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.
Discrete linear canonical wavelet transform; Multi-resolution approximation; Filter banks; Shift sampling; Denoising; Image fusion
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 [[
Chirplet transform (CT) was first proposed in [[
The short-time fractional fourier transform (STFrFT) introduced in [[
A novel fractional wavelet transform (NFrWT) based on the idea of the FrFT and the wavelet transform (WT) was proposed in [[
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.
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.
The GWT of x(t) with parameter M=(A,B,C,D) is defined as [[
where hM,a,b(t)=e−iA2Bt2−b2ψa,b(t) denotes generalized wavelets and ψa,b(t)=a−(1/2)ψt−ba denotes the scaled and shifted mother wavelet function ψ(t). It should be noticed that the dilation parameter and the shift parameter a,b∈ℝ. As a result, (
The signal analysis tool used in our paper is the LCT which is introduced as follows:
The LCT of signal x(t) with parameter M=(A,B,C,D) is defined as [[
where A,B,C,D∈ℝ with AD−BC=1, and the kernel KM(u,t)=1i2πBeiA2Bt2−1But+D2Bu2. The inverse LCT is x(t)=∫−∞+∞ℒM(x(t))(u)KM∗(u,t)du,b≠0Ae−iCA2t2ℒM(x(t))(At),b=0.
The convolution theorem of LCT is [[
and ℒMx(t)Θy(t)(u)=e−iD2Bu2XM(u)YM(u),
where Θ denotes the convolution for the LCT and ∗ denotes the conventional convolution for the FT.
The WD of X
The equation shows that the LCT performs a homogeneous linear mapping in the Wigner domain [[
According to (
With some modifications of the generalized wavelets defined in [[
Due to the complex amplitude eiA2Bba2, the DOLCWT can be obtained by a fast filter banks algorithm which we will explain in Section 4. Besides, the LCT of ψ
where Ψ
By the inner-product between the signal and the linear canonical wavelets, the LCWT of x(t) with parameter M=(A,B,C,D), therefore can be defined as WxM(a,b)=x(t),ψM,a,b(t)=e−iA2Bba2∫−∞+∞x(t)ψa,b∗(t)eiA2Bt2dt.
According to convolution theory of LCT, the definition of LCWT can be rewritten as WxM(a,b)=e−iA2Bγ2t2x(t)Θa−12ψ∗−tae−iA2Bt2
with γ=1−aa. Substituting (
where Γ=ai2πBeiγb,XM(u) is the LCT of x(t), and Ψ(u) is the Fourier transform (FT) of the conventional mother wavelet ψ(t).
What Wei et al. [[
The constant-Q property, linearity, time shifting property, scaling property, inner product property, and Parseval’s relation can be easily derived according to [[
If Cψ=∫−∞+∞Ψ(Ω)2ΩdΩ<∞, then x(t) can be derived from WxM(a,b), i.e., x(t)=2πCψ∫−∞+∞∫−∞+∞a−2WxM(a,b)ψM,a,b(t)dadb.
C
Like the conventional wavelet transform, the LCWT is a redundant representation with a redundancy characterized by reproducing kernel equation.
Suppose (a
where KψM(a0,b0;a,b) is the reproducing kernel with KψM(a0,b0;a,b)=2πCψψM,a,b(t),ψM,a0,b0(t)
Inserting the reconstruction formula (
The theorem is proved. □
The reproducing kernel KψM(a0,b0;a,b) measures the correlation of the two linear canonical wavelets, ψ
where ψ(t)∈W0⊂L2(ℝ), and find a set of orthonormal linear canonical wavelets ψ
The DLCWT of x(t) with parameter M=(A,B,C,D) can be defined as WxM(j,k)=x(t),ψM,j,k(t)=e−iA2Bk2∫−∞+∞x(t)ψj,k∗(t)eiA2Bt2dt=2−j/2e−iA2Bk2×∫−∞+∞x(t)ψ∗2−jt−keiA2Bt2dt,
where j∈ℤ and k∈ℤ.
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 [[
A sequence of closet subspaces VjM,j∈ℤ of L2(ℝ) is a multi-resolution approximation associated with LCT if the following six properties are satisfied:
∀(j,k)∈ℤ2,
x(t)∈VjM⇔xt−2jke−iAB2jk(t−2jk)∈VjM;
∀j∈ℤ, VjM⊃Vj+1M;
∀j∈ℤ, x(t)∈VjM⇔xt2e−i3A8Bt2∈Vj+1M;
limj→∞VjM=∩∞j=−∞VjM={0};
limj→−∞VjM=Closure∪∞j=−∞VjM=L2(ℝ);
There exists a basic function θ(t)∈V0⊂L2(ℝ) such that θM,0,k(t)=θ(t−k)e−iA2Bt2−k2,k∈ℤ is a Riesz basis of subspace V0M.
