Augmented Quaternion Research Articles

Novel Augmented Quaternion UKF for Enhanced Loosely Coupled GPS/INS Integration

Novel Augmented Quaternion UKF for Enhanced Loosely Coupled GPS/INS Integration

QDRJL: Quaternion dynamic representation with joint learning neural network for heart sound signal abnormal detection

At present, deep learning based heart sound diagnosis algorithms are mostly complex and large models for high accuracy, which are difficult to deploy on mobile devices due to the high number of parameters and large computational cost. The current mainstream approach for processing heart sound signals involves utilizing their Mel-frequency cepstral coefficients (MFCC) features. However, most existing methods have overlooked the multi-channel characteristics of MFCC. To address this issue, we propose a Quaternion Dynamic Representation with Joint Learning (QDRJL) neural network for learning MFCC multi-channel features. Our proposed approach combines quaternion dynamic convolution with dynamic weighting and the Quaternion Interior Learning Block (QILB). Finally, we present a global and energy joint learning branch for jointly learning MFCC features. The success of the proposed quaternion network depends on its ability to utilize the internal relations between quaternion-valued input features and the definition of the dynamic weight variables in the augmented quaternion domain. We assessed various state-of-the-art classification algorithms for detecting heart sounds and found that our proposed classifier achieved an accuracy of up to 97.2%, outperforming existing models. Our experimental evaluation, using the 2016 PhysioNet/CinC Challenge dataset, revealed that our model could reduce the number of network parameters to 25% due to quaternion properties.

RKHS Representations for Augmented Quaternion Random Signals: Application to Detection Problems

The reproducing kernel Hilbert space (RKHS) methodology has shown to be a suitable tool for the resolution of a wide range of problems in statistical signal processing both in the real and complex domains. It relies on the idea of transforming the original functional data into an infinite series representation by projection onto an specific RKHS, which usually simplifies the statistical treatment without any loss of efficiency. Moreover, the advantages of quaternion algebra over real-valued three and four-dimensional vector algebra in the modelling of multidimensional data have been proven useful in much relatively recent research. This paper accordingly proposes a generic RKHS framework for the statistical analysis of augmented quaternion random vectors, which provide a complete description of their second order characteristics. It will allow us to exploit the full advantages of the RKHS theory in widely linear processing applications, such as signal detection. In particular, we address the detection of a quaternion signal disturbed by additive Gaussian noise and the discrimination between two quaternion Gaussian signals in continuous time.

Augmented Quaternion ESPRIT-Type DOA Estimation With a Crossed-Dipole Array

In this letter, a quaternion-based ESPRIT-type algorithm called augmented quaternion ESPRIT (AQ-ESPRIT) is proposed for direction of arrival (DOA) estimation with a co-located crossed-dipole array. Firstly, two quaternion models are constructed by judiciously arranging the observed signals, and then concatenated to form a new AQ model. Secondly, the AQ signal subspace is estimated by applying the quaternionic eigenvalue decomposition to the resultant AQ covariance matrix. Finally, the derived AQ signal subspaces of different subarrays are employed to form the rotational invariance equation, which is then used to obtain the ultimate DOA estimates. Numerical results show that the proposed AQ-ESPRIT has a better performance in low signal-to-noise ratio scenarios.

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.

Augmented Quaternion Extreme Learning Machine

As an efficient training strategy for single hidden layer neural networks, extreme learning machine, and its variants have been widely used due to its fast learning speed and superior generalization performance. However, when used for quaternion signal processing, extreme learning machine cannot make full use of the cross-channel correlation and quaternion statistics and thus often provides only suboptimal solutions. To this end, in this paper, we extend the extreme learning machine to quaternion domain and then propose two augmented quaternion extreme learning machine models for quaternion signal processing. These two models incorporate the involutions of the network input and hidden nodes, respectively, and can fully capture the second-order statistics of quaternion signals. In order to overcome the possible overfitting problem, two corresponding regularized augmented algorithms are also derived with the help of the generalized HR calculus. The superiority of the proposed models is verified by the simulation results.

