Quaternion Signal Research Articles

Kernel quaternion principal component analysis and its application in RGB-D object recognition

While the existing quaternion principal component analysis (QPCA) is a linear tool developed mainly for processing linear quaternion signals, the quaternion representation (QR) used in QPCA creates redundancy when representing a color image signal of three components by a quaternion matrix having four components. In this paper, the kernel technique is used to improve the QPCA as kernel QPCA (KQPCA) for processing nonlinear quaternion signals; in addition, both RGB information and depth information are considered to improve QR for representing RGB-D images. The improved QR fully utilizes the four-dimensional quaternion domain. We first provide the basic idea of three types of our KQPCA and then propose an algorithm for RGB-D object recognition based on bidirectional two-dimensional KQPCA (BD2DKQPCA) and the improved QR. Experimental results on four public datasets demonstrate that the proposed BD2DKQPCA-based algorithm achieves the best performance among seventeen compared algorithms including other existing PCA-based algorithms, irrespective of RGB object recognition or RGB-D object recognition. Moreover, for all compared algorithms, consideration of both RGB and depth information is shown to achieve better performance in object recognition than considering only RGB information.

The geometry of proper quaternion random variables

Second order circularity, also called properness, for complex random variables is a well known and studied concept. In the case of quaternion random variables, some extensions have been proposed, leading to applications in quaternion signal processing (detection, filtering, estimation). Just like in the complex case, circularity for a quaternion-valued random variable is related to the symmetries of its probability density function. As a consequence, properness of quaternion random variables should be defined with respect to the most general isometries in 4D, i.e. rotations from SO(4). Based on this idea, we propose a new definition of properness, namely the (μ1, μ2)-properness, for quaternion random variables using invariance property under the action of the rotation group SO(4). This new definition generalizes previously introduced properness concepts for quaternion random variables. A second order study is conducted and symmetry properties of the covariance matrix of (μ1, μ2)-proper quaternion random variables are presented. Comparisons with previous definitions are given and simulations illustrate in a geometric manner the newly introduced concept.

Quaternionic Wiener–Khinchine Theorems and Spectral Representation of Convolution with Steerable Two-sided Quaternion Fourier Transform

In this paper we use the general two-sided quaternion Fourier transform (QFT) in order to derive Wiener–Khinchine theorems for the cross-correlation and for the auto-correlation of quaternion signals. Furthermore, we show how to derive a new four term spectral representation for the convolution of quaternion signals.

Cost-effective quaternion minimum mean square error estimation: From widely linear to four-channel processing

Widely linear estimation plays an important role in quaternion signal processing, as it caters for both proper and improper quaternion signals. However, widely linear algorithms are computationally expensive owing to the use of augmented variables and statistics. To reduce the computation cost while maintaining the performance level, we propose a four-channel estimation framework as an efficient alternative to quaternion widely linear estimation. This is achieved by using four linear models to estimate the four components of quaternion signals. We also show that any of the four channels is able to replace a strictly linear quaternion estimator when estimating strictly linear systems. The proposed method is shown to reduce computational complexity and provide more flexible algorithms, while preserving the physical meaning inherent in the quaternion domain. The proposed framework is next applied to quaternion minimum mean square error estimation to yield the reduced-complexity versions of the quaternion least mean square (QLMS), quaternion recursive least squares (QRLS), and quaternion nonlinear gradient decent (QNGD) algorithms. For the proposed QLMS algorithm, an adaptive step-size strategy is also explored. The effectiveness of the so introduced estimation techniques is validated by simulations on synthetic and real-world signals.

Widely linear estimation of quaternion signals with intermittent observations

This paper is concerned with the optimal quaternion estimation problem for linear discrete-time stochastic systems with intermittent observations by using a widely linear processing. The uncertainty of the observations is modeled by a Bernoulli distributed quaternion variable with known parameters. An augmented linear state–space model is developed to describe the evolution of both the quaternion state and their noisy observations. On the basis of that model, the optimal widely linear filter, predictor and smoothers (fixed-point and fixed-interval), that only depend on probabilities, are obtained via an innovation analysis approach. The special case of semi-widely linear estimators, which appears under Cη-properness, is also studied. Simulation examples demonstrate the effectiveness and applicability of the proposed estimators.

