Quaternion Multiplication Research Articles

Quaternion and Biquaternion Representations of Proper and Improper Transformations in Non-Cartesian Reference Systems

Quaternion and biquaternion symmetry transformations have been applied to non-Cartesian reference systems of direct and reciprocal crystal lattices. The transformations performed directly in the sets of crystal reference axes simplify the calculations, eliminate the need for orthogonalization, permit the use of crystallographic vectors for defining the directions of rotations and perform the computations directly in the crystal coordinates. The applications of the general quaternion transformations are envisioned for physical, chemical, crystallographic and engineering applications. The general quaternion multiplication rules for any symmetry-unrestricted lattices have been derived for the triclinic crystallographic system and have been applied to the biquaternion representations of all point-group symmetry elements, including the crystallographic hexagonal system. Cayley multiplication matrices for point-groups, based on the biquaternion symbols of proper and improper symmetry elements, have been exemplified.

Structure and Principles of Operation of a Quaternion VLSI Multiplier

The paper presents the original structure of a processing unit for multiplying quaternions. The idea of organizing the device is based on the use of fast Hadamard transform blocks. The operation principles of such a device are described. Compared to direct quaternion multiplication, the developed algorithm significantly reduces the number of multiplication and addition operations. Hardware implementations of the developed structure, in FPGA and ASIC, are presented. The FPGA blocks were implemented in the Vivado environment. The ASICs were designed using 130nm technology. The developed scripts in VHDL are available in the GitHub repository.

TELS: Learning time-evolving information and latent semantics using dual quaternion for temporal knowledge graph completion

In temporal knowledge graphs (TKGs), the status of facts is intricately tied to the dynamic and precise nature of temporal factors. Existing research merely treats time as supplementary information, without considering the latent semantic changes caused by the positional changes of entities within specific relations in a temporal context. Furthermore, due to the coarse granularity of timestamps in existing TKGs, the number of multiple relations pattern among entities significantly increases, limiting model performance. This paper proposes a Time-Evolving Information and Latent Semantics model (TELS), which represents facts as dual quaternion embeddings to provide a compact and elegant representation. Specifically, we use timestamp dual quaternions, transforming the entity and relation into temporal entity and temporal relation through dual quaternion multiplication. Besides, we introduce semantic-aware dual quaternion to capture the latent semantics arising from the positional changes of entities within specific relations. Next, TELS consists of two parts: (a) We use semantic-aware dual quaternions to perform transformations on head entity and tail entity respectively through dual quaternion multiplication. Next, we utilize temporal relation to transform head entity to tail entity through dual quaternion multiplication. (b) We adopt an evolutionary hierarchical factor to encapsulate the differences in modulus distribution between the temporal head entity and temporal tail entity. In this way, TELS not only uses dual quaternions to handle key patterns and multiple relations pattern, but also handles evolutionary hierarchical patterns by capturing the modulus distribution differences between temporal entities. Meanwhile, TELS learns semantic-aware dual quaternion embeddings to capture the latent semantics endowed by relations to entities. Empirically, TELS can boost the performance over seven temporal knowledge graph benchmarks.

Memristive Circuit of Quaternion Multiplication and Its Application in Aircraft Attitude Computation

Memristive Circuit of Quaternion Multiplication and Its Application in Aircraft Attitude Computation

Quaternion matrix completion using untrained quaternion convolutional neural network for color image inpainting

The use of quaternions as a novel tool for color image representation has yielded impressive results in color image processing. By considering the color image as a unified entity rather than separate color space components, quaternions can effectively exploit the strong correlation among the RGB channels, leading to enhanced performance. Especially, color image inpainting tasks are highly beneficial from the application of quaternion matrix completion techniques, in recent years. However, existing quaternion matrix completion methods suffer from two major drawbacks. First, it can be difficult to choose a regularizer that captures the common characteristics of natural images, and sometimes the regularizer that is chosen based on empirical evidence may not be the optimal or efficient option. Second, the optimization process of quaternion matrix completion models is quite challenging because of the non-commutativity of quaternion multiplication. To address the two drawbacks of the existing quaternion matrix completion approaches mentioned above, this paper tends to use an untrained quaternion convolutional neural network (QCNN) to directly generate the completed quaternion matrix. This approach replaces the explicit regularization term in the quaternion matrix completion model with an implicit prior that is learned by the QCNN. Extensive quantitative and qualitative evaluations demonstrate the superiority of the proposed method for color image inpainting compared with some existing quaternion-based and tensor-based methods.

