Quaternion Matrix Research Articles

A new multimedia cryptosystem using chaos, quaternion theory and modular arithmetic

Based on the combination of quaternion numbers, residual matrices, and chaotic attractors, a new cryptosystem is proposed for multimedia processing files such as images and audio. The key employed in this encryption schema consists of an image with a wide and sensitive range, obtained from the Julia Quaternion set rendered using a computational tool. Due to the use of quaternion matrices mixing the information between RGB layers and audio samples was possible, whereas using XOR operation and residual matrices modulus 257, added high sensitivity to small perturbations during encryption, key preparation and decryption processes, to such an extent that a minimal change in the image or in the audio leads to a totally different encryption result. The use of dynamic programming also reduced the processing time for matrix operations on the mathbb {Z}_{257} ring. To corroborate security of the algorithm, different tests were performed, including the National Institute of Standards and Technology test obtaining different indicators that were compared with other scientific references of similar works, finding behavioral patterns in accordance with those referenced works.

The η-Anti-Hermitian Solution to a System of Constrained Matrix Equations over the Generalized Segre Quaternion Algebra

In this paper, we propose three real representations of a generalized Segre quaternion matrix. We establish necessary and sufficient conditions for the existence of the η-anti-Hermitian solution to a system of constrained matrix equations over the generalized Segre quaternion algebra. We also obtain the expression of the general η-anti-Hermitian solution to the system when it is solvable. Finally, we provide a numerical example to verify the main results of this paper.

A color image contrast enhancement method based on improved PSO.

Image contrast enhancement uses the object intensity transformation function to maximize the amount of information to enhance an image. In this paper, the image enhancement problem is regarded as an optimization problem, and the particle swarm algorithm is used to obtain the optimal solution. First, an improved particle swarm optimization algorithm is proposed. In this algorithm, individual optimization, local optimization, and global optimization are used to adjust the particle's flight direction. In local optimization, the topology is used to induce comparison and communication between particles. The sparse penalty term in speed update formula is added to adjust the sparsity of the algorithm and the size of the solution space. Second, the three channels of the color images R, G, and B are represented by a quaternion matrix, and an improved particle swarm algorithm is used to optimize the transformation parameters. Finally, contrast and brightness elements are added to the fitness function. The fitness function is used to guide the particle swarm optimization algorithm to optimize the parameters in the transformation function. This paper verifies via two experiments. First, improved particle swarm algorithm is simulated and tested. By comparing the average values of the four algorithms under the three types of 6 test functions, the average value is increased by at least 15 times in the single-peak 2 test functions: in the multi-peak and multi-peak fixed-dimension 4 test functions, this paper can always search for the global optimal solution, and the average value is either the same or at least 1.3 times higher. Second, the proposed algorithm is compared with other evolutionary algorithms to optimize contrast enhancement, select images in two different data sets, and calculate various evaluation indicators of different algorithms under different images. The optimal value is the algorithm in this paper, and the performance indicators are at least a 5% increase and a minimum 15% increase in algorithm running time. Final results show that the effects the proposed algorithm have obvious advantages in both subjective and qualitative aspects.

Solving quaternion linear system $$AXB=E$$ based on semi-tensor product of quaternion matrices

Solving quaternion linear system $$AXB=E$$ based on semi-tensor product of quaternion matrices

W-MPD–N-DMP-solutions of constrained quaternion matrix equations

Abstract The solvability of several new constrained quaternion matrix equations is investigated, and their unique solutions are presented in terms of the weighted MPD inverse and weighted DMP inverse of suitable matrices. It is interesting to consider some exceptional cases of these new equations and corresponding solutions. Determinantal representations for the solutions of the equations as mentioned earlier are established as sums of appropriate minors. In order to illustrate the obtained results, a numerical example is shown.

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.

Quaternion-Type Catalan Transforms of the ρ-Fibonacci and ρ-Lucas Numbers

In this paper, we define a new sequence called the quaternion-type Catalan sequence and give generating function, exponential representation, quaternionic Catalan matrix and its some properties. Then, we introduce a quaternion-type Catalan transform, which is a generalization of the Catalan transform, and its inverse. Moreover, we apply the quaternion-type Catalan transforms to ρ -Fibonacci and ρ -Lucas sequences and derive some new sequences by using these quaternion-type Catalan transforms.

