Two-sided fractional quaternion Fourier transform and its application

Li, Zunfeng; Shi, Haipan; Ye, Qiao

doi:10.1186/s13660-021-02654-3

Cited by 5 publications

(4 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The QFRFT [22] is a generalization of the FRFT. Because quaternion multiplication does not satisfy the commutative law, there are three kinds the QFRFT: the left-sided QFRFT, the right-sided QFRFT, and the two-sided QFRFT.…”

Section: The Quaternion Fractional Fourier Transformmentioning

confidence: 99%

Convolution Theorem Associated with the QWFRFT

et al. 2023

View full text Add to dashboard Cite

The quaternion windowed fractional Fourier transform (QWFRFT) is a generalized form of the quaternion fractional Fourier transform (QFRFT), it plays a crucial role in signal processing for the analysis of multidimensional signals. In this paper, we first give the definition of the two-sided QWFRFT and some fundamental properties. Then, the quaternion convolution is proposed, and the relation between the quaternion convolution and the classical convolution is also given. Based on the quaternion convolution of the QWFRFT, relevant convolution theorems for the QWFRFT are studied. Moreover, the fast algorithm for QWFRFT is discussed. Finally, the complexity of QWFRFT and the quaternion windowed fractional convolution are given.

show abstract

Section: The Quaternion Fractional Fourier Transformmentioning

confidence: 99%

Convolution Theorem Associated with the QWFRFT

et al. 2023

View full text Add to dashboard Cite

show abstract

“…However, it wasn't recognized widely until 1993 in which FrFT was introduced independently by some groups. The main purpose of introducing the FrFT was to solve the differential equation in the mechanic of quantum and to interpret optics problems [31,32]. Since FrFT is a powerful time-frequency transform, it has great potential to be used in signal processing.…”

Section: Fractional Fourier Transformmentioning

confidence: 99%

“…The ath fractional-order of FrFT can be defined as its effect on classical FT eigenfunctions [31]. FrFT is defined as a linear operator corresponding to the counterclockwise rotation through an angle f a = p 2 in the time-frequency space [33,34].…”

Section: Fractional Fourier Transformmentioning

confidence: 99%

Phonocardiography-based mitral valve prolapse detection with using fractional fourier transform

Mehrabbeik¹,

Rashidi²,

Fallah³

et al. 2020

Biomed. Phys. Eng. Express

View full text Add to dashboard Cite

Mitral Valve Prolapse (MVP) is a common condition among people, which is often benign and does not need any serious treatment. However, this doesn’t mean that MVP can’t cause any problems. In malignant conditions, MVP can cause mitral failure and also heart failure. Early diagnosis of MVP is significantly important to control and reduce its complications. Since the phonocardiogram signal provides useful information about heart valves function, it can be used for MVP detection. To detect MVP, the signal was denoised and segmented into heart cycles and constant three-second pieces in the first and second approaches, respectively. Next, based on the Fractional Fourier Transform (FrFT), the desired features were extracted. Then, the extracted features were windowed by a Moving Logarithmic Median Window (MLMW) and optimum features were selected using Mahalanobis, Bhattacharyya, Canberra, and Minkowski distance criteria. Finally, using the selected features, classification was performed by using the K-Nearest Neighbor (KNN) and the Suppor Vector Machine (SVM) classifiers to find out whether a segment is prolapsed. The best results of the experiment on the collected database contain 15 prolapsed and 6 non-prolapsed subjects using the A-test method show 96.25 ± 2.43 accuracy, 98.5 ± 3.37 sensitivity, 94.0 ± 5.16 specificity, 96.0 ± 3.44 precision, 92.5 ± 4.86 kappa, and 96.6 ± 2.34 f-score with the SVM classifier.

show abstract

“…The quaternion number system was first described by Hamilton as a generalization of complex numbers [13,14]. In recent years, researchers have extended integral transforms into the quaternion algebra domain, leading to the development of theoretical frameworks such as quaternion Fourier transform (QFT) [15][16][17][18], quaternion fractional Fourier transform (QFRFT) [19], quaternion windowed fractional Fourier transform (QWFRFT) [20][21][22][23][24], quaternion linear canonical transform (QLCT) [25,26], and quaternion offset linear canonical transform [27]. Several important properties of QLCT have been investigated, including linearity, time shift, modulation, reconstruction formula, boundedness , and uncertainty principles in [28][29][30].…”

Section: Introductionmentioning

confidence: 99%

Reduced Biquaternion Windowed Linear Canonical Transform: Properties and Applications

Yang,

Feng,

Wang

et al. 2024

Mathematics

View full text Add to dashboard Cite

The quaternion windowed linear canonical transform is a tool for processing multidimensional data and enhancing the quality and efficiency of signal and image processing; however, it has disadvantages due to the noncommutativity of quaternion multiplication. In contrast, reduced biquaternions, as a special case of four-dimensional algebra, possess unique advantages in computation because they satisfy the multiplicative exchange rule. This paper proposes the reduced biquaternion windowed linear canonical transform (RBWLCT) by combining the reduced biquaternion signal and the windowed linear canonical transform that has computational efficiency thanks to the commutative property. Firstly, we introduce the concept of a RBWLCT, which can extract the time local features of an image and has the advantages of both time-frequency analysis and feature extraction; moreover, we also provide some fundamental properties. Secondly, we propose convolution and correlation operations for RBWLCT along with their corresponding generalized convolution, correlation, and product theorems. Thirdly, we present a fast algorithm for RBWLCT and analyze its computational complexity based on two dimensional Fourier transform (2D FTs). Finally, simulations and examples are provided to demonstrate that the proposed transform effectively captures the local RBWLCT-frequency components with enhanced degrees of freedom and exhibits significant concentrations.

show abstract

Two-sided fractional quaternion Fourier transform and its application

Cited by 5 publications

References 11 publications

Convolution Theorem Associated with the QWFRFT

Convolution Theorem Associated with the QWFRFT

Phonocardiography-based mitral valve prolapse detection with using fractional fourier transform

Reduced Biquaternion Windowed Linear Canonical Transform: Properties and Applications

Contact Info

Product

Resources

About