Image Restoration Problems Research Articles

New hybrid conjugate gradient method for nonlinear optimization with application to image restoration problems

New hybrid conjugate gradient method for nonlinear optimization with application to image restoration problems

Modified Double Inertial Extragradient-like Approaches for Convex Bilevel Optimization Problems with VIP and CFPP Constraints

Convex bilevel optimization problems (CBOPs) exhibit a vital impact on the decision-making process under the hierarchical setting when image restoration plays a key role in signal processing and computer vision. In this paper, a modified double inertial extragradient-like approach with a line search procedure is introduced to tackle the CBOP with constraints of the CFPP and VIP, where the CFPP and VIP represent a common fixed point problem and a variational inequality problem, respectively. The strong convergence analysis of the proposed algorithm is discussed under certain mild assumptions, where it constitutes both sections that possess a mutual symmetry structure to a certain extent. As an application, our proposed algorithm is exploited for treating the image restoration problem, i.e., the LASSO problem with the constraints of fractional programming and fixed-point problems. The illustrative instance highlights the specific advantages and potential infect of the our proposed algorithm over the existing algorithms in the literature, particularly in the domain of image restoration.

A New Hybrid Descent Algorithm for Large-Scale Nonconvex Optimization and Application to Some Image Restoration Problems

Conjugate gradient methods are widely used and attractive for large-scale unconstrained smooth optimization problems, with simple computation, low memory requirements, and interesting theoretical information on the features of curvature. Based on the strongly convergent property of the Dai–Yuan method and attractive numerical performance of the Hestenes–Stiefel method, a new hybrid descent conjugate gradient method is proposed in this paper. The proposed method satisfies the sufficient descent property independent of the accuracy of the line search strategies. Under the standard conditions, the trust region property and the global convergence are established, respectively. Numerical results of 61 problems with 9 large-scale dimensions and 46 ill-conditioned matrix problems reveal that the proposed method is more effective, robust, and reliable than the other methods. Additionally, the hybrid method also demonstrates reliable results for some image restoration problems.

Image restoration in frequency space using complex-valued CNNs.

Real-valued convolutional neural networks (RV-CNNs) in the spatial domain have outperformed classical approaches in many image restoration tasks such as image denoising and super-resolution. Fourier analysis of the results produced by these spatial domain models reveals the limitations of these models in properly processing the full frequency spectrum. This lack of complete spectral information can result in missing textural and structural elements. To address this limitation, we explore the potential of complex-valued convolutional neural networks (CV-CNNs) for image restoration tasks. CV-CNNs have shown remarkable performance in tasks such as image classification and segmentation. However, CV-CNNs for image restoration problems in the frequency domain have not been fully investigated to address the aforementioned issues. Here, we propose several novel CV-CNN-based models equipped with complex-valued attention gates for image denoising and super-resolution in the frequency domains. We also show that our CV-CNN-based models outperform their real-valued counterparts for denoising super-resolution structured illumination microscopy (SR-SIM) and conventional image datasets. Furthermore, the experimental results show that our proposed CV-CNN-based models preserve the frequency spectrum better than their real-valued counterparts in the denoising task. Based on these findings, we conclude that CV-CNN-based methods provide a plausible and beneficial deep learning approach for image restoration in the frequency domain.

A novel double inertial viscosity algorithm for convex bilevel optimization problems applied to image restoration problems

Convex bilevel optimization holds a crucial role in the decision-making process in a hierarchical setting, while image restoration plays a critical role in signal processing and computer vision. In this paper, a new double inertial viscosity approximation method with a linesearch technique is introduced for solving convex bilevel optimization problems and then its strong convergence is analyzed under moderate conditions. As an application, our proposed algorithm is employed for solving image restoration problems. The experiment results highlight the specific advantages and potential infect of the algorithm over the existing algorithms in the literature, particularly in the domain of image restoration.

