Asymptotic behaviour of the quaternion linear canonical transform and the Bochner–Minlos theorem

Kou, Kit Ian; Morais, J.

doi:10.1016/j.amc.2014.08.090

Cited by 48 publications

(40 citation statements)

References 27 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We reviews some important properties of the kernel function K A and the LCT L A ( f ) as follows : K A (− x ,− w ) = K A ( x , w ); K A (− x , w ) = K A ( x ,− w );

\overset{K_{A} (x, w)}{false} = K_{B} (x, w)

, where

B = ((), \begin{matrix} a & - b \\ - c & d \end{matrix})

;

\int_{double-struckR} K_{A_{1}} (x, w) K_{A_{2}} (w, y) dw = K_{A_{1} A_{2}} (x, y)

, where A 1 A 2 corresponds to matrix product and

A_{i} = ((), \begin{matrix} a_{i} & b_{i} \\ c_{i} & d_{i} \end{matrix}) (i = 1, 2)

with det( A i ) = 1; and

L_{A_{2}} (L_{A_{1}}) (f) = L_{A_{2} A_{1}} (f)

f \in L^{1} (double-struckR; double-struckC)

and

L_{A_{1}} (f) \in L^{1} (double-struckR; double-struckC)

L_{A^{- 1}} (L_{A}) (f) = f

, where

A^{- 1} = ((), \begin{matrix} d & - b \\ - c & a \end{matrix})

…”

Section: Preliminariesmentioning

confidence: 99%

“…This completes the proof. □Remark In Proposition 5.5 of , the authors miss the condition

b_{1} b_{2} = \frac{1}{4 π^{2}}

, and the conclusion that the functionals L i j ( μ ), L j i ( μ ) are all positive semi‐definite does not hold without this condition.…”

Section: The Quaternion Linear Canonical Transforms Of a Borel Probabmentioning

confidence: 99%

“…Let s be the space of quaternion sequences

s : = \{{p_{n}} : \lim_{n \to \infty} n^{t} p_{n} = 0, \forall t \in N_{+}\},

where

p_{n} \in double-struckH

and let

s_{m} : = {{p_{n}} : ∥ p ∥_{m}^{2} : = \sum_{n = 0}^{\infty} {(1 + n^{2})}^{m} | p_{n} |^{2} < + \infty}, m \in N_{+} .

From , we have the following property.Proposition

s = ⋂_{m \in {double-struckN}_{+}} s_{m}

, and s is a Fr

é

chet space.…”

Section: The Quaternion Linear Canonical Transforms Of a Borel Probabmentioning

confidence: 99%

See 2 more Smart Citations

Herglotz's theorem and quaternion series of positive term

Kou

Liu

Tao

2016

Math Methods in App Sciences

Self Cite

View full text Add to dashboard Cite

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.

show abstract

“…We reviews some important properties of the kernel function K A and the LCT L A ( f ) as follows : K A (− x ,− w ) = K A ( x , w ); K A (− x , w ) = K A ( x ,− w );

\overset{K_{A} (x, w)}{false} = K_{B} (x, w)

, where

B = ((), \begin{matrix} a & - b \\ - c & d \end{matrix})

;

\int_{double-struckR} K_{A_{1}} (x, w) K_{A_{2}} (w, y) dw = K_{A_{1} A_{2}} (x, y)

, where A 1 A 2 corresponds to matrix product and

A_{i} = ((), \begin{matrix} a_{i} & b_{i} \\ c_{i} & d_{i} \end{matrix}) (i = 1, 2)

with det( A i ) = 1; and

L_{A_{2}} (L_{A_{1}}) (f) = L_{A_{2} A_{1}} (f)

f \in L^{1} (double-struckR; double-struckC)

and

L_{A_{1}} (f) \in L^{1} (double-struckR; double-struckC)

L_{A^{- 1}} (L_{A}) (f) = f

, where

A^{- 1} = ((), \begin{matrix} d & - b \\ - c & a \end{matrix})

…”

Section: Preliminariesmentioning

confidence: 99%

“…This completes the proof. □Remark In Proposition 5.5 of , the authors miss the condition

b_{1} b_{2} = \frac{1}{4 π^{2}}

, and the conclusion that the functionals L i j ( μ ), L j i ( μ ) are all positive semi‐definite does not hold without this condition.…”

Section: The Quaternion Linear Canonical Transforms Of a Borel Probabmentioning

confidence: 99%

See 1 more Smart Citation

Herglotz's theorem and quaternion series of positive term

Kou

Liu

Tao

2016

Math Methods in App Sciences

Self Cite

View full text Add to dashboard Cite

show abstract

“…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 . The authors defined the generalized quaternion analytic signal (GQAS) associated with QLCTs, which is defined by an original signal with its generalized quaternion partial and total Hilbert transforms.…”

Section: Introductionmentioning

confidence: 99%

Phase‐based edge detection algorithms

Kou

2017

Math Methods in App Sciences

Self Cite

View full text Add to dashboard Cite

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.

show abstract

“…This extension is constructed by substituting the kernel of the QFT with the kernel of the LCT. A number of useful properties of the QLCT have been investigated including shift, orthogonality relation, reconstruction formula, and Heisenberg uncertainty principle (see, for example, [4][5][6] and the references given therein).…”

Section: Introductionmentioning

confidence: 99%

Relation between Quaternion Fourier Transform and Quaternion Wigner-Ville Distribution Associated with Linear Canonical Transform

Bahri

Fatimah

2017

Journal of Applied Mathematics

View full text Add to dashboard Cite

The quaternion Wigner-Ville distribution associated with linear canonical transform (QWVD-LCT) is a nontrivial generalization of the quaternion Wigner-Ville distribution to the linear canonical transform (LCT) domain. In the present paper, we establish a fundamental relationship between the QWVD-LCT and the quaternion Fourier transform (QFT). Based on this fact, we provide alternative proof of the well-known properties of the QWVD-LCT such as inversion formula and Moyal formula. We also discuss in detail the relationship among the QWVD-LCT and other generalized transforms. Finally, based on the basic relation between the quaternion ambiguity function associated with the linear canonical transform (QAF-LCT) and the QFT, we present some important properties of the QAF-LCT.

show abstract

Asymptotic behaviour of the quaternion linear canonical transform and the Bochner–Minlos theorem

Cited by 48 publications

References 27 publications

Herglotz's theorem and quaternion series of positive term

Herglotz's theorem and quaternion series of positive term

Phase‐based edge detection algorithms

Relation between Quaternion Fourier Transform and Quaternion Wigner-Ville Distribution Associated with Linear Canonical Transform

Contact Info

Product

Resources

About