Bernstein inequalities for quaternionic polynomials in the setting of generalized polynomials

Recently, several authors proved a Bernstein type inequality for so called quaternionic unilateral polynomials. Although these polynomials are not generalized complex polynomials between normed spaces, it will be proved in this contribution that their restrictions to each complex plane satisfy a Bernstein inequality for the real Fréchet derivative. This result will imply Bernstein's inequality for quaternionic unilateral polynomials which in addition holds for all complex norms in which the unit closed ball contains the real unit quaternion. The second main contribution consists in providing Bernstein-type inequalities for so called quaternionic canonical generalized polynomials. These polynomials have a factorized form which, due to the lack of commutativity for quaternionic multiplication leads to a completely different type of investigation comparing to the other class of quaternionic unilateral polynomials. However, Bernstein's inequality is proved for some remarkable classes of such polynomials of degree less than or equal to 3 and for a class of polynomials of arbitrary degree having at most two distinct roots.

Exponential synchronization of a class of quaternion-valued neural network with time-varying delays: A Matrix Measure Approach

This article investigates the exponential synchronization (ES) of quaternion-valued neural networks (QVNNs) with time-varying delays. Global ES has been studied using matrix measure techniques. Due to the incompatibility of quaternion multiplication, the QVNN system is first separated into four real-valued systems. Then ES is established in the master-response systems of QVNN by constructing a state feedback controller. The stability criteria for the error system of QVNN is then provided using the existing lemmas and the Halanay inequality. Additionally, it is no longer assumed that the activation functions are differentiable, which make the analytical process more concise. The viability of our findings are demonstrated using two numerical examples. The present theoretical findings have been confirmed in the first case, while the second example includes the use of QVNNs connected to associative memory to demonstrate their ability to restore real color image patterns properly.

Reduced Biquaternion Windowed Linear Canonical Transform: Properties and Applications

The quaternion windowed linear canonical transform is a tool for processing multidimensional data and enhancing the quality and efficiency of signal and image processing; however, it has disadvantages due to the noncommutativity of quaternion multiplication. In contrast, reduced biquaternions, as a special case of four-dimensional algebra, possess unique advantages in computation because they satisfy the multiplicative exchange rule. This paper proposes the reduced biquaternion windowed linear canonical transform (RBWLCT) by combining the reduced biquaternion signal and the windowed linear canonical transform that has computational efficiency thanks to the commutative property. Firstly, we introduce the concept of a RBWLCT, which can extract the time local features of an image and has the advantages of both time-frequency analysis and feature extraction; moreover, we also provide some fundamental properties. Secondly, we propose convolution and correlation operations for RBWLCT along with their corresponding generalized convolution, correlation, and product theorems. Thirdly, we present a fast algorithm for RBWLCT and analyze its computational complexity based on two dimensional Fourier transform (2D FTs). Finally, simulations and examples are provided to demonstrate that the proposed transform effectively captures the local RBWLCT-frequency components with enhanced degrees of freedom and exhibits significant concentrations.

A SMARANDACHE CURVE ACCORDING TO THE DARBOUX FRAME AND THE KINEMATICS OF THIS CURVE IN MINKOWSKI SPACE

In this study, the curvatures and frames of the Smarandache curve formed by a timelike curve with unit velocity taken on a timelike surface according to the Darboux frame in Semi-Euclidean space were created. Then, the image curve on the screw surface was obtained by applying translational and rotational motion (screw motion) using the quaternion multiplication under the same frame of this curve..Finally, the curvature and torsion of the obtained image curve were calculated

Rational torsion points on abelian surfaces with quaternionic multiplication

Abstract Let A be an abelian surface over ${\mathbb {Q}}$ whose geometric endomorphism ring is a maximal order in a non-split quaternion algebra. Inspired by Mazur’s theorem for elliptic curves, we show that the torsion subgroup of $A({\mathbb {Q}})$ is $12$ -torsion and has order at most $18$ . Under the additional assumption that A is of $ {\mathrm{GL}}_2$ -type, we give a complete classification of the possible torsion subgroups of $A({\mathbb {Q}})$ .