General two-sided quaternion Fourier transform, convolution and Mustard convolution

In this paper we use the general two-sided quaternion Fourier transform (QFT), and relate the classical convolution of quaternion-valued signals over \({{\mathbb R}^2}\) with the Mustard convolution. A Mustard convolution can be expressed in the spectral domain as the point wise product of the QFTs of the factor functions. In full generality do we express the classical convolution of quaternion signals in terms of finite linear combinations of Mustard convolutions, and vice versa the Mustard convolution of quaternion signals in terms of finite linear combinations of classical convolutions.

A Modified Uncertainty Principle for Two-Sided Quaternion Fourier Transform

This paper proposes a new uncertainty principle for the two-sided quaternion Fourier transform. This uncertainty principle describes that the spread of a quaternion-valued function and its two-sided quaternion Fourier transform (QFT) are inversely proportional. We obtain a tighter lower bound about the product of the spread of quaternion signal in the QFT domain. As a consequence, we show that the quaternionic Gabor filters minimize the uncertainty.

Connecting spatial and frequency domains for the quaternion Fourier transform

The quaternion Fourier transform (qFT) is an important tool in multi-dimensional data analysis, in particular for the study of color images. An important problem when applying the qFT is the mismatch between the spatial and frequency domains: the convolution of two quaternion signals does not map to the pointwise product of their qFT images. The recently defined ‘Mustard’ convolution behaves nicely in the frequency domain, but complicates the corresponding spatial domain analysis.The present paper analyses in detail the correspondence between classical convolution and the new Mustard convolution. In particular, an expression is derived that allows one to write classical convolution as a finite linear combination of suitable Mustard convolutions. This result is expected to play a major role in the further development of quaternion image processing, as it yields a formula for the qFT spectrum of the classical convolution.

Semi-Widely Simulation and Estimation of Continuous-Time <formula formulatype="inline"><tex Notation="TeX">$\C^{\eta}$</tex></formula>-Proper Quaternion Random Signals

The repercussions of quaternion $\BBC^{\eta}$ -properness on the continuous-time simulation and estimation problems are studied. As a first step, a semi-widely linear quaternion Karhunen–Loeve expansion is derived. Then this series representation is used in both problems to give the solutions. The simulation technique proposed is useful for transformations of nonstationary Gaussian quaternion signals. The estimation problem is also analyzed in a continuous-discrete setting and by considering two different formulations: the Cambanis formulation and the signal plus noise model. Initially, the estimators are developed on the basis of a semi-widely linear processing. Afterwards, the solutions are improved by incorporating the information supplied by the square quaternion observations and by using a semi-widely nonlinear processing. In this last scenario, we assume Gaussian observations and unknown probability distributions of the signal of interest.

Semi-widely linear estimation of [formula omitted]-proper quaternion random signal vectors under Gaussian and stationary conditions

Recursive algorithms for estimating quaternion signal vectors are provided. Assuming stationarity, Gaussianity and Cη-properness, two estimation problems are treated: the prediction and fixed-point smoothing problems. The problems are formulated in a very general way and solved following a semi-widely linear processing. The suggested solutions are derived taking additionally into account the information supplied by a square version of the quaternion observations. This extra information makes it possible to give improved estimators which outperform the corresponding full-widely linear estimators that ignore such information. A numerical example pertaining to the nonlinear prediction and fixed-point smoothing problems illustrates the application of the algorithms.

Optimization in Quaternion Dynamic Systems: Gradient, Hessian, and Learning Algorithms.

The optimization of real scalar functions of quaternion variables, such as the mean square error or array output power, underpins many practical applications. Solutions typically require the calculation of the gradient and Hessian. However, real functions of quaternion variables are essentially nonanalytic, which are prohibitive to the development of quaternion-valued learning systems. To address this issue, we propose new definitions of quaternion gradient and Hessian, based on the novel generalized Hamilton-real (GHR) calculus, thus making a possible efficient derivation of general optimization algorithms directly in the quaternion field, rather than using the isomorphism with the real domain, as is current practice. In addition, unlike the existing quaternion gradients, the GHR calculus allows for the product and chain rule, and for a one-to-one correspondence of the novel quaternion gradient and Hessian with their real counterparts. Properties of the quaternion gradient and Hessian relevant to numerical applications are also introduced, opening a new avenue of research in quaternion optimization and greatly simplified the derivations of learning algorithms. The proposed GHR calculus is shown to yield the same generic algorithm forms as the corresponding real- and complex-valued algorithms. Advantages of the proposed framework are illuminated over illustrative simulations in quaternion signal processing and neural networks.

