Quaternion Matrix Research Articles

Elliptic Quaternion Matrices: A MATLAB Toolbox and Applications for Image Processing

In this study, we developed a MATLAB 2024a toolbox that performs advanced algebraic calculations in the algebra of elliptic numbers and elliptic quaternions. Additionally, we introduce color image processing methods, such as principal component analysis, image compression, image restoration, and watermarking, based on singular-value decomposition theory for elliptic quaternion matrices; we added these to the newly developed toolbox. The experimental results demonstrate that elliptic quaternionic methods yield better image analysis and processing performance compared to other hypercomplex number-based methods.

[formula omitted]-norm regularized quaternion matrix completion using sparse representation and approximate QSVD

Color image completion is a challenging problem in computer vision. Despite the effectiveness of low-rank quaternion matrix completion (LRQMC) methods, their reliance on quaternion singular value decompositions (QSVD) makes them computationally expensive. We propose a novel method based on quaternion Qatar Riyal decomposition (QQR) and quaternion L2,1-norm, named QLNM-QQR, which reduces computational complexity by avoiding the need for calculating the QSVD of large quaternion matrices. Furthermore, we introduce two improvements: IRQLNM-QQR, which utilizes iteratively reweighted quaternion L2,1-norm minimization, and QLNM-QQR-SR, which integrates sparse regularization. Additionally, an analysis of the convergence of the proposed algorithms is conducted. Our experiments show that IRQLNM-QQR outperforms QLNM-QQR on both natural and medical color images. The QLNM-QQR-SR method achieves higher PSNR and SSIM values than state-of-the-art methods, with PSNR averaging 4-8 dB higher than non-sparse methods and 0.4 dB higher than the sparse-based TNN-SR method.

A new system of sylvester-like matrix equations with arbitrary number of equations and unknowns over the quaternion algebra

In this paper, we derive some practical necessary and sufficient conditions for the existence of a solution to a new system of coupled two-sided Sylvester-like matrix equations with arbitrary number of equations and unknowns over the quaternion algebra. As applications, we give some practical necessary and sufficient conditions for the existence of an η-Hermitian solution to a system of quaternion matrix equations in terms of ranks. We also use graphs to represent linear mappings associated with some Sylvester-type systems. As a special case of the main theorem, we prove a conjecture is correct, which was proposed in [Linear Algebra Appl. 2016;496:549–593]. The main findings of this paper widely extend almost all the known results in the literature.

Solving the Dual Generalized Commutative Quaternion Matrix Equation AXB = C

Dual generalized commutative quaternions have broad application prospects in many fields. Additionally, the matrix equation AXB=C has important applications in mathematics and engineering, especially in control systems, economics, computer science, and other disciplines. However, research on the matrix equation AXB=C over the dual generalized commutative quaternions remains relatively insufficient. In this paper, we derive the necessary and sufficient conditions for the solvability of the dual generalized commutative quaternion matrix equation AXB=C. Furthermore, we provide the general solution expression for this matrix equation, when it is solvable. Finally, a numerical algorithm and an example are provided to confirm the reliability of the main conclusions.

Pyramid quaternion discrete cosine transform based ConvNet for cancelable face recognition

The current face scanning era can quickly and conveniently attain identity authentication, but face images imply sensitive information simultaneously. Under such context, we introduce a novel cancelable face recognition methodology by using quaternion transform based convolutional network. Firstly, face images in different modalities (e.g., RGB and depth or near-infrared) are encoded into full quaternion matrix for synchronous processing. Based on the designed multiresolution quaternion singular value decomposition, we can obtain pyramid representation. Then they are transformed through random projection for making the process noninvertible. Even if the feature template is compromised, a new one can be generated. Subsequently, a three-stream convolutional network is developed to learn features, where predefined filters are stemmed from quaternion two-dimensional discrete cosine transform basis. Extensive experiments on the TIII-D, NVIE and CASIA datasets have demonstrated that the proposed method obtains competitive performance, also satisfies redistributable and irreversible.

The c-numerical range of a quaternion skew-Hermitian matrix is convex

We show that the c-numerical range of a non-scalar skew-Hermitian quaternion matrix is convex. In fact, included in our result is that the c-numerical range of a skew-Hermitian matrix is a rotation invariant subset of the quaternions with zero real parts.

Elliptic Quaternion Matrices: Theory and Algorithms

In this study, we obtained results for the computation of eigen-pairs, singular value decomposition, pseudoinverse, and the least squares problem for elliptic quaternion matrices. Moreover, we established algorithms based on these results and provided illustrative numerical experiments to substantiate the accuracy of our conclusions. In the experiments, it was observed that the p-value in the algebra of elliptic quaternions directly affects the performance of the problem under consideration. Selecting the optimal p-value for problem-solving and the elliptic behavior of many physical systems make this number system advantageous in applied sciences.