An IDFPM-based algorithm without Lipschitz continuity to constrained nonlinear equations for sparse signal and blurred image restoration problems

The derivative-free projection method (DFPM) is widely used to solve constrained nonlinear equations. To guarantee the convergence of the derivative-free projection method, the mapping should be Lipschitz continuous, which is a strict requirement in theory. Hence, it is interesting to design the new DFPM that possesses nice convergence under weaker theoretical hypothesis. In this paper, we propose an inertial DFPM-based algorithm (named IDFPM), in which the inertial extrapolation step is embedded in the design for the search direction. The global convergence of the proposed algorithm is obtained without the Lipschitz continuity of the mapping. Numerical experiments are carried out for two kinds of problems. The one consists of eight test problems from classical literature and the compressed sensing model. The numerical results show that the proposed algorithm is promising.

Novel Conjugate Gradient Methods for Image Restoration

The image restoration problem can be solved with the right techniques and tools. Although the quality of images may be lowered throughout the data collecting procedure, there are ways to do so. Clarity may be enhanced as a result of the procedure. It is the goal of image restoration to restore the original image from the damaged data. Here are some examples of applications in forensic research, picture restoration, medical imaging, astronomical imaging, and more. It can be advantageous to enhance image quality as much as possible, which justifies the expense and difficulty of the restoration methods. The conjugate gradient method is a numerical optimization technique that places emphasis on the conjugate of the coefficient in order to determine the optimal solution. The current study introduces a novel quadratic function-based coefficient conjugate gradient methodology for the removal of impulse noise from pictures. Global convergence has been shown to occur with the suggested method. The numerical results indicate that the proposed methods have high efficiency and global convergence in image restoration problems compared to the Fletcher-Reevess (FR) method.

A Second-Order Continuous-Time Dynamical System for Solving Sparse Image Restoration Problems

A Second-Order Continuous-Time Dynamical System for Solving Sparse Image Restoration Problems

An improved descent Perry‐type algorithm for large‐scale unconstrained nonconvex problems and applications to image restoration problems

AbstractConjugate gradient methods are much effective for large‐scale unconstrained optimization problems by their simple computations and low memory requirements. The Perry conjugate gradient method has been considered to be one of the most efficient methods in the context of unconstrained minimization. However, a globally convergent result for general functions has not been established yet. In this paper, an improved three‐term Perry‐type algorithm is proposed which automatically satisfies the sufficient descent property independent of the accuracy of line search strategy. Under the standard Wolfe line search technique and a modified secant condition, the proposed algorithm is globally convergent for general nonlinear functions without convexity assumption. Numerical results compared with the Perry method for stability, two modified Perry‐type conjugate gradient methods and two effective three‐term conjugate gradient methods for large‐scale problems up to 300,000 dimensions indicate that the proposed algorithm is more efficient and reliable than the other methods for the testing problems. Additionally, we also apply it to some image restoration problems.

An efficient hybrid conjugate gradient method with an adaptive strategy and applications in image restoration problems

In this study, we introduce a novel hybrid conjugate gradient method with an adaptive strategy called asHCG method. The asHCG method exhibits the following characteristics. (i) Its search direction guarantees sufficient descent property without dependence on any line search. (ii) It possesses strong convergence for the uniformly convex function using a weak Wolfe line search, and under the same line search, it achieves global convergence for the general function. (iii) Employing the Armijo line search, it provides an approximate guarantee for worst-case complexity for the uniformly convex function. The numerical results demonstrate promising and encouraging performances in both unconstrained optimization problems and image restoration problems.

A family of spectral conjugate gradient methods with strong convergence and its applications in image restoration and machine learning

In this paper, we propose a family of spectral conjugate gradient methods for solving unconstrained optimization problems. Specifically, we provide two classes of bounded spectral parameters to be chosen, design a new truncation scheme of the non-negative conjugate parameter and set a restart procedure in our search direction. Independently of the specific spectral parameter, conjugate parameter and line search criterion, we prove that our proposed family satisfies the sufficient descent condition. We also prove its strong convergence under mild assumptions and the weak Wolfe line search. Numerical comparisons with other methods demonstrate the outstanding performances of our algorithm for solving medium–large-scale unconstrained optimization, image restoration and machine learning problems.

