Image Restoration Problems Research Articles

The global convergence of spectral RMIL conjugate gradient method for unconstrained optimization with applications to robotic model and image recovery.

In 2012, Rivaie et al. introduced RMIL conjugate gradient (CG) method which is globally convergent under the exact line search. Later, Dai (2016) pointed out abnormality in the convergence result and thus, imposed certain restricted RMIL CG parameter as a remedy. In this paper, we suggest an efficient RMIL spectral CG method. The remarkable feature of this method is that, the convergence result is free from additional condition usually imposed on RMIL. Subsequently, the search direction is sufficiently descent independent of any line search technique. Thus, numerical experiments on some set of benchmark problems indicate that the method is promising and efficient. Furthermore, the efficiency of the proposed method is demonstrated on applications arising from arm robotic model and image restoration problems.

Modified accelerated Bregman projection methods for solving quasi-monotone variational inequalities

In this paper, we introduce three new inertial-like Bregman projection methods with a nonmonotone adaptive step-size for solving quasi-monotone variational inequalities in real Hilbert spaces. Under some suitable conditions, the weak convergence of these methods is proved without the prior knowledge of the Lipschitz constant of the operator and the strong convergence of some proposed methods under a strong quasi-monotonicity assumption of the mapping is also provided. Finally, several numerical experiments and applications in image restoration problems are provided to illustrate the performance of the proposed methods.

An Accelerated Three-Term Extension of a Descent Nonlinear Conjugate Gradient Method

In an effort to make modification on the classical Polak–Ribière–Polyak method, Wei et al. suggested an efficient nonlinear conjugate gradient method which possesses the sufficient descent property when the line search fulfills the strong Wolfe conditions (by restricting the line search parameters). Here, we develop a three-term extension of the method which satisfies the sufficient descent condition independent of the line search. Also, under a backtracking Armijo-type line search, we establish global convergence of the method without convexity assumption. Using a scalar approximation of the Hessian of the objective function, we suggest an acceleration scheme that can be used in the iterative line search methods of unconstrained optimization. At last, practical merits of the proposed method are investigated by numerical experiments on a set of CUTEr test functions as well as the well-known image restoration problem. The results show numerical efficiency of the method.

A modified PRP-type conjugate gradient algorithm with complexity analysis and its application to image restoration problems

In this paper, a modified conjugate gradient method is proposed for nonconvex optimization. This method possesses the sufficient descent property independent of any line search. The global convergence property of the algorithm is established under the Wolfe line search strategy or the Armijo line search condition, respectively. Additionally, the complexity analysis of the proposed algorithm is investigated. To reach the point with the norm of the gradient below ɛ, the worst-case complexity bound matches that of the gradient method. Moreover, the obtained numerical results demonstrate that the modified method is effective for large-scale optimization problems and image restoration problems.

Structure preserving quaternion full orthogonalization method with applications

AbstractThis article proposes a structure‐preserving quaternion full orthogonalization method (QFOM) for solving quaternion linear systems arising from color image restoration. The method is based on the quaternion Arnoldi procedure preserving the quaternion Hessenberg form. Combining with the preconditioning techniques, we further derive a variant of the QFOM for solving the linear systems, which can greatly improve the rate of convergence of QFOM. Numerical experiments on randomly generated data and color image restoration problems illustrate the effectiveness of the proposed algorithms in comparison with some existing methods.

S-Iteration inertial subgradient extragradient method for variational inequality and fixed point problems

In developing iterative methods for approximating solutions of optimization problems, one of the major goals of researchers is to construct efficient iterative schemes. Over the years, different techniques have been devised by authors to achieve this goal. In this paper, we study a non-Lipschitz pseudomonotone variational inequality problem with common fixed points constraint of Bregman quasi-nonexpansive mappings. We introduce a new iterative method for approximating the solution of this problem in a more general framework of reflexive Banach spaces. Our method employs several techniques to achieve high level of efficiency. One of the techniques employed is the S-iteration process, a method known to be highly efficient in comparison with several of the well-known methods. Moreover, our proposed algorithm utilizes the inertial technique and a non-monotonic self-adaptive step size to guarantee high rate of convergence and easy implementation. Unlike several of the existing results on variational inequality problem with non-Lipschitz operator, the design of our method does not involve any linesearch technique. We obtain strong convergence result for the proposed algorithm without the sequentially weakly continuity condition often assumed by authors to guarantee convergence when solving pseudomonotone variational inequality problems. Furthermore, we apply our result to study utility-based bandwidth allocation problem and image restoration problem. Finally, we present several numerical experiments to demonstrate the efficiency of our proposed method in comparison with existing state-of-the-art methods. Our result extends and improves several of the recently announced results in this direction in the literature.

