Color Image Restoration Research Articles

Quaternion-Aware Low-Rank Prior for Blind Color Image Deblurring

Blind image deblurring is a critical and challenging task in the field of imaging science due to its severe ill-posedness. Appropriate prior information and regularizations are normally introduced to alleviate this problem. Inspired by the fact that the matrix representing a natural image is intrinsically low-rank or approximately low-rank, we employ the low-rank matrix approximation (LRMA) approach for tackling blind image deblurring problems with unknown kernels. When applied to color image restoration tasks, making use of the quaternion representation in the hypercomplex domain enables us to better illustrate the inner relationships among color channels and thus more accurately characterize color image structure. Following this idea, we develop a novel model for color image blind deblurring by implementing the quaternion representation to the LRMA method. This proposed model facilitates better results for blur kernel estimation through preserving the sharper color intermediate latent image, which is first implemented for addressing the blind color image deblurring problem. Extensive numerical experiments demonstrate that our proposed quaternion-aware low-rank prior model greatly improves the performance when compared with the conventional low-rank based scheme and outperforms some of the state-of-the-art methods in terms of some criteria and visual quality.

Quaternion Nuclear Norm Minus Frobenius Norm Minimization for color image reconstruction

Color image restoration methods typically represent images as vectors in Euclidean space or combinations of three monochrome channels. However, they often overlook the correlation between these channels, leading to color distortion and artifacts in the reconstructed image. To address this, we present Quaternion Nuclear Norm Minus Frobenius Norm Minimization (QNMF), a novel approach for color image reconstruction. QNMF utilizes quaternion algebra to capture the relationships among RGB channels comprehensively. By employing a regularization technique that involves nuclear norm minus Frobenius norm, QNMF approximates the underlying low-rank structure of quaternion-encoded color images. Theoretical proofs are provided to ensure the method’s mathematical integrity. Demonstrating versatility and efficacy, the QNMF regularizer excels in various color low-level vision tasks, including denoising, deblurring, inpainting, and random impulse noise removal, achieving state-of-the-art results.

Color image restoration by filtering methods: a review

Digital images are corrupted with noise, and image denoising is an important step in image processing modules. In this review, the latest developments in filtering methods for color image restoration are analyzed. These algorithms are compared in terms of objective image quality measures and divided into major classes, such as spatial domain, switching and wavelet filtering methods. These classes are based on the particular methodology used in image denoising algorithms and further subdivided to show their classification in terms of noise models utilized, application style, and stages the filters applied in images. In particular, we present a review of filtering methods in color image denoising, published over the past two decades. Our classification and succinct descriptions of color image restoration by these mathematical filtering techniques and their characterizations can help choose the appropriate ones for various downstream image processing tasks.

Quaternion weighted Schatten [formula omitted]-norm minimization for color image restoration with convergence guarantee

Over the past decades, the rank approximation issue have been extensively investigated, among which weighted nuclear norm minimization (WNNM) and weighted Schatten p-norm minimization (WSNM) are two prevailing methods and have shown great superiority in various image restoration (IR) problems. However, for the complicated color image restoration (CIR) problems, traditional WNNM/WSNM method only processes three color channels individually and fails to consider their cross-channel correlations. Very recently, a quaternion-based WNNM approach (QWNNM) has been developed to mitigate this issue, which is capable of representing the color image as a whole in the quaternion domain. Despite QWNNM’s empirical success, its convergence behavior has not been rigorously studied. The main contributions of this paper are twofold. Firstly, we extend the WSNM into quaternion domain and correspondingly propose a novel quaternion-based WSNM model (QWSNM) for tackling the CIR problems. Extensive experiments on two representative CIR tasks, including color image denoising and deblurring, demonstrate that the proposed QWSNM method performs favorably against many state-of-the-art alternatives, in both quantitative and qualitative evaluations. Secondly, we provide a preliminary theoretical convergence analysis. By modifying the quaternion alternating direction method of multipliers (QADMM) through a simple continuation strategy, we theoretically prove the fixed-point convergence property of the iterative sequences generated by QWNNM and QWSNM. The source code of our algorithm can be available at the website: https://github.com/qiuxuanzhizi/QWSNM.

