An alternating inertial forward-backward-forward algorithm for solving monotone inclusion problem and its applications

<abstract><p>In this article, we propose a strongly convergent preconditioning method for finding a zero of the sum of two monotone operators. The proposed method combines a preconditioning approach with the robustness of the Krasnosel'skiĭ-Mann method. We show the strong convergence result of the sequence generated by the proposed method to a solution of the monotone inclusion problem. Finally, we provide numerical experiments on the convex minimization problem.</p></abstract>

Introduction: Tremendous developments in multimedia technology have promoted a massive amount of research in image and video processing. As imaging technologies are rapidly increasing, it is becoming essential to use images in almost every application in our day-today life. Materials and Methods: This paper presents a comparative analysis of various image restoration approaches, ranging from fundamental methods to advanced techniques. These approaches aim to improve the quality of images that have been degraded during acquisition or transmission. A brief overview of the image restoration approaches is mentioned in the paper, which are as follows: Wiener Filter: The Wiener filter is a classical approach used for image restoration. It is a linear filter that minimizes the mean square error between the original image and the restored image. Inverse Filter: The inverse filter is another traditional restoration technique. It attempts to invert the degradation process to recover the original image. However, inverse filtering is highly sensitive to noise and tends to amplify noise artifacts. Linear and Nonlinear Filtering: These methods involve applying linear or nonlinear filters to the degraded image to enhance its quality. Linear filters, such as Gaussian filters, can effectively reduce noise but may blur the image. Nonlinear filters, such as median filters, can preserve edges while reducing noise. Compressive Sensing (CS) Restoration Approaches: Compressive sensing is a signal processing technique that exploits the sparsity of signals or images to reconstruct them from fewer measurements. CS-based restoration methods aim to recover high-quality images from compressed or incomplete measurements. Neural Networks Approaches: With the advancements in deep learning, neural networks have been widely used for image restoration tasks. Convolutional neural networks (CNNs) and generative adversarial networks (GANs) have shown promising results in restoring degraded images by learning from large datasets. Results: The paper likely provides a detailed analysis and comparison of these approaches, highlighting their strengths, weaknesses, and performance in different scenarios Conclusion: This paper aims to improve the quality of restored images. Several image restoration approaches have been compared, and they have exhibited enhanced performance compared to several existing image restoration approaches.

In this article, we establish a new Mann-type method combining both inertial terms and errors to find a fixed point of a nonexpansive mapping in a Hilbert space. We show strong convergence of the iterate under some appropriate assumptions in order to find a solution to an investigative fixed point problem. For the virtue of the main theorem, it can be applied to an approximately zero point of the sum of three monotone operators. We compare the convergent performance of our proposed method, the Mann-type algorithm without both inertial terms and errors, and the Halpern-type algorithm in convex minimization problem with the constraint of a non-zero asymmetric linear transformation. Finally, we illustrate the functionality of the algorithm through numerical experiments addressing image restoration problems.

A new approach for image restoration by cellular neural network (CNN) is developed in this paper. Based on the statistical characteristics of Gibbs image model and the analysis of maximum entropy (ME) image restoration, a reasonable template for binary image restoration is proposed. To process multilevel image, a multi-layer cellular neural network is employed and an extensive algorithm for multilevel image restoration is proposed. The results of computer simulation prove the effectiveness of this approach and show that we can get the effective template of CNN for some special image questions if we apply the statistical characteristics of Gibbs image model and analyse the physical meaning of the questions. Copyright © 1999 John Wiley & Sons, Ltd.

In this paper, a Kalman filter-based approach for adaptive restoration of video images acquired by an in-vehicle camera in foggy conditions is proposed. In order to realize Kalman filter-based restoration, the proposed method regards the intensities in each frame as elements of the state variable of the Kalman filter and designs the following two models for restoration of foggy images. The first one is an observation model, which represents a fog deterioration model. The second one is a non-linear state transition model, which represents the target frame in the original video image from its previous frame based on motion vectors. By utilizing the observation and state transition models, the correlation between successive frames can be effectively utilized for restoration. Further, the proposed method introduces a new estimation scheme of the parameter, which determines the deterioration characteristic in foggy conditions, into the Kalman filter algorithm. Consequently, since automatic determination of the fog deterioration model, which specifies the observation model, from only the foggy images is realized, the accurate restoration can be achieved. Experimental results show that the proposed method has better performance than that of the traditional method based on the fog deterioration model.

