Quaternion Algebra Research Articles

One-dimensional quaternion Laplace transform: Properties and its application to quaternion-valued differential equations

This paper is devoted to the study of the quaternion Laplace transform, which is a natural generalization of the classical Laplace transform using the quaternion algebra. We find that some of its properties such as derivative, convolution and correlation theorems are quite different from the corresponding properties of the classical Laplace transform. Other properties like linearity, shifting, scaling and derivative are also studied in detail. Finally, the utility of the quaternion Laplace transform for solving quaternion-valued differential equations is presented.

Quaternion Factorization Machines: A Lightweight Solution to Intricate Feature Interaction Modeling.

Due to the sparsity of available features in web-scale predictive analytics, combinatorial features become a crucial means for deriving accurate predictions. As a well-established approach, a factorization machine (FM) is capable of automatically learning high-order interactions among features to make predictions without the need for manual feature engineering. With the prominent development of deep neural networks (DNNs), there is a recent and ongoing trend of enhancing the expressiveness of FM-based models with DNNs. However, though better results are obtained with DNN-based FM variants, such performance gain is paid off by an enormous amount (usually millions) of excessive model parameters on top of the plain FM. Consequently, the heavy parameterization impedes the real-life practicality of those deep models, especially efficient deployment on resource-constrained Internet of Things (IoT) and edge devices. In this article, we move beyond the traditional real space where most deep FM-based models are defined and seek solutions from quaternion representations within the hypercomplex space. Specifically, we propose the quaternion factorization machine (QFM) and quaternion neural factorization machine (QNFM), which are two novel lightweight and memory-efficient quaternion-valued models for sparse predictive analytics. By introducing a brand new take on FM-based models with the notion of quaternion algebra, our models not only enable expressive inter-component feature interactions but also significantly reduce the parameter size due to lower degrees of freedom in the hypercomplex Hamilton product compared with real-valued matrix multiplication. Extensive experimental results on three large-scale datasets demonstrate that QFM achieves 4.36% performance improvement over the plain FM without introducing any extra parameters, while QNFM outperforms all baselines with up to two magnitudes' parameter size reduction in comparison to state-of-the-art peer methods.

Quaternion algebra approach to nonlinear Schrödinger equations with nonvanishing boundary conditions

In this article we apply quaternionic linear algebra and quaternionic linear system theory to develop the inverse scattering transform theory for the nonlinear Schrödinger equation with nonvanishing boundary conditions. We also determine its soliton solutions by using triplets of quaternionic matrices.

On singular value decomposition for split quaternion matrices and applications in split quaternionic mechanics

In the theoretical studies and numerical computations of split quaternionic quantum mechanics, the singular value decomposition theorem of split quaternion matrices plays an essential role. In general the problem of singular value decomposition over split quaternion algebra has hitherto remained tangential for split quaternion matrices. In this paper, by means of a real representation matrix of a split quaternion matrix, the singular value decomposition of split quaternion matrices is studied, the singular value decomposition theorem and the corresponding algorithm for split quaternion matrices are given. Finally, three applications for solving the split quaternion least squares problem and the wave separation problem are given based on the singular value decomposition over split quaternion algebra.

Gelfand–Kirillov dimension and the p-adic Jacquet–Langlands correspondence

Abstract We bound the Gelfand–Kirillov dimension of unitary Banach space representations of p-adic reductive groups, whose locally analytic vectors afford an infinitesimal character. We use the bound to study Hecke eigenspaces in completed cohomology of Shimura curves and p-adic Banach space representations of the group of units of a quaternion algebra over ℚ p {\mathbb{Q}_{p}} appearing in the p-adic Jacquet–Langlands correspondence, deducing finiteness results in favorable cases.

Splitting of differential quaternion algebras

We study differential splitting fields of quaternion algebras with derivations. A quaternion algebra over a field k is always split by a quadratic extension of k. However, a differential quaternion algebra need not be split over any algebraic extension of k. We use solutions of certain Riccati equations to provide bounds on the transcendence degree of splitting fields of a differential quaternion algebra.

