Quaternionic Theory Research Articles

Image analysis by fractional-order weighted spherical Bessel-Fourier moments

Moment and moment invariants as effective feature descriptors have been widely applied in image analysis, pattern recognition and computer vision applications. Scholars have demonstrated that fractional-order orthogonal moments outperform their corresponding moments in representing the fine details of a given image. To this end, we propose a new fractional-order orthogonal moments based on fractional-order weighted spherical Bessel polynomial in this paper, which is named as Fractional-order weighted Spherical Bessel-Fourier Moments (FrSBFMs). Moreover, to address the issue of color image analysis, FrSBFMs are integrated with quaternion theory to develop Quaternion FrSBFMs (QFrSBFMs). In addition, the rotation invariance and parameter α analysis of FrSBFMs and QFrSBFMs are discussed in detail. The experimental results show the effectiveness of the proposed FrSBFMs and QFrSBFMs in terms of image reconstruction capability, pattern recognition accuracy and zero-watermark verification.

Book review: “Hurwitz’s Lectures on the Number Theory of Quaternions” by Nicola Oswald and Jörn Steuding

Book review: “Hurwitz’s Lectures on the Number Theory of Quaternions” by Nicola Oswald and Jörn Steuding

Real and complex solutions of the total least squares problem in commutative quaternionic theory

Real and complex solutions of the total least squares problem in commutative quaternionic theory

Synchronization Analysis for Quaternion-Valued Delayed Neural Networks with Impulse and Inertia via a Direct Technique

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.

Dual quaternion theory over HGC numbers

Knowing the applications of quaternions in various fields, such as robotics, navigation, computer visualization and animation, in this study, we give the theory of dual quaternions considering Hyperbolic-Generalized Complex (HGC) numbers as coefficients via generalized complex and hyperbolic numbers. We account for how HGC number theory can extend dual quaternions to HGC dual quaternions. Some related theoretical results with HGC Fibonacci/Lucas numbers are established, including their dual quaternions. Given HGC Fibonacci/Lucas numbers, their special matrix correspondences have been identified and these are carried out to HGC Fibonacci/Lucas dual quaternions. Furthermore, we provide a more accurate way to quickly calculate HGC Fibonacci numbers and associate this with HGC generalized Fibonacci numbers. For implementation, we produce an algorithm in Maple. Lastly, we put the theory into practice.

Quaternion Chromaticity Contrast Preserving Decolorization Method Based on Adaptive Singular Value Weighting

Color image decolorization can not only simplify the complexity of image processing and analysis, improving computational efficiency, but also help to preserve the key information of the image, enhance visual effects, and meet various practical application requirements. However, with existing decolorization methods it is difficult to simultaneously maintain the local detail features and global smooth features of the image. To address this shortcoming, this paper utilizes singular value decomposition to obtain the hierarchical local features of the image and utilizes quaternion theory to overcome the limitation of existing color image processing methods that ignore the correlation between the three channels of the color image. Based on this, we propose a singular value adaptive weighted fusion quaternion chromaticity contrast preserving decolorization method. This method utilizes the low-rank matrix approximation principle to design a singular value adaptive weighted fusion strategy for the three channels of the color image and implements image decolorization based on singular value adaptive weighting. To address the deficiency of the decolorization result obtained in this step, which cannot maintain global smoothness characteristics well, a contrast preserving decolorization algorithm based on quaternion chromaticity distance is further proposed, and the global weighting strategy obtained by this algorithm is integrated into the image decolorization based on singular value adaptive weighting. The experimental results show that the decolorization method proposed in this paper achieves excellent results in both subjective visual perception and objective evaluation metrics.

Cubic Sublattices

A sublattice of the three-dimensional integer lattice Z3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {Z}}^3$$\\end{document} is called cubic sublattice if there exists a basis of the sublattice whose elements are pairwise orthogonal and of equal lengths. We show that for an integer vector v∈Z3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extbf{v}}\\in {\\mathbb {Z}}^3$$\\end{document} whose squared length is divisible by d2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d^2$$\\end{document}, there exists a cubic sublattice containing v\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extbf{v}}$$\\end{document} with edge length d. This improves one of the main result of a paper of Goswick et al. (J. Number Theory 132(1), 37–53 (2012)), where similar theorem was proved by using the decomposition theory of Hurwitz integral quaternions. We give an elementary proof heavily using cross product. This method allows us to characterize the cubic sublattices.

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.

Quasi-synchronization of discrete-time fractional-order quaternion-valued memristive neural networks with time delays and uncertain parameters

In this paper, quasi-synchronization of discrete-time fractional-order quaternion-valued memristive neural networks (DTFQMNNs) with time delays and uncertain parameters is investigated. Firstly, some related preliminaries are introduced and two new inequalities are obtained based on nabla fractional difference, quaternion theory and nabla h-Laplace transform. Then, a quaternion-valued controller with time delay is designed to realize quasi-synchronization goal, and some sufficient criteria are derived to guarantee quasi-synchronization of DTFQMNNs by using our created inequalities and some analysis techniques. Finally, a numerical example is provided to verify the correctness of our theoretical results.

