Quaternionic Theory Research Articles

Quaternion methods and models of regular celestial mechanics and astrodynamics

This paper is a review, which focuses on our work, while including an analysis of many works of other researchers in the field of quaternionic regularization. The regular quaternion models of celestial mechanics and astrodynamics in the Kustaanheimo-Stiefel (KS) variables and Euler (Rodrigues-Hamilton) parameters are analyzed. These models are derived by the quaternion methods of mechanics and are based on the differential equations of the perturbed spatial two-body problem and the perturbed spatial central motion of a point particle. This paper also covers some applications of these models. Stiefel and Scheifele are known to have doubted that quaternions and quaternion matrices can be used efficiently to regularize the equations of celestial mechanics. However, the author of this paper and other researchers refuted this point of view and showed that the quaternion approach actually leads to efficient solutions for regularizing the equations of celestial mechanics and astrodynamics.This paper presents convenient geometric and kinematic interpretations of the KS transformation and the KS bilinear relation proposed by the present author. More general (compared with the KS equations) quaternion regular equations of the perturbed spatial two-body problem in the KS variables are presented. These equations are derived with the assumption that the KS bilinear relation was not satisfied. The main stages of the quaternion theory of regularizing the vector differential equation of the perturbed central motion of a point particle are presented, together with regular equations in the KS variables and Euler parameters, derived by the aforementioned theory. We also present the derivation of regular quaternion equations of the perturbed spatial two-body problem in the Levi-Civita variables and the Euler parameters, developed by the ideal rectangular Hansen coordinates and the orientation quaternion of the ideal coordinate frame.This paper also gives new results using quaternionic methods in the perturbed spatial restricted three-body problem.

Three dimensional stabilization controller based on improved quaternion transformation for fixed-wing UAVs

To realize the three-dimensional stabilization control of fixed-wing unmanned aerial vehicles (UAVs), we analyze the nonholonomic characteristics, constraint non-integrability, and controllability of UAVs. To simplify the trigonometric function term in the dynamics of the UAV and avoid the singularity problem of the Euler angle in describing attitude, we use the quaternion theory to transform the dynamics of the UAV to avoid the complex trigonometric function derivation, which makes the dynamic matrix more concise. Based on this, a continuous periodic time-varying controller (CPTVC) is designed, and the effectiveness of the controller is proved using the homogeneous method. Finally, the results of the hardware in a loop simulation indicated that the exponential stability provided by the feedback controller can realize the three-dimensional stabilization of any initial position.

Quaternion discrete fractional Krawtchouk transform and its application in color image encryption and watermarking

This paper extends the traditional discrete fractional Krawtchouk transform (DFrKT) with quaternion theory to quaternion discrete fractional Krawtchouk transform (QDFrKT) for the purpose of color image processing. Also, the implementation of QDFrKT is given from two points of view, i.e. direct computation and calculation with quaternion symplectic form. The detailed mathematical derivation together with definition and some properties of quaternion algebra are provided. In order to evaluate the applicability of the proposed QDFrKT method, two applications are investigated, including color image encryption and color image watermarking. For color image encryption, experiments are conducted on key sensitivity, histogram analysis, and noise attacks; while for color image watermarking, experiments are conducted on watermarking invisibility and the robustness on attacks of media filter, average filter, Gaussian filter, and geometric attacks. Moreover, with the imperfect decryption property of the encryption algorithm, a potential application of the encryption strategy for designing watermarking algorithm is discussed. The promising experiment results demonstrate the effectiveness and efficiency of our proposed QDFrKT based color image encryption and watermarking techniques.