Similarity and consimilarity of hyper‐dual generalized quaternions

The aim of this paper is to investigate similarity and consimilarity of hyper‐dual generalized quaternions and their matrices. For this purpose, we give different conjugates according to the generalized quaternionic units . We present ‐consimilarity of hyper‐dual generalized quaternions and their matrices except hyper‐dual ‐quaternions. For the generalization consisting of hyper‐dual coefficients quaternion and split quaternion, we search ‐consimilarity and ‐consimilarity with the help of ‐conjugate and ‐conjugate. We also give ‐coneigenvalues and ‐coneigenvectors of the matrices of these generalizations. In addition, we examine right coneigenvalue problem in generalized quaternion matrices for real and split quaternions. The complex matrix representation obtained through the complex adjoint matrix representation of this generalization is introduced, and its properties are presented. Besides, we give algebraic methods for the concept of right coneigenvalues and coneigenvectors for matrices, which are the generalization of real quaternion and split quaternion.

Quaternion optimized model with sparseness for color image recovery

This paper presents a novel approach for sparse regularization of low-rank quaternion matrix optimization problems. Quaternion matrices, which extend the concept of complex numbers to four dimensions, have shown promising applications in various fields. In this work, we exploit the inherent sparsity present in different signal types, such as audio formats and images, when represented in their respective bases. By introducing a sparse regularization term in the optimization objective. We propose a regularization technique that promotes sparsity in the Quaternion Discrete Cosine Transform (QDCT) domain for efficient and accurate solutions. By combining low-rank restriction with sparsity, the optimized model is updated using a two-step Alternating Direction Method of Multipliers (ADMM) algorithm. Experimental results on color images demonstrate the effectiveness of the proposed method, which outperforms existing relative methods. This superior performance underscores its potential for applications in computer vision and related fields.

Global quaternion generalized minimal residual method for generalized coupled Sylvester quaternion matrix equations with application to colour image encryption and decryption

ABSTRACT Sylvester matrix equations play important roles in control and system theory. We consider a class of the generalized coupled Sylvester quaternion matrix equations, which includes a lot of special matrix equations such as Stein matrix equation, Lyapunov matrix equation and Sylvester matrix equation on the quaternion skew field. First, the quaternion Krylov subspace and its F-orthogonal basis are constructed via a new quaternion matrix product and the global quaternion Arnoldi process. Then a global quaternion generalized minimal residual (Gl-QGMRES) method is proposed for solving the generalized coupled Sylvester quaternion matrix equations. The convergence of the proposed method is theoretically analysed. Finally, the effectiveness and feasibility of the proposal are verified by some numerical experiments. As an application of Gl-QGMRES method, we establish a kind of colour image encryption and decryption by embedding watermark images.

Solving the QLY Least Squares Problem of Dual Quaternion Matrix Equation Based on STP of Dual Quaternion Matrices

Dual algebra plays an important role in kinematic synthesis and dynamic analysis, but there are still few studies on dual quaternion matrix theory. This paper provides an efficient method for solving the QLY least squares problem of the dual quaternion matrix equation AXB+CYD≈E, where X, Y are unknown dual quaternion matrices with special structures. First, we define a semi-tensor product of dual quaternion matrices and study its properties, which can be used to achieve the equivalent form of the dual quaternion matrix equation. Then, by using the dual representation of dual quaternion and the GH-representation of special dual quaternion matrices, we study the expression of QLY least squares Hermitian solution of the dual quaternion matrix equation AXB+CYD≈E. The algorithm is given and the numerical examples are provided to illustrate the efficiency of the method.

On rank‐revealing QR factorizations of quaternion matrices

On rank‐revealing QR factorizations of quaternion matrices

Quaternion-based 2D-DOST and stacked principal component analysis network for multimodal face recognition

Face recognition is widely utilized and has become a ubiquitous part of daily lives. However, original face images are often sensitive and unauthorized access could imperil personal information. To protect privacy and security of multimodal face data, we propose an original approach that integrates differential privacy and a lightweight convolutional neural network with unsupervised predefined quaternion-type filters. Firstly, different modalities of face images are banded together by means of full quaternion matrix representation for simultaneously processing. Then the proposed quaternion two-dimensional discrete orthogonal Stockwell transform is exploited for implementing difference privacy, which is disturbed by virtue of Laplace and exponential mechanism. Afterwards, we devise a stacked quaternion two-dimensional principal component analysis network with three-stages for learning representative features. The proposed network model can enhance the discriminative characteristics through inserting nonlinearity, together with union of feature maps in different levels to achieve information complementation. Experiments conducted on four multimodal face datasets unveil that the proposed method achieves superior recognition performance in comparison with other state-of-the-art approaches.