Spin and angular momentum in quaternionic quantum mechanics

We present two novel solutions of real Hilbert state quaternionic quantum mechanics . Firstly, we observe that the angular momentum operator admits two different classes of physically non-equivalent free particles. As a second result, we study the Larmor precession to observe that it has a quaternionic solution where a novel phenomenological interpretation is possible, as well as a different spin is possible, and these results may encourage experimental and theoretical investigations of the quaternionic theory.

Kinematics Modeling and Singularity Analysis of a 6-DOF All-Metal Vibration Isolator Based on Dual Quaternions

Driven by the need for impact resistance and vibration reduction for mechanical devices in extreme environments, an all-metal vibration isolator with 6-degree-of-freedom (6-DOF) motion that is horizontally symmetrical was developed. Its stiffness and damping ability are provided by oblique springs in symmetrical arrangement and a metal–rubber elasto-porous damper. The spring is symmetrically distributed in the center axis of the support load surface. It is necessary to investigate the kinematics and the singularity before conducting multi-body dynamics analysis of the vibration isolator. Based on the theory of dual quaternions, the forward kinematics equations of the isolator were successively derived for theoretical kinematics modeling. In addition, an enhanced Broyden numerical iterative algorithm was developed and applied to the numerical solution of the forward kinematics equations of the vibration isolator. Compared with the traditional rotation-matrix method and Newton–Raphson method, the computational efficiency of the enhanced Broyden numerical iterative algorithm was increased by 680% and 290%, respectively. This was due to the enhanced algorithm without the calculations of any inverse matrix and forward kinematics equations. Finally, according to the forward kinematics Jacobian matrix, the position-singularity trajectory at a given orientation and the orientation-singularity space at a given position are calculated, which provides a basis for the algorithm of the 6-DOF vibration isolator to avoid singular positions and orientations.

Dual quaternion-based kinematic modeling for decoupling identification of geometric errors of rotary axes in five-axis platforms

Identification of the geometric errors of rotary axes is important for deciding the machining accuracy of five-axis platforms. In this paper, a programmable identification method is proposed to decouple the position-dependent and position-independent geometric errors (PDGEs and PIGEs) of the rotation axis. First, the kinematics model of the five-axis platform is developed by dual quaternions theory, and then the physical significance of left multiplication and right multiplication of PDGEs error model is analyzed. The motion difference between the base coordinate system and its local coordinate system is compared by simulation analysis. The geometric relationship between rotation direction and motion elements is established based on the error model, which is used to design an optimization algorithm for identifying the orientation errors of PIGEs. Additionally, the effect of linear axis PIGEs is eliminated by equivalent replacement, and a sample regression model is developed to identify the offset errors of PIGEs. The PDGEs are separated considering the Motion First principle. Finally, the proposed model and identification method were verified on a five-axis dispensing platform.

Quasi-projective synchronization of discrete-time fractional-order quaternion-valued neural networks

In this article, without decomposing the quaternion-valued neural networks (QVNNs) into two complex-valued subsystems or four real-valued subsystems, quasi-projective synchronization of discrete-time fractional-order QVNNs is investigated. To this end, the sign function for quaternion number is introduced and some related properties are given. Then, two inequalities are built according to the nabla fractional difference and quaternion theory. Subsequently, a simple linear quaternion-valued controller is designed, and some synchronization conditions are given by means of our created inequalities. Finally, numerical simulations are given to prove the feasibility and correctness of the theoretical results.

Algebraic methods for equality constrained least squares problems in commutative quaternionic theory

This paper, by means of two matrix representations of a commutative quaternion matrix, studies the relationship between the solutions of commutative quaternion equality constrained least squares (LSE) problems and that of complex and real LSE problems and derives two algebraic methods for finding the solutions of equality constrained least squares problems in commutative quaternionic theory.

Quaternion Capsule Neural Network With Region Attention for Facial Expression Recognition in Color Images

Facial expression recognition (FER) is an essential subject of computer vision and human-computer interaction. It has been reported that many factors are closely related to the FER performance such as the pose, facial muscle variations, and the ignored color information in facial images. In this study, we propose a quaternion capsule neural network (Q-CapsNet) with region attention mechanism for FER in color images. The proposed Q-CapsNet is an end-to-end deep learning framework, which adopts the concept of quaternion theory to the Capsule Neural Network and further uses an attention mechanism to focus on the facial region of interests (ROIs). The Q-CapsNet addresses the FER problems in the following aspects. First, the internal dependencies between color channels are captured by the quaternion technique, which is of great importance in color image processing. Second, the Q-CapsNet could disentangle latent geometry information from facial images by quaternion capsules and quaternion routing algorithm. Third, inspired by the fact that facial expressions are mainly determined by several vital facial regions, the region attention mechanism is introduced to extract emotional features from the ROIs of the face. The proposed Q-CapsNet is evaluated on four public color FER datasets including MMI, Oulu-CASIA, RAF-DB, and SFEW. The comprehensive experiments and visualization results demonstrate that the proposed network outperforms comparative models and the state-of-the-art FER methods.

