Quaternion Signal Research Articles

Affine phase retrieval of quaternion signals

Quaternion algebra H is an extension of the complex number field, which is a noncommutative associative algebra. In recent years, quaternionic Fourier analysis has interested some mathematicians due to its applications in signal analysis and image processing. This paper addresses quaternionic affine phase retrieval (QAPR) in quaternion Euclidean spaces H M , which aims to exactly recover a signal in H M from the magnitudes of its affine measurements. We introduce the concepts of QAPR and phaselift operator in H M . Then, we characterize QAPR in terms of the real Jacobian matrix, prove that 5M is the minimal measurement number for QAPR in H M , study the stability of QAPR-sequences, and use the phaselift techniques to give some sufficient conditions on QAPR for H M which provide us with a method to construct QAPR-sequences.

Spectrum of quaternion signals associated with quaternion linear canonical transform

In recent years quaternion linear canonical transform (QLCT) has emerged due to its applications in various fields, including image and signal processing. This article discusses two spectrum-related theorems (real Paley–Wiener and Boas type). The real Paley–Wiener type theorem is formulated to describe the character of a compactly supported two-sided quaternion linear canonical transformed (QLCT) signal. The Boas-type theorem is also discussed to explain the property of right-sided QLCT of signals that vanish in the neighborhood of origin. Some potential applications of these theorems on some particular quaternion-type operators are also discussed.

Optical multi-image encryption scheme with chaotic apertured quaternion fractional Mellin transform

To enhance the encryption quality and the transmission efficiency, an optical compression and encryption scheme for multiple images based on chaotic apertured quaternion fractional Mellin transform (QFrMT) is designed. The apertured optical systems and the quaternions with several imaginary components guarantee our designed scheme harder for the attackers to acquire valuable information. The QFrMT with the chaotic aperture is proposed to eliminate potential insecurity resulting from the known-plaintext attack and the phase-retrieval attack. The quaternion signal containing the main information of the plaintext images is converted into the chaotic apertured QFrMT domain, and then the resulting amplitude is encrypted with the chaotic mask constructed by the truncated phase in the fractional Fourier plane for nonlinear processing and information integrity. The chaos-based non-conventional diffusion is applied to further enhance the robustness against attacks. Extensive experiments results verify the security and efficiency of the presented optical multi-image encryption scheme.

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.

Adaptive reweighted quaternion sparse learning for data recovery and classification

Sparse representation (SR) methods in quaternion space have been attracting increasing interests recently. However, most existing quaternion SR methods adopt the quaternion ℓ1 norm, which penalizes all the entries of the quaternion sparse vector equally and ignores the differences and significance of different entries. Ideally, the entries with large magnitude should be less penalized while those with small magnitude (such as zero entries) should be more penalized. Therefore, we propose an Adaptive Weighted Quaternion Sparse Representation (AWQSR) method in this paper, which can learn weights for distinct entries of the quaternion sparse entries in an adaptive manner. Due to the noncommutativity of quaternion multiplication, it is difficult to tackle the resulting optimization problem of AWQSR. For this reason, we devise an effective iteratively reweighted optimization algorithm based on quaternion operators. To further improve the classification performance, we also develop a Supervised AWQSR based Classification (SAWQSRC) method by leveraging the label information of training samples to learn discriminative weights. Theoretical analysis of SAWQSRC has also been established to show that SAWQSRC succeeds in classification under appropriate conditions. The experiments on simulated data and real data prove the validity of the proposed methods for quaternion signal recovery and classification.

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.

Some Essential Relations for the Quaternion Quadratic-Phase Fourier Transform

Motivated by the fact that the quaternion Fourier transform is a powerful tool in quaternion signal analysis, here, we study the quaternion quadratic-phase Fourier transform, which is a generalized version of the quaternion Fourier transform. We first give a definition of the quaternion quadratic-phase Fourier transform. We derive in detail some essential properties related to this generalized transformation. We explore how the quaternion quadratic-phase Fourier transform is related to the quaternion Fourier transform. It is shown that this relation allows us to obtain several versions of uncertainty principles concerning the quaternion quadratic-phase Fourier transform.