Zeroing neural networks for computing quaternion linear matrix equation with application to color restoration of images

<abstract><p>The importance of quaternions in a variety of fields, such as physics, engineering and computer science, renders the effective solution of the time-varying quaternion matrix linear equation (TV-QLME) an equally important and interesting task. Zeroing neural networks (ZNN) have seen great success in solving TV problems in the real and complex domains, while quaternions and matrices of quaternions may be readily represented as either a complex or a real matrix, of magnified size. On that account, three new ZNN models are developed and the TV-QLME is solved directly in the quaternion domain as well as indirectly in the complex and real domains for matrices of arbitrary dimension. The models perform admirably in four simulation experiments and two practical applications concerning color restoration of images.</p></abstract>

Randomized Quaternion Matrix UTV Decomposition and its Applications in Quaternion Matrix Optimization

Randomized Quaternion Matrix UTV Decomposition and its Applications in Quaternion Matrix Optimization

Computing quaternion matrix pseudoinverse with zeroing neural networks

<abstract><p>In recent years, it has become essential to compute the time-varying quaternion (TVQ) matrix Moore-Penrose inverse (MP-inverse or pseudoinverse) to solve time-varying issues in a range of disciplines, including engineering, physics and computer science. This study examines the problem of computing the TVQ matrix MP-inverse using the zeroing neural network (ZNN) approach, which is nowadays considered a cutting edge technique. As a consequence, three new ZNN models are introduced for computing the TVQ matrix MP-inverse in the literature for the first time. Particularly, one model directly employs the TVQ input matrix in the quaternion domain, while the other two models, respectively, use its complex and real representations. In four numerical simulations and a real-world application involving robotic motion tracking, the models exhibit excellent performance.</p></abstract>

Robust dual color images watermarking scheme with hyperchaotic encryption based on quaternion DFrAT and genetic algorithm

A robust dual color images watermarking algorithm is designed based on quaternion discrete fractional angular transform (QDFrAT) and genetic algorithm. To guarantee the watermark security, the original color watermark image is encrypted with a 4D hyperchaotic system. A pure quaternion matrix is acquired by performing the discrete wavelet transform (DWT), the block division and the discrete cosine transform on the original color cover image. The quaternion matrix is operated by the QDFrAT to improve the robustness and the security of the watermarking scheme with the optimal transform angle and the fractional order. Then the singular value matrix is obtained by the quaternion singular value decomposition (QSVD) to further enhance the scheme’s stability. The encryption watermark is also processed by DWT and QSVD. Afterward, the singular value matrix of the encryption watermark is embedded into the singular value matrix of the host image by the optimal scaling factor. Moreover, the values to balance imperceptibility and robustness are optimized with a genetic algorithm. It is shown that the proposed color image watermarking scheme performs well in imperceptibility, security, robustness and embedding capacity.

An EKF-Based Attitude Estimator for Eliminating the Effect of Magnetometer Measurements on Pitch and Roll Angles

Nowadays, magnetic and inertial measurement units (MIMUs) have become useful tools for determining the attitude of moving objects. An attitude calculation algorithm (ACA) is programed into an MIMU, which determines the attitude by fusing three built-in sensors, namely, gyroscope, accelerometer, and magnetometer. Attitude is often represented by Euler angles, quaternion, or rotation matrix. Despite that great development has been achieved, how to eliminate the negative effect of magnetic disturbance on the pitch and roll estimation is still difficult for an ACA. This article aims to propose an Extended Kalman Filter (EKF)-based approach for problem solving. Inspired by the fact that the Euler angles are computed by using some elements of the rotation matrix, a diagonalizable regulation matrix (DRM) is multiplied to the right half part of the gain matrix (which is multiplied to the residual of the magnetometer measurement in the update stage of EKF). The determination of DRM is based on the principle that these elements can be appropriately regulated by the DRM so as to make the magnetometer measurement only participate in the computation of azimuth. No further revision to EKF is made except the Kalman gain matrix, hence the proposed EKF behaves similar as traditional EKF, except its ability of eliminating the negative effect of magnetic disturbance on the pitch and roll estimation. The superiority of the proposed EKF is validated by comparing it with four other competing algorithms through simulation and experimental experiments.

