95 research outputs found
Modeling of widely-linear quaternion valued systems using hypercomplex algorithms
The data-driven optimal modeling and identification of widely-linear quaternion-valued synthetic systems is achieved by using a quaternion-valued gradient based algorithms. To account rigorously for the second-order statistics of the quaternion system, the quaternion least mean square (QLMS) and widely linear quaternion least mean square (WL-QLMS) were selected. The QLMS is shown to successfully model the quaternion-valued systems and the WL-QLMS is able to model both quaternion and widely-linear quaternion valued systems taking into account the full second-order statistics of the system. Analysis has proven that both algorithms are able to adapt to non-stationary nature of the systems. This approach is supported by simulations of various synthetic systems
An Invitation to Hypercomplex Phase Retrieval: Theory and Applications
Hypercomplex signal processing (HSP) provides state-of-the-art tools to
handle multidimensional signals by harnessing intrinsic correlation of the
signal dimensions through Clifford algebra. Recently, the hypercomplex
representation of the phase retrieval (PR) problem, wherein a complex-valued
signal is estimated through its intensity-only projections, has attracted
significant interest. The hypercomplex PR (HPR) arises in many optical imaging
and computational sensing applications that usually comprise quaternion and
octonion-valued signals. Analogous to the traditional PR, measurements in HPR
may involve complex, hypercomplex, Fourier, and other sensing matrices. This
set of problems opens opportunities for developing novel HSP tools and
algorithms. This article provides a synopsis of the emerging areas and
applications of HPR with a focus on optical imaging.Comment: 10 pages, 4 figures, 2 table
Filtering and Tracking with Trinion-Valued Adaptive Algorithms
A new model for three-dimensional processes based on the trinion algebra is introduced for the first time. Compared
with the pure quaternion model, the trinion model is more compact and computationally more efficient, while having similar or
comparable performance in terms of adaptive linear filtering. Moreover, the trinion model can effectively represent the general
relationship of state evolution in Kalman filtering, where the pure quaternion model fails. Simulations on real-world wind
recordings and synthetic data sets are provided to demonstrate the potentials of this new modeling method
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 , i.e. rotations from .
Based on this idea, we propose a new definition of properness, namely the
-properness, for quaternion random variables using invariance
property under the action of the rotation group . 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 -proper quaternion random variables are
presented. Comparisons with previous definitions are given and simulations
illustrate in a geometric manner the newly introduced concept.Comment: 14 pages, 3 figure
PHNNs: Lightweight Neural Networks via Parameterized Hypercomplex Convolutions
Hypercomplex neural networks have proven to reduce the overall number of parameters while ensuring valuable performance by leveraging the properties of Clifford algebras. Recently, hypercomplex linear layers have been further improved by involving efficient parameterized Kronecker products. In this article, we define the parameterization of hypercomplex convolutional layers and introduce the family of parameterized hypercomplex neural networks (PHNNs) that are lightweight and efficient large-scale models. Our method grasps the convolution rules and the filter organization directly from data without requiring a rigidly predefined domain structure to follow. PHNNs are flexible to operate in any user-defined or tuned domain, from 1-D to nD regardless of whether the algebra rules are preset. Such a malleability allows processing multidimensional inputs in their natural domain without annexing further dimensions, as done, instead, in quaternion neural networks (QNNs) for 3-D inputs like color images. As a result, the proposed family of PHNNs operates with 1/n free parameters as regards its analog in the real domain. We demonstrate the versatility of this approach to multiple domains of application by performing experiments on various image datasets and audio datasets in which our method outperforms real and quaternion-valued counterparts
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