Quaternion Matrix Research Articles

Semi‐tensor product of quaternion matrices and its application

Unlike real or complex numbers, quaternion multiplication does not satisfy the commutative law. This makes it impossible to generalize some conclusions on semi‐tensor product to quaternion directly, which is valid on real or complex number fields. In this paper, we define semi‐tensor product of quaternion matrices and study the properties of semi‐tensor product of quaternion matrices, and the conclusion is applied to the color digital image restoration model.

A novel algebraic approach for the Schrödinger equation in split quaternionic mechanics

In the theoretical exploration and numerical computation of split quaternionic quantum mechanics, an important goal is to solve the Schrödinger equation ∂∂t|f〉=−A|f〉 with A an i-Hermitian split quaternion matrix, and |f〉 an eigenstate to A. The split quaternionic Schrödinger equation is a fundamental equation in split quaternionic mechanics and plays an indispensable role. It is known that the problems of split quaternion Schrödinger equation can be discretized into the related eigen-problems of split quaternion matrix Aα=αλ. This paper studies the problem of split quaternion Schrödinger equation based on an isomorphic mapping, derives a novel approach for Schrödinger equation in split quaternion mechanics. This paper also shows an algebraic approach to the eigen-problems for i-Hermitian split quaternion matrices. The algebraic approach will have potential applications in the research field of split quaternionic mechanics.

Quaternion-based dynamic mode decomposition for background modeling in color videos

Scene Background Initialization (SBI) is one of the challenging problems in computer vision. Dynamic mode decomposition (DMD) is a recently proposed method to robustly decompose a video sequence into the background model and the corresponding foreground part. However, this method needs to convert the color image into a grayscale image for processing, which leads to the neglect of the coupling information between the three channels of the color image. In this study, we propose a quaternion-based DMD (Q-DMD), which extends the DMD by quaternion matrix analysis, so as to ultimately preserve the inherent color structure of the color image and the color video. We exploit the standard eigenvalues of the quaternion matrix to compute its spectral decomposition and calculate the corresponding Q-DMD modes and eigenvalues. The results on the publicly available benchmark datasets prove that our Q-DMD outperforms the exact DMD method, and experiment results also demonstrate that the performance of our approach is comparable to that of the state-of-the-art ones.

Color Image Inpainting via Robust Pure Quaternion Matrix Completion: Error Bound and Weighted Loss

In this paper, we study color image inpainting as a pure quaternion matrix completion problem. In the literature, the theoretical guarantee for quaternion matrix completion is not well-established. Our main aim is to propose a new minimization problem with an objective combining nuclear norm and a quadratic loss weighted among three channels. To fill the theoretical vacancy, we obtain the error bound in both clean and corrupted regimes, which relies on some new results of quaternion matrices. A general Gaussian noise is considered in robust completion where all observations are corrupted. Motivated by the error bound, we propose to handle unbalanced or correlated noise via a cross-channel weight in the quadratic loss, with the main purpose of rebalancing noise level, or removing noise correlation. Extensive experimental results on synthetic and color image data are presented to confirm and demonstrate our theoretical findings.

The implicitly restarted multi-symplectic block-Lanczos method for large-scale Hermitian quaternion matrix eigenvalue problem and applications

The large-scale Hermitian quaternion eigenvalue problem has become the key issue of color image recognition. However, there are still lack of efficient iterative methods of computing its partial eigenpairs. To solve this problem, a new implicitly restarted multi-symplectic block-Lanczos method is proposed with generalizing the block and implicit restarting techniques to quaternion matrix eigenvalue computation. The proposed algorithm is applied to color image identification and approximation, and the experimental results demonstrate its efficiency and advantages to the existing algorithms.

Algebraic methods for equality constrained least squares problems in commutative quaternionic theory

This paper, by means of two matrix representations of a commutative quaternion matrix, studies the relationship between the solutions of commutative quaternion equality constrained least squares (LSE) problems and that of complex and real LSE problems and derives two algebraic methods for finding the solutions of equality constrained least squares problems in commutative quaternionic theory.

Singular Values of Dual Quaternion Matrices and Their Low-Rank Approximations

The dual quaternion matrix is an important and powerful tool for the study of multi-agent formation control. However, many basic properties of dual quaternion matrices are still not studied, due to the difficulty incurred by the non-commutativity of multiplication and existence of zero divisors of dual quaternion numbers. In this paper, we study some basic properties of dual quaternion matrices, including the polar decomposition theorem, the minimax principle and Weyl’s type monotonicity inequality for singular values, spectral norm and the Pythagoras theorem. Based upon these, we further present the best low-rank approximations for dual quaternion matrices, and discuss the relationship between the approximation degree characterizations of the best low-rank approximations in the sense of Frobenius norm and spectral norm, in addition to proving a fundamental property of the best low-rank approximation of dual quaternion matrices in a given subspace. These results are of importance to the applications in rigid body motion and data reduction.

