Envelope detection using generalized analytic signal in 2D QLCT domains

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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.

Section: Quaternion Analytic Signal and Quaternion Hardy Spacementioning

confidence: 99%

“…Definition (Kou et al ) The generalized quaternion partial Hilbert transforms (GQPHTs)

{scriptH}_{1}^{A_{i}}, false(i = 1, 2 false)

along the x ‐axis or y ‐axis and total Hilbert transform (GQTHT)

{scriptH}_{2}^{A_{1} A_{2}}

along the x ‐axis and y ‐axis of

f : R^{2} \to double-struckH

associated with QLCT, are given by

{scriptH}_{1}^{A_{1}} false[f false(\cdot, y false) false] false(x false) : = \frac{1}{π} p . v . e^{- boldi \frac{a_{1} x^{2}}{2 b_{1}}} \int_{R} \frac{e^{boldi \frac{a_{1} t^{2}}{2 b_{1}}} f false(t, y false)}{x - t} d t,

{scriptH}_{1}^{A_{2}} false[f false(x, \cdot false) false] false(y false) : = \frac{1}{π} p . v . \int_{R} \frac{f false(x, t false) e^{boldj \frac{a_{2} t^{2}}{2 b_{2}}}}{y - t} d t e^{- boldj \frac{a_{2} y^{2}}{2 b_{2}}},

{scriptH}_{2}^{A_{1}, A_{2}} false[f false] false(x, y false) : = \frac{1}{π^{2}} p . v . e<...>

…”

Section: Quaternion Analytic Signal and Quaternion Hardy Spacementioning

confidence: 99%

“…Definition The GQAS f q of a quaternionic signal f can be defined as

f_{A_{1}, A_{2}} (x, y) : = f (x_{1}, x_{2}) + i H_{1}^{A_{1}} [f (\cdot, x_{2})] (x_{1}) + H_{1}^{A_{2}} [f (x_{1}, \cdot)] (x_{2}) j + i H_{2}^{A_{1}, A_{2}} (x_{1}, x_{2}) j,

provided that Equations to are all well defined.…”

Section: Quaternion Analytic Signal and Quaternion Hardy Spacementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Phase‐based edge detection algorithms

Kou

2017

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Quaternion Fourier and linear canonical inversion theorems

Kou

2016

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The quaternion Fourier transform (QFT) is one of the key tools in studying color image processing. Indeed, a deep understanding of the QFT has created the color images to be transformed as whole, rather than as color separated component. In addition, understanding the QFT paves the way for understanding other integral transform, such as the quaternion fractional Fourier transform, quaternion linear canonical transform, and quaternion Wigner–Ville distribution. The aim of this paper is twofold: first to provide some of the theoretical background regarding the quaternion bound variation function. We then apply it to derive the quaternion Fourier and linear canonical inversion formulas. Secondly, to provide some in tuition for how the quaternion Fourier and linear canonical inversion theorems work on the absolutely integrable function space. Copyright © 2016 John Wiley & Sons, Ltd.

Sharper uncertainty principles in quaternionic Hilbert spaces

Ren

2019

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