Quaternion Extreme Learning Machine Based on Real Augmented Representation

Widely linear modeling is an important quaternion signal processing technique for capturing the complete secondorder statistics of quaternion signals. However, the algorithms based on widely linear modeling are computationally expensive due to the augmented variables and statistics. In this letter, a fast estimation technique based on a real augmented representation of widely linear modeling is proposed in the context of quaternion extreme learning machine with augmented hidden layer (QELMAH), resulting in a reduction of almost 93% of multiplications and 75% of additions. An equivalence between the proposed algorithm and the original QELMAH is theoretically established by proving that the trained networks with the proposed algorithm and the original one mathematically implement identical mapping. Such a technique is also applicable to the widely linear quaternion recursive least squares algorithm. The theoretical analysis and the effectiveness of the proposed algorithms are validated by simulations on two benchmark problems.

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.

Uncertainty principles and applications of quaternion windowed linear canonical transform

Quaternion signal processing is frequently used in color image processing. The quaternion windowed linear canonical transform (QWLCT), a generalization of the windowed linear canonical transform (WLCT), has a wide range of application domains, including signal processing and optics. In this paper, we study QWLCT-based characterization range, reproducing kernel, one-one map, Donoho-Stark inequality and Pitt’s inequality. Some useful uncertainty principles (UP) like Heisenberg UP, Lieb UP, and local UP are discussed. Moreover, some applications associated with QWLCT in linear time-varying (TV) systems are explained in detail.

Double-Color-Image Compression-Encryption Algorithm Based on Quaternion Multiple Parameter DFrAT and Feature Fusion with Preferable Restoration Quality.

To achieve multiple color images encryption, a secure double-color-image encryption algorithm is designed based on the quaternion multiple parameter discrete fractional angular transform (QMPDFrAT), a nonlinear operation and a plaintext-related joint permutation-diffusion mechanism. QMPDFrAT is first defined and then applied to encrypt multiple color images. In the designed algorithm, the low-frequency and high-frequency sub-bands of the three color components of each plaintext image are obtained by two-dimensional discrete wavelet transform. Then, the high-frequency sub-bands are further made sparse and the main features of these sub-bands are extracted by a Zigzag scan. Subsequently, all the low-frequency sub-bands and high-frequency fusion images are represented as three quaternion signals, which are modulated by the proposed QMPDFrAT with three quaternion random phase masks, respectively. The spherical transform, as a nonlinear operation, is followed to nonlinearly make the three transform results interact. For better security, a joint permutation-diffusion mechanism based on plaintext-related random pixel insertion is performed on the three intermediate outputs to yield the final encryption image. Compared with many similar color image compression-encryption schemes, the proposed algorithm can encrypt double-color-image with higher quality of image reconstruction. Numerical simulation results demonstrate that the proposed double-color-image encryption algorithm is feasibility and achieves high security.

Geometric Analysis of Signals for Inference of Multiple Faults in Induction Motors.

Multiple fault identification in induction motors is essential in industrial processes due to the high costs that unexpected failures can cause. In real cases, the motor could present multiple faults, influencing systems that classify isolated failures. This paper presents a novel methodology for detecting multiple motor faults based on quaternion signal analysis (QSA). This method couples the measured signals from the motor current and the triaxial accelerometer mounted on the induction motor chassis to the quaternion coefficients. The QSA calculates the quaternion rotation and applies statistics such as mean, variance, kurtosis, skewness, standard deviation, root mean square, and shape factor to obtain their features. After that, four classification algorithms are applied to predict motor states. The results of the QSA method are validated for ten classes: four single classes (healthy condition, unbalanced pulley, bearing fault, and half-broken bar) and six combined classes. The proposed method achieves high accuracy and performance compared to similar works in the state of the art.

Commutative quaternion algebra and DSP fundamental properties: Quaternion convolution and Fourier transform

In recent years, quaternion algebra has been applied more and more in different applications. A primary motivation for investigating quaternion algebras is to extend the classical signal processing to quaternion signal processing theory. However, this algebra has non-commutative multiplication, leading to the uncertainty of such essential operations as the convolution and Fourier transform. As a result, it became difficult to process quaternion signals in the frequency domain. In this work, the arithmetic of quaternion numbers is analyzed using the Cayley-Dickson construction, but with an associative and commutative operation of multiplication. We call it the (2,2)-model of quaternions. This model's main properties of quaternions are described, including inverse, division, and conjugate operations. In the (2,2)-model, only two exponential functions with pure quaternion units exit, and therefore, only two types of the quaternion discrete Fourier transform (QDFT) can be defined. One of the crucial benefits of this model is that the unique definition of convolution will help reduce the complexity of quaternion convolution operation calculated using both types of QDFTs.

