Quaternion Algebra Research Articles

Optimal spinor selectivity for quaternion Bass orders

Let A be a quaternion algebra over a number field F, and O be an OF-order of full rank in A. Let K be a quadratic field extension of F that embeds into A, and B be an OF-order in K. Suppose that O is a Bass order that is well-behaved at all the dyadic primes of F. We provide a necessary and sufficient condition for B to be optimally spinor selective for the genus of O. This partially generalizes previous results on optimal (spinor) selectivity by C. Maclachlan (2008) [21] for Eichler orders of square-free levels, and independently by M. Arenas et al. (2018) [1] and by J. Voight (2021) [27, Chapter 31] for Eichler orders of arbitrary levels.

A Quaternion Gated Recurrent Unit Neural Network for Sensor Fusion

Recurrent Neural Networks (RNNs) are known for their ability to learn relationships within temporal sequences. Gated Recurrent Unit (GRU) networks have found use in challenging time-dependent applications such as Natural Language Processing (NLP), financial analysis and sensor fusion due to their capability to cope with the vanishing gradient problem. GRUs are also known to be more computationally efficient than their variant, the Long Short-Term Memory neural network (LSTM), due to their less complex structure and as such, are more suitable for applications requiring more efficient management of computational resources. Many of such applications require a stronger mapping of their features to further enhance the prediction accuracy. A novel Quaternion Gated Recurrent Unit (QGRU) is proposed in this paper, which leverages the internal and external dependencies within the quaternion algebra to map correlations within and across multidimensional features. The QGRU can be used to efficiently capture the inter- and intra-dependencies within multidimensional features unlike the GRU, which only captures the dependencies within the sequence. Furthermore, the performance of the proposed method is evaluated on a sensor fusion problem involving navigation in Global Navigation Satellite System (GNSS) deprived environments as well as a human activity recognition problem. The results obtained show that the QGRU produces competitive results with almost 3.7 times fewer parameters compared to the GRU.

The continuous quaternion algebra-valued wavelet transform and the associated uncertainty principle

This article aims to extend the wavelet transform to quaternion algebra using the kernel of the two-sided quaternion Fourier transform. We study some fundamental properties of this extension such as scaling, translation, rotation, and Parseval’s identity, then we derive the associated Heisenberg–Pauli–Weyl uncertainty principle UP. Finally, using the quaternion Fourier representation of the CQWT we generalize the logarithmic UP to the CQWT domain.

On application of algebra of quaternions II

On application of algebra of quaternions II

On application of algebra of quaternions I

On application of algebra of quaternions I

Dual Hyperquaternion Poincaré Groups

A new representation of the Poincaré groups in n dimensions via dual hyperquaternions is developed, hyperquaternions being defined as a tensor product of quaternion algebras (or a subalgebra thereof). This formalism yields a uniquely defined product and simple expressions of the Poincaré generators, with immediate physical meaning, revealing the algebraic structure independently of matrices or operators. An extended multivector calculus is introduced (allowing a possible sign change of the metric or of the exterior product). The Poincaré groups are formulated as a dual extension of hyperquaternion pseudo-orthogonal groups. The canonical decomposition and the invariants are discussed. As concrete example, the 4D Poincaré group is examined together with a numerical application. Finally, the hyperquaternion representation is compared to the quantum mechanical one. Potential applications include in particular, moving reference frames and computer graphics.

Augmented Online Sequential Quaternion Extreme Learning Machine

Online sequential extreme learning machine (OS-ELM) is one of the most popular real-time learning strategy for feedforward neural networks with single hidden layer due to its fast learning speed and excellent generalization ability. When dealing with quaternion signals, traditional real-valued learning models usually provide only suboptimal solutions compared with their quaternion-valued counterparts. However, online sequential quaternion extreme learning machine (OS-QELM) model is still lacking in literature. To fill this gap, this paper aims to establish a framework for the derivation and the design of OS-QELM. Specifically, we first derive a standard OS-QELM, and then propose two augmented OS-QELM models which can capture the complete second-order statistics of noncircular quaternion signals. The corresponding regularized models and two approaches to reducing the computational complexity are also derived and discussed respectively. Benefiting from the quaternion algebra and the augmented structure, the proposed models exhibit superiority over OS-ELM in simulation results on several benchmark quaternion regression problems and colour face recognition problems.

Homotopy nilpotency of some homogeneous spaces

Let {mathbb {K}}={mathbb {R}},,{mathbb {C}}, the field of reals or complex numbers and {mathbb {H}}, the skew {mathbb {R}}-algebra of quaternions. We study the homotopy nilpotency of the loop spaces Omega (G_{n,m}({mathbb {K}})), Omega (F_{n;n_1,ldots ,n_k}({mathbb {K}})), and Omega (V_{n,m}({mathbb {K}})) of Grassmann G_{n,m}({mathbb {K}}), flag F_{n;n_1,ldots ,n_k}({mathbb {K}}) and Stiefel V_{n,m}({mathbb {K}}) manifolds. Additionally, homotopy nilpotency classes of p-localized Omega (G^+_{n,m}({mathbb {K}})_{(p)}) and Omega (V_{n,m}({mathbb {K}})_{(p)}) for certain primes p are estimated, where G^+_{n,m}({mathbb {K}})_{(p)} is the oriented Grassmann manifolds. Further, the homotopy nilpotency classes of loop spaces of localized homogeneous spaces given as quotients of exceptional Lie groups are investigated as well.