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.

Pitt's inequality and the uncertainty principle associated with the quaternion Fourier transform

The quaternion Fourier transform – a generalized form of the classical Fourier transform – has been shown to be a powerful analyzing tool in image and signal processing. This paper investigates Pitt's inequality and uncertainty principle associated with the two-sided quaternion Fourier transform. It is shown that by applying the symmetric form f=f1+if2+f3j+if4j of quaternion from Hitzer and the novel module or Lp-norm of the quaternion Fourier transform fˆ, then any nonzero quaternion signal and its quaternion Fourier transform cannot both be highly concentrated. Two part results are provided, one part is Heisenberg–Weyl's uncertainty principle associated with the quaternion Fourier transform. It is formulated by using logarithmic estimates which may be obtained from a sharp of Pitt's inequality; the other part is the uncertainty principle of Donoho and Stark associated with the quaternion Fourier transform.

Asymptotic behaviour of the quaternion linear canonical transform and the Bochner–Minlos theorem

There have been numerous proposals in the literature to generalize the classical Fourier transform by making use of the Hamiltonian quaternion algebra. The present paper reviews the quaternion linear canonical transform (QLCT) which is a generalization of the quaternion Fourier transform and it studies a number of its properties. In the first part of this paper, we establish a generalized Riemann–Lebesgue lemma for the (right-sided) QLCT. This lemma prescribes the asymptotic behaviour of the QLCT extending and refining the classical Riemann–Lebesgue lemma for the Fourier transform of 2D quaternion signals. We then introduce the QLCT of a probability measure, and we study some of its basic properties such as linearity, reconstruction formula, continuity, boundedness, and positivity. Finally, we extend the classical Bochner–Minlos theorem to the QLCT setting showing the applicability of our approach.

Non-harmonic quaternion Fourier transform and uncertainty principle

The quaternionic Fourier transform (QFT) plays a vital role in the representation of signals and transforms a quaternionic 2 D signal into a quaternion-valued frequency domain signal. Associated with a kind of phase functions satisfying certain assumption, this paper generalizes the QFT to a kind of non-harmonic quaternionic Fourier transform (NQFT) and derives several important properties such as the scalar Plancherel and Parseval theorems, partial derivative, specific shift. The uncertainty principle for QFT is generalized for the NQFT. It is shown that only a Gaussian-type quaternion signal minimizes the uncertainty.

Prediction of wide-sense stationary quaternion random signals

An efficient widely linear prediction algorithm is introduced for the class of wide-sense stationary quaternion signals. Specifically, using second order statistics information in the quaternion domain, a multivariate Durbin–Levison-like algorithm is derived. The proposed solution can be applied under a very general formulation of the problem, allowing for the estimation of a function of the quaternion signal which is observed through a system with both additive/multiplicative noises.

Detection of continuous-time quaternion signals in additive noise

Different kinds of quaternion signal detection problems in continuous-time by using a widely linear processing are dealt with. The suggested solutions are based on an extension of the Karhunen-Loeve expansion to the quaternion domain which provides uncorrelated scalar real-valued random coefficients. This expansion presents the notable advantage of transforming the original four-dimensional eigen problem to a one-dimensional problem. Firstly, we address the problem of detecting a quaternion deterministic signal in quaternion Gaussian noise and a version of Pitcher’s Theorem is given. Also the particular case of a general quaternion Wiener noise is studied and an extension of the Cameron-Martin formula is presented. Finally, the problem of detecting a quaternion random signal in quaternion white Gaussian noise is tackled. In such a case, it is shown that the detector depends on the quaternion widely linear estimator of the signal.

Method of obtaining rotation-invariant images of spatial group point objects

This paper considers the problem of obtaining the shape of spatial group point objects using the principal component method, which allows one to find a rotation-invariant representation of quaternion signals defining spatial group point objects. An advantage of this approach is the possibility of estimating rotation parameters in the absence of information on the numbering of points in the object.

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.