Convex and non-convex adaptive TV regularizations for color image restoration

Convex and non-convex adaptive TV regularizations for color image restoration

Color restoration of images through high order zeroing neural networks

One of the fundamental tasks in pattern recognition is color image restoration. Every color image has three channels in the RGB color space, in contrast to grayscale images. The restoration of color images is typically far more challenging than that of grayscale images because of the internal relationships among the three channels. Given that the color image restoration can be represented as a dynamic problem with quaternion matrices, a new high order zeroing neural network (HZNN) model is developed to tackle this issue. Specifically, the time-varying quaternion matrix linear equations can be solved using the HZNN design, which is a member of the family of zeroing neural network (ZNN) models that correlate to hyperpower iterative techniques. In a realistic color image restoration application, the HZNN design outperforms the ZNN design, although both approaches work amazingly well.

Group sparse representation and saturation-value total variation based color image denoising under multiplicative noise

<abstract><p>In this article, we propose a novel group-based sparse representation (GSR) model for restoring color images in the presence of multiplicative noise. This model consists of a convex data-fidelity term, and two regularizations including GSR and saturation-value-based total variation (SVTV). The data-fidelity term is suitable for handling heavy multiplicative noise. GSR enables the retention of textures and details while sufficiently removing noise in smooth regions without producing the staircase artifacts engendered by total variation-based models. Furthermore, we introduce a multi-color channel-based GSR that involves coupling between three color channels. This avoids the generation of color artifacts caused by decoupled color channel-based methods. SVTV further improves the visual quality of restored images by diminishing certain artifacts induced by patch-based methods. To solve the proposed nonconvex model and its subproblem, we exploit the alternating direction method of multipliers, which contributes to an efficient iterative algorithm. Numerical results demonstrate the outstanding performance of the proposed model compared to other existing models regarding visual aspect and image quality evaluation values.</p></abstract>

TWO ALGEBRAIC ALGORITHMS FOR THE LU DECOMPOSITION OF COMMUTATIVE QUATERNION MATRICES AND THEIR APPLICATIONS

The paper proposes two algebraic algorithms for the LU decomposition of com- mutative quaternion matrices. The first of them is an algebraic algorithm based on a complex representation matrix, and the second is a complex structure-preserving algorithm based on Gaussian elimination of commutative quaternion matrices. With their help, it was established that the LU decomposition of commutative quaternion matrices is not unique. The paper also presents the results of numerical implementations of a mathematical model for color image restoration using the proposed algorithms, which demonstrated the higher efficiency of the first algorithm.

Color image restoration with mixed Gaussian–Cauchy noise and blur

Color image restoration with mixed Gaussian–Cauchy noise and blur

A new interpolation algorithm based on Hibbard-Laroche algorithm and its superiority

In order to optimize the possible problems and improvements in the existing color image restoration interpolation algorithms, we conduct research based on the existing bilinear interpolation method, cok algorithm and Hibbard-Laroche algorithm. Our method is to use our own comparison method to compare different types of images through three algorithms to find the advantages and disadvantages and to some extent combine the advantages of bilinear interpolation and Hibbard-Laroche algorithm to try to innovate a new algorithm to compare with the existing three algorithms. The results show that the existing three algorithms have their own advantages in different scenarios, and the new algorithm is superior to the existing algorithms in terms of clarity and color restoration accuracy in most scenarios. However, due to the large computational complexity, the operation speed is slow.

Color image restoration using DSS-NL-mapping-based multi-noiseNet CNN model

Color image restoration using DSS-NL-mapping-based multi-noiseNet CNN model

A Class of Priors for Color Image Restoration Parameterized by Lie Groups Acting on Pixel Values

A Class of Priors for Color Image Restoration Parameterized by Lie Groups Acting on Pixel Values

High-quality color image restoration from a disturbed graded-index imaging system by deep neural networks.