Performance Analysis of Quaternion-Valued Adaptive Filters in Nonstationary Environments

Quaternion adaptive filters have been widely used for the processing of three-dimensional (3-D) and 4-D phenomena, but complete analysis of their performance is still lacking, partly due to the cumbersomeness of multivariate quaternion analysis. This causes difficulties in both understanding their behavior and designing optimal filters. Based on a thorough exploration of the augmented statistics of quaternion random vectors, this paper extends an analysis framework for real-valued adaptive filters to the mean and mean square convergence analyses of general quaternion adaptive filters in nonstationary environments. The extension is nontrivial, considering the noncommutative quaternion algebra, only recently resolved issues with quaternion gradient, and the multidimensional augmented quaternion statistics. Also, for rigor, in order to model a nonstationary environment, the system weights are assumed to vary according to a first-order random-walk model. Transient and steady-state performance of a general class of quaternion adaptive filters is provided by exploiting the augmented quaternion statistics. An innovative quaternion decorrelation technique allows us to develop intuitive closed-form expressions for the performance of quaternion least mean square (QLMS) filters with Gaussian inputs, which provide new insights into the relationship between the filter behavior and the complete second-order statistics of the input signal, that is, quaternion noncircularity. The closed-form expressions for the performance of strictly linear, semiwidely linear, and widely linear QLMS filters are investigated in detail, while numerical simulations for the three classes of QLMS filters with correlated Gaussian inputs support the theoretical analysis.

Generating Augmented Quaternion Random Variable With Generalized Gaussian Distribution

There is a need to generate quaternion valued processes with a given distribution ranging from super-Gaussian, Gaussian to sub-Gaussian with different degrees of properness for numerous practical applications, such as modeling the signal source, testing nonlinear adaptive filter models, watermarking, generating $\mathbb {Q}$ -proper and $\mathbb {Q}$ -improper processes, the synthesis of color images, and testing the algorithms of step detection. Therefore, in this paper, the alternative quaternion generalized Gaussian probability density function is derived for an augmented quaternion valued variable, and the procedure for generating the augmented quaternion valued random variables is defined for this distribution. Additionally, the 3-D generalized Gaussian distribution probability density function parameterized by the shape parameter $p$ and the covariance matrix $C$ is revised.

A novel scheme on multi-channel mechanical fault signal diagnosis based on augmented quaternion singular spectrum analysis

In this paper, a novel multi-channel mechanical failure signal classification method based on augmented quaternion singular spectrum analysis (AQSSA) is proposed. Quaternion is used to couple four channels signal, and the quaternion trajectory matrixes can be developed as augmented quaternion matrix by using the feature of the quaternion. The singular value sequence including characteristic information can be extracted by quaternion singular value decomposition (QSVD) of the augmented trajectory matrix using its covariance matrix. The method of traditional singular spectrum analysis (SSA) can only analyze the single channel signal, however, AQSSA can fully use the correlation of multi-channel and reduce the loss of the effective information. Additionally, the main singular values are defined by some methods such as difference spectrum aimed, which has the limitation that major singular values can’t be obtained under the high background noise. Thus, a concept of partial mean of singular value sequence is proposed, and it can be set as the standard of evaluating the trend of singular value sequence. In order to testify the performance of the proposed method, the numerical simulation signal and the fault vibration signal of bearing are simultaneously adopted to verify its effectiveness. The results indicate that the effectiveness of mechanical fault classification by the proposed method is superior to the traditional SSA method and the method of permutation entropy.

