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Dynamic Sylvester equation (DSE) problems have drawn a lot of interest from academics due to its importance in science and engineering. Due to this, the quest for the quaternion DSE (QDSE) solution is the subject of this work. This is accomplished using the zeroing neural network (ZNN) technique, which has achieved considerable success in tackling time-varying issues. Keeping in mind that the original ZNN can handle QDSE successfully in a noise-free environment, but it might not work in a noisy one, and the noise-resilient ZNN (NZNN) technique is also utilized. In light of that, one new ZNN model is introduced to solve the QDSE problem and one new NZNN model is introduced to solve the QDSE problem under different types of noises. Two simulation experiments and one application to control of the sine function memristor (SFM) chaotic system show that the models function superbly.

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.

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.

In the Standard Model, ad hoc hypotheses assume the existence of three generations of point-like leptons and quarks, which possess a point-like structure and follow the Dirac equation involving four anti-commutative matrices. In this work, we consider the sedenion hypercomplex algebra as an extension of the Standard Model and show its close link to SU(5), which is the underlying symmetry group for the grand unification theory (GUT). We first consider the direct-product quaternion model and the eight-element octonion algebra model. We show that neither the associative quaternion model nor the non-associative octonion model could generate three fermion generations. Instead, we show that the sedenion model, which contains three octonion sub-algebras, leads naturally to precisely three fermion generations. Moreover, we demonstrate the use of basis sedenion operators to construct twenty-four 5 × 5 generalized lambda matrices representing SU(5) generators, in analogy to the use of octonion basis operators to generate Gell-Mann’s eight 3 × 3 lambda-matrix generators for SU(3). Thus, we provide a link between the sedenion algebra and Georgi and Glashow’s SU(5) GUT model that unifies the electroweak and strong interactions for the Standard Model’s elementary particles, which obey SU(3)⊗SU(2)⊗U(1) symmetry.

A knowledge graph is a structured semantic network designed to describe physical entities and relations in the world. A comprehensive and accurate knowledge graph is essential for tasks such as knowledge inference and recommendation systems, making link prediction a popular problem for knowledge graph completion. However, existing approaches struggle to model complex relations among entities, which severely hampers their ability to complete knowledge graphs effectively. To address this challenge, we propose a novel hierarchical multi-head attention network embedding framework, called RiQ-KGC, which integrates different-grained contextual information of knowledge graph triples and models quaternion rotation relations between entities. Furthermore, we propose a relation instantiation method for alleviating the difficulty of expressing complex relations between entities. To enhance the expressiveness of relation representation, the relation is integrated by Transformer to obtain multi-hop neighbor information, so that one relation can be embedded into different embeddings according to different entities. Experimental results on four datasets demonstrate that RiQ-KGC exhibits strong competitiveness compared to state-of-the-art models in link prediction, while the ablation experiments reveal that the proposed relation instantiation method achieves great performance.

We study the Rayleigh quotient of a Hermitian matrix with quaternionic coefficients and prove its main properties. As an application, we give some relationships between left and right eigenvalues of Hermitian and symplectic matrices. [ABSTRACT FROM AUTHOR]

Linear coupled matrix equations are widely utilized in applications, including stability analysis of control systems and robust control. In this paper, we establish the necessary and sufficient conditions for the consistency of the system of linear coupled matrix equations and derive an expression of the corresponding general solution (where it is solvable) over quaternion. Additionally, we investigate the necessary and sufficient conditions for the system of linear coupled matrix equations with construct to have a solution and derive a formula of its general solution (where it is solvable). Finally, an algorithm and an example were provided in order to further illustrate the primary outcomes of this paper. [ABSTRACT FROM AUTHOR]

In this paper, the adaptive synchronization problem of quaternion-valued Cohen–Grossberg neural networks (QVCGNNs), with and without known parameters, is investigated. On the basis of constructing an appropriate Lyapunov function, and utilizing parameter identification theory and decomposition methods, two effective adaptive feedback schemes are proposed, to guarantee the realization of global synchronization of CGQVNNs. The control gain of the above schemes can be obtained using the Matlab LMI toolbox. The theoretical results presented in this work enrich the literature exploring the adaptive synchronization problem of quaternion-valued neural networks (QVNNs). Finally, the reliability of the theoretical schemes derived in this work is shown in two interesting numerical examples. [ABSTRACT FROM AUTHOR]