Imaging transmission plays an important role in endoscopic clinical diagnosis involved in modern medical treatment. However, image distortion due to various reasons has been a major obstacle to state-of-art endoscopic development. Here, as a preliminary study we demonstrate ultra-efficient recovery of exemplary 2D color images transmitted by a disturbed graded-index (GRIN) imaging system through the deep neural networks (DNNs). Indeed, the GRIN imaging system can preserve analog images through the GRIN waveguides with high quality, while the DNNs serve as an efficient tool for imaging distortion correction. Combining GRIN imaging systems and DNNs can greatly reduce the training process and achieve ideal imaging transmission. We consider imaging distortion under different realistic conditions and use both pix2pix and U-net type DNNs to restore the images, indicating the suitable network in each condition. This method can automatically cleanse the distorted images with superior robustness and accuracy, which can potentially be used in minimally invasive medical applications.

Numerical considerations of block GMRES methods when applied to linear discrete ill-posed problems

Linear systems of equations with a matrix whose singular values decay to zero with increasing index number without a significant gap are commonly referred to as linear discrete ill-posed problems. We are interested in solving large systems of this kind when the right-hand side has k>1 column vectors. The systems may be regarded as one system of equations with a block vector (with k columns) as the right-hand side and then solved by a block iterative method, or as k linear systems of equations (one for each right-hand side vector) that can be solved independently. Thus, the solution is a block vector with k columns. In many applications, including the restoration of color images, the right-hand side represents measurements that are contaminated by errors. Block iterative methods compute all columns of the solution block vector simultaneously. We will illustrate the performance of standard block GMRES methods and global GMRES methods, which also are block methods, and show that they may determine computed solutions of lower quality than when each column of the solution block vector is computed independently by a “standard” iterative method. We introduce a new local block GMRES method that can overcome the problems associated with block GMRES methods applied to linear discrete ill-posed problems.

Correction: Saturation-value based higher-order regularization for color image restoration

Correction: Saturation-value based higher-order regularization for color image restoration

Saturation-value based higher-order regularization for color image restoration

Saturation-value based higher-order regularization for color image restoration

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.

A new image restoration model associated with special elliptic quaternionic least-squares solutions based on LabVIEW

In this paper, we take advantage of the elliptic complex matrix representation of elliptic quaternion matrices. Then we obtain the methods of the elliptic quaternionic least-squares solution, the pure elliptic quaternionic least-squares solution, and the real least-squares solution with the least norm of the elliptic quaternion matrix equation AX=B. We also apply the newly obtained method of the pure elliptic quaternionic least-squares solution with the least norm to the color image restoration based on the LabVIEW program. In this context, we propose a new image restoration model called “ELSI image restoration model” associated with elliptic quaternionic least-squares solutions.

A new method based on semi-tensor product of matrices for solving reduced biquaternion matrix equation $ \sum\limits_{p = 1}^l A_pXB_p = C $ and its application in color image restoration

<abstract><p>In this paper, semi-tensor product of real matrices is extended to reduced biquaternion matrices, and then some new conclusions of the reduced biquaternion matrices under the vector operator are proposed using semi-tensor product of reduced biquaternion matrices, so that the reduced biquaternion matrix equation $ \sum\limits_{p = 1}^l A_pXB_p = C $ can be transformed into a reduced biquaternion linear equations, then the expression of the least squares solution of the equation is obtained using the $ \mathcal{L}_\mathcal{C} $-representation and Moore-Penrose inverse. The necessary and sufficient conditions for the compatibility and the expression of general solutions of the equation are obtained, and the minimal norm solutions are also given. Finally, our proposed method of solving the reduced biquaternion matrix equation is applied to color image restoration.</p></abstract>

Preconditioned Bminpert Algorithms for Matrix Equation <i>AX</i> = <i>B</i> and Their Applications in Color Image Restoration

The purpose of this paper is to show the preconditioned BMinPert algorithm and analyse the practical implementation. Then a posteriori backward error for BGMRES is given. Furthermore, we discuss their applications in color image restoration. The key differences between BMinPert and other methods such as BFGMRES-S(m, pf), GsGMRES and BGMRES are illustrated with numerical experiments which expound the advantages of BMinPert in the presence of sensitive data with ill-condition problems.