Quaternion-Based Bilinear Factor Matrix Norm Minimization for Color Image Inpainting

Miao, Jifei; Kou, Kit Ian

doi:10.1109/tsp.2020.3025519

Cited by 37 publications

(33 citation statements)

References 48 publications

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“…As such, Q-ADMM appears as a versatile algorithm for quaternion-domain optimization. Note that there have been several attempts to formulate ADMM for quaternion-domain optimization problems: they either rely on a real augmented formulation [37], [38] or are particularly designed for specific applications [27], [28]. In constrast, this paper introduces a general Q-ADMM framework by leveraging the proposed quaternion convex optimization framework.…”

Section: Quaternion Alternating Direction Methods Of Multipliersmentioning

confidence: 99%

A general framework for constrained convex quaternion optimization

Flamant,

Miron,

Brie

2021

Preprint

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This paper introduces a general framework for solving constrained convex quaternion optimization problems in the quaternion domain. To soundly derive these new results, the proposed approach leverages the recently developed generalized HR-calculus together with the equivalence between the original quaternion optimization problem and its augmented realdomain counterpart. This new framework simultaneously provides rigorous theoretical foundations as well as elegant, compact quaternion-domain formulations for optimization problems in quaternion variables. Our contributions are threefold: (i) we introduce the general form for convex constrained optimization problems in quaternion variables, (ii) we extend fundamental notions of convex optimization to the quaternion case, namely Lagrangian duality and optimality conditions, (iii) we develop the quaternion alternating direction method of multipliers (Q-ADMM) as a general purpose quaternion optimization algorithm. The relevance of the proposed methodology is demonstrated by solving two typical examples of constrained convex quaternion optimization problems arising in signal processing. Our results open new avenues in the design, analysis and efficient implementation of quaternion-domain optimization procedures.

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Section: Quaternion Alternating Direction Methods Of Multipliersmentioning

confidence: 99%

A general framework for constrained convex quaternion optimization

Flamant,

Miron,

Brie

2021

Preprint

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“…However, for other values of l, the orthogonality is generally not satisfied. From equation (12), for example, we can find that VT L UL−1 ⊗ . .…”

Section: Decompositionmentioning

confidence: 99%

“…The authors are with the Department of Mathematics, Faculty of Science and Technology, University of Macau, Macau 999078, China (e-mail: jifmiao@163.com; kikou@umac.mo) [10], [11], color image inpainting [12], [13] and so on. However, most of the existing quaternion-based methods are limited to one-dimensional quaternion vector or two-dimensional quaternion matrix.…”

Section: Introductionmentioning

confidence: 99%

Quaternion higher-order singular value decomposition and its applications in color image processing

Miao¹,

Kou

2021

Preprint

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Higher-order singular value decomposition (HOSVD) is one of the most efficient tensor decomposition techniques. It has the salient ability to represent highdimensional data and extract features. In more recent years, the quaternion has proven to be a very suitable tool for color pixel representation as it can well preserve cross-channel correlation of color channels. Motivated by the advantages of the HOSVD and the quaternion tool, in this paper, we generalize the HOSVD to the quaternion domain and define quaternion-based HOSVD (QHOSVD). Due to the non-commutability of quaternion multiplication, QHOSVD is not a trivial extension of the HOSVD. They have similar but different calculation procedures. The defined QHOSVD can be widely used in various visual data processing with color pixels. In this paper, we present two applications of the defined QHOSVD in color image processing-multi-focus color image fusion and color image denoising. The experimental results on the two applications respectively demonstrate the competitive performance of the proposed methods over some existing ones.Index Terms-Quaternion higher-order singular value decomposition (QHOSVD), quaternion tensor, multi-focus color image fusion, color image denoising.1 We will introduce the concept of all-orthogonality hereinafter. For more information about HOSVD, readers can refer to [1].

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“…Base on the above property and theorem, the QSVD can be obtained by computing the classical SVD of the complex matrix A c , and more details can be found in [18]. For detailed introduction of quaternion algebra, please refer to [12,20].…”

Section: And All Singular Valuesmentioning

confidence: 99%

“…More precisely, in [17], all the nonnegative singular val-ues of the quaternion matrix been adjusted to get better results, besides, it is necessary for this strategies to compute a large quaternion singular value decomposition (QSVD) in every step. The computations of large-size QSVD in each iteration is expensive, therefore the authors in [18] based on quaternion double Frobenius norm (Q-DFN), quaternion double nuclear norm (Q-DNN), and quaternion Frobenius/nuclear norm (Q-FNN), designed a novel LRQMC method by recovering two smaller factor quaternion matrices to obtain the recovered results. In this LRQMC method, the process of large-scale QSVD has been avoided, so this strategy would reduce the time consumption.…”

Section: Introductionmentioning

confidence: 99%

Weighted Truncated Nuclear Norm Regularization for Low-Rank Quaternion Matrix Completion

Yang¹,

Kou²,

Miao³

2021

Preprint

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In recent years, quaternion matrix completion (QMC) based on low-rank regularization has been gradually used in image de-noising and de-blurring. Unlike low-rank matrix completion (LRMC) which handles RGB images by recovering each color channel separately, the QMC models utilize the connection of three channels by processing them as a whole. Most of the existing quaternion-based methods formulate low-rank QMC (LRQMC) as a quaternion nuclear norm (a convex relaxation of the rank) minimization problem. The main limitation of these approaches is that the singular values being minimized simultaneously so that the low-rank property could not be approximated well and efficiently. To achieve a more accurate low-rank approximation, the matrix-based truncated nuclear norm has been proposed and also been proved to have the superiority. In this paper, we introduce a quaternion truncated nuclear norm (QTNN) for LRQMC and utilize the alternating direction method of multipliers (ADMM) to get the optimization. We further propose weights to the residual error quaternion matrix during the update process for accelerating the convergence of the QTNN method with admissible performance. The weighted method utilizes a concise gradient descent strategy which has a theoretical guarantee in optimization. The effectiveness of our method is illustrated by experiments on real visual data sets.

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Quaternion-Based Bilinear Factor Matrix Norm Minimization for Color Image Inpainting

Cited by 37 publications

References 48 publications

A general framework for constrained convex quaternion optimization

A general framework for constrained convex quaternion optimization

Quaternion higher-order singular value decomposition and its applications in color image processing

Weighted Truncated Nuclear Norm Regularization for Low-Rank Quaternion Matrix Completion

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