Wide-sense Markov signals on the tessarine domain. A study under properness conditions

The quaternion algebra is not always the best choice for processing 4D hypercomplex signals. This paper aims to explore tessarines as an alternative algebra to solve the estimation problem. More concretely, wide-sense Markov signals in the tessarine domain are introduced and their properties under properness properties are analyzed. Firstly, the T2-properness condition in the tessarine setting is defined and then, the linear estimation problem under tessarine processing is addressed. The equivalence between the optimal estimator based on tessarine widely linear processing and the one based on tessarine T2 processing is proved, thus attaining a notable reduction in computational burden. Next, the Ti-proper wide-sense Markov signals, i=1,2, are defined and a forwards representation for modeling them is suggested. Finally, the estimation problem with intermittent observations for this class of signals is tackled. Specifically, based on the forwards representation, two algorithms for the problems of filtering, prediction and fixed-interval smoothing are devised. Numerical simulations are developed where the superiority of the Ti estimators, i=1,2, over their counterparts in the quaternion domain is shown.

A new eigenspace characterization of split-quaternion involutions

It is well known that a general quaternion algebra over a field F of characteristic different from 2 is either a division algebra or is split, which means isomorphic to the algebra of -matrices with entries from F. Since we have characterized in [Lawson J, Kizil E. Characterization of automorphic and anti-automorphic involutions of the quaternions. Linear Multilin Algebra, Published online. 04/2020] automorphic and anti-automorphic involutions of the division algebra of Hamilton's quaternions, we treat in this paper split-quaternions and obtain characterizations of the class of involutions of the algebra of split quaternions both in terms of inner automorphisms and of involution eigenspaces. These descriptions carry over to involutive automorphisms.

A central quaternionic Nullstellensatz

Let I be a proper left ideal in the ring H[x1,…,xn] of polynomials in n central variables over the quaternion algebra H. Then there exists a point a=(a1,…,an)∈Hn with aiaj=ajai for all i,j, such that every polynomial in I vanishes at a. This generalizes a theorem of Jacobson, who proved the case n=1. Moreover, a polynomial f∈H[x1,…,xn] vanishes at all common zeroes of polynomials in I if and only if f belongs to the intersection of all completely prime left ideals that contain I – a notion introduced by Reyes in 2010.

Kinematic modeling of Rytov's law and electromagnetic curves in the optical fiber based on elliptical quaternion algebra

In this paper, we aim a kinematic method to investigate the behavior of polarized light in an optical fiber. Geometric phase equations are obtained using elliptic quaternions. The behavior of polarized light is studied for three conditions where the angle between the polarization vector (electric field) and the Darboux frame fields are constant on the ellipsoid. For these conditions, the polarization vector experiences a homothetic motion on the ellipsoid. Moreover, this motion can be defined by the Fermi Walker parallelism rule. Also, the traced curves of the polarization vector (Rytov curves) are calculated by homothetic motions and Elliptic quaternions. Additionally, the magnetic vector field related to the polarization vector is derived. Then the characterizations of the electromagnetic curve are obtained. To reinforce the theory, the homothetic motions of the polarization vector on the ellipsoid are visualized by giving examples for each case.

Quaternion Processing Techniques for Color Synthesized NDT Thermography

Infrared thermography is a widely used technology that has been successfully applied to many and varied applications. These applications include the use as a non-destructive testing tool to assess the integrity state of materials. The current level of development of this application is high and its effectiveness is widely verified. There are application protocols and methodologies that have demonstrated a high capacity to extract relevant information from the captured thermal signals and guarantee the detection of anomalies in the inspected materials. However, there is still room for improvement in certain aspects, such as the increase of the detection capacity and the definition of a detailed characterization procedure of indications, that must be investigated further to reduce uncertainties and optimize this technology. In this work, an innovative thermographic data analysis methodology is proposed that extracts a greater amount of information from the recorded sequences by applying advanced processing techniques to the results. The extracted information is synthesized into three channels that may be represented through real color images and processed by quaternion algebra techniques to improve the detection level and facilitate the classification of defects. To validate the proposed methodology, synthetic data and actual experimental sequences have been analyzed. Seven different definitions of signal-to-noise ratio (SNR) have been used to assess the increment in the detection capacity, and a generalized application procedure has been proposed to extend their use to color images. The results verify the capacity of this methodology, showing significant increments in the SNR compared to conventional processing techniques in thermographic NDT.