Quantitative analysis of multi-optical length NIR spectroscopy based on quaternion parallel feature extraction method

Near infrared spectroscopy (NIR) using multi-optical lengths is a very effective technique for quantitative analysis of mixture composition detection. Traditional chemometric methods build multi-optical lengths NIR quantitative model by representing multi-optical lengths NIR spectral data serially, which will also introduce noise and affect the predictive ability. In this paper, we demonstrate for the first time a parallel express and feature extraction method based on quaternion signal processing techniques, which can greatly improve the performance of the multi-optical lengths NIR quantitative model. We measured the NIR spectra of the mixture solutions of Intra-lipid and India ink under 9 optical lengths that were 1, 1.15, 1.3, 1.45, 1.6, 1.75, 1.9, 2.05, and 2.2 mm, then we randomly selected three optical lengths NIR spectra to constitute an integral representation model by a pure quaternion matrix. Baseline correction pretreatment and data enhancement were performed and followed by Quaternion principal component analysis (QPCA) to obtain quaternion feature vectors. Support vector regression (SVR) was then employed to establish the quantitative model of Intra lipid in the mixture solution with volume percent from 6% to 10%. Compared with PCA-SVR for single optical length NIR data method, QPCA-SVR method for random three optical lengths NIR data significantly improved the accuracy of quantification. The determination coefficients of prediction (Rp2) are in the range of 0.9713 to 0.9882, which is remarkably higher than that of the PCA-SVR method for single optical length NIR data. The root mean square error of prediction (RMSEP) are in the range of 0.1365 to 0.2215, which is appreciably lower than that of the PCA-SVR method. The research results show that the QPCA method can effectively improve the accuracy of multi-optical lengths NIR quantitative analysis, and has huge potential in multi-optical lengths quantitative applications.

An Information-Theoretic Perspective on Proper Quaternion Variational Autoencoders.

Variational autoencoders are deep generative models that have recently received a great deal of attention due to their ability to model the latent distribution of any kind of input such as images and audio signals, among others. A novel variational autoncoder in the quaternion domain , namely the QVAE, has been recently proposed, leveraging the augmented second order statics of -proper signals. In this paper, we analyze the QVAE under an information-theoretic perspective, studying the ability of the -proper model to approximate improper distributions as well as the built-in -proper ones and the loss of entropy due to the improperness of the input signal. We conduct experiments on a substantial set of quaternion signals, for each of which the QVAE shows the ability of modelling the input distribution, while learning the improperness and increasing the entropy of the latent space. The proposed analysis will prove that proper QVAEs can be employed with a good approximation even when the quaternion input data are improper.

Augmented Online Sequential Quaternion Extreme Learning Machine

Online sequential extreme learning machine (OS-ELM) is one of the most popular real-time learning strategy for feedforward neural networks with single hidden layer due to its fast learning speed and excellent generalization ability. When dealing with quaternion signals, traditional real-valued learning models usually provide only suboptimal solutions compared with their quaternion-valued counterparts. However, online sequential quaternion extreme learning machine (OS-QELM) model is still lacking in literature. To fill this gap, this paper aims to establish a framework for the derivation and the design of OS-QELM. Specifically, we first derive a standard OS-QELM, and then propose two augmented OS-QELM models which can capture the complete second-order statistics of noncircular quaternion signals. The corresponding regularized models and two approaches to reducing the computational complexity are also derived and discussed respectively. Benefiting from the quaternion algebra and the augmented structure, the proposed models exhibit superiority over OS-ELM in simulation results on several benchmark quaternion regression problems and colour face recognition problems.

