Quaternion-valued Signals Research Articles

Hyperbolic linear canonical transforms of quaternion signals and uncertainty

This paper is concerned with Linear Canonical Transforms (LCTs) associated with two-dimensional quaternion-valued signals defined in an open rectangle of the Euclidean plane endowed with a hyperbolic measure, which we call Quaternion Hyperbolic Linear Canonical Transforms (QHLCTs). These transforms are defined by replacing the Euclidean plane wave with a corresponding hyperbolic relativistic plane wave in one dimension multiplied by quadratic modulations in both the hyperbolic spatial and frequency domains, giving the hyperbolic counterpart of the Euclidean LCTs. We prove the fundamental properties of the partial QHLCTs and the right-sided QHLCT by employing hyperbolic geometry tools and establish main results such as the Riemann–Lebesgue Lemma, the Plancherel and Parseval Theorems, and inversion formulas. The analysis is carried out in terms of novel hyperbolic derivative and hyperbolic primitive concepts, which lead to the differentiation and integration properties of the QHLCTs. The results are applied to establish two quaternionic versions of the Heisenberg uncertainty principle for the right-sided QHLCT. These uncertainty principles prescribe a lower bound on the product of the effective widths of quaternion-valued signals in the hyperbolic spatial and frequency domains. It is shown that only hyperbolic Gaussian quaternion functions minimize the uncertainty relations.

Uncertainty Principle for the Quaternion Linear Canonical Transform in Terms of Covariance

An uncertainty principle (UP), which offers information about a signal and its Fourier transform in the time-frequency plane, is particularly powerful in mathematics, physics and signal processing community. Under the polar coordinate form of quaternion-valued signals, the UP of the two-sided quaternion linear canonical transform (QLCT) is strengthened in terms of covariance. The condition giving rise to the equal relation of the derived result is obtained as well. The novel UP with covariance can be regarded as one in a tighter form related to the QLCT. It states that the product of spreads of a quaternion-valued signal in the spatial domain and the QLCT domain is bounded by a larger lower bound.

Discrete uncertainty principle in quaternion setting and application in signal reconstruction

In this paper, a novel discrete uncertainty principle associated with discrete Quaternion Fourier transform is established to give the relationship between the nonzero numbers of the discrete quaternion-valued signals and their Quaternion Fourier transforms. We obtain that the product of the numbers of nonzero elements of a sequence [Formula: see text], [Formula: see text] and its Quaternion Fourier transform is no less than [Formula: see text] and the result is sharp. Then we extend the uncertainty principle of discrete signals proved by Donoho and Starkin to two dimensional case. It suggests how sparsity helps in the recovery of missing frequency. The experimental results on the recovery of Lena also demonstrate the necessity of the two-dimensional discrete uncertainty principle associated with Quaternion Fourier transforms.

On uncertainty principle for the two-sided quaternion linear canonical transform

The quaternion linear canonical transform (QLCT), as a generalized form of the quaternion Fourier transform, is a powerful analyzing tool in image and signal processing. In this paper, we propose five different forms of uncertainty principles for the two-sided QLCT, including logarithmic uncertainty principle, Heisenberg-type uncertainty principle, local uncertainty principle, Benedicks–Amrein–Berthier uncertainty principle and entropic uncertainty principle. These consequences actually describe the quantitative relationships of a quaternion-valued signal in arbitrary two different QLCT domains, and they have great applications in signal recovery and physical quantum.

Near orthogonal discrete quaternion Fourier transform components via an optimal frequency rescaling approach

The quaternion-valued signals consist of four signal components. The discrete quaternion Fourier transform is to map these four signal components in the time domain to that in the frequency domain. These four signal components in the frequency domain are called the discrete quaternion Fourier transform components. There are a total of 16 inner products among any two discrete quaternion Fourier transform components. The total orthogonal error among the discrete quaternion Fourier transform components is defined based on these 16 inner products. This study aims to find the optimal quaternion number in the discrete quaternion Fourier transforms so that the total orthogonal errors among the discrete quaternion Fourier transform components are minimised. It is worth noting that finding the optimal quaternion number in the discrete quaternion Fourier transform is equivalent to finding the optimal rescaling factors. Since the discrete quaternion Fourier transform components are expressed in terms of the high-order polynomials of the trigonometric functions of the rescaling factors, this optimisation problem is non-convex. To address this problem, a two-stage approach is employed for finding the solution to the optimisation problem. The comparison results show that the authors proposed method outperforms the existing methods in terms of achieving the low total orthogonal error among the discrete quaternion Fourier transform components.

