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Discrete Linear Canonical Wavelet Transform and Its Applications Jiatong Wang, Yue Wang, Weijiang Wang and Shiwei Ren*

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Discrete Linear Canonical Wavelet Transform and Its Applications Jiatong Wang, Yue Wang, Weijiang Wang and Shiwei Ren* Wang et al. EURASIP Journal on Advances in Signal EURASIP Journal on Advances Processing (2018) 2018:29 https://doi.org/10.1186/s13634-018-0550-z in Signal Processing RESEARCH Open Access Discrete linear canonical wavelet transform and its applications Jiatong Wang, Yue Wang, Weijiang Wang and Shiwei Ren* 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. Keywords: Discrete linear canonical wavelet transform, Multi-resolution approximation, Filter banks, Shift sampling, Denoising, Image fusion 1 Introduction original window function (i.e., mother chirplet) [13]. Due The linear canonical transform (LCT), the generalization to the chirping operation, the users are available to new of the Fourier transform (FT), the fractional Fourier trans- degrees of freedom in shaping the time-frequency cells form (FrFT), the Fresnel transform and the scaling opera- with respect to the other TFRs. However, as the non- tions, has been found useful in many applications such as orthogonality between the chirplet with different chirp optics [1, 2] and signal processing [3–11]. Higher concen- rates, the CT is very redundant which makes the compu- tration and lower sampling rate make the LCT more com- tational complexity too high. petent to resolve non-stationary signals. However, due to The short-time fractional fourier transform (STFrFT) theglobalkernelituses,theLCTcanonlyrevealtheover- introduced in [14] is regarded as a kind of time-fractional- all linear canonical domain (LCD)-frequency contents. Fourier-domain-frequency representation. It plays a pow- Therefore, the LCT is not competent in those scenarios erful role in the 2D analysis of the chirp signals because which require the signal processing tools to display the the short-time fractional Fourier domain support is com- time and LCD-frequency information jointly. pact when the matched order STFrFT is taken. However, Chirplet transform (CT) was first proposed in [12]to the continuous STFrFT is highly redundant on its 2D solve this problem. Like the other time-frequency repre- plane (t, u) in signal reconstruction, and its computational sentations (TFRs), the CT projects the input signal onto complexity is high. a set of functions that are all obtained by modifying an A novel fractional wavelet transform (NFrWT) based on the idea of the FrFT and the wavelet transform (WT) was *Correspondence: [email protected] proposed in [15, 16]. It takes the fractional convolution School of Information and Electronics, Beijing Institute of Technology, South between the signal and the conventional wavelets which ZhongGuanCun Street, Haidian District, 100081 Beijing, China © The Author(s). 2018 Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Wang et al. EURASIP Journal on Advances in Signal Processing (2018) 2018:29 Page 2 of 18 makes the NFrWT available with tunable time and frac- 2 Methods tional Fourier domain frequency resolutions and constant The aim of this paper is to eliminate the redundancy of Q-factor. However, in the process of the multi-resolution the GWT. First, we modify the definition of the GWT analysis of the discrete NFrWT, the coefficients of each slightly without having any effect on the partition of time- layer need to be chirp modulated and de-modulated with LCD-frequency plane. Then, we discrete the continuous different chirp rates in different layers (see Fig. 1). Such dilation parameter and shift parameter to construct a kind of operation simply increases the complexity of the set of orthonormal linear canonical wavelets. Finally, we NFrWT which makes it hardly to use in practice. Similar exploit two-channel conjugate orthogonal mirror filter idea of taking the linear canonical convolution between banks to compute this novel discrete orthonormal trans- the signal and