A System of Sylvester-type Quaternion Matrix Equations with Ten Variables

A System of Sylvester-type Quaternion Matrix Equations with Ten Variables

Quaternion matrix decomposition and its theoretical implications

This paper proposes a novel matrix rank-one decomposition for quaternion Hermitian matrices, which admits a stronger property than the previous results in Ai W et al (Math Progr 128(1):253–283, 2011), Huang Y, Zhang S (Math Oper Res 32(3):758–768, 2007), Sturm JF, Zhang S (Math Oper Res 28(2):246–267 2003). The enhanced property can be used to drive some improved results in joint numerical range, \({\mathcal {S}}\)-Procedure and quadratically constrained quadratic programming (QCQP) in the quaternion domain, demonstrating the capability of our new decomposition technique.

Crosstalk Defect Detection Method Based on Salient Color Channel Frequency Domain Filtering.

Display crosstalk defect detection is an important link in the display quality inspection process. We propose a crosstalk defect detection method based on salient color channel frequency domain filtering. Firstly, the salient color channel in RGBY is selected by the maximum relative entropy criterion, and the color quaternion matrix of the displayed image is formed with the Lab color space. Secondly, the image color quaternion matrix is converted into the logarithmic spectrum in the frequency domain through the hyper-complex Fourier transform. Finally, Gaussian threshold band-pass filtering and hyper-complex inverse Fourier transform are used to separate the low-contrast defects and background of the display image. The experimental results show that the accuracy of the proposed algorithm reaches 96% for a variety of crosstalk defect detection. Compared with the current advanced defect detection algorithms, the effectiveness of the proposed method for low-contrast crosstalk defect detection is confirmed.

The matrices of Pauli quaternions, their De Moivre’s and Euler’s formulas

In this paper, we obtain Euler’s and De Moivre’s formulas for the [Formula: see text] matrix representation of Pauli quaternions. Moreover, we provide De Moivre’s formula for the light-like Pauli quaternions. Additionally, we give the nth roots of the matrix representation of Pauli quaternions. Moreover, we exemplify some of the results with illustrative examples to support the main formulas.

An Efficient LiDAR Point Cloud Map Coding Scheme Based on Segmentation and Frame-Inserting Network.

In this article, we present an efficient coding scheme for LiDAR point cloud maps. As a point cloud map consists of numerous single scans spliced together, by recording the time stamp and quaternion matrix of each scan during map building, we cast the point cloud map compression into the point cloud sequence compression problem. The coding architecture includes two techniques: intra-coding and inter-coding. For intra-frames, a segmentation-based intra-prediction technique is developed. For inter-frames, an interpolation-based inter-frame coding network is explored to remove temporal redundancy by generating virtual point clouds based on the decoded frames. We only need to code the difference between the original LiDAR data and the intra/inter-predicted point cloud data. The point cloud map can be reconstructed according to the decoded point cloud sequence and quaternion matrices. Experiments on the KITTI dataset show that the proposed coding scheme can largely eliminate the temporal and spatial redundancies. The point cloud map can be encoded to 1/24 of its original size with 2 mm-level precision. Our algorithm also obtains better coding performance compared with the octree and Google Draco algorithms.

Sampling expansions associated with quaternion difference equations

Starting with a quaternion difference equation with boundary conditions, a parameterized sequence that is complete in the finite-dimensional quaternion Hilbert space is derived. By employing the parameterized sequence as the kernel of the discrete transform, we construct a quaternion function space whose elements have sampling expansions. It is shown that the sample points of the sampling expansions are the eigenvalues of an associated quaternion difference operator. For the operator that has multiple standard eigenvalues, we provide a method to find the ‘missing sample points’. Through formulating the boundary-value problems, we make a connection between a class of tridiagonal quaternion matrices and the polynomials with quaternion coefficients. We prove that for a tridiagonal symmetric quaternion matrix, one can always associate a quaternion characteristic polynomial whose roots are the eigenvalues of the matrix. By representing colour image pixels as quaternions, an application of the proposed sampling theorem in colour image encryption is discussed. Throughout the paper, examples are given to illustrate the results.

Solving Quaternion Linear System Based on Semi-Tensor Product of Quaternion Matrices

In this paper, we use semi-tensor product of quaternion matrices, L-representation of quaternion matrices, and GH-representation of special quaternion matrices such as quaternion (anti)-centrosymmetric matrices to solve the special solutions of quaternion matrix equation. Based on semi-tensor product of quaternion matrices and the structure matrix of the multiplication of quaternion, we propose the vector representation operation conclusion of quaternion matrices, and study the different matrix representations of quaternion matrices. Then the problem of the quaternion matrix equation is transformed into the corresponding problem in the real number fields by using vector representation and L-representation of quaternion matrices, combined with the special structure of (anti)-centrosymmetric matrices, the independent elements are extracted by GH-representation method, so as to reduce the number of variables to be calculated and improve the calculation accuracy. Finally, the effectiveness of the method is verified by numerical examples, and the time comparison with the two existing algorithms is carried out. The algorithm in this paper is also applied in a centrosymmetric color digital image restoration model.