Two families of self-adjusting spectral hybrid DL conjugate gradient methods and applications in image denoising

The study in this paper introduces two families of self-adjusting spectral hybrid DL conjugate gradient methods to solve unconstrained optimization problems. The search directions in the proposed methods are improved by integrating the negative spectral gradient and the last search direction that has convex combination structure. One component of composite conjugate parameter in the search direction is a hybridization of DL-like conjugate parameter and another flexible conjugate parameter. It is proved that the search directions in the two families of newly proposed methods always automatically satisfy the sufficient descent property, which is independent of choosing the specific conjugate parameter and line search. With general nonconvex objective functions, the convergence results under the weak Wolfe line search are obtained. Numerical experiments verify the efficiency and applicability of our developed methods by comparison with some existing methods for solving unconstrained optimization and image restoration problems.

Inertial projection and contraction methods for solving variational inequalities with applications to image restoration problems

In this paper, we introduce two inertial self-adaptive projection and contraction methods for solving the pseudomonotone variational inequality problem with a Lipschitz-continuous mapping in real Hilbert spaces. The adaptive stepsizes provided by the algorithms are simple to update and their computations are more efficient and flexible. Also we prove some weak and strong convergence theorems without prior knowledge of the Lipschitz constant of the mapping. Finally, we present some numerical experiments to demonstrate the effectiveness of the proposed algorithms by comparisons with related methods and some applications of the proposed algorithms to the image deblurring problem.

An improved conjugate gradient method for solving unconstrained optimisation and image restoration problems

An improved conjugate gradient method for solving unconstrained optimisation and image restoration problems

Conditional deep learning approach for the Helsinki deblur challenge 2021

Image captured by an optical lens-based imaging device usually has some part that appears out of focus and blurry. Such degradation is known as defocus blur. It happens when the object being captured is too far away from (or close to) the camera lens. Blurring removal is also helpful for other down-stream tasks, such as optical character recognition (OCR). Nowadays the deep learning is a prominent solution to practical image deblurring using supervised learning from a paired dataset. The Helsinki Deblur Challenge 2021 (HDC2021) provided the paired images taken by two identical cameras of the same target but with different conditions. The paper details the 2nd place solution to the challenge with discussion. Although the solution was designed to participate with the challenge, the two-stage framework and the way of incorporating the blur level information into the network architecture can be applied to other image restoration problems.

An Improved Conjugate Gradient Method for Solving Unconstrained Optimization and Image Restoration Problems

An Improved Conjugate Gradient Method for Solving Unconstrained Optimization and Image Restoration Problems

Iterative manner involving sunny nonexpansive retractions for nonlinear operators from the perspective of convex programming as applicable to differential problems, image restoration and signal recovery

<abstract><p>In this paper, using sunny nonexpansive retractions which are different from the metric projection in Banach spaces, we develop the $ CR $-iteration algorithm in view of two quasi-nonexpansive nonself mappings and also give the convergence analysis for the proposed method in the setting of uniformly convex Banach spaces. Furthermore, our results can be applied for the purpose of finding common zeros of accretive operators, convexly constrained least square problems and convex minimization problems. Regarding application, some numerical experiments involving real-world problems are provided, with focus on differential problems, image restoration problems and signal recovery problems.</p></abstract>

Retinex-Based Variational Framework for Low-Light Image Enhancement and Denoising

Low-light image enhancement is an important task in the domain of computer vision. Images taken under insufficient lighting conditions manifest low visibility and unknown noises which disrupt image contents and pose considerable challenges for low-light image enhancement. Most of Retinex-based methods usually attempt to design different priors on the gradient of both illumination and reflectance. However, noises can be involved in the Retinex-based models. To address the problem, we explore the problem of low-light image restoration through joint contrast enhancement and denoising. We propose a Retinex-based variational model for low-light image enhancement that effectively generates a noise-free image, yet proves to generalize well to diverse light-conditions. First, we present a simple constraint on the fidelity term between the fractional derivative of an observed image and the fractional derivative of the recomposed one which is the product of the reflectance and illumination. This strategy aims to model spatial consistency to preserve natural variation. Second, we introduce a weighted regularization term for the reflectance that can remove noise with a adaptive texture map. We evaluate our proposed approach using three challenging datasets: NPE, LOL and GladNet. Extensive experiments demonstrate that our proposed method outperforms other competing methods in terms of visual quality and quantitative comparisons.