Generalized Quaternions and Matrix Algebra

In this paper, we established the connection between generalized quaternion algebra and real (complex) matrix algebras by using Hamilton operators. We obtained real and complex matrices corresponding to the real and complex basis of the generalized quaternions. Also, we investigated the basis features of real and complex matrices. We get Pauli matrices corresponding to generalized quaternions. Then, we have shown that the algebra produced by these matrices is isomorphic to the Clifford algebra Cl(E_αβ^3) produced by generalized space E_αβ^3.
 Finally, we studied the relations among the symplectic matrices group corresponding to generalized unit quaternions, generalized unitary matrices group, and generalized orthogonal matrices group.

Compatibility of theta lifts and tempered condition

AbstractIn this note, assuming the nonvanishing result of explicit theta correspondence for the symplectic–orthogonal dual pair over quaternion algebra $\mathbb {H}$ , we show that, for metapletic–orthogonal dual pair over $\mathbb {R}$ and the symplectic–orthogonal dual pair over quaternion algebra $\mathbb {H}$ , the theta correspondence is compatible with tempered condition by directly estimating the matrix coefficients, without using the classification theorem.

Smoothing method for unit quaternion time series in a classification problem: an application to motion data

Smoothing orientation data is a fundamental task in different fields of research. Different methods of smoothing time series in quaternion algebras have been described in the literature, but their application is still an open point. This paper develops a smoothing approach for smoothing quaternion time series to obtain good performance in classification problems. Starting from an existing method which involves an angular velocity transformation of unit quaternion time series, a new method which employ the logarithm function to transform the quaternion time series to a real three-dimensional time series is proposed. Empirical evidences achieved on real data set and artificially noisy data sets confirm the effectiveness of the proposed method compared with the classical approach based on angular velocity transformation. The R functions developed for this paper will be provided in a Github repository.

Epsilon Dichotomy for Linear Models: The Archimedean Case

Abstract Let $G=\textrm {GL}_{2n}({\mathbb {R}})$ or $G=\textrm {GL}_n({\mathbb {H}})$ and $H=\textrm {GL}_n({\mathbb {C}})$ regarded as a subgroup of $G$. Here, ${\mathbb {H}}$ is the quaternion division algebra over ${\mathbb {R}}$. For a character $\chi $ on ${\mathbb {C}}^\times $, we say that an irreducible smooth admissible moderate growth representation $\pi $ of $G$ is $\chi _H$-distinguished if $\operatorname {Hom}_H(\pi , \chi \circ \det _H)\neq 0$. We compute the root number of a $\chi _H$-distinguished representation $\pi $ twisted by the representation induced from $\chi $. This proves an Archimedean analogue of the conjecture by Prasad and Takloo-Bighash (J. Reine Angew. Math., 2011). The proof is based on the analysis of the contribution of $H$-orbits in a flag manifold of $G$ to the Schwartz homology of principal series representations. A large part of the argument is developed for general real reductive groups of inner type. In particular, we prove that the Schwartz homology $H_\ast (H, \pi \otimes \chi )$ is finite-dimensional and hence it is Hausdorff for a reductive symmetric pair $(G, H)$ and a finite-dimensional representation $\chi $ of $H$.

Characteristic of Quaternion Algebra Over Fields

Quaternion is an extension of the complex number system. Quaternion are discovered by formulating 4 points in 4-dimensional vector space using the cross product between two standard vectors. Quaternion algebra over a field is a 4-dimensional vector space with bases and the elements of the algebra are members of the field. Each element in quaternion algebra has an inverse, despite the fact that the ring is not commutative. Based on this, the purpose of this study is to obtain the characteristics of split quaternion algebra and determine how it interacts with central simple algebra. The research method used in this paper is literature study on quaternion algebra, field and central simple algebra. The results of this study establish the equivalence of split quaternion algebra as well as the theorem relating central simple algebra and quaternion algebra. The conclusion obtained from this study is that split quaternion algebra has five different characteristics and quaternion algebra is a central simple algebra with dimensions less than equal to four.

An explicit comparison of anticyclotomic p-adic L-functions for Hida families

We provide comparison results for anticyclotomic [Formula: see text]-adic [Formula: see text]-functions attached to Hida families of modular forms. The main result is a comparison between the anticyclotomic restriction of the three variable [Formula: see text]-adic [Formula: see text]-function introduced by Skinner and Urban, and the anticyclotomic [Formula: see text]-function constructed by means of [Formula: see text]-adic families of Gross points in the setting of definite quaternion algebras. Several auxiliary results involving anticyclotomic [Formula: see text]-adic [Formula: see text]-functions introduced by Chida-Hsieh and Büyükboduk–Lei are also proved.