A novel fixed-time error-monitoring neural network for solving dynamic quaternion-valued Sylvester equations

This paper addresses the dynamic quaternion-valued Sylvester equation (DQSE) using the quaternion real representation and the neural network method. To transform the Sylvester equation in the quaternion field into an equivalent equation in the real field, three different real representation modes for the quaternion are adopted by considering the non-commutativity of quaternion multiplication. Based on the equivalent Sylvester equation in the real field, a novel recurrent neural network model with an integral design formula is proposed to solve the DQSE. The proposed model, referred to as the fixed-time error-monitoring neural network (FTEMNN), achieves fixed-time convergence through the action of a state-of-the-art nonlinear activation function. The fixed-time convergence of the FTEMNN model is theoretically analyzed. Two examples are presented to verify the performance of the FTEMNN model with a specific focus on fixed-time convergence. Furthermore, the chattering phenomenon of the FTEMNN model is discussed, and a saturation function scheme is designed. Finally, the practical value of the FTEMNN model is demonstrated through its application to image fusion denoising.

The polarization quaternion and its applications: a joint representationof the Q/U Stokes parameters and E/B mode polarizations

In this work, we point out that the Q/U Stokes parameters and E/B modepolarizations are the four components of a unique quaternion, which describes atthe same time the directions and the parity states of spherical linearpolarizations. We then point out that, with this polarization quaternion, themathematical form of all Q/U and E/B transforms are greatly simplified, to anextent that requires only one quaternion multiplication for each transform. Apreliminary application of the polarization quaternion is shown as an example todetect peculiar pixel domain patterns within the E- and B-families, which arethe former and latter halves of the polarization quaternion.

Adaptive reweighted quaternion sparse learning for data recovery and classification

Sparse representation (SR) methods in quaternion space have been attracting increasing interests recently. However, most existing quaternion SR methods adopt the quaternion ℓ1 norm, which penalizes all the entries of the quaternion sparse vector equally and ignores the differences and significance of different entries. Ideally, the entries with large magnitude should be less penalized while those with small magnitude (such as zero entries) should be more penalized. Therefore, we propose an Adaptive Weighted Quaternion Sparse Representation (AWQSR) method in this paper, which can learn weights for distinct entries of the quaternion sparse entries in an adaptive manner. Due to the noncommutativity of quaternion multiplication, it is difficult to tackle the resulting optimization problem of AWQSR. For this reason, we devise an effective iteratively reweighted optimization algorithm based on quaternion operators. To further improve the classification performance, we also develop a Supervised AWQSR based Classification (SAWQSRC) method by leveraging the label information of training samples to learn discriminative weights. Theoretical analysis of SAWQSRC has also been established to show that SAWQSRC succeeds in classification under appropriate conditions. The experiments on simulated data and real data prove the validity of the proposed methods for quaternion signal recovery and classification.

Modeling Noncommutative Composition of Relations for Knowledge Graph Embedding

Knowledge Graph Embedding (KGE) is a powerful way to express Knowledge Graphs (KGs), which can help machines learn patterns hidden in the KGs. Relation patterns are useful hidden patterns, and they usually assist machines to predict unseen facts. Many existing KGE approaches can model some common relation patterns like symmetry/antisymmetry, inversion, and commutative composition patterns. However, most of them are weak in modeling noncommutative composition patterns. It means these approaches can not distinguish a lot of composite relations like “father’s mother” and “mother’s father”. In this work, we propose a new KGE method called QuatRotatScalE (QRSE) to overcome this weakness, since it utilizes rotation and scaling transformations of quaternions to design the relation embedding. Specifically, we embed the relations and entities into a quaternion vector space under the difference norm KGE framework. Since the multiplication of quaternions does not satisfy the commutative law, QRSE can model noncommutative composition patterns naturally. The experimental results on the synthetic dataset also support that QRSE has this ability. In addition, the experimental results on real-world datasets show that QRSE reaches state-of-the-art in link prediction problem.

