Discrete linear canonical wavelet transform and its applications
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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.
Keywords
Discrete linear canonical wavelet transform Multi-resolution approximation Filter banks Shift sampling Denoising Image fusionAbbreviations
- CT
Chirplet transform
- CT
Contourlet
- CVT
Curvelet
- DLCWT
Discrete LCWT
- DOLCWT
Discrete orthogonal LCWT
- DTFT
Discrete time Fourier transform
- DWT
Discrete wavelet transform
- FT
Fourier transform
- FrFT
Fractional Fourier transform
- GWT
Generalized wavelet transform
- LCD
Linear canonical domain
- LCT
Linear canonical transform
- LCWT
Linear canonical wavelet transform
- LP
Laplacian pyramid
- MRA
Multi-resolution approximation
- MI
Mutual information NFrWT: Novel fractional wavelet transform
- NMSE
Normalized mean-square error
- STFrFT
Short-time FrFT
- SNR
Signal-to-noise ratio
- TFR
Tim-frequency representation
- WT
Wavelet transform
- WD
Wigner distribution
1 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. However, due to the global kernel it uses, the LCT can only reveal the overall linear canonical domain (LCD)-frequency contents. Therefore, the LCT is not competent in those scenarios which require the signal processing tools to display the time and LCD-frequency information jointly.
Chirplet transform (CT) was first proposed in [12] to solve this problem. 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 original window function (i.e., mother chirplet) [13]. Due to the chirping operation, the users are available to new degrees of freedom in shaping the time-frequency cells with respect to the other TFRs. However, as the non-orthogonality between the chirplet with different chirp rates, the CT is very redundant which makes the computational complexity too high.
The short-time fractional fourier transform (STFrFT) introduced in [14] is regarded as a kind of time-fractional-Fourier-domain-frequency representation. It plays a powerful role in the 2D analysis of the chirp signals because the short-time fractional Fourier domain support is compact when the matched order STFrFT is taken. However, the continuous STFrFT is highly redundant on its 2D plane (t,u) in signal reconstruction, and its computational complexity is high.
In this paper, we propose the discrete linear canonical wavelet transform (DLCWT) to solve these problems. In order to eliminate the redundancy, the multi-resolution approximation (MRA) associated with the LCT is proposed, and the construction of a Riesz basis or an orthogonal basis is derived. Furthermore, to reduce the computational complexity, a fast algorithm of DOLCWT is proposed based on the relationship between the discrete orthogonal LCWT (DOLCWT) and the two-channel filter banks associated with the LCT. As a kind of time-LCD-frequency representation, the proposed DLCWT allows multi-scale analysis and the signal reconstruction without redundancy. Finally, three applications are discussed to verify the effectiveness of our proposed method.
The rest of this paper is organized as follows. In Section 2, the goals and methodologies of our paper are presented. The LCT is introduced as well. In Section 3, the theories of the continuous LCWT are proposed, including the physical explanation and the reproducing kernel. In Section 4, the theories of the DLCWT are proposed, including the definition of multi-resolution approximation, the necessary and sufficient conditions to generate a Riesz basis or an orthonormal basis, and the fast algorithm that computes the DOLCWT. In Section 5, three applications are discussed, including shift sampling in multi-resolution subspaces, denoising of non-stationary signals, and multi-focus image fusion. Finally, the Conclusions is presented in Section 6.
2 Methods
The aim of this paper is to eliminate the redundancy of the GWT. First, we modify the definition of the GWT slightly without having any effect on the partition of time-LCD-frequency plane. Then, we discrete the continuous dilation parameter and shift parameter to construct a set of orthonormal linear canonical wavelets. Finally, we exploit two-channel conjugate orthogonal mirror filter banks to compute this novel discrete orthonormal transform with lower computational complexity. The following is the definition of the GWT.