the conventional wavelets was introduced form with lower computational complexity. The following in [17]. However, the GWT is lack of reasonable physi- is the definition of the GWT. cal explanation. The continuous GWT is highly redundant as well. 2.1 The generalized wavelet transform In this paper, we propose the discrete linear canonical The GWT of x(t) with parameter M = (A, B, C, D) is wavelet transform (DLCWT) to solve these problems. In defined as [17] order to eliminate the redundancy, the multi-resolution +∞ approximation (MRA) associated with the LCT is M( ) = ( ) ∗ ( ) Wx a, b x t hM,a,b t dt (1) proposed, and the construction of a Riesz basis or an −∞ orthogonal basis is derived. Furthermore, to reduce the ( ) = −i A (t2−b2)ψ ( ) computational complexity, a fast algorithm of DOLCWT where hM,a,b t e 2B a,b t denotes generalized ψ ( ) = −(1/2)ψ t−b is proposed based on the relationship between the discrete wavelets and a,b t a a denotes the scaled orthogonal LCWT (DOLCWT) and the two-channel and shifted mother wavelet function ψ(t). It should be filter banks associated with the LCT. As a kind of time- noticed that the dilation parameter and the shift parame- LCD-frequency representation, the proposed DLCWT ter a, b ∈ R.Asaresult,(1) is highly redundant when it is allows multi-scale analysis and the signal reconstruction used in signal decomposition and reconstruction. without redundancy. Finally, three applications are dis- The signal analysis tool used in our paper is the LCT cussed to verify the effectiveness of our proposed method. which is introduced as follows: The rest of this paper is organized as follows. In Section 2, the goals and methodologies of our paper are 2.2 The linear canonical transform presented. The LCT is introduced as well. In Section 3,the The LCT of signal x(t) with parameter M = (A, B, C, D) is theories of the continuous LCWT are proposed, includ- defined as [18] ing the physical explanation and the reproducing kernel. X (u) = L (x(t))(u) In Section 4, the theories of the DLCWT are proposed, M M including the definition of multi-resolution approxima- +∞ ( ) ( ) = = √−∞ x t KM u, t dt, B 0 tion, the necessary and sufficient conditions to generate a i CD u2 (2) De 2 x(Du), B = 0 Riesz basis or an orthonormal basis, and the fast algorithm that computes the DOLCWT. In Section 5, three appli- where A, B, C, D ∈ R with AD − BC = 1, and the kernel cations are discussed, including shift sampling in multi- i A t2− 1 ut+ D u2 K (u, t) = √ 1 e 2B B 2B . The inverse LCT is resolution subspaces, denoising of non-stationary signals, M i2πB and multi-focus image fusion. Finally, the Conclusions is +∞ L ( ( ))( ) ∗ ( ) = presented in Section 6. ( ) = √−∞ M x t u KM u, t du, b 0 x t −i CA t2 .(3) Ae 2 LM(x(t))(At), b = 0 The convolution theorem of LCT is [19] 1 −i A t2 i A t2 i A t2 [xy] (t) = √ e 2B x(t)e 2B ∗ y(t)e 2B (t) i2πB (4) and −i D u2 LM x(t)y(t) (u) = e 2B XM(u)YM(u),(5) Fig. 1 The flow chart of coefficient decomposition of one layer DFrWT where denotes the convolution for the LCT and ∗ [15] denotes the conventional convolution for the FT. Wang et al. EURASIP Journal on Advances in Signal Processing (2018) 2018:29 Page 3 of 18 The WD of XM computed with arguments (u, v) is equal where M,a,b(u) is the LCT of the linear canonical wavelet to the WD of x computed with arguments (t, ω): ψM,a,b(t), (u) is the FT of the conventional mother +∞ wavelet ψ(t). ( ω) = 2iuv (ε) ∗ ( − ε) −2ivε ε X t, 2e XM XM 2u e d .(6) By the inner-product between the signal and the linear −∞ canonical wavelets, the LCWT of x(t) with parameter The equation shows that the LCT performs a homoge- M = (A, B, C, D), therefore can be defined as neous linear mapping in the Wigner domain [20]: M( ) = ( ) ψ ( ) Wx a, b x t , M,a,b t u AB t 2 +∞ = · −i A b ∗ A 2 .(7)= 2B a ( )ψ ( ) i 2B t v CD ω e x t a,b t e dt. (10) −∞ According to (7), the LCD-frequency u is rotated by an According to convolution theory of LCT, the definition θ (θ) = / angle with tan B A in the time-frequency plane of LCWT can be rewritten as (see Fig.
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