GCN Katsayılı Dual Kuaterniyonlar için Cebirsel Yapı

In this paper, we explain how dual quaternion theory can extend to dual quaternions with generalized complex number (GCN) components. More specifically, we algebraically examine this new type dual quaternion and give several matrix representations both as a dual quaternion and as a GCN.

Quaternionic scalar field in the real Hilbert space

Using the complex Klein–Gordon field as a model, we quantize the quaternionic scalar field in the real Hilbert space. The Lagrangian formulation has accordingly been obtained, as well as the hamiltonian formulation, and the energy and charge operators. Conversely to the complex case, the quaternionic quantization admits two quantization schemes, concerning either two or four components. Therefore, the quaternionic field permits a richer structure of states, if compared to the complex scalar field case. Moreover, the quaternionic theory admits as a further novel feature a non-associative algebraic structure in their complex components, something not observed in the complex case.

New Sampling Expansion Related to Derivatives in Quaternion Fourier Transform Domain

The theory of quaternions has gained a firm ground in recent times and is being widely explored, with the field of signal and image processing being no exception. However, many important aspects of quaternionic signals are yet to be explored, particularly the formulation of Generalized Sampling Expansions (GSE). In the present article, our aim is to formulate the GSE in the realm of a one-dimensional quaternion Fourier transform. We have designed quaternion Fourier filters to reconstruct the signal, using the signal and its derivative. Since derivatives contain information about the edges and curves appearing in images, therefore, such a sampling formula is of substantial importance for image processing, particularly in image super-resolution procedures. Moreover, the presented sampling expansion can be applied in the field of image enhancement, color image processing, image restoration and compression and filtering, etc. Finally, an illustrative example is presented to demonstrate the efficacy of the proposed technique with vivid simulations in MATLAB.

A Musculoskeletal Multibody Algorithm Based on a Novel Rheonomic Constraints Definition Applied to the Lower Limb.

In this paper, a multibody model was developed in the framework of biotribology of lower limb artificial joints. The presented algorithm performs the inverse dynamics of musculoskeletal systems with the aim to achieve a tool for the calculation of the joint reaction forces. The revolute joint, the cam joint, the spherical joint and the free joint were considered in the analyzed lower limb system by introducing a novel analytical formulation of the rheonomic constraint equations based on the quaternions theory. Within the kinematical analysis, the curved muscle paths were modeled by simulating their geodesic wrapping over bony surfaces while the muscle actuations were formulated through the Hill muscle model. The developed theoretical model was developed in matlab environment allowing to follow the classical musculoskeletal analysis pipeline: kinematical analysis, inverse dynamics, and static optimization, applied to the lower limb during the gait kinematics. The validation of the results was obtained by comparing the calculated hip joint reactions with the ones obtained in vivo by Bergmann and calculated by Opensim software, showing a satisfactory agreement. The proposed model and algorithm represent a fully open and controllable synovial joint tribological configuration generator tool, useful to be coupled with numerical lubrication/contact models in the framework of the in silico artificial joints tribological optimization.

Privacy-Preserving Color Image Feature Extraction by Quaternion Discrete Orthogonal Moments

To implement image storage and computation in cloud servers without violating users’ privacy, privacy-preserving feature extraction has been a new research interest. The existing works are mainly designed for grayscale images. For color images, they tend to convert them to grayscale images or obtain the results of the combination of single-channel processes. While the capabilities of features extracted from the encrypted color images will be affected if color information and inter-relationship between color channels are ignored. To fully preserve features of color images, we introduce quaternion theory to encode each color image and propose an improved vector homomorphic encryption scheme (IVHE) to encrypt quaternion-based color images. IVHE helps protect image content and keep vector characteristics of color images. Based on IVHE, the framework for feature extraction of privacy-preserving Quaternion Discrete Orthogonal Moments (PPQDOMs) is presented. Theoretical analyses prove that Quaternion Discrete Orthogonal Moments (QDOMs) can be extracted from the encrypted color images by PPQDOMs. Furthermore, we apply three common Discrete Orthogonal Moments to the proposed framework to evaluate its performance. Experimental results demonstrate that the proposed framework can protect color image content and perform well compared to QDOMs in image reconstruction and image recognition.