Singular value decomposition of quaternion matrices: a new tool for vector-sensor signal processing

Bihan, Nicolas Le; Mars, Jérôme

doi:10.1016/j.sigpro.2004.04.001

Cited by 335 publications

(157 citation statements)

References 15 publications

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“…The augmented quaternion-valued covariance matrix of an augmented random vector q a = [q T q ıT q T q κT ] T see also (9) , is therefore given by,…”

Section: B Duality Between Quaternionic and Quadrivariate Statisticsmentioning

confidence: 99%

“…Based on (9) and (18), it is the augmented random vector x a = [x T x ıT x T x κT ] T that contains all the required second order statistical information. Then, the quaternion widely linear model (QWL) can be constructed as…”

Section: Quaternion Widely Linear Modelmentioning

confidence: 99%

“…To make the Gaussian distribution (31) cater for both Q-proper and Q-improper signals, we need to express it in terms of the augmented quaternion valued vector q a (9). To this end, the determinant of the quadrivariate covariance C R and the quadratic function (32) need to be further investigated.…”

Section: Variablesmentioning

confidence: 99%

“…where A is given in (9). From (11), det(A) = 16 N and since det(A −1 ) = det(A) −1 , the above expression can be further simplified to…”

Section: Variablesmentioning

confidence: 99%

“…Applications of quaternions include those in information forensics [1], instrumentation [2], communications [3], robotics [4], neural networks [5], and seismics & oceanics [6]. In the signal processing community, quaternions have been employed in Kalman filtering [7], the well-known MUSIC technique [8], singular value decomposition for vector sensing [9] and the least-mean-square estimation [10].…”

Section: Introductionmentioning

confidence: 99%

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Augmented second-order statistics of quaternion random signals

2011

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“…The augmented quaternion-valued covariance matrix of an augmented random vector q a = [q T q ıT q T q κT ] T see also (9) , is therefore given by,…”

Section: B Duality Between Quaternionic and Quadrivariate Statisticsmentioning

confidence: 99%

Section: Quaternion Widely Linear Modelmentioning

confidence: 99%

Section: Variablesmentioning

confidence: 99%

“…where A is given in (9). From (11), det(A) = 16 N and since det(A −1 ) = det(A) −1 , the above expression can be further simplified to…”

Section: Variablesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Augmented second-order statistics of quaternion random signals

2011

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Generalized sampling expansions associated with quaternion Fourier transform

Dong

Kou

2017

Math Methods in App Sciences

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Quaternion‐valued signals along with quaternion Fourier transforms (QFT) provide an effective framework for vector‐valued signal and image processing. However, the sampling theory of quaternion‐valued signals has not been well developed. In this paper, we present the generalized sampling expansions associated with QFT by using the generalized translation and convolution. We show that a σ‐bandlimited quaternion‐valued signal in QFT sense can be reconstructed from the samples of output signals of M linear systems based on QFT. Quaternion linear canonical transform is a generalization of QFT with six parameters. Using the relationship between QFT, we derive the sampling formula for σ‐bandlimited quaternion‐valued signal in quaternion linear canonical transform sense. Examples are given to illustrate our results. Copyright © 2017 John Wiley & Sons, Ltd.

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Solvability conditions and general solutions to some quaternion matrix equations

Wang

2021

Math Methods in App Sciences

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Since quaternion and quaternion matrices have applications in color image processing, 1 quantum mechanics, 2 signal processing, 3 etc., research on quaternion matrix equations has been actively ongoing for more recent years. Among concerns with a quaternion matrix equation are solvability conditions, general solutions, minimum norm least squares solutions, various symmetric solutions, the maximal and minimal ranks of solutions, etc. (e.g., ).Rodman gave a necessary and sufficient conditions for the existence of a unique solution to the quaternion matrix equation AX + XB = C. 5, page 117 Wang 6 presented bisymmetric and centrosymmetric solutions to some systems of quaternion matrix equations. Kyrchei 7 provided the minimum norm least squares solutions of some quaternion matrix Abbreviations: ANA, anti-nuclear antibodies; APC, antigen-presenting cells; IRF, interferon regulatory factor.

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Singular value decomposition of quaternion matrices: a new tool for vector-sensor signal processing

Cited by 335 publications

References 15 publications

Augmented second-order statistics of quaternion random signals

Augmented second-order statistics of quaternion random signals

Generalized sampling expansions associated with quaternion Fourier transform

Solvability conditions and general solutions to some quaternion matrix equations

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