2.1 The generalized wavelet transform
where \({h_{M,a,b}}(t) = {e^{- i\frac {A}{{2B}}\left ({t^{2}} - {b^{2}}\right)}}{\psi _{a,b}}(t)\) denotes generalized wavelets and \({\psi _{a,b}}(t) = {a^{- (1/2)}}\psi \left (\frac {{t - b}}{a}\right)\) denotes the scaled and shifted mother wavelet function ψ(t). It should be noticed that the dilation parameter and the shift parameter \(a, b \in \mathbb {R}\). As a result, (1) is highly redundant when it is used in signal decomposition and reconstruction.
The signal analysis tool used in our paper is the LCT which is introduced as follows:
2.2 The linear canonical transform
where Θ denotes the convolution for the LCT and ∗ denotes the conventional convolution for the FT.
3 The proposed continuous LCWT and its reproducing kernel
3.1 Definition of the continuous LCWT
where Ψ_{ M,a,b }(u) is the LCT of the linear canonical wavelet ψ_{ M,a,b }(t), Ψ(u) is the FT of the conventional mother wavelet ψ(t).
where \(\Gamma = \sqrt {\frac {a}{{i2\pi B}}} {e^{i\gamma b}}, {X_{M}}(u)\) is the LCT of x(t), and Ψ(u) is the Fourier transform (FT) of the conventional mother wavelet ψ(t).
The constant-Q property, linearity, time shifting property, scaling property, inner product property, and Parseval’s relation can be easily derived according to [17]. We will not provide the details here.
C_{ ψ }<∞ is called the admissibility condition of the LCWT which coincides with the admissibility condition defined in WT. It implies that not any \(\psi _{M,a,b}(t) \in {L^{2}}(\mathbb {R})\) could be the linear canonical wavelet unless the admissibility condition of the LCWT is satisfied.
3.2 Reproducing kernel and corresponding equation
Like the conventional wavelet transform, the LCWT is a redundant representation with a redundancy characterized by reproducing kernel equation.
Theorem 1
Proof
The theorem is proved. □
where \(\psi (t) \in {W_{0}} \subset {L^{2}}(\mathbb {R})\), and find a set of orthonormal linear canonical wavelets ψ_{ M,j,k }(t) to make \({K_{{\psi _{M}}}}({a_{0}},{b_{0}}; a,b) = \delta (a - {a_{0}},b - {b_{0}})\) hold and eliminate the redundancy induced from the continuous LCWT.
4 The proposed DLCWT and its fast algorithm
where \(j \in \mathbb {Z}\) and \(k \in \mathbb {Z}\).
4.1 Multi-resolution approximation associated with LCT
The theory of multi-resolution approximation associated with LCT is first proposed here since it sets the ground for the DLCWT and the construction of orthogonal linear canonical wavelets. According to the definition of multi-resolution approximation in [16], we give the following definition.
Definition 1
- 1)
\(\forall (j,k) \in {\mathbb {Z}^{2}}\),
\(x(t) \in V_{j}^{M} \Leftrightarrow x\left (t - {2^{j}}k\right){e^{- i\frac {A}{B}{2^{j}}k(t - {2^{j}}k)}} \in V_{j}^{M}\);
- 2)
\(\forall j \in \mathbb {Z}\), \(V_{j}^{M} \supset V_{j + 1}^{M}\);
- 3)
\(\forall j \in \mathbb {Z}\), \(x(t) \in V_{j}^{M} \Leftrightarrow x\left (\frac {t}{2}\right){e^{- i\frac {{3A}}{{8B}}{t^{2}}}} \in V_{j + 1}^{M}\);
- 4)
\(\underset {j \to \infty }{\lim } {V_{j}^{M}} = \underset {j=-\infty }{\overset {\infty }{\cap }} {V_{j}^{M}} = \{ 0\} \);
- 5)
\(\underset {j \to - \infty }{\lim } {V_{j}^{M}} = {\text {Closure}}\left (\underset {{j = - \infty }}{\overset {\infty }{\cup }} {V_{j}^{M}}\right) = {L^{2}}(\mathbb {R})\);
- 6)
There exists a basic function \(\theta (t) \in V_{0} \subset {L^{2}}(\mathbb {R})\) such that \(\left \{ {\theta _{M,0,k}}(t) = \theta (t - k){e^{- i\frac {A}{{2B}}\left ({t^{2}} - {k^{2}}\right)}}, k \in \mathbb {Z}\right \}\) is a Riesz basis of subspace \(V_{0}^{M}\).