We introduce a framework based on Rockafellar’s perturbation theory to analyze and solve general nonsmooth convex minimization and monotone inclusion problems involving nonlinearly composed functions as well as linear compositions. Such problems have been investigated only from a primal perspective and only for nonlinear compositions of smooth functions in finite-dimensional spaces in the absence of linear compositions. In the context of Banach spaces, the proposed perturbation analysis serves as a foundation for the construction of a dual problem and of a maximally monotone Kuhn–Tucker operator, which is decomposable as the sum of simpler monotone operators. In the Hilbertian setting, this decomposition leads to a block-iterative primal-dual algorithm that fully splits all the components of the problem and appears to be the first proximal splitting algorithm for handling nonlinear composite problems. Various applications are discussed. Funding: The work of L. M. Briceño-Arias was supported by Agencia Nacional de Investigación y Desarrollo-Chile [Grant Fondo Nacional de Desarrollo Científico y Tecnológico 1190871, Grant Centro de Modelamiento Matemático ACE210010, Grant Centro de Modelamiento Matemático FB210005, and basal Funds for Centers of Excellence], and the work of P. L. Combettes was supported by the National Science Foundation [Grant DMS-1818946].

Reconstruction of depth from 2D images is an important research issue in computer vision. Depth from defocus (DFD) technique uses space varying blurring of an image as a cue in reconstructing the 3D structure of a scene. In this paper we explore the regularization based approach for simultaneous estimation of depth and image restoration from defocused observations. We are given two defocused observations of a scene that are captured with different camera parameters. Our method consists of two steps. First we obtain the initial estimates for the depth as well as for the focused image. In the second step we refine the solution by using a fast optimization technique. Here we use the classic depth recovery method due to Subbarao for obtaining the initial depth map andWeiner filter approach for initial image restoration. Since the problem we are solving is ill-posed and does not yield unique solution, it is necessary to regularize the solution by imposing additional constraint to restrict the solution space. The regularization is performed by imposing smoothness constraint only. However, for preserving the depth and image intensity discontinuities, they are identified prior to the minimization process from initial estimates of the depth map and the restored image. The final solution is obtained by using computationally efficient gradient descent algorithm, thus avoiding the need for computationally taxing algorithms. The depth as well as intensity edge details of the final solution correspond to those obtained using the initial estimates. The experimental results indicate that the quality of the restored image is good even under severe space-varying blur conditions.

In this article, the problem of solving a strongly monotone variational inequality problem over the solution set of a monotone inclusion problem in the setting of real Hilbert spaces is considered. To solve this problem, two methods, which are improvements and modifications of the Tseng splitting method, and projection and contraction methods, are presented. These methods are equipped with inertial terms to improve their speed of convergence. The strong convergence results of the suggested methods are proved under some standard assumptions on the control parameters. Also, strong convergence results are achieved without prior knowledge of the operator norm. Finally, the main results of this research are applied to solve bilevel variational inequality problems, convex minimization problems, and image recovery problems. Some numerical experiments to show the efficiency of our methods are conducted.

Through exploiting the image nonlocal self-similarity (NSS) prior by clustering similar patches to construct patch groups, recent studies have revealed that structural sparse representation (SSR) models can achieve promising performance in various image restoration tasks. However, most existing SSR methods only exploit the NSS prior from the input degraded (internal) image, and few methods utilize the NSS prior from external clean image corpus; how to jointly exploit the NSS priors of internal image and external clean image corpus is still an open problem. In this paper, we propose a novel approach for image restoration by simultaneously considering internal and external nonlocal self-similarity (SNSS) priors that offer mutually complementary information. Specifically, we first group nonlocal similar patches from images of a training corpus. Then a group-based Gaussian mixture model (GMM) learning algorithm is applied to learn an external NSS prior. We exploit the SSR model by integrating the NSS priors of both internal and external image data. An alternating minimization with an adaptive parameter adjusting strategy is developed to solve the proposed SNSS-based image restoration problems, which makes the entire algorithm more stable and practical. Experimental results on three image restoration applications, namely image denoising, deblocking and deblurring, demonstrate that the proposed SNSS produces superior results compared to many popular or state-of-the-art methods in both objective and perceptual quality measurements.