Unit Quaternion Description Method for Detecting High-Temperature Geothermal Well Drilling Conditions

When the typically utilized method for detecting the drilling conditions of high-temperature geothermal wells is applied, the detection takes a long time, the detection results are inconsistent with the actual conditions, and there are problems such as low detection efficiency and large detection deviation. Therefore, a method for detecting the drilling conditions of high-temperature geothermal wells described by a unit quaternion is proposed. Based on quaternion theory, the quaternion model of the position and attitude is constructed to obtain the drilling attitude. According to the analysis results and the basic principle of kernel principal component analysis, a model is built to realize the detection of high-temperature geothermal well drilling conditions. The experimental results show that in many iterations, the time required is stable and lower than that of other comparison methods, and the detection errors are all lower than 10%. The proposed method has high detection efficiency and low detection errors.

Robust zero-watermarking scheme based on novel quaternion radial fractional Charlier moments

In this paper, we propose a zero-watermarking scheme based on a new set of quaternion moments called Quaternion Radial Fractional Charlier Moments (QRFrCMs) for the copyright protection of color medical images. The proposed moments are developed from a new type of discrete orthogonal polynomials called Fractional Charlier Polynomials (FrCPs) and from quaternion theory. The robustness of the proposed scheme is ensured thanks to robustness of the proposed quaternion moments against image processing attacks and geometric attacks where the BCR is greater than 98%. In addition, the scheme uses the fractional order of QRFrCMs and a chaotic system based on two-dimensional CML (2DCML) using mixed linear-nonlinear coupling to provide a high level of security. Experimental results show the superiority of the proposed scheme over other recent schemes in terms of robustness against geometric attacks and common image processing attacks. The proposed scheme can be used for the secure transmission of color medical images over insecure networks.

Analytical solution of uncertainty with the GUM method for a dynamic stereo vision measurement system.

Directed at the strong correlation among the input parameters and long measurement chain, which are difficult for uncertainty analysis with the guide of the expression of uncertainty in the measurement (GUM) method, a novel dynamic stereo vision measurement system based on the quaternion theory is presented to reduce the orthogonality restrictions of shafting manufacturing and application. According to the quaternion theory in the kinematic model of the cameras and the analytical solution of uncertainty with the GUM method, the complete, detailed, and continuous uncertainty results of the full-scale measurement space can be obtained. Firstly, one-dimensional turntables and rigid connections are utilized to form the motion cores and the automatic control carriers in the system. Secondly, the novel measurement model is used in the measurement process to shorten the calibration and measurement chains. Once the system based on the novel measurement model is set up, the analytical solution of uncertainty is utilized in the accuracy process. During the analysis process, the strong correlation among the extrinsic parameters is decoupled by introducing virtual circles and the measurement strategy with the GUM method. Through analyzing the relationship among the attitude angles, the major factors which influence the uncertainties in each axis and the final uncertainty are clarified. Moreover, the analytical continuous uncertainty maps for the uncertainties along each axis, combined standard uncertainty, and the expanded uncertainty are illustrated and the uncertainty variation tendency is declared. Finally, the analytical solution of uncertainty with the GUM method proposed in this paper predicts the uncertainty in the full-scale space and provides a new idea of the uncertainty analysis for the complicated combined measurement system.

DOA and Polarization Estimation for Non-Circular Signals in 3-D Millimeter Wave Polarized Massive MIMO Systems

In this article, a multiple signal classification (MUSIC) based algorithm is proposed for two-dimensional (2-D) direction-of-arrival (DOA) and polarization estimation of non-circular signals in three-dimensional (3-D) millimeter wave polarized massive multiple-input-multiple-output (MIMO) systems. The traditional MUSIC-based algorithms can estimate either the DOA and polarization for circular signals or the DOA for non-circular signals by using spectrum search. By contrast, based on the quaternion theory, a novel algorithm named quaternion non-circular MUSIC (QNC-MUSIC) is proposed for parameter estimation of non-circular signals with high estimation accuracy. Moreover, only the DOA estimation needs spectrum search, and the polarization estimation has a closed-form expression. First, the DOA estimation can be achieved based on the derivation principle. Then the closed-form expression of the polarization estimation can be obtained based on the chain rule of the derivation w.r.t. the polarization parameters. In addition, the computational complexity analysis shows that compared with the conventional DOA and polarization estimation algorithms, our proposed QNC-MUSIC has much lower computational complexity, especially when the source number is large. The stochastic Cramér-Rao Bound (CRB) for the estimation of the 2-D DOA and polarization parameters of the non-circular signals is derived as well. Finally, numerical examples are provided to demonstrate that the proposed algorithms can improve the parameter estimation performance when large-scale/massive MIMO systems are employed.