Multi-image encryption scheme with quaternion discrete fractional Tchebyshev moment transform and cross-coupling operation

With quaternion theory, the traditional discrete fractional Tchebyshev transform is extended to the quaternion algebra domain for multi-image processing. A new multi-image encryption scheme based on quaternion discrete fractional Tchebyshev moment transform (QDFrTMT) and the cross-coupling chaotic system is suggested. The original images are first confused by fractal sorting square matrix and sine-logistic exponential chaotic map, and then the resulting image is divided into four matrices. With the quaternion symplectic form representation, a quaternion signal can be formed by the divided matrices, which can reduce the number of the discrete fractional Tchebyshev transforms used and enhance the computational efficiency. Subsequently, the quaternion array is encrypted with the proposed QDFrTMT. To overcome the weakness of the permutation-diffusion operation based on a single chaotic map, a Logistic-sine exponential chaotic map and a piece-wise linear chaotic map are cross-coupled. The final ciphertext images can be acquired by carrying out the dual-layer encryption processes in horizontal and vertical directions. Numerical simulations and security analyses verify the effectiveness of the multi-image encryption algorithm and its strong ability to counteract common attacks.

A Novel Approach for Quaternion Algebra Based JSEG Color Texture Segmentation

In this work, a novel colour quantization approach has been applied to the JSEG colour texture segmentation using quaternion algebra. As a rule, the fundamental vectors of the colour space are derived by inverting the three RGB colour directions in the complex hyperplanes. In the proposed system, colour is represented as a quaternion because quaternion algebra provides a very intuitive means of working with homogeneous coordinates. This representation views a colour pixel as a point in the three-dimensional space. A novel quantization approach that makes use of projective geometry and level set methods has been produced as a consequence of the suggested model. The JSEG colour texture segmentation will use this technique. The new colour quantization approach utilises the binary quaternion moment preserving thresholding methodology, and is therefore a splintering clustering method. This method is used to segment the colour clusters found inside the RGB cube and the colour consistency throughout the spectrum and in the space are both considered. The results of the segmentation are compared with JSEG as well as with the most recent standard segmentation techniques. These comparisons show that the suggested quantization technique makes JSEG segmentation more robust.

Small diameters and generators for arithmetic lattices in textrm{SL}_2(mathbb {R}) and certain Ramanujan graphs

We show that arithmetic lattices in textrm{SL}_{2}(mathbb {R}), stemming from the proper units of an Eichler order in an indefinite quaternion algebra over mathbb {Q}, admit a ‘small’ covering set. In particular, we give bounds on the diameter if the quotient space is co-compact. Consequently, we show that these lattices admit small generators. Our techniques also apply to definite quaternion algebras where we show Ramanujan-strength bounds on the diameter of certain Ramanujan graphs without the use of the Ramanujan bound.

Quaternion-Valued Correlation Learning for Few-Shot Semantic Segmentation

Few-shot segmentation (FSS) aims to segment unseen classes given only a few annotated samples. Encouraging progress has been made for FSS by leveraging semantic features learned from base classes with sufficient training samples to represent novel classes. The correlation-based methods lack the ability to consider interaction of the two subspace matching scores due to the inherent nature of the real-valued 2D convolutions. In this paper, we introduce a quaternion perspective on correlation learning and propose a novel Quaternion-valued Correlation Learning Network (QCLNet), with the aim to alleviate the computational burden of high-dimensional correlation tensor and explore internal latent interaction between query and support images by leveraging operations defined by the established quaternion algebra. Specifically, our QCLNet is formulated as a hyper-complex valued network and represents correlation tensors in the quaternion domain, which uses quaternion-valued convolution to explore the external relations of query subspace when considering the hidden relationship of the support sub-dimension in the quaternion space. Extensive experiments on the PASCAL-5 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>i</i></sup> and COCO-20 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>i</i></sup> datasets demonstrate that our method outperforms the existing state-of-the-art methods effectively.