Geographic registration (geo-registration) is a crucial foundation for augmented reality (AR) map applications. However, existing methods encounter difficulties in aligning spatial data with the ground surface in complex outdoor scenarios. These challenges make it difficult to accurately estimate the geographic north orientation. Consequently, the accuracy and robustness of these methods are limited. To overcome these challenges, this paper proposes a rotation-invariant estimation method for high-precision geo-registration in AR maps. The method introduces several innovations. Firstly, it improves the accuracy of generating heading data from low-cost hardware by utilizing Real-Time Kinematic GPS and visual-inertial fusion. This improvement contributes to the increased stability and precise alignment of virtual objects in complex environments. Secondly, a fusion method combines the true-north direction vector and the gravity vector to eliminate alignment errors between geospatial data and the ground surface. Lastly, the proposed method dynamically combines the initial attitude relative to the geographic north direction with the motion-estimated attitude using visual-inertial fusion. This approach significantly reduces the requirements on sensor hardware quality and calibration accuracy, making it applicable to various AR precision systems such as smartphones and augmented reality glasses. The experimental results show that this method achieves AR geo-registration accuracy at the 0.1-degree level, which is about twice as high as traditional AR geo-registration methods. Additionally, it exhibits better robustness for AR applications in complex scenarios. [ABSTRACT FROM AUTHOR]

Summary: In this paper, the global exponential synchronization of quaternion‐valued memristor‐based neural networks with time‐varying delays is discussed. Firstly, by using the differential inclusion theory and the set‐valued map theory, the discontinuous quaternion‐valued memristive neural networks is transformed into an uncertain system with interval parameters. A novel controller is designed to achieve the control goal. With the ω$$ \omega $$‐measure method and Halanay inequality, the criterion for global exponential synchronization of the quaternion‐valued memristive neural networks is given. At last, a numerical simulation is given to prove the validity of the main results. [ABSTRACT FROM AUTHOR]

This paper investigates the fixed-time passivity of coupled quaternion-valued neural networks (CQVNNs) with multiple delayed couplings. Different from the finite-time passivity proposed in Wang et al. (IEEE Trans Cybern 50(5):2014–2025, 2020), a novel concept of fixed-time passivity is considered in this paper, where the settling time is independent on initial value. Compared with the existing literature, in this work we extend the passivity problem of CNNs from real field to quaternion field. In particular, the multiple coupling delays and discontinuous activation functions are considered, which have not been investigated in previous papers yet. By designing appropriate controllers and applying inequality technique, sufficient conditions are obtained correspondingly for three different types of fixed-time passivity of CQVNNs. Finally, simulations are given to substantiate the correctness of the theoretical results. [ABSTRACT FROM AUTHOR]

To ensure the smoothness of spatial Pythagorean-hodograph (PH) curve and minimize its rotation, spatial quintic PH curve construction was proposed. Quartic Bézier curve was constructed by quaternion method based on the given four spatial control pouts and the endpoint attributes. Quintic PH curve of Interpolated endpoint was obtained by integrating the quartic Bézier curve. Control points of PH curve can be obtained by inserting two new knots into the original control points, and both control points can be expressed by quaternion operation The PH curve contains many degrees off freedom, which can be used for optimization and adjustment. On the one hand, free parameters can be used to optimize the fairness and minimum rotation of the curve according to the curve geometry. On the other hand, reasonable are length can also be restricted by assigning a corresponding free parameter. Moreover, the PH curve is Better curve, which has the freedom to realize the smoothness splicing of these piecewise curves. [ABSTRACT FROM AUTHOR]

Recent visual-based depression recognition methods mostly use hand-crafted features with information lost in color channels, or deep network features with a limited performance from the finite data. In this paper, we propose a method called Local Quaternion and Global Deep Network (LQGDNet) which can combine advantages from hand-crafted and deep features. Specifically, the Quaternion XOR Asymmetrical Regional Local Gradient Coding (XOR-AR-LGC) is first designed, which encodes the facial images with local textures in the quaternion domain to keep the dependence of color channels, and integrated into the Quaternion Feature Extractor (QFE). To the best of our knowledge, it is the first attempt to use a quaternion-based method for facial depression recognition. Second, we design the Local Quaternion Representation Module (LQRM) composed of Local Deep Feature Extractor (LDFE) and QFE to output local quaternion facial features. Third, global deep facial features are encoded from the Global Deep Representation Module (GDRM) with the deep convolutional neural network. Finally, the LQGDNet integrates LQRM and GDRM with the local quaternion and global deep features and predicts the depression score. The experimental results on AVEC 2013 and AVEC 2014 show the superiority of our method compared to the state-of-the-art approaches. [ABSTRACT FROM AUTHOR]