Modeling of Inverse Kinematic of 3-DoF Robot, Using Unit Quaternions and Artificial Neural Network

SUMMARYThis paper presents a novel method for modeling a 3-degree of freedom open kinematic chain using quaternions algebra and neural network to solve the inverse kinematic problem. The structure of the network was composed of 3 hidden layers with 25 neurons per layer and 1 output layer. The network was trained using the Bayesian regularization backpropagation. The inverse kinematic problem was modeled as a system of six nonlinear equations and six unknowns. Finally, both models were tested using a straight path to compare the results between the Newton–Raphson method and the network training.

A Survey on Quaternion Algebra and Geometric Algebra Applications in Engineering and Computer Science 1995–2020

Geometric Algebra (GA) has proven to be an advanced language for mathematics, physics, computer science, and engineering. This review presents a comprehensive study of works on Quaternion Algebra and GA applications in computer science and engineering from 1995 to 2020. After a brief introduction of GA, the applications of GA are reviewed across many fields. We discuss the characteristics of the applications of GA to various problems of computer science and engineering. In addition, the challenges and prospects of various applications proposed by many researchers are analyzed. We analyze the developments using GA in image processing, computer vision, neurocomputing, quantum computing, robot modeling, control, and tracking, as well as improvement of computer hardware performance. We believe that up to now GA has proven to be a powerful geometric language for a variety of applications. Furthermore, there is evidence that this is the appropriate geometric language to tackle a variety of existing problems and that consequently, step-by-step GA-based algorithms should continue to be further developed. We also believe that this extensive review will guide and encourage researchers to continue the advancement of geometric computing for intelligent machines.

Perturbation analysis of matrices over a quaternion division algebra

Perturbation analysis of matrices over a quaternion division algebra

Biologically-Inspired Legged Robot Locomotion Controlled With a BCI by Means of Cognitive Monitoring

Brain-computer interfaces (BCI) are a mechanism to record the electrical signals of the brain and translate them into commands to operate an output device like a robotic system. This article presents the development of a real-time locomotion system of a hexapod robot with bio-inspired movement dynamics inspired in the stick insect and tele-operated by cognitive activities of motor imagination. Brain signals are acquired using only four electrodes from a BCI device and sent to computer equipment for processing and classification by the iQSA method based on quaternion algebra. A structure consisting of three main stages are proposed: (1) signal acquisition, (2) data analysis and processing by the iQSA method, and (3) bio-inspired locomotion system using a Spiking Neural Network (SNN) with twelve neurons. An off-line training stage was carried out with data from 120 users to create the necessary decision rules for the iQSA method, obtaining an average performance of 97.72%. Finally, the experiment was implemented in real-time to evaluate the performance of the entire system. The recognition rate to achieve the corresponding gait pattern is greater than 90% for BCI, and the time delay is approximately from 1 to 1.5 seconds. The results show that all the subjects could generate their desired mental activities, and the robotic system could replicate the gait pattern in line with a slight delay.

AutoAssociative Memory for Face Recognition using High-Dimensional Octonion Data

Uniqueness or individuality of an individual face is the representation of one’s identity. Recent research in color image processing focusing on octonion algebra, which is the generalization of complex and quaternion algebra. In this paper, the face of an individual is used for identification making automatically. For this, an autoassociative memory is used for the recognition of images or videos with 8-dimensions. Octonion algebra is not commutative and not associative but it is an alternate algebra for representing the phase and angle of images effectively. The prospective experiments are conducted on color images for image reconstruction, denoising and efficiently retrieving blurred images in true color images.

Fascinating number sequences from fourth order difference equation via quaternion algebras

Fascinating number sequences from fourth order difference equation via quaternion algebras

Quaternionic analysis, representation theory and physics II

We develop further quaternionic analysis introducing left and right doubly regular functions. We derive Cauchy-Fueter type formulas for these doubly regular functions that can be regarded as another counterpart of Cauchy's integral formula for the second order pole, in addition to the one studied in the first paper with the same title. We also realize the doubly regular functions as a subspace of the quaternionic-valued functions satisfying a Euclidean version of Maxwell's equations for the electromagnetic field. Then we return to the study of the original quaternionic analogue of Cauchy's second order pole formula and its relation to the polarization of vacuum. We find the decomposition of the space of quaternionic-valued functions into irreducible components that include the spaces of doubly left and right regular functions. Using this decomposition, we show that a regularization of the vacuum polarization diagram is achieved by subtracting the component corresponding to the one-dimensional subrepresentation of the conformal group. After the regularization, the vacuum polarization diagram is identified with a certain second order differential operator which yields a quaternionic version of Maxwell equations. Next, we introduce two types of quaternionic algebras consisting of spaces of scalar-valued and quaternionic-valued functions. We emphasize that these algebra structures are invariant under the action of the conformal Lie algebra. This uses techniques from our study of the vacuum polarization diagram. These algebras are not associative, but we can define an infinite family of n-multiplications, and we conjecture that they have structures of weak cyclic A-infinity algebras. We also conjecture the relation between the multiplication operations of the scalar and non-scalar quaternionic algebras with the n-photon Feynman diagrams in the scalar and ordinary conformal QED.