where X_{ M }(u) denotes the LCT of x(t). Since \(V_{j}^{M} \subset {L^{2}}(\mathbb {R})\) and \(V_{j+1}^{M} \subset {L^{2}}(\mathbb {R})\) denote the subspace of all functions bandlimited to the interval [−2^{−j}πB,+2^{−j}πB] and [−2^{−(j+1)}πB,+2^{−(j+1)}πB] in the LCT domain separately, therefore \(x\left ({\frac {t}{2}} \right){e^{- i\frac {{3A}}{{8B}}{t^{2}}}} \in V_{j+1}^{M}\) according to (20).
Theorem 2
\(\{ {\theta _{M,0,k}}(t), k \in \mathbb {Z}\}\) is a Riesz basis of the subspace \(V_{0}^{M}\) if and only if \(\{ \theta (t-k),k \in \mathbb {Z}\}\) forms a Riesz basis of the subspace V_{0} with θ(t)∈V_{0} as the basis function.
Proof
where \(\tilde C_{M}(u)\) denotes the DTLCT of c_{ k } with a period of 2πB, and \(\hat {\theta }\left (\frac {u}{B}\right)\) denotes the FT of θ(t) with its argument scaled by \(\frac {1}{B}\).
On the other hand, if (25) holds, then (24) can be obtained. If x(t)=0, then according to (24), for ∀k,c_{ k }=0. \(\left \{ r (t - k){e^{- i\frac {A}{B}k\left (t - \frac {k}{2}\right)}}, k \in \mathbb {Z}\right \}\) is therefore linear independent with each other. \(\big \{ {\theta _{M,0,k}}(t) = \theta (t - k) {e^{- i\frac {A}{{2B}}\left ({t^{2}} - {k^{2}}\right)}}, k \in \mathbb {Z}\big \}\) is a Riesz basis of the subspace \(V_{0}^{M}\).
holds (see Theorem 3.4 [23] for details), where u^{′}=u/B∈[−π,π].
The theorem is proved. □
In particular, the family \(\{ {\theta _{M,0,k}}(t), k \in \mathbb {Z}\}\) is an orthonormal basis of the space \(V_{j}^{M}\) if and only if P=Q=1. Theorem 2 implies that \(V_{j}^{M}\) are actually the chirp-modulated shift-invariant subspaces of \(L^{2}(\mathbb {R})\), because they are spaces in which the generators are modulated by chirps and then translated by integers [24, 25, 26].
The following theorem provides the condition to construct an orthogonal basis of each space \(V_{j}^{M}\) by dilating, translating, and chirping the scaling function ϕ(t)∈V_{0}.
Theorem 3
Define \(\left \{ V_{j}^{M}\right \},j \in \mathbb {Z}\) as a sequence of closet subspaces, and \(\{{\phi _{M,j,k}}(t), j,k \in \mathbb {Z}\} \) as a set of scaling functions. If \(\{\phi _{j,k} (t),j,k \in \mathbb {Z}\} \) is an orthonormal basis of the subspace V_{ j }, then for all \(j \in \mathbb {Z}\), ϕ_{ M,j,k } forms an orthonormal basis of subspace \(V_{j}^{M}\).
Proof
where \({e_{k}} = {c_{k}}{e^{j\frac {A}{{2B}}{k^{2}}}}\), and \(\tilde E({e^{i\omega }})\) is the DTFT of e_{ k }. Notice that Φ(u) and \(\hat \theta (u)\) are the FT of ϕ(t) and θ(t), respectively.
If \(\{ {2^{- j/2}}\phi ({2^{- j}}t - k),k \in \mathbb {Z} \} \) is an orthonormal basis of the subspace V_{ j }, then the FT of ϕ(t) definitely makes (31) hold. Therefore, \(\left \{{\phi _{M,0,k}}(t) = \phi (t - k){e^{- i\frac {A}{{2B}}({t^{2}} - {k^{2}})}},k \in \mathbb {Z}\right \}\) forms an orthonormal basis of the subspace \(V_{0}^{M}\).