A direct analysis method to state bounding for high‐order quaternion Hopfield neural networks with mixed delays and bounded disturbances

The main research in this paper is the state bounding problem of high‐order quaternion Hopfield neural networks (HOQHNNs) with time‐varying discrete delays, unbounded distributed delays, and bounded disturbances. In the research process, firstly, the considered HOQHNNs are decomposed into four equivalent real‐valued Hopfield neural networks (RHNNs) to solve the problem that quaternion multiplication is not commutative, and then, a sufficient condition related to time delays is given to ensure that the state trajectories that are contained in or globally exponentially convergent into a cuboid. Thereby, the corresponding reachable set estimation is obtained. Numerical examples are provided to illustrate validity of theoretical results. At present, the research on HOQHNNs with mixed time delays adopts Lyapunov–Krasovskii functional (LKF) method, which needs to not only choose the appropriate LKF but also take into account both conservatism and computational complexity. Besides, there is no literature to study the state bounding of HOQHNNs with mixed delays. The method suggested in the present text has three merits: (1) According to the definition of state bounding straightway, it has no use any LKF, which avoids massive calculations and solutions of high‐dimensional matrix inequalities; (2) it is available not only to HOQHNNs but also to RHNNs and complex‐valued Hopfield neural networks (CHNNs); and (3) make up for the blank of state bounding research of HOQHNNs.

МОДЕЛЬ РУХУ БЕЗПІЛОТНИХ ЛІТАЛЬНИХ АПАРАТІВ НА ОСНОВІ АЛГЕБРИ ДУАЛЬНИХ КВАТЕРНІОНІВ

The widespread use of unmanned aerial vehicles during warfare has intensified the problem of their management, especially when they are used in large groups. One of the main tasks is to ensure coordinated movement of the group's aircraft in space. Optimizing the movement of each device of the group in three-dimensional space is expedient to use mathematical models. The movement of any unmanned aerial vehicle can be presented as a combination of translational and rotational movements, and its speed as a combination of translational and rotational velocities. Previously, these movements were modeled separately using a system of differential equations or quaternions. In this article, a mathematical model of rotational and translational movements of an aircraft based on the algebra of dual quaternions is developed. Dual quaternions consisting of eight scalars are a compact representation of rigid transformations in space. Therefore, their properties determine the advantage in the course of motion simulation, as they reduce the amount of calculations. Thus, with the help of one dual quaternion, it is possible to provide both translational and rotational motions at once, and the operation of non-commutative multiplication of dual quaternions is used to simulate the movement. The model assumes that the real part of the dual quaternion determines the orientation of the UAV in space, and the dual part determines its position in three-dimensional space. In order to connect aircraft coordinate systems with the model, expressions for the transition from aircraft orientation angles (roll, yaw, and pitch) to dual quaternion parameters and vice versa are obtained. The functionality of the proposed model was confirmed using the developed software for modeling the coordinated movement of aircraft. The software is adapted for graphical display of a large number of aircraft in web browsers with WebGl support. Keywords: motion modeling; rotational and translational movement; unmanned aerial vehicles; quaternions; dual quaternions; algebra of quaternions.

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.

Settling-Time Estimation for Finite-Time Stabilization of Fractional-Order Quaternion-Valued Fuzzy NNs

This article solves the problems of finite-time control and settling-time estimation for fractional-order quaternion-valued fuzzy neural networks (FQFNNs) with time delays. A novel fractional differential inequality is established to estimate the settling time of the addressed system, which is more general and less conservative than the existing results. In addition, owing to the noncommutativity of multiplication of quaternions, the decomposition method is usually adopted to discuss the finite-time stabilization (FTS) of quaternion-valued neural networks, which inevitably doubles the dimensionality of the system and brings a great computational burden. To avoid the aforementioned issues, some new properties of the quaternion-valued signum function are presented for exploring the FTS of FQFNNs by the direct quaternion method without any decomposition. Then, 1-norm and 2-norm control strategies are designed to stabilize the addressed system in finite time, and several sufficient criteria are derived to ensure the FTS of FQFNNs. Finally, the validity of the obtained theoretical results and the superiority of the proposed estimation method are illustrated by numerical simulations.

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

Higher-order singular value decomposition (HOSVD) is one of the most efficient tensor decomposition techniques. It has the salient ability to represent high-dimensional data and extract features. On the other hand, 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. Theoretically, QHOSVD is a proper tensor generalization of the quaternion singular value decomposition (QSVD), and a proper quaternion generalization of the standard HOSVD. From the application point of view, the defined QHOSVD can be widely used in various visual data processing with color pixels. As examples, 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.

Construction of dual-generalized complex Fibonacci and Lucas quaternions

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.