A novel quaternion linear matrix equation solver through zeroing neural networks with applications to acoustic source tracking

<abstract><p>Due to its significance in science and engineering, time-varying linear matrix equation (LME) problems have received a lot of attention from scholars. It is for this reason that the issue of finding the minimum-norm least-squares solution of the time-varying quaternion LME (ML-TQ-LME) is addressed in this study. This is accomplished using the zeroing neural network (ZNN) technique, which has achieved considerable success in tackling time-varying issues. In light of that, two new ZNN models are introduced to solve the ML-TQ-LME problem for time-varying quaternion matrices of arbitrary dimension. Two simulation experiments and two practical acoustic source tracking applications show that the models function superbly.</p></abstract>

Dual quaternion matrices in multi-agent formation control

Dual quaternion matrices in multi-agent formation control

On Third Order Bronze Fibonacci Quaternions

In this study, we define third order bronze Fibonacci quaternions. We obtain the generating functions, the Binet's formula and some properties of these quaternions. We give d'Ocagne's-like and Cassini's-like identity and we use q-determinants for quaternionic matrices to give the Cassini's identity for third order bronze Fibonacci quaternions.

Key frame extraction based on quaternion Fourier transform with multiple features fusion

Key frame extraction, an important technique to improve video viewing efficiency and reduce video redundancy, has attracted considerable attention. Notwithstanding, existing key frame extraction for surveillance video not only fails to extract the local detail information of the target accurately but also preprocess the image with grayscale, so the integrity of the image information is also damaged. In order to solve the above-described problems, the paper presents a key frame extraction method based on the quaternion Fourier transform with the multiple features fusion. Initially, the method extracts dynamic and static features that can characterize the color image information separately. Then, the quaternion matrix is used to perform the quaternion Fourier transform on the different features to obtain the fused phase spectrum of the Fourier transform. Immediately after, the information containing the overall structure of the image is filtered by Gaussian filtering, and the quaternion Fourier inverse transform is performed to obtain the fused feature map. Finally, an adaptive key frame filtering criterion, which uses the fused feature map to extract the key frames is constructed in this paper for accurately extracting key frames. Notably, experimental results indicate that the proposed method ensures the integrity of the image information while accurately capturing the global and local motion state of the target.

Matrix Equation’s Reflexive and Anti-Reflexive Solutions over Quaternions

We consider when the quaternion matrix equation AXB+CXD=E has a reflexive (or anti-reflexive) solution with respect to a given generalized reflection matrix. We adopt a real representation method to derive the solutions when it is solvable. Moreover, we obtain the explicit expressions of the least-squares reflexive (or anti-reflexive) solutions.

Two novel numerical methods for the diagonalisation of a reduced biquaternion matrix in the reduced biquaternionic algebra

Unlike quaternions and split quaternions, reduced biquaternions satisfy the multiplication commutative rule and are commonly used in image processing, fuzzy recognition, image compression, Hopfield neural networks, and digital signal processing. However, although algebraic techniques have been developed for the diagonalisation of quaternion and split quaternion matrices, the diagonalisation of a reduced biquaternion matrix is yet to be studied. In this study, we derive sufficient and necessary conditions for the diagonalisation of a reduced biquaternion matrix and devise two numerical methods for the diagonalisation of a reduced biquaternion matrix. These methods were derived using complex and real representations in the reduced biquaternionic algebra.

Some Properties of the Solution to a System of Quaternion Matrix Equations

This paper investigates the properties of the ϕ-skew-Hermitian solution to the system of quaternion matrix equations involving ϕ-skew-Hermicity with four unknowns AiXi(Ai)ϕ+BiXi+1(Bi)ϕ=Ci,(i=1,2,3),A4X4(A4)ϕ=C4. We present the general ϕ-skew-Hermitian solution to this system. Moreover, we derive the β(ϕ)-signature bounds of the ϕ-skew-Hermitian solution X1 in terms of the coefficient matrices. We also give some necessary and sufficient conditions for the system to have β(ϕ)-positive semidefinite, β(ϕ)-positive definite, β(ϕ)-negative semidefinite and β(ϕ)-negative definite solutions.

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.