Hypercomplex Signals Research Articles

Hypercomplex Techniques in Signal and Image Processing Using Network Graph Theory: Identifying core research directions [Hypercomplex Signal and Image Processing

Hypercomplex Techniques in Signal and Image Processing Using Network Graph Theory: Identifying core research directions [Hypercomplex Signal and Image Processing

Widely Linear Adaptive Filtering Based on Clifford Geometric Algebra: A unified framework [Hypercomplex Signal and Image Processing

Widely Linear Adaptive Filtering Based on Clifford Geometric Algebra: A unified framework [Hypercomplex Signal and Image Processing

Demystifying the Hypercomplex: Inductive biases in hypercomplex deep learning [Hypercomplex Signal and Image Processing

Demystifying the Hypercomplex: Inductive biases in hypercomplex deep learning [Hypercomplex Signal and Image Processing

Hypercomplex Signal and Image Processing: Part 2 [From the Guest Editors

Hypercomplex Signal and Image Processing: Part 2 [From the Guest Editors

Quaternion Neural Networks: A physics-incorporated intelligence framework [Hypercomplex Signal and Image Processing

Quaternion Neural Networks: A physics-incorporated intelligence framework [Hypercomplex Signal and Image Processing

Understanding Vector-Valued Neural Networks and Their Relationship With Real and Hypercomplex-Valued Neural Networks: Incorporating intercorrelation between features into neural networks [Hypercomplex Signal and Image Processing

Understanding Vector-Valued Neural Networks and Their Relationship With Real and Hypercomplex-Valued Neural Networks: Incorporating intercorrelation between features into neural networks [Hypercomplex Signal and Image Processing

Augmented Statistics of Quaternion Random Variables: A lynchpin of quaternion learning machines [Hypercomplex Signal and Image Processing

Augmented Statistics of Quaternion Random Variables: A lynchpin of quaternion learning machines [Hypercomplex Signal and Image Processing

Quaternion-Based Arithmetic in Quantum Information Processing: A promising approach for efficient color quantum imaging [Hypercomplex Signal and Image Processing

Quaternion-Based Arithmetic in Quantum Information Processing: A promising approach for efficient color quantum imaging [Hypercomplex Signal and Image Processing

Geometric Algebra Quantum Convolutional Neural Network: A model using geometric (Clifford) algebras and quantum computing [Hypercomplex Signal and Image Processing

Geometric Algebra Quantum Convolutional Neural Network: A model using geometric (Clifford) algebras and quantum computing [Hypercomplex Signal and Image Processing

Holistic Processing of Color Images Using Novel Quaternion-Valued Wavelets on the Plane: A promising transformative tool [Hypercomplex Signal and Image Processing

Holistic Processing of Color Images Using Novel Quaternion-Valued Wavelets on the Plane: A promising transformative tool [Hypercomplex Signal and Image Processing

Split-Quaternions for Perceptual White Balance: A quantum information-based chromatic adaptation transform [Hypercomplex Signal and Image Processing

Split-Quaternions for Perceptual White Balance: A quantum information-based chromatic adaptation transform [Hypercomplex Signal and Image Processing

Hypercomplex Processing of Vector Field Seismic Data: Toward vector-valued signal processing [Hypercomplex Signal and Image Processing

Hypercomplex Processing of Vector Field Seismic Data: Toward vector-valued signal processing [Hypercomplex Signal and Image Processing

An octonion-based nonlinear echo state network for speech emotion recognition in Metaverse

While the Metaverse is becoming a popular trend and drawing much attention from academia, society, and businesses, processing cores used in its infrastructures need to be improved, particularly in terms of signal processing and pattern recognition. Accordingly, the speech emotion recognition (SER) method plays a crucial role in creating the Metaverse platforms more usable​ and enjoyable for its users. However, existing SER methods continue to be plagued by two significant problems in the online environment. The shortage of adequate engagement and customization between avatars and users is recognized as the first issue and the second problem is related to the complexity of SER problems in the Metaverse as we face people and their digital twins or avatars. This is why developing efficient machine learning (ML) techniques specified for hypercomplex signal processing is essential to enhance the impressiveness and tangibility of the Metaverse platforms. As a solution, echo state networks (ESNs), which are an ML powerful tool for SER, can be an appropriate technique to enhance the Metaverse’s foundations in this area. Nevertheless, ESNs have some technical issues restricting them from a precise and reliable analysis, especially in the aspect of high-dimensional data. The most significant limitation of these networks is the high memory consumption caused by their reservoir structure in face of high-dimensional signals. To solve all problems associated with ESNs and their application in the Metaverse, we have come up with a novel structure for ESNs empowered by octonion algebra called NO2GESNet. Octonion numbers have eight dimensions, compactly display high-dimensional data, and improve the network precision and performance in comparison to conventional ESNs. The proposed network also solves the weaknesses of the ESNs in the presentation of the higher-order statistics to the output layer by equipping it with a multidimensional bilinear filter. Three comprehensive scenarios to use the proposed network in the Metaverse have been designed and analyzed, not only do they show the accuracy and performance of the proposed approach, but also the ways how SER can be employed in the Metaverse platforms.