A transverse wave is a wave in which the particles are displaced perpendicular to the direction of the wave's advance. Examples of transverse waves include ripples on the surface of water and light waves. Polarization is one of the primary properties of transverse waves. Analysis of polarization states can reveal valuable information about the sources. In this paper, we propose a separable low-rank quaternion linear mixing model for polarized signals: we assume each column of the source factor matrix equals a column of the polarized data matrix and refer to the corresponding problem as separable quaternion matrix factorization (SQMF). We discuss some properties of the matrix that can be decomposed by SQMF. To determine the source factor matrix in quaternion space, we propose a heuristic algorithm called quaternion successive projection algorithm (QSPA) inspired by the successive projection algorithm. To guarantee the effectiveness of QSPA, a new normalization operator is proposed for the quaternion matrix. We use a block coordinate descent algorithm to compute nonnegative activation matrix in real number space. We test our method on the applications of polarization image representation and spectro-polarimetric imaging unmixing to verify its effectiveness. [ABSTRACT FROM AUTHOR]

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In this study, using the Leonardo numbers, we define a new type of quaternion that is called a Leonard Gaussian quaternion. We also give a negative-Leonardo Gaussian quaternion. These numbers are introduced from the set of complex numbers and quaternions. Moreover, we obtain the Binet’s formula, generating function formula, d’Ocagne’s identity, Catalan’s identity, Cassini’s identity, Honsberger’s identity, like-Vajda’s identity and some formulas for these new types of numbers. Morever, we give the matrix representation of the Leonardo Gaussian quaternion. [ABSTRACT FROM AUTHOR]

This paper introduces a new method for kinematic modeling of the robot arm by deriving a new elegant mathematical formula based on the axis vector with the tangent of the rotation angle. For this purpose, an innovative analytical quaternion is introduced through integration between Axis-Invariants and unit quaternion features named Ju-Gibbs quaternion, which expresses the body rotation with non-redundant parameters compared with the quaternions in literature. Two theorems based on the new form of the quaternion are developed and proved for the kinematic modeling of the robot arm. The first is attitude alignment, which is essential in multiaxial rotation systems. The second theorem for the wrist inverse kinematic (IK) solution is utilized to obtain the joint variables for the last joints of the end effector. In order to verify the effectiveness and accuracy of the proposed method, a numerical example and simulation of different structural configurations of robot and human arms are intensively studied. The novel quaternion provides a new tool for kinematic analysis and reduces the computational complexity of the kinematic solutions of the Robot-Arms wrist. Furthermore, the method laid a new foundation for the IKs of multi-axis systems based on Axis-Invariant and tangent quaternion. [ABSTRACT FROM AUTHOR]

The quaternion was introduced via the special unitary group in two dimensions and its properties were discussed systematically and rigorously. Some important results were educed such as the homomorphism of SU（2） onto SO（3）. Besides， the axis and the angle of the rotation were given by a quaternion， and the angular displacement of an infinitesimal rotation was derived smoothly.

With the rapid development and application of Internet technology in recent years, the issue of information security has received more and more attention. Digital steganography is used as a means of secure communication to hide information by modifying the carrier. However, steganography can also be used for illegal acts, so it is of great significance to study steganalysis techniques. The steganalysis technology can be used to solve the illegal steganography problem of computer vision and engineering applications technology. Most of the images in the Internet are color images, and steganalysis for color images is a very critical problem in the field of steganalysis at this stage. Currently proposed algorithms for steganalysis of color images mainly rely on the manual design of steganographic features, and the steganographic features do not fully consider the internal connection between the three channels of color images. In recent years, advanced steganography techniques for color images have been proposed, which brings more serious challenges to color image steganalysis. Quaternions are a good tool to represent color images, and the transformation of quaternions can fully exploit the correlation among color image channels. In this paper, we propose a color image steganalysis algorithm based on quaternion discrete cosine transform, firstly, the image is represented by quaternion, then the quaternion discrete cosine transform is applied to it, and the coefficients obtained from the transformation are extracted to design features of the coeval matrix. The experimental results show that the proposed algorithm works better than the typical color image steganalysis algorithm.

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.

