Discrete linear canonical wavelet transform and its applications

Jiatong Wang,Yue Wang,Shiwei Ren,Weijiang Wang

doi:10.1186/s13634-018-0550-z

Jiatong Wang, Yue Wang + Show 2 more

Open Access

https://doi.org/10.1186/s13634-018-0550-z

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Abstract

The continuous generalized wavelet transform (GWT) which is regarded as a kind of time-linear canonical domain (LCD)-frequency representation has recently been proposed. Its constant-Q property can rectify the limitations of the wavelet transform (WT) and the linear canonical transform (LCT). However, the GWT is highly redundant in signal reconstruction. The discrete linear canonical wavelet transform (DLCWT) is proposed in this paper to solve this problem. First, the continuous linear canonical wavelet transform (LCWT) is obtained with a modification of the GWT. Then, in order to eliminate the redundancy, two aspects of the DLCWT are considered: the multi-resolution approximation (MRA) associated with the LCT and the construction of orthogonal linear canonical wavelets. The necessary and sufficient conditions pertaining to LCD are derived, under which the integer shifts of a chirp-modulated function form a Riesz basis or an orthonormal basis for a multi-resolution subspace. A fast algorithm that computes the discrete orthogonal LCWT (DOLCWT) is proposed by exploiting two-channel conjugate orthogonal mirror filter banks associated with the LCT. Finally, three potential applications are discussed, including shift sampling in multi-resolution subspaces, denoising of non-stationary signals, and multi-focus image fusion. Simulations verify the validity of the proposed algorithms.

Highlights

The linear canonical transform (LCT), the generalization of the Fourier transform (FT), the fractional Fourier transform (FrFT), the Fresnel transform and the scaling operations, has been found useful in many applications such as optics [1, 2] and signal processing [3,4,5,6,7,8,9,10,11]
We propose the discrete linear canonical wavelet transform (DLCWT) to solve these problems
5 Simulations results and discussion we provide simulation results of three applications to illustrate the performance of the proposed DLCWT

Summary

Introduction

The linear canonical transform (LCT), the generalization of the Fourier transform (FT), the fractional Fourier transform (FrFT), the Fresnel transform and the scaling operations, has been found useful in many applications such as optics [1, 2] and signal processing [3,4,5,6,7,8,9,10,11]. Higher concentration and lower sampling rate make the LCT more competent to resolve non-stationary signals. Due to the global kernel it uses, the LCT can only reveal the overall linear canonical domain (LCD)-frequency contents. The LCT is not competent in those scenarios which require the signal processing tools to display the time and LCD-frequency information jointly. Like the other time-frequency representations (TFRs), the CT projects the input signal onto a set of functions that are all obtained by modifying an

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