The theorem is proved. □
the orthogonal projection of input signal x on \(V_{j-1}^{M}\) can be decomposed as the sum of orthogonal projections on \(V_{j}^{M}\) and \(W_{j}^{M}\).
4.2 Discrete orthogonal LCWT and its fast algorithm
In this section, we will give the relationship between the DOLCWT and the conjugate mirror filter banks associated with LCT, and the condition to construct the orthonormal linear canonical wavelets. These two-channel filter banks implement a fast computation of DOLCWT which only has O(N) computational complexity for signals of length N.
4.2.1 Relationship between DOLCWT and two-channel filter banks associated with LCT
As can be seen from (38), h_{M,0}(k) and h_{M,1}(k) are irrelevant to j, because of the complex amplitude we multiply to the mother linear canonical wavelet (see (8)). Moreover, it should be noticed that the sequence h_{0}(k) and h_{1}(k) are the conjugate mirror filters in the FT domain. Therefore, according to Zhao [7], h_{M,0}(k) and h_{M,1}(k) actually represent the two-channel filter banks in the LCT domain.
where H_{0}(u) and H_{1}(u) are the discrete time Fourier transform (DTFT) of h_{0}(k) and h_{1}(k), respectively.
holds.
Equation (47) indicates that when \(\{ {\psi _{M,j,k}}(t),k \in \mathbb {Z}\}\) forms an orthonormal basis for \(W^{M}_{j}\), h_{M,0}(k) and h_{M,1}(k) are actually the two-channel conjugate orthogonal mirror filter banks associated with the LCT.
Overall, the construction of the orthonormal linear canonical wavelets can be summarized in the following theorem.
Theorem 4
Define \(\left \{ V_{j}^{M}\right \},j \in \mathbb {Z}\) as a sequence of closet subspaces. \(W_{j}^{M}\) is the orthogonal complement of \(V_{j}^{M}\) in \(V_{j-1}^{M}\). If \(\{ {\phi _{M,j,k}}(t), j,k \in \mathbb {Z}\}\) is a set of orthonormal basis of \(V_{j}^{M}\), then \(\{{\psi _{M,j,k}}(t), j,k \in \mathbb {Z} \}\) is a set of orthonormal basis of \(W_{j}^{M}\) if and only if M satisfy (47), i.e., \(\{ \psi _{j,k}(t), j,k \in \mathbb {Z} \}\) is a set of orthonormal basis of W_{ j }.
4.2.2 Fast algorithm
An actual implementation of the MAR of LCWT requires computation of the inner products shown above, which is computationally rather involved. Therefore, in this section, we develop a fast filter bank algorithm associated with the LCT that computes the orthogonal linear canonical wavelet coefficients of a signal measured at a finite resolution.
where \(\bar h(k) = h(- k)\).
4.2.3 Computational complexity
Direct computation of (11) would involve O(N^{2}) operations per scale with N as the length of the input sequence. However, when using the fast algorithm shown in Fig. 4, the DOLCWT’s computational complexity depends on that of the linear canonical convolution. According to (4), (53a), and (53b), each takes O(N) time at the first level. Then, the downsampling operation splits the signal into two branches of size N/2. But the filter bank only recursively splits one branch convolved with h_{M,0}(n). This leads to a recurrence relation which conduces to an O(N) time for the entire operation. Furthermore, because the proposed fast filter bank algorithm can inherit the conventional lifting scheme, the computational complexity could be halved for long filters [27].
5 Simulations results and discussion
In this section, we provide simulation results of three applications to illustrate the performance of the proposed DLCWT.
5.1 Shift sampling in multi-resolution subspaces
The idea of sampling in multi-resolution subspaces is to find an invertible map \(\mathcal {T}\) between c_{ k } and samples {f(t_{ n })} where t_{ n } denotes the sampling times. To simplify the problem, in the rest of the section, we normalize the sampling interval as Δt=1.
by substituting (65) into (63).