Widely Linear Quaternion Unscented Kalman Filter for Quaternion-Valued Feedforward Neural Network

Recently, the quaternion-valued feedforward neural network (QFNN) has been developed to process three dimensional (3-D) and 4-D signals in quaternion domain, and the weight matrices and bias vectors of the QFNN were obtained based on the quaternion backward propagation (QBP) method. However, it should be noted that the QBP is a first-order quaternion gradient descent algorithm. The convergence speed of the QBP is usually slow and may not be very suitable to process nonstationary quaternion-valued signals. To address this problem, a widely linear quaternion unscented Kalman filter (WLQUKF) algorithm is proposed to train the QFNN. This is derived by utilizing some recent studies in the augmented quaternion statistics and the $\mathbb {HR}$ -calculus. With the augmented quaternion statistics, the WLQUKF is able to process general quaternion-valued noncircular, nonlinear, and nonstationary signals, effectively. Simulations on both benchmark circular and noncircular quaternion-valued signals, and on real-world quaternion-valued signals support the analysis.

Stokes parameters and DOAs estimation of partially polarized sources using a EM vector sensor

In this paper, the problem of polarization and direction-of-arrival (DOA) estimation using an electromagnetic (EM) vector sensor is addressed based on the augmented second-order statistics of quaternions. The augmented quaternion model of an EM vector sensor is introduced by employing the augmented second-order statistics of quaternions, which include the information in both the standard covariance and the pseudo-covariance. The fact is revealed that the Stokes parameters of an EM wave can be extracted from an augmented covariance matrix. An augmented quaternion-based approach is proposed for the Stokes parameters and DOA estimation of polarized EM waves. The proposed approach uses only information in three imaginary parts of both the standard covariance and the pseudo-covariance matrix. In theory, it is guaranteed to remove the effect of the additive noise which is uncorrelated between measurement channels. The proposed approach is applicable to both completely polarized and partially polarized EM waves. Simulations are used to verify the performance of the proposed approach.

Quaternion-valued echo state networks.

Quaternion-valued echo state networks (QESNs) are introduced to cater for 3-D and 4-D processes, such as those observed in the context of renewable energy (3-D wind modeling) and human centered computing (3-D inertial body sensors). The introduction of QESNs is made possible by the recent emergence of quaternion nonlinear activation functions with local analytic properties, required by nonlinear gradient descent training algorithms. To make QENSs second-order optimal for the generality of quaternion signals (both circular and noncircular), we employ augmented quaternion statistics to introduce widely linear QESNs. To that end, the standard widely linear model is modified so as to suit the properties of dynamical reservoir, typically realized by recurrent neural networks. This allows for a full exploitation of second-order information in the data, contained both in the covariance and pseudocovariances, and a rigorous account of second-order noncircularity (improperness), and the corresponding power mismatch and coupling between the data components. Simulations in the prediction setting on both benchmark circular and noncircular signals and on noncircular real-world 3-D body motion data support the analysis.

A class of quaternion Kalman filters.

The existing Kalman filters for quaternion-valued signals do not operate fully in the quaternion domain, and are combined with the real Kalman filter to enable the tracking in 3-D spaces. Using the recently introduced HR-calculus, we develop the fully quaternion-valued Kalman filter (QKF) and quaternion-extended Kalman filter (QEKF), allowing for the tracking of 3-D and 4-D signals directly in the quaternion domain. To consider the second-order noncircularity of signals, we employ the recently developed augmented quaternion statistics to derive the widely linear QKF (WL-QKF) and widely linear QEKF (WL-QEKF). To reduce computational requirements of the widely linear algorithms, their efficient implementation are proposed and it is shown that the quaternion widely linear model can be simplified when processing 3-D data, further reducing the computational requirements. Simulations using both synthetic and real-world circular and noncircular signals illustrate the advantages offered by widely linear over strictly linear quaternion Kalman filters.