Computationally Efficient Two-Dimensional DOA Estimation Algorithm Based on Quaternion Theory

In this letter, we present a novel computationally efficient DOA estimation algorithm based on quaternion theory for two-dimensional (2-D) direction-of-arrival (DOA) estimation. An orthogonal propagator method based on the cross-correlation of the quaternion models (OPM-CQM) is developed to alleviate the computation burden. To eliminate the effect of additive noise, we construct two quaternion-based signal models judiciously. Then, we obtain the statistics of the observed signals by performing the cross-correlation between the quaternion models. Meanwhile, the additive noise is eliminated without introducing other denoising methods. Moreover, the compact modeling approach based on quaternions provides a significant advantage to OPM-CQM in terms of computational effort. Simulations demonstrate that the proposed algorithm offers performance superiority in angular resolution compared with the non-quaternion schemes.

МОДЕЛЮВАННЯ ПРОСТОРОВИХ ІЗОТРОПНИХ КРИВИХ БЕЗЬЄ НА ОСНОВІ КРИВИХ ЗА ГОДОГРАФОМ ПІФАГОРА

Minimal surfaces are essential for geometric modeling, computer science, and for various branches of engineering, and the determination of the conditions for their construction has been studied in many works and includes a large number of methods. However, this topic has not yet been exhausted when using isotropic curves to construct minimal surfaces. These studies focus on methods for determining zero-length curves. For interactive control of such curves, it is advisable to use curves constructed on the basis of characteristic polygons, therefore, studies are based on Bézier curves. The paper highlights the experience of previous studies in the field of constructing spatial isotropic curves and plane Pythagorean-Hodograph curves. The modeling of spatial Pythagorean-Hodograph curves is based on the theory of quaternions. The authors of the paper present a different approach, namely, the construction of an isotropic spatial curve based on a plain Bézier Pythagorean-Hodograph curve. The transition to the spatial curve is carried out on the basis of determining the third coordinate from the condition that the length is equal to zero. In this case, a plain curve is built on the basis of real values, and the third coordinate will be purely imaginary. The authors carry out studies based on fifth-order curves. In this case, it is advisable to choose quadratic functions as basic polynomials. Three options are proposed for setting the initial values ​​of the coordinates for determining the Pythagorean-Hodograph curve. The problem of the boundary points of the curve and the determination of the intermediate points of the Bezier curve based on the given dependences have the most practical importance. To prove the reliability of the proposed provisions, the calculation of the points of the curve is performed and examples of simulated curves are given. Further research involves the use of the constructed curves for modeling minimal surfaces and isotropic portions of Bézier. Key words: isotropic curve, Bézier curve, Pythagorean-Hodograph curve.

Microstructural features effect on the evolution of cyclic damage for polycrystalline metals using a multiscale approach

In the context of polycrystalline metals, the damage process analysis is generally restricted to surface of a sample containing some hundreds of grains, due to the inherent difficulties microstructural analysis. Determination of the grain size influence, misorientation and neighboring grains effect remains difficult with experimental studies. In this work, the influence of microstructural characteristics of polycrystalline metals on the evolution of the damage process under cyclic loading is investigated. On the basis of the quaternion theory, the orientation of each (crystal) grain is represented by a single angle (theta). The relative misorientation of each grain is set by the ratio of its misorientation over the average value of polycrystalline aggregate. The relative volume of grain is used as a parameter representing the grain size. The damage evolution under cyclic loading is identified using the coupling of the crystal plasticity model with the Continuum Damage Mechanics (CDM) model. The damage evolution and its distribution on polycrystalline aggregate are calculated by means of numerical homogenization over each grain. The heterogeneity distribution and damage evolution for polycrystalline metals have been analyzed. The obtained results show that neighboring grains effects are larger and may change the tendency that larger grains with great misorientation are the most damaged.