Reconstruction performances comparison
Metrics | Definition | Reconstructed signal | Proposed method | Stern’s method [4] | Janssen’s method [28] |
---|---|---|---|---|---|
NMSE | \(\frac {{\left \|{\hat f(t) - f(t)} \right \|_{{L^{2}}}^{2}}}{\left \| {f(t)} \right \|_{L^{2}}^{2}}\) | Real part | − 27.1518 dB | − 24.7675 dB | 2.4739 dB |
Imaginary part | − 27.0527 dB | − 26.0184 dB | 3.0201 dB | ||
Normalized L^{ ∞ } error | \(\frac {{\left \| {\hat f(t) - f(t)} \right \|}_{L^{\infty }}}{{\left \| {f(t)}\right \|}_{L^{\infty }}}\) | Real part | − 26.4773 dB | − 5.9511 dB | 5.1289 dB |
Imaginary part | − 26.9728 dB | − 8.2247 dB | 6.0701 dB |
The simulations illustrate that the proposed sampling and reconstruction algorithm outperforms the conventional algorithm in [4] when we are given only finite numbers of samples. This is because the synthesis function S_{ u }(t) is compactly supported while the Sinc function used in [4] is slowly decayed. The Haar scaling function used here is rather simple (a rectangle in time domain) which causes some distortions to the signal in LCD. Therefore, some other scaling functions or wavelets with proper frequency shapes can be considered. The synthesis filters \(\big ({1}/{{{{\tilde \Phi }_{{u}}}\left (\frac {\omega }{B}\right)}}\big)\) may be found by using their Laurent series [31].
Besides, when using the algorithm in [28], the real part’s and the imaginary part’s NMSE’s are 2.4739 and 3.0201 dB, respectively. This is due to the fact that chirp signals are non-bandlimited in the FT domain but bandlimited in LCT domain. When applying the common sampling theorem to signals non-bandlimited in the FT domain may lead to wrong (or at least suboptimal) conclusions [5]. Therefore, our proposed algorithm can be found more applicable for non-stationary signal processing, such as radar chirp signals.
5.2 Denoising of non-stationary signals
The LCWT enjoys both high concentrations and tunable resolutions when dealing with chirp signals. The DOLCWT and its fast algorithm we propose eliminate the redundancy and imply that it is a potent signal processing tool. The LCWT-based denoising of chirp signals is investigated here to validate the theory proposed above.
- Step 1:
Choose a linear canonical wavelet, a level N and the threshold rule.
- Step 2:
Decide the matched-parameter M of LCWT.
- Step 3:
Compute the LCWT decomposition of the signal at N level and apply threshold rule to the detail coefficients.
- Step 4:
Compute the inverse LCWT to reconstruct the signal.
After digitalization, the length of the sequence is N=1024 and the sampling frequency is F_{ s }=100Hz. The phase parameters of the signal are k_{0}=3 and ω_{0}=1. The envelope parameters are t_{0}=5 and \(\sigma = \sqrt {0.5} \). The interference’s phase parameters are v=−0.3,u=6, and ω_{1}=10. The amplitude is a=0.1.
5.3 Multi-focus image fusion
The performance of the 2D LCWT-based fusion scheme is compared to the results obtained by applying the Laplacian pyramid (LP) [34], the discrete wavelet transform (DWT) [23], the Curvelet (CVT) [35], and the Contourlet (CT) [36] which are frequently used to perform image fusion task.
In particular, the filter-bank structure illustrated in Fig. 4 can be used to implement the orthogonal 2D LCWT. Note that both the linear canonical wavelet and the filter shown in (8) and (37) are complex. Hence, the coefficients of 2D LCWT are complex which makes the 2D LCWT two-times expansive.
Transform setting for the LP, the DWT, the CVT, and the CT (according to [37])
Transform | Filter | Levels | |
---|---|---|---|
LP | LeGall 5/3 | 4 | |
DWT | bior6.8 | 4 | |
CVT | 4 | ||
CT | CDF 9/7 | CDF 9/7 | [4 8 8 16] |
We choose five metrics recommended in [37] to quantitatively evaluate the fusion performance. They are mutual information (MI) [38], Q^{AB/F} [39], Q_{0},Q_{ W }, and Q_{ E } [40, 41]. The scores of all five evaluation metrics closer to 1 indicate a higher quality of the composite image.