Call for Papers: Special Issue on Hypercomplex Signal & Image Processing

Call for Papers: Special Issue on Hypercomplex Signal & Image Processing

Hypercomplex Low Rank Reconstruction for NMR Spectroscopy

Nuclear magnetic resonance (NMR) spectroscopy serves as an important tool to analyze chemicals and proteins in bioengineering. Multi-dimensional NMR offers a major improvement in resolution with multi-dimensional spectrum, but significantly increases data acquisition time and produces hypercomplex data that is difficult to be handled. To reduce this time, non-uniformly sampling can be applied to obtain undersampled data and using a reconstruction approach, such as the state-of-the-art low rank method, to remove the spectral artifacts introduced by undersampling. However, only complex format of signal, including the real and imaginary parts, is considered in previous low rank approach, which is less efficient when dealing with hypercomplex data that has multiple components. To solve this problem, a hypercomplex low rank model is proposed by introducing an adjoint matrix operation and then solved with a fast matrix factorization algorithm. This method explores redundant information among all the components of hypercomplex signal. Using simulated data and real protein data, we demonstrate that the proposed method provides a fast and high-fidelity reconstruction of hypercomplex spectroscopy in fast NMR.

Wide-sense Markov signals on the tessarine domain. A study under properness conditions

The quaternion algebra is not always the best choice for processing 4D hypercomplex signals. This paper aims to explore tessarines as an alternative algebra to solve the estimation problem. More concretely, wide-sense Markov signals in the tessarine domain are introduced and their properties under properness properties are analyzed. Firstly, the T2-properness condition in the tessarine setting is defined and then, the linear estimation problem under tessarine processing is addressed. The equivalence between the optimal estimator based on tessarine widely linear processing and the one based on tessarine T2 processing is proved, thus attaining a notable reduction in computational burden. Next, the Ti-proper wide-sense Markov signals, i=1,2, are defined and a forwards representation for modeling them is suggested. Finally, the estimation problem with intermittent observations for this class of signals is tackled. Specifically, based on the forwards representation, two algorithms for the problems of filtering, prediction and fixed-interval smoothing are devised. Numerical simulations are developed where the superiority of the Ti estimators, i=1,2, over their counterparts in the quaternion domain is shown.

Stronger uncertainty principles for hypercomplex signals

In this paper, for real para-vector-valued signals, we obtain stronger uncertainty principles in terms of covariance and absolute covariance based on Fourier transform in both directional and the spatial cases. We provide certain conditions that give rise to the equal relation between the two uncertainty principles. Examples are presented to verify the results.

Uncertainty principles for hypercomplex signals in the linear canonical transform domains

Linear canonical transforms (LCTs) are a family of integral transforms with wide application in optical, acoustical, electromagnetic, and other wave propagation problems. The Fourier and fractional Fourier transforms are special cases of LCTs. In this paper, we extend the uncertainty principle for hypercomplex signals in the linear canonical transform domains, giving the tighter lower bound on the product of the effective widths of complex paravector- (multivector-)valued signals in the time and frequency domains. It is seen that this lower bound can be achieved by a Gaussian signal. An example is given to verify the result.

Algebraic foundations of split hypercomplex nonlinear adaptive filtering

A split hypercomplex learning algorithm for the training of nonlinear finite impulse response adaptive filters for the processing of hypercomplex signals of any dimension is proposed. The derivation strictly takes into account the laws of hypercomplex algebra and hypercomplex calculus, some of which have been neglected in existing learning approaches (e.g., for quaternions). Already in the case of quaternions, we can predict improvements in performance of hypercomplex processes. The convergence of the proposed algorithms is rigorously analyzed. Copyright © 2012 John Wiley & Sons, Ltd.

Fractional Fourier transforms of hypercomplex signals

An overview is given to a new approach for obtaining generalized Fourier transforms in the context of hypercomplex analysis (or Clifford analysis). These transforms are applicable to higher-dimensional signals with several components and are different from the classical Fourier transform in that they mix the components of the signal. Subsequently, attention is focused on the special case of the so-called Clifford-Fourier transform where recently a lot of progress has been made. A fractional version of this transform is introduced and a series expansion for its integral kernel is obtained. For the case of dimension 2, also an explicit expression for the kernel is given.