The asymptotic synchronization of quaternion-valued delayed neural networks with impulses and inertia is studied in this article. Firstly, a convergence result on piecewise differentiable functions is developed, which is a generalization of the Barbalat lemma and provides a powerful tool for the convergence analysis of discontinuous systems. To achieve synchronization, a constant gain-based control scheme and an adaptive gain-based control strategy are directly proposed for response quaternion-valued models. In the convergence analysis, a direct analysis method is developed to discuss the synchronization without using the separation technique or reduced-order transformation. In particular, some Lyapunov functionals, composed of the state variables and their derivatives, are directly constructed and some synchronization criteria represented by matrix inequalities are obtained based on quaternion theory. Some numerical results are shown to further confirm the theoretical analysis.

This article indicates one of the ways to solve the generalized Cauchy – Riemann system for quaternionic functions in an eight-dimensional space. In previous works, some classes of solutions of this system were studied and it was stated that it is possible to use the method of generalized degrees to construct solutions of this system of differential equations. It is shown that the solution of the problem can be reduced to finding two arbitrary quaternionic harmonic functions in an eight-dimensional space. All 8 components of these functions $\varphi ,\psi$ must be harmonic functions, that is, be twice continuously differentiable over all 8 real variables $x_i$, $y_i$, where $i = \overline {1,4} $ solutions of the Laplace equation. In this article, the parametric method of generalized degrees is considered, which is applicable to individual equations of the second and higher orders.

Toward the quadcopter unmanned aerial vehicle (UAV) attitude measurement problem, to improve the accuracy of acquisition of vehicle attitude parameters, and ensure accuracy of subsequent attitude control, the Quaternion-based Unscented Kalman filter (QUKF) data fusion method is presented. This attitude measurement system uses STM32F103 as the central controller, MPU6050 with integrated accelerometer and gyroscope, and magnetometer HMC5883l as the measurement sensor. Coordinate Rotation Relationships for the Attitude Heading Reference System (AHRS) in Quaternions, combining Unscented Kalman Filter (UKF) to Fuse Low-Cost Attitude Measurement Systems, tracking estimation of the genuine attitude of the vehicle. High-precision sensor measurements as real values, by comparing with Extended Kalman Filter (EKF), Complementary Filter (CF), and genuine values to validate and analyze the effectiveness of the algorithm applied to the low-cost attitude measurement system. Experimental results show that the low-cost attitude measurement system using quaternions as state variables combined with UKF can accurately estimate attitude information, providing precise attitude information for subsequent attitude control of UAVs.

In this study, two models of multilayer quaternionic feedforward neural networks are presented. Whereas the first model is based on quaternion algebra, the second model uses split quaternion algebra. For both quaternionic neural networks, a learning algorithm was derived using an adaptation of the extended Kalman filter. In addition, to analyze the performance of these two neural network models, they were applied to address the problem of enhancing low-light color images, which for this work consists particularly in the recovery of illuminated color images by quaternionic neural network processing from underexposed images. The quaternion neural network enhances images in the RGB color space (Euclidean metric), whereas the split quaternion neural network enhances images in the HSV color space (Minkowski metric). From the results, we can observe that the split quaternion neural network using the HSV color model shows advantages that were not previously published and were not shown by the quaternion neural network using the RGB color model. Therefore, this article introduces a novel quaternionic neural network that uses the Minkowski metric for color image processing, which can be advantageously used by practitioners interested in working with the HSV color model.

A high level of computation is required for edge detection in color images captured by unmanned aerial vehicles (UAVs) to address issues, such as noise, distortion, and information loss. Thus, an edge detection method for UAV-captured color images based on the improved whale optimization algorithm (WOA) is proposed in this study. In this method, the color image pixels are represented by quaternions, and the global random position variables and information exchange mechanism are introduced into the random walk foraging formula of the WOA. Further, a random disturbance factor is also introduced into the predator-prey formula of the spiral bubble net. The proposed improved WOA is then used to obtain the preliminary edge of the UAV-captured color image. An edge-point classification method using the radius of the shortest distance between the whale and the current global optimum in each iteration is presented to enhance a preliminary edge. The experimental results show that the proposed edge detection method has the advantages of strong denoising, fast speed, and good quality.