ON BÄCKLUND TRANSFORMATIONS WITH SPLIT QUATERNIONS

The present paper deals with the introduction of Bäcklund Transformations with split quaternions in Minkowski space. Firstly, we tersely summarized the basic concepts of split quaternion theory and Bishop Frames of non-null curves in Minkowski space. Then, for Bäcklund transformations defined with each case of non-null curves, we give relationships between Bäcklund transformations and split quaternions. It is also presented some special propositions for transformations constructed with split quaternions. At the end, results obtained with the mathematical model have been evaluated.

Low-Complexity Channel Estimation for Circular and Noncircular Signals in Virtual MIMO Vehicle Communication Systems

In intelligent transportation systems (ITSs), the communication between the base station (BS) and the vehicles can form a virtual multiple-input multiple-out put (VMIMO) vehicle communication system. This paper addresses the issue of channel estimation in a virtual MIMO system that simultaneously adopt a smart antenna (SA). First, based on quaternion theory, a novel algorithm named quaternion noncircular MUSIC (QNC-MUSIC) is proposed, and direction of arrival (DOA) estimation is achieved for circular and noncircular signals. Then, a dimension reduction procedure is performed for the parameter search by employing the partial derivative of the spectrum function with respect to the noncircular phase; thus, a novel dimensional reduction noncircular MUSIC (DRNC-MUSIC) algorithm with a low computational complexity is proposed for DOA and polarization estimation of circular signals and noncircular signals. Finally, numerical examples are provided to demonstrate the parameter estimation performance of the proposed algorithm.

Research on Color Image Watermarking Algorithm Based on Quaternion Fourier Transform

This paper mainly proposes a color image digital watermarking algorithm based on quaternion Fourier transform (QFT). We can describe color image as a quaternion matrix, then we can use QFT to get its frequency domain information. By using Fourier transform, we can embed the watermarking information into the frequency domain information. By using inverse QFT, we can get a color image that contain watermarking information. All of the steps can be reversible, so the watermarking extraction process is opposite to the embedding process. The advantage of processing color image by quaternion theory is that three color channels of the image can be treated as a whole.

Algebraic techniques for least squares problems in commutative quaternionic theory

Due to the rise of commutative quaternion in Hopfield neural networks, digital signal, and image processing, one encounters the approximate solution problems of the commutative quaternion linear equations and . This paper, by means of real representation and complex representation of commutative quaternion matrices, introduces concepts of norms of commutative quaternion matrices and derives two algebraic techniques for finding solutions of least squares problems in commutative quaternionic theory.

Augmented Quaternion MUSIC Method for Near-Field Noncircular Sources With a COLD Array

Based on the quaternion theory, a novel algorithm named non-circular augmented quaternion MUSIC (NCAQ-MUSIC) is proposed for DOA and range estimation of noncircular signals impinging on a concentered orthogonal loop and dipole (COLD) array. Firstly, based on the augmented quaternion, the proposed algorithm uses the noncircular characteristic of the signals to achieve the virtual array expansion; secondly, the DOA and range parameters can be completely separated in the principle of rank reduction, and finally, the parameters of DOA and range are estimated through one dimensional search. Compared with direct mutil-dimensional (M-D) searching algorithms, the proposed method merely requires several one-dimension (1-D) spectral peak search which does not need parameter pairing. Simulation results verify the performance promotion of the proposed approach.

Algebraic techniques for least squares problem over generalized quaternion algebras: A unified approach in quaternionic and split quaternionic theory

This paper aims to present, in a unified manner, algebraic techniques for least squares problem in quaternionic and split quaternionic mechanics. This paper, by means of a complex representation and a real representation of a generalized quaternion matrix, studies generalized quaternion least squares (GQLS) problem, and derives two algebraic methods for solving the GQLS problem. This paper gives not only algebraic techniques for least squares problem over generalized quaternion algebras, but also a unification of algebraic techniques for least squares problem in quaternionic and split quaternionic theory.