Uncertainty Principles for the Two-Sided Quaternion Linear Canonical Transform

The quaternion linear canonical transform (QLCT), as a generalized form of the quaternion Fourier transform, is a powerful analyzing tool in image and signal processing. In this paper, we propose three different forms of uncertainty principles for the two-sided QLCT, which include Hardy’s uncertainty principle, Beurling’s uncertainty principle and Donoho–Stark’s uncertainty principle. These consequences actually describe the quantitative relationships of the quaternion-valued signal in arbitrary two different QLCT domains, which have many applications in signal recovery and color image analysis. In addition, in order to analyze the non-stationary signal and time-varying system, we present Lieb’s uncertainty principle for the two-sided short-time quaternion linear canonical transform (SQLCT) based on the Hausdorff–Young inequality. By adding the nonzero quaternion-valued window function, the two-sided SQLCT has a great significant application in the study of signal local frequency spectrum. Finally, we also give a lower bound for the essential support of the two-sided SQLCT.

Quaternionic shearlet transform

The shearlet transform has been shown to be a valuable and powerful time–frequency analyzing tool for optics and non-stationary signal processing. In this article, we propose a novel transform called quaternionic shearlet transform which is designed to represent quaternion-valued signals at different scales, locations and orientations. We investigate the fundamental properties of quaternionic shearlet transform including Parseval's formula, Moyal's principle, inversion formula, and characterization of its range using the machinery of quaternion Fourier transform and quaternion convolution. We conclude our investigation by deriving an analogue of the classical Heisenberg–Pauli–Weyl uncertainty inequality and the associated logarithmic version for the quaternionic shearlet transform.

Novel quaternion‐valued least‐mean kurtosis adaptive filtering algorithm based on the GHR calculus

A novel quaternion-valued least-mean kurtosis (QLMK) adaptive filtering algorithm is proposed for three- and four-dimensional processes by using the recent generalised Hamilton-real (GHR) calculus. The proposed QLMK algorithm based GHR calculus minimises the negated kurtosis of the error signal as a cost function in the quaternion domain, thus provides an elegant way to solve a trade-off problem between the convergence rate and steady-state error. Moreover, the proposed QLMK algorithm has naturally a robust behaviour for a wide range of noise signals due to its kurtosis-based cost function. Furthermore, the steady-state performance of the proposed QLMK algorithm is analysed to obtain convergence and misadjustment conditions. The comprehensive simulation results on benchmark and real-world problems show that the use of this cost function defined by the quaternion statistics in the proposed QLMK algorithm allows us to process quaternion-valued signals and thus, significantly enhances the performance of the adaptive filter in terms of both the steady-state error and the convergence rate, as compared with the quaternion-valued least-mean-square algorithm based on the recent GHR calculus.

Simultaneous diagonalisation of the covariance and complementary covariance matrices in quaternion widely linear signal processing

Recent developments in quaternion-valued widely linear processing have established that the exploitation of complete second-order statistics requires consideration of both the standard covariance and the three complementary covariance matrices. Although such matrices have a tremendous amount of structure and their decomposition is a powerful tool in a variety of applications, the non-commutative nature of the quaternion product has been prohibitive to the development of quaternion uncorrelating transforms. To this end, we introduce novel techniques for a simultaneous decomposition of the covariance and complementary covariance matrices in the quaternion domain, whereby the quaternion version of the Takagi factorisation is explored to diagonalise symmetric quaternion-valued matrices. This gives new insights into the quaternion uncorrelating transform (QUT) and forms a basis for the proposed quaternion approximate uncorrelating transform (QAUT) which simultaneously diagonalises all four covariance matrices associated with improper quaternion signals. The effectiveness of the proposed uncorrelating transforms is validated by simulations on both synthetic and real-world quaternion-valued signals.

Plancherel theorem and quaternion Fourier transform for square integrable functions

ABSTRACTThe quaternion Fourier transform (QFT), a generalization of the classical 2D Fourier transform, plays an increasingly active role in particular signal and colour image processing. There tends to be an inordinate degree of interest placed on the properties of QFT. The classical convolution theorem and multiplication formula are only suitable for 2D Fourier transform of complex-valued signal, and do not hold for QFT of quaternion-valued signal. The purpose of this paper is to overcome these problems and establish the Plancherel and inversion theorems of QFT in the square integrable signals space . First, we investigate the behaviours of QFT in the integrable signals space . Next, we deduce the energy preservation property which extends functions from to space. Moreover, some other important properties such as modified multiplication formula are also analyzed for QFT.

Novel Sampling Formulas Associated with Quaternionic Prolate Spheroidal Wave functions

The Whittaker-Shannon-Kotel'nikov (WSK) sampling theorem provides a reconstruction formula for the bandlimited signals. In this paper, a novel kind of the WSK sampling theorem is established by using the theory of quaternion reproducing kernel Hilbert spaces. This generalization is employed to obtain the novel sampling formulas for the bandlimited quaternion-valued signals. A special case of our result is to show that the 2D generalized prolate spheroidal wave signals obtained by Slepian can be used to achieve a sampling series of cube-bandlimited signals. The solutions of energy concentration problems in quaternion Fourier transform are also investigated.