Quaternion convolutional neural networks for hyperspectral image classification

Quaternion convolutional neural networks (QCNNs) can capture quaternion features, which contain not only the contextual information among quaternion feature units but also utilize the quaternion algebra inside the quaternion feature units to express structural information. However, building efficient QCNNs for hyperspectral image (HSI) classification is a challenge due to the lack of methods to map real features into quaternion features and the missing key quaternion modules. This paper proposes methodologies to build QCNNs specifically for HSI classification and designs several key quaternion modules. Firstly, a novel quaternion feature encoder is presented to map real HSI features into quaternion features. Secondly, a paradigm is designed to conveniently modify classical convolutional neural networks to implement QCNNs for HSI classification. Thirdly, we propose a new separable quaternion convolutional neural network (SQNet) based on presented advanced quaternion modules, such as separable quaternion convolutions and quaternion attention mechanisms. In addition, the computational rules in quaternion neurons are improved to enhance the adaptability of QCNNs. Experiments conducted on four widely used HSI datasets indicate that the implemented QCNNs and the proposed SQNet achieve satisfactory results for HSI classification with limited training samples.

A robust and simple collision avoidance approach for pick and place robot applications

Purpose The aim of this study is to create a robust and simple collision avoidance approach based on quaternion algebra for vision-based pick and place applications in manufacturing industries, specifically for use with industrial robots and collaborative robots (cobots). Design/methodology/approach In this study, an approach based on quaternion algebra is developed to prevent any collision or breakdown during the movements of industrial robots or cobots in vision system included pick and place applications. The algorithm, integrated into the control system, checks for collisions before the robot moves its end effector to the target position during the process flow. In addition, a hand–eye calibration method is presented to easily calibrate the camera and define the geometric relationships between the camera and the robot coordinate systems. Findings This approach, specifically designed for vision-based robot/cobot applications, can be used by developers and robot integrator companies to significantly reduce application costs and the project timeline of the pick and place robotics system installation. Furthermore, the approach ensures a safe, robust and highly efficient application for robotics vision applications across all industries, making it an ideal solution for various industries. Originality/value The algorithm for this approach, which can be operated in a robot controller or a programmable logic controller, has been tested as real-time in vision-based robotics applications. It can be applied to both existing and new vision-based pick and place projects with industrial robots or collaborative robots with minimal effort, making it a cost-effective and efficient solution for various industries.

Theories and applications associated with biquaternion linear canonical transform

The quaternion linear canonical transform (QLCT) has been widely used in color image processing. Biquaternion is a more generalized algebra of quaternion, which has attracted scholars' research interest in recent years. In this paper, a new transform is proposed called the biquaternion linear canonical transforms (BiQLCTs). Due to the noncommutativity of biquaternion algebra multiplication, there are three different types of the BiQLCTs: Left‐sided BiQLCT, right‐sided BiQLCT, and two‐side BiQLCT. The transforms are the extension of the complex linear canonical transforms. Then, the relationships between the three kinds of transforms are obtained. Next, based on the right‐side biquaternion linear canonical transform (RBiQLCT), some general properties of this transform are proved. Moreover, the convolution and correlation theorems of the RBiQLCT are studied. As an application, according to the convolution operator and convolution theorem, the biquaternion linear time‐invariant system is analyzed. Finally, the Heisenberg uncertainty principle for the RBiQLCT is established.

Robust Image Watermarking With Quaternion Fractional-Order Polar Harmonic-Fourier Moments Based On Wavelet Transformation: Resistance Against Rotation Attacks

This study presents a zero-watermarking algorithm that can resist rotation attacks. The algorithm uses quaternion fractional-order polar harmonic-Fourier moments (QFr-PHFMs) based on wavelet-transformation. First, the wavelet-transformation is applied to each component of the host image, which is in RGB three-channel color. The low-frequency sub-bands of each component are then extracted and represented using quaternion algebra. Multiple QFr-PHFMs are calculated, and the invariants of the QFr-PHFMs are utilized to establish the watermark system. The watermark extraction process is also simplified. The detection of the image requires a two-level wavelet transformation, followed by the calculation of multiple invariant moments of the low-frequency sub-image. The experimental results are shown and compared with similar methods. Simulations show that this method can produce high-quality visual effects and withstand noise, filtering, JPEG compression, and cropping attacks.