A class of quaternion valued affine projection algorithms

The strictly linear quaternion valued affine projection algorithm (QAPA) and its widely linear counterpart (WLQAPA) are introduced, in order to provide fast converging stochastic gradient learning in the quaternion domain, for the processing of both second order circular (proper) and second order noncircular (improper) signals. This is achieved based on the recent advances in augmented quaternion statistics, which employs all second order information available, together with the associated widely linear models and through performing rigorous gradient calculation (HR-calculus). Further, mean square error analysis is performed based on the energy conservation principle, which provides a theoretical justification for the WLQAPA offering enhanced steady state performance for quaternion noncircular (improper) signals, a typical case in real world scenarios. Simulations on benchmark circular and noncircular signals, and on noncircular real world 4D wind and 3D body motion data support the analysis.

Optimal geometric motion planning for a spin-stabilized spacecraft

A method requiring low-computational overhead is presented which generates low-torque reference motions between arbitrary orientations for a spin-stabilized spacecraft. The initial stage solves a constrained optimal control problem deriving analytical solutions for a class of smooth and feasible reference motions. Specifically, for a quadratic cost function an application of Pontryagin’s maximum principle leads to a completely integrable Hamiltonian system that is, exactly solvable in closed-form, expressed in terms of several free parameters. This is shown to reduce the complexity of a practical motion planning problem from a constrained functional optimization problem to an unconstrained parameter optimization problem. The generated reference motions are then tracked using an augmented quaternion feedback law, consisting of the sum of a proportional plus derivative term and a term to compensate nonlinear dynamics. The method is illustrated with an application to re-point a spin-stabilized agile micro-spacecraft using zero propellant. The low computational overhead of the method enhances its suitability for on-board motion generation.

Fast Independent Component Analysis Algorithm for Quaternion Valued Signals

An extension of the fast independent component analysis algorithm is proposed for the blind separation of both Q-proper and Q-improper quaternion-valued signals. This is achieved by maximizing a negentropy-based cost function, and is derived rigorously using the recently developed HR calculus in order to implement Newton optimization in the augmented quaternion statistics framework. It is shown that the use of augmented statistics and the associated widely linear modeling provides theoretical and practical advantages when dealing with general quaternion signals with noncircular (rotation-dependent) distributions. Simulations using both benchmark and real-world quaternion-valued signals support the approach.

Quaternion-Valued Nonlinear Adaptive Filtering

A class of nonlinear quaternion-valued adaptive filtering algorithms is proposed based on locally analytic nonlinear activation functions. To circumvent the stringent standard analyticity conditions which are prohibitive to the development of nonlinear adaptive quaternion-valued estimation models, we use the fact that stochastic gradient learning algorithms require only local analyticity at the operating point in the estimation space. It is shown that the quaternion-valued exponential function is locally analytic, and, since local analyticity extends to polynomials, products, and ratios, we show that a class of transcendental nonlinear functions can serve as activation functions in nonlinear and neural adaptive models. This provides a unifying framework for the derivation of gradient-based learning algorithms in the quaternion domain, and the derived algorithms are shown to have the same generic form as their real- and complex-valued counterparts. To make such models second-order optimal for the generality of quaternion signals (both circular and noncircular), we use recent developments in augmented quaternion statistics to introduce widely linear versions of the proposed nonlinear adaptive quaternion valued filters. This allows full exploitation of second-order information in the data, contained both in the covariance and pseudocovariances to cater rigorously for second-order noncircularity (improperness), and the corresponding power mismatch in the signal components. Simulations over a range of circular and noncircular synthetic processes and a real world 3-D noncircular wind signal support the approach.

Quaternion-valued short-term joint forecasting of three-dimensional wind and atmospheric parameters

This work introduces novel methodology for the simultaneous modelling and forecasting of three-dimensional wind field. This is achieved based on a quaternion wind model, which by virtue of its division algebra accounts naturally for the coupling between the three wind dimensions. To fully exploit the available second order statistics, we employ the newly developed augmented quaternion statistics and perform prediction based on the widely linear model. The proposed quaternion domain processing also facilitates the fusion of external atmospheric parameters, such as air temperature, yielding improved forecasts. Simulations for wind regimes with different dynamics and over a range of prediction horizons, together with the fusion of air temperature, support the approach.