Tikhonov regularization with conjugate gradient least squares method for large-scale discrete ill-posed problem in image restoration

Image restoration is a large-scale discrete ill-posed problem, which can be transformed into a Tikhonov regularization problem that can approximate the original image. Kronecker product approximation is introduced into the Tikhonov regularization problem to produce an alternative problem of solving the generalized Sylvester matrix equation, reducing the scale of the image restoration problem. This paper considers solving this alternative problem by applying the conjugate gradient least squares (CGLS) method which has been demonstrated to be efficient and concise. The convergence of the CGLS method is analyzed, and it is demonstrated that the CGLS method converges to the least squares solution within the finite number of iteration steps. The effectiveness and superiority of the CGLS method are verified by numerical tests.

Pre-conditioning CQ algorithm for solving the split feasibility problem and its application to image restoration problem

In this study, we define two pre-conditioning CQ algorithms for the split feasibility problem by using self-adaptive and line-search techniques in the real Hilbert space. One of the self-adaptive techniques is used to obtain optimal step size. These techniques do not require prior knowledge of the operator norm or estimation of the matrix norm. We also establish weak convergence theorems with respect to particular norms. As an application, numerical experiments show that the proposed algorithm is effective in image restoration problem.

Modification of a conjugate gradient approach for convex constrained nonlinear monotone equations with applications in signal recovery and image restoration

Conjugate gradient (CG )methods have received widespread use in recent years for recovering damaged signals in compressive sensing. This study attempts to create a CG method that is more effective than the recent existing scheme for recovering disrupted signals. The newly created method is a modification of the method proposed by Halilu et al. (2021). Because the authors’ CG parameter can be undefined while solving general functions. As a result, to accomplish the desired result, a new CG method combined with the projection scheme of Solodov and Svaiter (1999) is presented. In addition, under mild conditions, the proposed method converges globally. Numerical experiments conducted with recent methods show the effectiveness of the proposed method. Furthermore, the proposed method is applied efficiently to address the signal recovery and image restoration problems, yielding a better result than some existing methods in the literature.

Principal Uncertainty Quantification With Spatial Correlation for Image Restoration Problems.

Uncertainty quantification for inverse problems in imaging has drawn much attention lately. Existing approaches towards this task define uncertainty regions based on probable values per pixel, while ignoring spatial correlations within the image, resulting in an exaggerated volume of uncertainty. In this paper, we propose PUQ (Principal Uncertainty Quantification) - a novel definition and corresponding analysis of uncertainty regions that takes into account spatial relationships within the image, thus providing reduced volume regions. Using recent advancements in generative models, we derive uncertainty intervals around principal components of the empirical posterior distribution, forming an ambiguity region that guarantees the inclusion of true unseen values with a user-defined confidence probability. To improve computational efficiency and interpretability, we also guarantee the recovery of true unseen values using only a few principal directions, resulting in more informative uncertainty regions. Our approach is verified through experiments on image colorization, super-resolution, and inpainting; its effectiveness is shown through comparison to baseline methods, demonstrating significantly tighter uncertainty regions.

Multi-task learning with self-learning weight for image denoising

BackgroundImage denoising technology removes noise from the corrupted image by utilizing different features between image and noise. Convolutional neural network (CNN)-based algorithms have been the concern of the recent progress on diverse image restoration problems and become an efficient solution in image denoising.ObjectiveAlthough a quite number of existing CNN-based image denoising methods perform well on the simplified additive white Gaussian noise (AWGN) model, their performance often degrades severely on the real-world noisy images which are corrupted by more complicated noise.MethodsIn this paper, we utilized the multi-task learning (MTL) framework to integrate multiple loss functions for collaborative training of CNN. This approach aims to improve the denoising performance of CNNs on real-world images with non-Gaussian noise. Simultaneously, to automatically optimize the weights of individual sub-tasks within the MTL framework, we incorporated a self-learning weight layer into the CNN.ResultsExtensive experiments demonstrate that our approach effectively enhances the denoising performance of CNN-based image denoising algorithms on real-world images. It reduces excessive image smoothing, improves quantitative metrics, and enhances visual quality in the restored images.ConclusionOur method shows the effectiveness of the improved performance of denoising CNNS for real-world image denoising processing.