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.

Channel equalization and beamforming for quaternion-valued wireless communication systems

Quaternion-valued wireless communication systems have been studied in the past. Although progress has been made in this promising area, a crucial missing link is lack of effective and efficient quaternion-valued signal processing algorithms for channel equalization and beamforming. With most recent developments in quaternion-valued signal processing, in this work, the author fills the gap to solve the problem by studying two quaternion-valued adaptive algorithms: one is the reference signal based quaternion-valued least mean square (QLMS) algorithm and the other one is the quaternion-valued constant modulus algorithm (QCMA). The quaternion-valued Wiener solution for possible block-based calculation is also derived. Simulation results are provided to show the working of the system.

Late Glacial and Holocene environmental history of the Ethiopian Highlands inferred from a 12 m long sediment record from Dendi crater lakes

Quaternion-valued wireless communication systems have been studied in the past. Although progress has been made in this promising area, a crucial missing link is lack of effective and efficient quaternion-valued signal processing algorithms for channel equalization and beamforming. With most recent developments in quaternion-valued signal processing, in this work, the author fills the gap to solve the problem by studying two quaternion-valued adaptive algorithms: one is the reference signal based quaternion-valued least mean square (QLMS) algorithm and the other one is the quaternion-valued constant modulus algorithm (QCMA). The quaternion-valued Wiener solution for possible block-based calculation is also derived. Simulation results are provided to show the working of the system.

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.

Properties of a general quaternion-valued gradient operator and its applications to signal processing

The gradients of a quaternion-valued function are often required for quaternionic signal processing algorithms. The HR gradient operator provides a viable framework and has found a number of applications. However, the applications so far have been limited to mainly real-valued quaternion functions and linear quaternionvalued functions. To generalize the operator to nonlinear quaternion functions, we define a restricted version of the HR operator, which comes in two versions, the left and the right ones. We then present a detailed analysis of the properties of the operators, including several different product rules and chain rules. Using the new rules, we derive explicit expressions for the derivatives of a class of regular nonlinear quaternion-valued functions, and prove that the restricted HR gradients are consistent with the gradients in the real domain. As an application, the derivation of the least mean square algorithm and a nonlinear adaptive algorithm is provided. Simulation results based on vector sensor arrays are presented as an example to demonstrate the effectiveness of the quaternion-valued signal model and the derived signal processing algorithm.

The Quaternion Domain Fourier Transform and its Properties

So far quaternion Fourier transforms have been mainly defined over \({\mathbb{R}^2}\) as signal domain space. But it seems natural to define a quaternion Fourier transform for quaternion valued signals over quaternion domains. This quaternion domain Fourier transform (QDFT) transforms quaternion valued signals (for example electromagnetic scalar-vector potentials, color data, space-time data, etc.) defined over a quaternion domain (space-time or other 4D domains) from a quaternion position space to a quaternion frequency space. The QDFT uses the full potential provided by hypercomplex algebra in higher dimensions and may moreover be useful for solving quaternion partial differential equations or functional equations, and in crystallographic texture analysis. We define the QDFT and analyze its main properties, including quaternion dilation, modulation and shift properties, Plancherel and Parseval identities, covariance under orthogonal transformations, transformations of coordinate polynomials and differential operator polynomials, transformations of derivative and Dirac derivative operators, as well as signal width related to band width uncertainty relationships.

Some Results on the Lattice Parameters of Quaternionic Gabor Frames

Gabor frames play a vital role not only in modern harmonic analysis but also in several fields of applied mathematics, for instances, detection of chirps, or image processing. In this work we present a non-trivial generalization of Gabor frames to the quaternionic case and give new density results. The key tool is the two-sided windowed quaternionic Fourier transform (WQFT). As in the complex case, we want to write the WQFT as an inner product between a quaternion-valued signal and shifts and modulates of a real-valued window function. We demonstrate a Heisenberg uncertainty principle and for the results regarding the density, we employ the quaternionic Zak transform to obtain necessary and sufficient conditions to ensure that a quaternionic Gabor system is a quaternionic Gabor frame. We conclude with a proof that the Gabor conjecture does not hold true in the quaternionic case.

Tighter Uncertainty Principles Based on Quaternion Fourier Transform

The quaternion Fourier transform (QFT) and its properties are reviewed in this paper. Under the polar coordinate form for quaternion-valued signals, we strengthen the stronger uncertainty principles in terms of covariance for quaternion-valued signals based on the right-sided quaternion Fourier transform in both the directional and the spatial cases. We also obtain the conditions that give rise to the equal relations of two uncertainty principles. Examples are given to verify the results.

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.