Global Dissipativity Analysis and Stability Analysis for Fractional-Order Quaternion-Valued Neural Networks With Time Delays

This article studies dissipativity analysis of fractional-order quaternion-valued neural networks (FOQVNNs) with time delays. Two specific activation functions are considered along with common bounded and activation functions of Lipschitz-kind. Since quaternion multiplication is not commutative, we must divide the model, which is evaluated by quaternion, into four elements that are real-valued elements. On the basis of the construction of novel Lyapunov functional, and applying fractional-calculus theory, new criteria for the test of the global dissipativity and exponential stability of FOQVNNs model are established. FOQVNNs have also been suggested to provide global dissipativity and exponential stability, whereas nonlinear complex activation functions are constrained by the usage of linear matrix inequality methods, which utilize quaternion matrices and positive quaternion definite matrices. Finally, the effectiveness and superiority of the proposed approach is validated through numerical examples.

Consistency and General Solutions to Some Sylvester-like Quaternion Matrix Equations

This article makes use of simultaneous decomposition of four quaternion matrixes to investigate some Sylvester-like quaternion matrix equation systems. We present some useful necessary and sufficient conditions for the consistency of the system of quaternion matrix equations in terms of the equivalence form and block matrixes. We also derive the general solution to the system according to the partition of the coefficient matrixes. As an application of the system, we present some practical necessary and sufficient conditions for the consistency of a ϕ-Hermitian solution to the system of quaternion matrix equations in terms of the equivalence form and block matrixes. We also provide the general ϕ-Hermitian solution to the system when the equation system is consistent. Moreover, we present some numerical examples to illustrate the availability of the results of this paper.

The Solvability of a System of Quaternion Matrix Equations Involving ϕ-Skew-Hermicity

Let H be the real quaternion algebra and Hm×n denote the set of all m×n matrices over H. For A∈Hm×n, we denote by Aϕ the n×m matrix obtained by applying ϕ entrywise to the transposed matrix AT, where ϕ is a non-standard involution of H. A∈Hn×n is said to be ϕ-skew-Hermicity if A=−Aϕ. In this paper, we provide some necessary and sufficient conditions for the existence of a ϕ-skew-Hermitian solution to the system of quaternion matrix equations with four unknowns AiXi(Ai)ϕ+BiXi+1(Bi)ϕ=Ci,(i=1,2,3),A4X4(A4)ϕ=C4.

Multiple color image encryption based on cascaded quaternion gyrator transforms

In this paper, we propose a novel encryption algorithm using cascaded quaternion gyrator transforms that allows protecting multiple color images with efficiency at once. The originality is reflected in the integration of quaternion algebra with a cascaded encryption structure. First, image RGB channels are constituted into three imaginary parts of a quaternion matrix. A cascaded encryption structure is constructed to cooperate with the vortex phase mask, which is easy to transmit and saves storage. Then, a quaternion gyrator transform is conducted, after which the phase is extracted to serve as the decryption key while the amplitude is used as the real part of the following quaternion matrix. Finally, the amplitude of the last gyrator transform is scrambled to obtain the ciphertext. All phase matrices are further modulated by a secondary image phase mask for additional security. Numerical simulations have been conducted to verify the validity of the proposed algorithm. Experiments on several data test set demonstrate the robustness of the proposed method to various attacks.

An Efficient Method for Split Quaternion Matrix Equation X − Af(X)B = C

In this paper, we consider the split quaternion matrix equation X−Af(X)B=C, f(X)∈{X,XH,XiH,XjHXkH}. The H representation method has the characteristics of transforming a matrix with a special structure into a column vector with independent elements. By using the real representation of split quaternion matrices, H representation method, the Kronecker product of matrices and the Moore-Penrose generalized inverse, we convert the split quaternion matrix equation into the real matrix equation, and derive the sufficient and necessary conditions and the general solution expressions for the (skew) bisymmetric solution of the original equation. Moreover, we provide numerical algorithms and illustrate the efficiency of our method by two numerical examples.

A New Method of Solving Special Solutions of Quaternion Generalized Lyapunov Matrix Equation

In this paper, we study the bisymmetric and skew bisymmetric solutions of quaternion generalized Lyapunov equation. With the help of semi-tensor product of matrices, some new conclusions on the expansion rules of row and column of matrix product on quaternion matrices are proposed and applied to the calculation of quaternion matrix equation. Using the H-representation method, the independent elements are extracted according to the structural characteristics of bisymmetric matrix and skew bisymmetric matrix, so as to simplify the operation process. Finally, it is compared with the real vector representation method of quaternion matrix equation to illustrate the effectiveness and superiority of the proposed method.