Real-World Image Super-Resolution by Exclusionary Dual-Learning

Real-world image super-resolution is a practical image restoration problem that aims to obtain high-quality images from in-the-wild input, has recently received considerable attention with regard to its tremendous application potentials. Although deep learning-based methods have achieved promising restoration quality on real-world image super-resolution datasets, they ignore the relationship between L1- and perceptual- minimization and roughly adopt auxiliary large-scale datasets for pre-training. In this paper, we discuss the image types within a corrupted image and the property of perceptual- and Euclidean- based evaluation protocols. Then we propose a method, Real-World image Super-Resolution by Exclusionary Dual-Learning (RWSR-EDL) to address the feature diversity in perceptual- and L1- based cooperative learning. Moreover, a noise-guidance data collection strategy is developed to address the training time consumption in multiple datasets optimization. When an auxiliary dataset is incorporated, RWSR-EDL achieves promising results and repulses any training time increment by adopting the noise-guidance data collection strategy. Extensive experiments show that RWSR-EDL achieves competitive performance over state-of-the-art methods on four in-the-wild image super-resolution datasets.

Image restoration by using a modified proximal point algorithm

<abstract><p>In this paper, we establish a modified proximal point algorithm for solving the common problem between convex constrained minimization and modified variational inclusion problems. The proposed algorithm base on the proximal point algorithm in <sup>[<xref ref-type="bibr" rid="b19">19</xref>]</sup> and the method of Khuangsatung and Kangtunyakarn in <sup>[<xref ref-type="bibr" rid="b21">21</xref>]</sup> by using suitable conditions in Hilbert spaces. The proposed algorithm is not only presented in this article; however has also been demonstrated to generate a robust convergence theorem. The proposed algorithm could be used to solve image restoration problems where the images have suffered a variety of blurring operations. Additionally, we contrast the signal-to-noise ratio (SNR) of the proposed algorithm against that of Khuangsatung and Kangtunyakarn's method in <sup>[<xref ref-type="bibr" rid="b21">21</xref>]</sup> in order to compare image quality.</p></abstract>

Convex minimization problems based on an accelerated fixed point algorithm with applications to image restoration problems

Convex minimization problems based on an accelerated fixed point algorithm with applications to image restoration problems

AN ACCELERATED HALPERN-TYPE ALGORITHM FOR SOLVING VARIATIONAL INCLUSION PROBLEMS WITH APPLICATIONS

An inertial Halpern-type forward-backward iterative algorithm for approximating a zero of sum of two accretive operators is introduced and studied. Strong convergence theorem is established in a uniformly convex and q-uniformly smooth real Banach space. The convergence result obtained is applied to convex minimization and image restoration problems. Furthermore, numerical experiments are carried out on some classical test images and personal images degraded with motion blur and random noise. Finally, numerical illustrations in the Banach space, L5([−1, 1]) are presented to support the main theorem.

A fast inertial primal–dual algorithm to composite optimization models with application to image restoration problems

In this paper, we consider a fast inertial primal–dual algorithm (FIPD) for finding minimization problems of the sum of a smooth function with Lipschitzian gradient, a non-smooth proximable function, and linear composite functions. We not only prove the convergence of the proposed algorithm, but also prove that the new algorithm can achieve the worst-case o(1/k2) optimal convergence rate in terms of objective function value. This work brings together and notably extends several classical splitting schemes, like the primal–dual splitting method (PDS) proposed by Condat, the algorithm which is presented by Chambolle and Pock, as well as the recent modified fast iterative shrinkage-thresholding algorithm. The efficiency of the proposed method is demonstrated on image denoising and image deblurring. Numerical results show that our iterative algorithm (FIPD) performs better than the original one (PDS).

A three-term conjugate gradient algorithm with restart procedure to solve image restoration problems

Based on the Liu–Storey conjugate gradient method, a three-term conjugate gradient method with restart procedure is proposed for unconstrained optimization. The designed restart condition is chosen according to the Liu–Storey conjugate parameter, and the restart direction consists of the first and third items in the non-restart direction, which can be picked flexibly. Without depending on any line search, the resulting direction satisfies the sufficient descent condition. Moreover, the proposed method possesses the global convergence under usual assumptions and the weak Wolfe line search. Finally, medium–large-scale numerical experiments are conducted for solving unconstrained optimization and image restoration problems, and the numerical results show that the proposed method is promising and practical, even compared with the state-of-the-art methods.

An Algorithm for Solving Common Points of Convex Minimization Problems with Applications

In algorithm development, symmetry plays a vital part in managing optimization problems in scientific models. The aim of this work is to propose a new accelerated method for finding a common point of convex minimization problems and then use the fixed point of the forward-backward operator to explain and analyze a weak convergence result of the proposed algorithm in real Hilbert spaces under certain conditions. As applications, we demonstrate the suggested method for solving image inpainting and image restoration problems.