The efficient solution of the time-varying quaternion matrix inverse (TVQ-INV) is a challenging but crucial topic due to the significance of quaternions in many disciplines, including physics, engineering, and computer science. The main goal of this research is to employ the higher-order zeroing neural network (HZNN) strategy to address the TVQ-INV problem. HZNN is a family of zeroing neural network models that correlates to the hyperpower family of iterative methods with adjustable convergence order. Particularly, three novel HZNN models are created in order to solve the TVQ-INV both directly in the quaternion domain and indirectly in the complex and real domains. The noise-handling version of these models is also presented, and the performance of these models under various types of noises is theoretically and numerically tested. The effectiveness and practicality of these models are further supported by their use in robotic motion tracking. According to the principal results, each of these six models can solve the TVQ-INV effectively, and the HZNN strategy offers a faster convergence rate than the conventional zeroing neural network strategy. [ABSTRACT FROM AUTHOR]

We derive the quaternions expressing the rotation transformation between celestial and terrestrial reference systems as functions of the parameters giving the position of the Celestial Intermediate Pole (CIP) in those two systems, and of the rotation angle. A first version is associated with the common Earth Orientation Parameters (EOP) recommended by the International Earth Rotation and Reference System Service (IERS) and IAU; a second version uses the direction cosines of the CIP in the International Terrestrial Reference System (ITRS) instead of the conventional pole coordinates. Whereas matrix and quaternion methods are numerically comparable, the quaternion formulation offers a very elegant and concise analytical representation, that does not exist for Earth rotation matrix and that can be easily programmed. In our view, this quaternion representation reinforces the scope of the nowadays parameters adopted for describing Earth rotation. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, a class of global finite-time stability problem for quaternion-valued neural networks with time-varying delays are investigated by adopting an extended modification Lyapunov–Razumikhin (L–R) method and a new upper bounds estimation of system solution in terms of convergence rate was obtained. Firstly, a new extended method of L–R is proposed to solve the general difficulty to find a proper Lyapunov functional. Then, a new suitable controller is designed, the new conditions of inequalities global finite-time stability are obtained via combining with the former proposed L–R method in the separated real-valued system. Finally, for purpose of verifying the availability of the theorem presented, two given illustrative examples are shown. [ABSTRACT FROM AUTHOR]

Copyright of Machine Tool & Hydraulics is the property of Guangzhou Mechanical Engineering Research Institute (GMERI) and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Images are a convenient way for humans to obtain information and knowledge, but they are often destroyed throughout the collection or distribution process. Therefore, image processing evolves as the need arises, and color image processing is a broad and active field. A color image includes three distinct but closely related channels (red, green, and blue (RGB)). Compared to directly expressing color images as vectors or matrices, the quaternion representation offers an effective alternative. There are several papers and works on this subject, as well as numerous definitions, hypotheses, and methodologies. Our observations indicate that the quaternion representation method is effective, and models and methods based on it have rapidly developed. Hence, the purpose of this paper is to review and categorize past methods, as well as study their efficacy and computational examples. We hope that this research will be helpful to academics interested in quaternion representation. [ABSTRACT FROM AUTHOR]

Quaternion is a four-dimensional and an extension of the complex number system. It is often viewed from various fields, such as analysis, algebra, and geometry. Several applications of quaternions are related to an object's rotation and motion in three-dimensional space in the form of a differential equation. In this paper, we do a systematic literature review on the development of quaternion differential equations. We utilize PRISMA (preferred reporting items for systematic review and meta-analyses) framework in the review process as well as content analysis. The expected result is a state-of-the-art and the gap of concepts or problems that still need to develop or answer. It was concluded that there are still some opportunities to develop a quaternion differential equation using a quaternion function domain. [ABSTRACT FROM AUTHOR]

The aim of this paper is to construct dual-generalized complex Fibonacci and Lucas quaternions. It examines the properties both as dual-generalized complex number and as quaternion. Additionally, general recurrence relations, Binet's formulas, Tagiuri's (or Vajda's like), Honsberger's, d'Ocagne's, Cassini's and Catalan's identities are obtained. A series of matrix representations of these special quaternions is introduced. Finally, the multiplication of dual-generalized complex Fibonacci and Lucas quaternions are also expressed as their different matrix representations.

Aimed at the modeling error of the integrated strapdown inertial navigation system(SINS)/global positioning system (GPS) navigation system and the particle degradation problem of particle filter(PF), an in-flight alignment method of integrated SINS/GPS navigation system based on the combined PF-UKF filter is proposed, in combiation with the unscented Kalman filter(UKF). First, the attitude angle is replaced by the error quaternion. The position and velocity differences between SINS and GPS are selected as the observation variables. In addition, a novel error equation of the integrated navigation system is established. Moreover, the sampled particles are divided into random particles and deterministic particles in the proposed combined PF-UKF filter. The random particles are collected by probability density function, and the determined particles are the state values obtained by collecting sigma point of UKF algorithm. Therefore, the proposed method can effectively reduce the complexity of PF and the degree of particle degradation. The simulation results show that compared with the UKF algorithm, the proposed method can effectively improve the error accuracy of integrated navigation system with a better robustness.