In this paper, we consider error estimation for image restoration problems based on generalized Bregman distances. This error estimation technique has been used to derive convergence rates of variational regularization schemes for linear and nonlinear inverse problems by the authors before (cf. Burger in Inverse Prob 20: 1411–1421, 2004; Resmerita in Inverse Prob 21: 1303–1314, 2005; Inverse Prob 22: 801–814, 2006), but so far it was not applied to image restoration in a systematic way. Due to the flexibility of the Bregman distances, this approach is particularly attractive for imaging tasks, where often singular energies (non-differentiable, not strictly convex) are used to achieve certain tasks such as preservation of edges. Besides the discussion of the variational image restoration schemes, our main goal in this paper is to extend the error estimation approach to iterative regularization schemes (and time-continuous flows) that have emerged recently as multiscale restoration techniques and could improve some shortcomings of the variational schemes. We derive error estimates between the iterates and the exact image both in the case of clean and noisy data, the latter also giving indications on the choice of termination criteria. The error estimates are applied to various image restoration approaches such as denoising and decomposition by total variation and wavelet methods. We shall see that interesting results for various restoration approaches can be deduced from our general results by just exploring the structure of subgradients.

Noise affects the recovery effect. So this paper proposed an improved method for image restoration. In this method, first of degraded images using wavelet transform to remove noise while fuzzy POCS algorithm to measure the reliability of the wavelet coefficients, so while eliminating noise can incorporate more reliable information. Experimental results show that the proposed approach in image restoration can effectively eliminate noise and improve image clarity. Keywords-wavelet transform; fuzzy POCS; image restoration; reliability

We introduce a penalty term-based splitting algorithm with inertial effects designed for solving monotone inclusion problems involving the sum of maximally monotone operators and the convex normal cone to the (nonempty) set of zeros of a monotone and Lipschitz continuous operator. We show weak ergodic convergence of the generated sequence of iterates to a solution of the monotone inclusion problem, provided a condition expressed via the Fitzpatrick function of the operator describing the underlying set of the normal cone is verified. Under strong monotonicity assumptions we can even show strong nonergodic convergence of the iterates. This approach constitutes the starting point for investigating from a similar perspective monotone inclusion problems involving linear compositions of parallel-sum operators and, further, for the minimization of a complexly structured convex objective function subject to the set of minima of another convex and differentiable function.

We introduce a new iterative method for finding a common element of the set of solutions of an equilibrium problem and the set of zeros of the sum of maximal monotone operators, and we obtain strong convergence theorems in Hilbert spaces. We also apply our results to the variational inequality and convex minimization problems. Our results extend and improve the recent result of Takahashi et al. (2012).

Sparse coding (SC) models have been proven as powerful tools applied in image restoration tasks, such as patch sparse coding (PSC) and group sparse coding (GSC). However, these two kinds of SC models have their respective drawbacks. PSC tends to generate visually annoying blocking artifacts, while GSC models usually produce over-smooth effects. Moreover, conventional minimization-based convex regularization was usually employed as a standard scheme for estimating sparse signals, but it cannot achieve an accurate sparse solution under many realistic situations. In this paper, we propose a novel approach for image restoration via simultaneous patch-group sparse coding (SPG-SC) with dual-weighted minimization. Specifically, in contrast to existing SC-based methods, the proposed SPG-SC conducts the local sparsity and nonlocal sparse representation simultaneously. A dual-weighted minimization-based non-convex regularization is proposed to improve the sparse representation capability of the proposed SPG-SC. To make the optimization tractable, a non-convex generalized iteration shrinkage algorithm based on the alternating direction method of multipliers (ADMM) framework is developed to solve the proposed SPG-SC model. Extensive experimental results on two image restoration tasks, including image inpainting and image deblurring, demonstrate that the proposed SPG-SC outperforms many state-of-the-art algorithms in terms of both objective and perceptual quality.

A novel anisotropic diffusion-based image denoising and restoration approach is proposed in this paper. A variational model for image restoration is introduced first, then the corresponding Euler-Lagrange equation being determined. A nonlinear parabolic PDE model is then obtained from this equation. It is based on a novel edge-stopping function and conductance parameter. A serious mathematical treatment is performed on this second-order anisotropic diffusion scheme, its well-possedness being investigated. Then, a consistent explicit numerical approximation scheme based on the finite difference method is developed for the proposed PDE model.