Quaternion-MUSIC for near-field strictly noncircular sources with large-scale polarization array

In this paper, a novel localization method is proposed for DOA, range and polarization estimation of near-field noncircular sources in massive multiple-input-multiple-output (MIMO) systems. Compared with traditional MUSIC-based algorithms, the proposed algorithms can separate the polarization parameters from the spatial spectrum function, avoiding the four-dimensional (4-D) spectrum search and realizing the fast localization of the near-field source with high accuracy. First, the dimension-reduced MUSIC (DR-MUSIC) algorithm is proposed for DOA and range estimation with low computational complexity, and given a closed-form expression of polarization estimation. Next, based on the quaternion theory, a novel algorithm named quaternion non-circular MUSIC (QNC-MUSIC) is proposed for parameter estimation of non-circular signals with high estimation accuracy. In addition, the analysis of the computational complexity and simulations of the proposed method are provided, showing that the proposed method yields a better performance than DR-MUSIC in massive MIMO systems.

Universal approach to derivation of quaternion rotation formulas

This paper introduces and defines the quaternion with a brief insight into its properties and algebra. The main part of this paper is devoted to the derivation of basic equations of the vector rotation around each rotationalx, y, zaxis. Then, the equations of generalized quaternion rotation and express the general rotation operator is derived. Finally the utilization of equations is demonstrated on a simple example. For purposes of simplicity the quaternions theory is demonstrated around thez-axis byγangle. For the purpose of this paper, the fact that the subspace of vector quaternions may be regarded as being equivalent to the ordinary is used.

A Computational Geometric Approach for Motion Generation of Spatial Linkages With Sphere and Plane Constraints

This paper studies the problem of spatial linkage synthesis for motion generation from the perspective of extracting geometric constraints from a set of specified spatial displacements. In previous work, we have developed a computational geometric framework for integrated type and dimensional synthesis of planar and spherical linkages, the main feature of which is to extract the mechanically realizable geometric constraints from task positions, and thus reduce the motion synthesis problem to that of identifying kinematic dyads and triads associated with the resulting geometric constraints. The proposed approach herein extends this data-driven paradigm to spatial cases, with the focus on acquiring the point-on-a-sphere and point-on-a-plane geometric constraints which are associated with those spatial kinematic chains commonly encountered in spatial mechanism design. Using the theory of kinematic mapping and dual quaternions, we develop a unified version of design equations that represents both types of geometric constraints, and present a simple and efficient algorithm for uncovering them from the given motion.

A two-stage switching vector median filter based on quaternion for removing impulse noise in color images

This paper presents a novel two-stage filtering algorithm for removing impulse noise in color images. Quaternion theory is used to represent the intensity and chromaticity differences of two color pixels. Use of quaternion treats color pixels as vectors and processes color images as single unit rather than as separated color components. This preserves the existing correlation and three dimensional vector natures of the color channels. In the first stage of noise detection, the color pixels are sorted and assigned a rank based on the aggregated sum of color pixel differences with other pixels inside the filtering window. The central pixel is considered as probably corrupted by an impulse if its rank is bigger than a predefined rank. In the second stage, the probably corrupted candidate is again checked for an edge or an impulse by using four Laplacian convolution kernels. If the minimum difference of these four convolution is larger than a predefined threshold, then the central pixel is regarded as an impulse. For filtering, we extend the size of the sliding window to cover more pixels information. The noisy pixel is replaced by output of weighted vector median filter implemented using the quaternion distance. More weight is assigned to those pixels belonging to the direction of minimum difference. Experimental results indicate the improved performance of the proposed filter in suppressing the impulse noise while retaining the original image details comparing against other well-known filters.