On Uncertainty Principle for Quaternionic Linear Canonical Transform

Yang

Zou

2017

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The classical uncertainty principle of harmonic analysis states that a nontrivial function and its Fourier transform cannot both be sharply localized. It plays an important role in signal processing and physics. This paper generalizes the uncertainty principle for measurable sets from complex domain to hypercomplex domain using quaternion algebras, associated with the Quaternion Fourier transform. The performance is then evaluated in signal recovery problems where there is an interplay of missing and timelimiting data.

Section: Uncertainty Principlesmentioning

confidence: 99%

Uncertainty principle for measurable sets and signal recovery in quaternion domains

Yang

Zou

2017

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“…Because of the non‐commutative property of multiplication of quaternions, there are many different types of QLCTs : left‐sided QLCTs, right‐sided QLCTs and two‐sided QLCTs. For simplicity of sake, in this paper, we restrict our attention to the two‐sided QLCTs of 2D quaternion‐valued signals.…”

Section: The Quaternion Linear Canonical Transformmentioning

confidence: 99%

Herglotz's theorem and quaternion series of positive term

Liu

Tao

2016

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The present paper first introduces the notion of quaternion infinite series of positive term and establishes its several tests. Next, we give the definitions of the positive‐definite quaternion sequence and the positive semi‐definite quaternion function, and we extend the classical Herglotz's theorem to the quaternion linear canonical transform setting. Then we investigate the properties of the two‐sided quaternion linear canonical transform, such as time shift characteristics and differential characteristics. Finally, we derive its several basic properties of the quaternion linear canonical transform of a probability measure, in particular, and establish the Bochner–Minlos theorem. Copyright © 2016 John Wiley & Sons, Ltd.

“…It suppresses the negative frequency components in the QFT domains. Researchers generalized the QFTs to the quaternion linear canonical transforms (QLCTs). The QFTs, quaternion fractional Fourier transform are the special cases of QLCTs .…”

Section: Introductionmentioning

confidence: 99%

“…Researchers generalized the QFTs to the quaternion linear canonical transforms (QLCTs). The QFTs, quaternion fractional Fourier transform are the special cases of QLCTs . The QLCT has more degree of freedoms; therefore, it has shown to be more flexible for signal processing .…”

Section: Introductionmentioning

confidence: 99%

Phase‐based edge detection algorithms

2017

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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.