Incremental quaternion singular value decomposition and its application for low rank quaternion matrix completion

Incremental quaternion singular value decomposition and its application for low rank quaternion matrix completion

A Fixed-Time Noise-Tolerance ZNN Model for Time-Variant Inequality-Constrained Quaternion Matrix Least-Squares Problem.

Presently, numerical algorithms for solving quaternion least-squares problems have been intensively studied and utilized in various disciplines. However, they are unsuitable for solving the corresponding time-variant problems, and thus few studies have explored the solution to the time-variant inequality-constrained quaternion matrix least-squares problem (TVIQLS). To do so, this article designs a fixed-time noise-tolerance zeroing neural network (FTNTZNN) model to determine the solution of the TVIQLS in a complex environment by exploiting the integral structure and the improved activation function (AF). The FTNTZNN model is immune to the effects of initial values and external noise, which is much superior to the conventional zeroing neural network (CZNN) models. Besides, detailed theoretical derivations about the global stability, the fixed-time (FXT) convergence, and the robustness of the FTNTZNN model are provided. Simulation results indicate that the FTNTZNN model has a shorter convergence time and superior robustness compared to other zeroing neural network (ZNN) models activated by ordinary AFs. At last, the construction method of the FTNTZNN model is successfully applied to the synchronization of Lorenz chaotic systems (LCSs), which shows the practical application value of the FTNTZNN model.

Logarithmic norm minimization of quaternion matrix decomposition for color image sparse representation

Logarithmic norm minimization of quaternion matrix decomposition for color image sparse representation

SE(3) Synchronization by eigenvectors of dual quaternion matrices

Abstract In synchronization problems, the goal is to estimate elements of a group from noisy measurements of their ratios. A popular estimation method for synchronization is the spectral method. It extracts the group elements from eigenvectors of a block matrix formed from the measurements. The eigenvectors must be projected, or ‘rounded’, onto the group. The rounding procedures are constructed ad hoc and increasingly so when applied to synchronization problems over non-compact groups. In this paper, we develop a spectral approach to synchronization over the non-compact group $\mathrm{SE}(3)$, the group of rigid motions of $\mathbb{R}^{3}$. We based our method on embedding $\mathrm{SE}(3)$ into the algebra of dual quaternions, which has deep algebraic connections with the group $\mathrm{SE}(3)$. These connections suggest a natural rounding procedure considerably more straightforward than the current state of the art for spectral $\mathrm{SE}(3)$ synchronization, which uses a matrix embedding of $\mathrm{SE}(3)$. We show by numerical experiments that our approach yields comparable results with the current state of the art in $\mathrm{SE}(3)$ synchronization via the spectral method. Thus, our approach reaps the benefits of the dual quaternion embedding of $\mathrm{SE}(3)$ while yielding estimators of similar quality.

Generalized phase retrieval in quaternion Euclidean spaces

Recently, quaternionic Fourier analysis has received increasing attention due to its applications in signal analysis and image processing. This paper addresses quaternionic generalized phase retrieval (QGPR) problem in quaternion Euclidean spaces . We introduce the concept of QGPR which aims to reconstruct a signal in from the quadratic measurements , where each is an self‐adjoint quaternion matrix. We characterize QGPR sequences in terms of their real Jacobian matrices, prove that the set of QGPR sequences is an open set in some sense, and present some phaselift‐based sufficient conditions on QGPR which gives a method to construct QGPR sequences.

Fay Identities of Pfaffian Type for Hyperelliptic Curves

We establish identities of Pfaffian type for the theta function associated with twice or half the period matrix of a hyperelliptic curve. They are implied by the large size asymptotic analysis of exact Pfaffian identities for expectation values of ratios of characteristic polynomials in ensembles of orthogonal or quaternionic self-dual random matrices. We show that they amount to identities for the theta function with the period matrix of a hyperelliptic curve, and in this form we reprove them by direct geometric methods.

Inverses and Determinants of Arrowhead and Diagonal-Plus-Rank-One Matrices over Associative Algebras

This article considers arrowhead and diagonal-plus-rank-one matrices in Fn×n where F∈{R,C,H} and where H is a noncommutative algebra of quaternions. We provide unified formulas for fast determinants and inverses for considered matrices. The formulas are unified in the sense that the same formula holds in both commutative and noncommutative associative fields or algebras, with noncommutative examples being matrices of quaternions and block matrices. Each formula requires O(n) arithmetic operations, as does multiplication of such matrices with a vector. The formulas are efficiently implemented using the polymorphism or multiple-dispatch feature of the Julia programming language.