An Efficient Inexact Gauss–Seidel-Based Algorithm for Image Restoration with Mixed Noise

A challenge in image restoration is to recover a clear image from the blurry observation in the presence of different types of noise. There are few works addressing image deblurring under mixed noise. To handle this issue, we propose a general model based on classical wavelet tight frame regularization. We utilize a convexity-preserving term to obtain a component-wise convex model under a mild condition. Indeed, to reduce the cost of solving subproblems, the inexact Gauss–Seidel-based majorized semi-proximal alternating direction method of multipliers (sGS-imsPADMM) with relative error control is developed. Besides, the global convergence of sGS-imsPADMM is demonstrated. Numerical results for the image restoration problems show that the proposed model and solving approach are superior to some state-of-the-art methods both in numerical analysis and visual quality.

Knowledge-driven deep learning for fast MR imaging: Undersampled MR image reconstruction from supervised to un-supervised learning.

Deep learning (DL) has emerged as a leading approach in accelerating MRI. It employs deep neural networks to extract knowledge from available datasets and then applies the trained networks to reconstruct accurate images from limited measurements. Unlike natural image restoration problems, MRI involves physics-based imaging processes, unique data properties, and diverse imaging tasks. This domain knowledge needs to be integrated with data-driven approaches. Our review will introduce the significant challenges faced by such knowledge-driven DL approaches in the context of fast MRI along with several notable solutions, which include learning neural networks and addressing different imaging application scenarios. The traits and trends of these techniques have also been given which have shifted from supervised learning to semi-supervised learning, and finally, to unsupervised learning methods. In addition, MR vendors' choices of DL reconstruction have been provided along with some discussions on open questions and future directions, which are critical for the reliable imaging systems.

Pretraining a foundation model for generalizable fluorescence microscopy-based image restoration.

Fluorescence microscopy-based image restoration has received widespread attention in the life sciences and has led to significant progress, benefiting from deep learning technology. However, most current task-specific methods have limited generalizability to different fluorescence microscopy-based image restoration problems. Here, we seek to improve generalizability and explore the potential of applying a pretrained foundation model to fluorescence microscopy-based image restoration. We provide a universal fluorescence microscopy-based image restoration (UniFMIR) model to address different restoration problems, and show that UniFMIR offers higher image restoration precision, better generalization and increased versatility. Demonstrations on five tasks and 14 datasets covering a wide range of microscopy imaging modalities and biological samples demonstrate that the pretrained UniFMIR can effectively transfer knowledge to a specific situation via fine-tuning, uncover clear nanoscale biomolecular structures and facilitate high-quality imaging. This work has the potential to inspire and trigger new research highlights for fluorescence microscopy-based image restoration.

Impulse Noise Image Restoration Using Nonconvex Variational Model and Difference of Convex Functions Algorithm.

In this article, the problem of impulse noise image restoration is investigated. A typical way to eliminate impulse noise is to use an L1 norm data fitting term and a total variation (TV) regularization. However, a convex optimization method designed in this way always yields staircase artifacts. In addition, the L1 norm fitting term tends to penalize corrupted and noise-free data equally, and is not robust to impulse noise. In order to seek a solution of high recovery quality, we propose a new variational model that integrates the nonconvex data fitting term and the nonconvex TV regularization. The usage of the nonconvex TV regularizer helps to eliminate the staircase artifacts. Moreover, the nonconvex fidelity term can detect impulse noise effectively in the way that it is enforced when the observed data is slightly corrupted, while is less enforced for the severely corrupted pixels. A novel difference of convex functions algorithm is also developed to solve the variational model. Using the variational method, we prove that the sequence generated by the proposed algorithm converges to a stationary point of the nonconvex objective function. Experimental results show that our proposed algorithm is efficient and compares favorably with state-of-the-art methods.