Uncertainty principle for measurable sets and signal recovery in quaternion domains

Kou, Kit Ian; Yang, Yan; Zou, Cuiming

doi:10.1002/mma.4271

Cited by 17 publications

(10 citation statements)

References 32 publications

(75 reference statements)

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“…In [36,50], the authors studied the problem of signal recovery by uncertainty principles. The paper [37] evaluated in signal recovery problems where there is an interplay of missing and time-limiting data. The correlative result has been applied to the window Fourier transform (WFT) as well in [49].…”

Section: Numerical Example and Potential Applicationmentioning

confidence: 99%

“…In signal processing, an uncertainty principle states that the product of the variances of the signal in the time and frequency domains has a lower bound. There are many different kinds of uncertainty principles, for instance, Heisenberg uncertainty principle [22,30,31,39], Logarithmic uncertainty principle [32], Hardy's uncertainty principle [39,40], Beurling's uncertainty principle [50], Lieb uncertainty principle [7,35], Donoho-Stark's uncertainty principle [36][37][38]. The Lieb uncertainty principle for the WLCT has been discussed in [7], which takes the LCT version as one of its special cases.…”

Section: Introductionmentioning

confidence: 99%

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Uncertainty principle for the two-sided quaternion windowed linear canonical transform

Gao¹,

Li²

2021

Preprint

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In this paper, we investigate the (two-sided) quaternion windowed linear canonical transform (QWLCT) and study the uncertainty principles associated with the QWLCT. Firstly, several important properties of the QWLCT such as bounded, shift, modulation, orthogonality relation, are presented based on the spectral representation of the quaternionic linear canonical transform (QLCT). Secondly, Pitt's inequality and Lieb inequality for the QWLCT are explored. Moreover, we study different kinds of uncertainty principles for the QWLCT, such as Logarithmic uncertainty principle, Entropic uncertainty principle, Lieb uncertainty principle and Donoho-Stark's uncertainty principle. Finally, we give a numerical example and a potential application in signal recovery by using Donoho-Stark's uncertainty principle associated with the QWLCT.

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Section: Numerical Example and Potential Applicationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Uncertainty principle for the two-sided quaternion windowed linear canonical transform

Gao¹,

Li²

2021

Preprint

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“…Recently, the UP for signals in the quaternionic setting has obtained much attentions; see also Yang et al for a tighter version. The UP in quaternionic quantum mechanics has also been investigated (see, eg, Horwitz and Biedenharn and Muraleetharan, Thirulogasanthar and Sabadini).…”

Section: Introductionmentioning

confidence: 99%

Sharper uncertainty principles in quaternionic Hilbert spaces

Ren

2019

Math Methods in App Sciences

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Hilbert spaces is established, which generalizes the result of Goh-Micchelli. It turns out that there appears an additional term given by a commutator that reflects the feature of quaternions. The result is further strengthened when one operator is self-adjoint, which extends under weaker conditions the uncertainty principle of Dang-Deng-Qian from complex numbers to quaternions. In particular, our results are applied to concrete settings related to quaternionic Fock spaces, quaternionic periodic functions, quaternion Fourier transforms, quaternion linear canonical transforms, and nonharmonic quaternion Fourier transforms. KEYWORDSHilbert space, quaternion, uncertainty principle MSC CLASSIFICATIONRecently, the UP for signals in the quaternionic setting has obtained much attentions 7-17 ; see also Yang et al 18 for a tighter version. The UP in quaternionic quantum mechanics has also been investigated (see, eg, Horwitz and Biedenharn 19 and Muraleetharan, Thirulogasanthar and Sabadini 20 ). It is known that quantum mechanics can be formulated on Hilbert spaces over the reals R, complex numbers C, and quaternions H, respectively; see Birkhoff, 21 Solèr, 22 Holland, 23 and Piron. 24 Furthermore, the former two formulations are equivalent in the sense of Stueckelberg. 25 The quaternionic operator plays a vital role in the formulation of quantum mechanics. 26,27 However, there exist some difficulties in dealing with quaternionic quantum mechanics. One of the difficulties is that how to establish a well-behaved spectral theory for quaternionic linear operators. To determine the eigenvalues for the quaternionic linear operator A, Math Meth Appl Sci. 2020;43:1608-1630. wileyonlinelibrary.com/journal/mma

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“…Alternative higher dimensional extension in several complex variables was considered in other studies, which is to yield a one‐quadrant analytic signal by the mean of partial and total Hilbert transforms, namely, quaternion analytic signal. Motivated by the two‐sided quaternion Fourier transform (QFT), which have been found widely applications in high‐dimensional signal and color image processing . The quaternion analytic signal is defined by the original signal with its quaternion partial and total Hilbert transform.…”

Section: Introductionmentioning

confidence: 99%

“…Motivated by the two-sided quaternion Fourier transform (QFT), which have been found widely applications in high-dimensional signal and color image processing. [27][28][29][30][31][32] The quaternion analytic signal [19][20][21][22] is defined by the original signal with its quaternion partial and total Hilbert transform. It suppresses the negative frequency components in the QFT domains.…”

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confidence: 99%

Phase‐based edge detection algorithms

Kou

2017

Math Methods in App Sciences

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Quaternion analytic signal is regarded as a generalization of analytic signal from 1D to 4D space. It is defined by an original signal with its quaternion partial and total Hilbert transforms. The quaternion analytic signal provides the signal features representation, such as the local amplitude and local phase angle, the latter includes the structural information of the original signal. The aim of the present study is twofold. Firstly, it attempts to analyze the Plemelj‐Sokhotzkis formula associated with quaternion Fourier transform and quaternion linear canonical transform. With these formulae, we show that the quaternion analytic signals are the boundary values of quaternion Hardy functions in the upper half space of 2 complex variables space. Secondly, the quaternion analytic signal can be extended to the quaternion Hardy function in the upper half space of 2 complex variables space. Two novel types of phase‐based edge detectors are proposed, namely, quaternion differential phase angle and quaternion differential phase congruency methods. In terms of peak signal‐to‐noise ratio and structural similarity index measure, comparisons with competing methods on real‐world images consistently show the superiority of the proposed methods.

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Uncertainty principle for measurable sets and signal recovery in quaternion domains

Cited by 17 publications

References 32 publications

Uncertainty principle for the two-sided quaternion windowed linear canonical transform

Uncertainty principle for the two-sided quaternion windowed linear canonical transform

Sharper uncertainty principles in quaternionic Hilbert spaces

Phase‐based edge detection algorithms

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