24 research outputs found
Clifford Multivector Toolbox (for MATLAB)
matlab ® is a numerical computing environment oriented towards manipulation of matrices and vectors (in the linear algebra sense, that is arrays of numbers). Until now, there was no comprehensive toolbox (software library) for matlab to compute with Clifford algebras and matrices of multivectors. We present in the paper an account of such a toolbox, which has been developed since 2013, and released publically for the first time in 2015. The paper describes the major design decisions made in implementing the toolbox, gives implementation details, and demonstrates some of its capabilities, up to and including the LU decomposition of a matrix of Clifford multivectors
Fundamental representations and algebraic properties of biquaternions or complexified quaternions
The fundamental properties of biquaternions (complexified quaternions) are
presented including several different representations, some of them new, and
definitions of fundamental operations such as the scalar and vector parts,
conjugates, semi-norms, polar forms, and inner and outer products. The notation
is consistent throughout, even between representations, providing a clear
account of the many ways in which the component parts of a biquaternion may be
manipulated algebraically
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
The development of the quaternion wavelet transform
The purpose of this article is to review what has been written on what other authors have called quaternion wavelet transforms (QWTs): there is no consensus about what these should look like and what their properties should be. We briefly explain what real continuous and discrete wavelet transforms and multiresolution analysis are and why complex wavelet transforms were introduced; we then go on to detail published approaches to QWTs and to analyse them. We conclude with our own analysis of what it is that should define a QWT as being truly quaternionic and why all but a few of the “QWTs” we have described do not fit our definition
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
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
Semi-automated testing of proportional counters
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