Fusion results for multi-focus image pairs: LP, DWT, CVT
Transform | MI(mean/std) | Q^{AB/F}(mean/std) | Q_{0}(mean/std) | Q_{ W }(mean/std) | Q_{ E }(mean/std) | |
---|---|---|---|---|---|---|
LP | 0.4908/0.0931 | 0.7084/0.0625 | 0.8071/0.0977 | 0.9287/0.0186 | 0.7317/0.0699 | |
DWT | 0.4865/0.1001 | 0.6986/0.0685 | 0.7876/0.1052 | 0.9265/0.0193 | 0.7327/0.0719 | |
CVT | 0.4937/0.0874 | 0.7097/0.0619 | 0.7968/0.1053 | 0.9288/0.0193 | 0.7314/0.0828 | |
CT | 0.4797/0.0933 | 0.6838/0.0716 | 0.7804/0.1054 | 0.9271/0.0189 | 0.7284/0.0726 |
Fusion results for multi-focus image pairs: the proposed LCWT
Filters | A/B | MI(mean/std) | Q^{AB/F}(mean/std) | Q_{0}(mean/std) | Q_{ W }(mean/std) | Q_{ E }(mean/std) |
bior6.8 | 42 | 0.4984/0.0996 | 0.7026/0.0671 | 0.7930/0.1044 | 0.9184/0.0276 | 0.7277/0.0739 |
bior6.8 | 48 | 0.4971/0.0997 | 0.7041/0.0665 | 0.7935/0.1032 | 0.9244/0.0205 | 0.7320/0.0728 |
db1 | 1 | 0.4863/0.0888 | 0.6878/0.0646 | 0.8673/0.0685 | 0.9340/0.0139 | 0.7021/0.0765 |
db1 | 9 | 0.4859/0.0886 | 0.6876/0.0647 | 0.8661/0.0693 | 0.9340/0.0139 | 0.7044/0.0755 |
db13 | 49 | 0.4892/0.1012 | 0.6991/0.0695 | 0.7788/0.1059 | 0.9286/0.0185 | 0.7382/0.0788 |
db13 | 50 | 0.4903/0.1027 | 0.6995/0.0694 | 0.7809/0.1079 | 0.9286/0.0187 | 0.7380/0.0789 |
rbio1.3 | 53 | 0.4941/0.0980 | 0.7007/0.0666 | 0.8132/0.0950 | 0.9304/0.0175 | 0.7265/0.0713 |
6 Conclusions
In this paper, the theories of DLCWT and multi-resolution approximation associated with LCT are proposed to eliminate the redundancy of the continuous LCWT. In order to reduce the computational complexity of DOLCWT, a fast filter banks algorithm associated with LCT is derived. Three potential applications are discussed as well, including shift sampling in multi-resolution subspaces, denoising of non-stationary signals, and multi-focus image fusion.
Further improvements of our proposed methods include the lifting scheme [42] to accelerate the fast filter banks algorithm, the periodic non-uniform sampling of signals in multi-resolution subspaces associated with the DLCWT, etc. Potential applications include single-image super-resolution reconstruction [43], blind reconstruction of multi-band signal in LCT domain [44, 45], multi-channel SAR imaging [11, 46], speech recovery [33], estimations of the time-of-arrival, and pulse width of chirp signals [14], etc.
Notes
Acknowledgements
The authors thank the National Natural Science Foundation of China for their supports for the research work. The authors are also grateful for the anonymous reviewers for their insightful comments and suggestions, which helped improve the quality of this paper significantly.
Funding
This work was supported by the National Natural Science Foundation of China (Grant No. 61271113).
Authors’ contributions
JW is the first author of this paper. His main contributions include (1) the basic idea, (2) the derivation of equations, (3) computer simulations, and (4) writing of this paper. YW is the second author whose main contribution includes checking simulations. WW is the third author and his main contribution includes refining the whole paper. SR is the corresponding author of this paper whose main contribution includes analyzing the basic idea. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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