Color image encryption scheme based on quaternion discrete multi-fractional random transform and compressive sensing

A color image compression-encryption algorithm by combining quaternion discrete multi-fractional random transform with compressive sensing is investigated, in which the chaos-based fractional orders greatly improve key sensitivity. The original color image is compressed and encrypted with the assistance of compressive sensing, in which the partial Hadamard matrix adopted as a measurement matrix is constructed by iterating Chebyshev map instead of utilizing the entire Guassian matrix as a key. The sparse images are divided into 12 sub-images and then represented as three quaternion signals, which are modulated by the quaternion discrete multi-fractional random transform. The image blocking and the quaternion representation make the proposed cryptosystem avoid additional data extension existing in many transform-based methods. To further improve the level of security, the plaintext-related key streams generated by the 2D logistic-sine-coupling map are adopted to diffuse and confuse the intermediate results simultaneously. Consequently, the final ciphertext image is attained. Simulation results reveal that the proposed cryptosystem is feasible with high security and has strong robustness against various attacks.

Automatic Model Determination for Quaternion NMF

Nonnegative Matrix Factorization (NMF) is a well-known method for Blind Source Separation (BSS). Recently, BSS for polarized signals in spectropolarimetric data, containing both polarization and spectral information, was introduced. This information was encoded in 4-dimensional Stokes vectors represented by quaternion numbers. In the proposed Quaternion NMF (QNMF), the common challenge of determining the (usually) unknown number of quaternion signals remained unaddressed. Estimating the number of signals (aka model determination) is important, since an underestimation of this number results in poor source separation and omission of signals, while overestimation leads to extraction of noisy signals without physical meaning. Here, we introduce a method for determining the number of polarized signals in spectropolarimetric data, named QNMF $k$ . QNMF $k$ integrates: (a) Quaternion Alternating Direction Method of Multipliers (QADMM) implemented for QNMF, (b) random resampling of the initial quaternion data, and (c) custom clustering of sets of QADMM solutions with same number of sources, $k$ , needed to estimate the stability of the solutions. The appropriate latent dimension is determined based on the stability of the solutions. We demonstrate that, without any prior information, QNMF $k$ accurately extracts the correct number of signals used to generate synthetic quaternion datasets and a benchmark spectropolarimetric data.

Quaternion block sparse representation for signal recovery and classification

This paper presents a quaternion block sparse representation (QBSR) method for structural sparse signal recovery in quaternion space. Due to the noncommutativity of quaternion multiplication, conventional optimization algorithms originally designed for real-valued and complex-valued signal recovery problems are not applicable to the optimization problem of QBSR. To combat this problem, we leverage several quaternion operators and devise an effective algorithm for QBSR within the ADMM (Alternating Direction Method of Multipliers) framework. The second contribution of this work is to develop a QBSR based classifier referred to as QBSRC for quaternion data classification with application to color face recognition. Compared with real-valued representation based classifiers handling multiple color channels of a color image independently, QBSRC treats a color image as a quaternion signal and represents it in a holistic manner. The third contribution is to provide the theoretical analysis of QBSRC and rigorously prove that QBSRC is guaranteed to succeed in classification of any new test sample under appropriate condition. Experiments on both synthetic and real-world datasets demonstrate the effectiveness of the proposed method for quaternion signal recovery and classification.

Multi-image compression-encryption scheme based on quaternion discrete fractional Hartley transform and improved pixel adaptive diffusion

A new multi-image encryption scheme based on quaternion discrete fractional Hartley transform (QDFrHT) and an improved pixel adaptive diffusion is proposed, which can simultaneously increase the encryption capacity and reduce the consumption of keys. In the proposed scheme, the QDFrHT is proposed to generalize the discrete fractional Hartley transform to quaternion transform domain and then applied to multi-image encryption. The original images are compressed into four fusion images by discrete cosine transform (DCT) and Zigzag operations and then the resulting four images are represented as quaternion algebra. Afterward, the quaternion signal is processed with the proposed QDFrHT and the double random phase encoding technique. To enhance the security of the algorithm, the plaintext-related pixel adaptive diffusion and the pixel scrambling operation based on chaotic systems are followed to obtain the final encryption image. Different from the traditional cryptosystem whose secret keys are dependent on the plaintext, the designed cryptosystem ensures the sensitivity of the cryptosystem to plaintext while the selection of the secret keys is independent on plaintext by introducing an improved diffusion mechanism. Numerical experiments verify the feasibility and the efficiency of the proposed multi-image compression and encryption scheme.