each entry of N is independently drawn from this distribution.

Figure 1. Comparison of image restoration results for the Flower image. The original image (a) is degraded by motion blur (b). Restoration results obtained by TZQ2020 (c), PKKK2021 (d), and the proposed Algorithm 1 (e) are presented for comparison.

Figure 1. Comparison of image restoration results for the Flower image. The original image (a) is degraded by motion blur (b). Restoration results obtained by TZQ2020 (c), PKKK2021 (d), and the proposed Algorithm 1 (e) are presented for comparison.

Figure 2. Flower. Quantitative comparison of ISNR and SSIM for different methods under motion blur. The top row shows ISNR curves (full and zoomed views), while the bottom row shows SSIM curves.

Figure 2. Flower. Quantitative comparison of ISNR and SSIM for different methods under motion blur. The top row shows ISNR curves (full and zoomed views), while the bottom row shows SSIM curves.

Figure 3. Comparison of image restoration results for the cat image. The original image (a) is degraded by average blur (b). Restoration results obtained by TZQ2020 (c), PKKK2021 (d), and the proposed Algorithm 1 (e) are presented for comparison.

Figure 3. Comparison of image restoration results for the cat image. The original image (a) is degraded by average blur (b). Restoration results obtained by TZQ2020 (c), PKKK2021 (d), and the proposed Algorithm 1 (e) are presented for comparison.

Figure 4. Cat. Quantitative comparison of ISNR and SSIM for different methods under average blur. The top row shows ISNR curves (full and zoomed views), while the bottom row shows SSIM curves.

Figure 4. Cat. Quantitative comparison of ISNR and SSIM for different methods under average blur. The top row shows ISNR curves (full and zoomed views), while the bottom row shows SSIM curves.

Table 3. ISNR and SSIM values for the “Flower”image obtained by Algorithm 1, TZQ2020, and PKKK2021 at different iterations.

(k)	ISNR			SSIM
	Algorithm 1	TZQ2020	PKKK2021	Algorithm 1	TZQ2020	PKKK2021
1	−10.7700	−14.1610	−15.7440	0.7669	0.7294	0.6995
10	1.9331	0.8802	−0.6077	0.8530	0.8171	0.8031
50	7.0753	5.4464	6.2996	0.9353	0.9271	0.9280
100	8.4886	5.6962	8.1462	0.9471	0.9472	0.9449
200	9.7794	5.7255	9.6314	0.9532	0.9313	0.9525

Table 3. ISNR and SSIM values for the “Flower”image obtained by Algorithm 1, TZQ2020, and PKKK2021 at different iterations.

(k)	ISNR			SSIM
	Algorithm 1	TZQ2020	PKKK2021	Algorithm 1	TZQ2020	PKKK2021
1	−10.7700	−14.1610	−15.7440	0.7669	0.7294	0.6995
10	1.9331	0.8802	−0.6077	0.8530	0.8171	0.8031
50	7.0753	5.4464	6.2996	0.9353	0.9271	0.9280
100	8.4886	5.6962	8.1462	0.9471	0.9472	0.9449
200	9.7794	5.7255	9.6314	0.9532	0.9313	0.9525

Table 4. ISNR and SSIM values for the “Cat”image obtained by Algorithm 1, TZQ2020, and PKKK2021 at different iterations.

(k)	ISNR			SSIM
	Algorithm 1	TZQ2020	PKKK2021	Algorithm 1	TZQ2020	PKKK2021
1	$-13.3190$	$-16.7810$	$-18.3900$	$0.6283$	$0.5998$	$0.5780$
10	$1.3875$	$0.8198$	$0.2304$	$0.6983$	$0.6804$	$0.6636$
50	$3.0833$	$2.5646$	$2.8072$	$0.7830$	$0.7585$	$0.7691$
100	$3.5384$	$2.6064$	$3.4060$	$0.8044$	$0.7606$	$0.7981$
200	4.0209	$2.6247$	$3.9507$	0.8247	$0.7616$	$0.8218$

Table 4. ISNR and SSIM values for the “Cat”image obtained by Algorithm 1, TZQ2020, and PKKK2021 at different iterations.

(k)	ISNR			SSIM
	Algorithm 1	TZQ2020	PKKK2021	Algorithm 1	TZQ2020	PKKK2021
1	$-13.3190$	$-16.7810$	$-18.3900$	$0.6283$	$0.5998$	$0.5780$
10	$1.3875$	$0.8198$	$0.2304$	$0.6983$	$0.6804$	$0.6636$
50	$3.0833$	$2.5646$	$2.8072$	$0.7830$	$0.7585$	$0.7691$
100	$3.5384$	$2.6064$	$3.4060$	$0.8044$	$0.7606$	$0.7981$
200	4.0209	$2.6247$	$3.9507$	0.8247	$0.7616$	$0.8218$

5. Conclusions

In this paper, we proposed a new inertial forward–backward–forward algorithm (Algorithm 1) combined with a moving point projection technique for solving monotone inclusion problems in real Hilbert spaces. The proposed method integrates the advantages of inertial extrapolation and projection-based techniques, while avoiding the need to compute projections onto the intersection of multiple sets as required in some classical hybrid projection methods. Under suitable assumptions on the operators and control parameters, we established the weak convergence of the sequence generated by Algorithm 1 to a solution of the monotone inclusion problem (3).

To demonstrate the practical performance of the proposed method, numerical experiments were conducted on image restoration problems formulated as LASSO problems (29). The parameter settings and experimental configurations are summarized in Table 1 and Table 2, respectively. The visual restoration results shown in Figure 1 and Figure 3 indicate that Algorithm 1 produces clearer reconstructed images. In addition, the quantitative comparisons reported in Table 3 and Table 4, together with the convergence behavior illustrated in Figure 2 and Figure 4, confirm that the proposed method achieves superior performance in terms of the improvement in signal-to-noise ratio (ISNR) and the structural similarity index measure (SSIM) when compared with the TZQ2020 and PKKK2021 algorithms.

Overall, the proposed algorithm provides an effective and reliable approach for solving monotone inclusion problems and shows strong potential for practical applications in image restoration.