Condition (
Replacing t2 with t
where X
{θM,0,k(t),k∈ℤ} is a Riesz basis of the subspace V0M if and only if {θ(t−k),k∈ℤ} forms a Riesz basis of the subspace V
if θM,0,k(t)=θ(t−k)e−iA2Bt2−k2,k∈ℤ is a Riesz basis of the subspace V0M, then for ∀x(t)∈V0M, we have x(t)=∑k∈ℤckθ(t−k)e−iA2Bt2−k2.
After taking the LCT on both sides of (
where C~M(u) denotes the DTLCT of c
According to the Parseval’s relation associated with LCT, x(t)2=XM(u)2=∫−∞+∞C~M(u)2θ̂uB2du=∫02πBC~M(u)2∑k=−∞+∞θ̂uB+2kπ2du.
Since x(t)∈L2(ℝ), it can be easily obtained that Px(t)2≤∫02πBC~M(u)2du=∑k=−∞+∞ck2≤Qx(t)2
and 1Q≤∑k=−∞+∞θ̂uB+2kπ2≤1P.
On the other hand, if (
This is to say, θM,0,k(t)=θ(t−k)e−iA2Bt2−k2,k∈ℤ will be a Riesz basis of the subspace V0M, if and only if there exist constants P>0 and Q>0 such that (
holds (see Theorem 3.4 [[
The theorem is proved. □
In particular, the family {θM,0,k(t),k∈ℤ} is an orthonormal basis of the space VjM if and only if P=Q=1. Theorem 2 implies that VjM are actually the chirp-modulated shift-invariant subspaces of L2(ℝ), because they are spaces in which the generators are modulated by chirps and then translated by integers [[
The following theorem provides the condition to construct an orthogonal basis of each space VjM by dilating, translating, and chirping the scaling function ϕ(t)∈V
Define VjM,j∈ℤ as a sequence of closet subspaces, and {ϕM,j,k(t),j,k∈ℤ} as a set of scaling functions. If {ϕj,k(t),j,k∈ℤ} is an orthonormal basis of the subspace V
First, it is easy to find that ϕM,j,k∈VjM. From it, we have ϕM,0,0=∑kckθM,0,k(t).
Taking the LCT with M=(A,B,C,D) on both sides of (
where ek=ckejA2Bk2, and E~(eiω) is the DTFT of e
If ϕM,0,k(t)=ϕ(t−k)e−iA2B(t2−k2),k∈ℤ forms an orthonormal basis of V0M, according to Theorem 2, we have ∑k=−∞∞Φ(u/B+2kπ)2=1.
Applying (
As ∑k=−∞∞θ̂(u/B+2kπ)2 is limited, combining (
If {2−j/2ϕ(2−jt−k),k∈ℤ} is an orthonormal basis of the subspace V
Moreover, it is easy to prove that for ∀j,k1,k2∈ℤ, ϕM,j,k1(t),ϕM,j,k2(t)=δ(k1−k2).
The theorem is proved. □
Thus, according to Theorems 2 and 3, one can use the mother wavelet ψ(t)∈W
is an orthonormal basis of WjM. As WjM is the orthogonal complement of VjM in Vj−1M, i.e., WjM⊥VjM
and Vj−1M=VjM⊕WjM,
the orthogonal projection of input signal x on Vj−1M can be decomposed as the sum of orthogonal projections on VjM and WjM.
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.
Since both ψ
and ψM,j,0(t)=∑k=−∞∞hM,1(k)ϕM,j−1,k(t)
with hM,0(k)=h0(k)e−iA2Bk2
and hM,1(k)=h1(k)e−iA2Bk2.
Equation (
and hM,1(k)=ψM,j,0(·),ϕM,j−1,k(·).
As can be seen from (
Assume that j=1. By taking the LCT of both sides of (
and Ψ(u/B)=12H1(u/2B)Φ(u/2B),
where H
According to the orthogonality of {ϕM,0,k(t),k∈ℤ}, we have ∑k=−∞∞H0(u/2B+kπ)2Φ(u/2B+kπ)2=2.
Since H
Notice that ∑p=−∞∞Φ(u/B+2pπ)2=1 and ∑p=−∞∞Φ(u/B+2pπ+π)2=1, it is easy to find that H0(u/B)2+H0(u/B+π)2=2.
Similar with {ϕM,0,k(t),k∈ℤ}, the relationship H1(u/B)2+H1(u/B+π)2=2
holds.
Moreover, because W0M and V0M are orthogonal with each other, ψM,0,k(t),k∈ℤ and ϕM,0,k(t),k∈ℤ are orthogonal, i.e., ϕM,0,k1(t),ψM,0,k2(t)=0
for ∀k1,k2∈ℤ, and it is easy to verify that ∑k=−∞∞Φ(u/B+2kπ)Ψ∗(u/B+2kπ)=0.
Therefore, substituting (39a) and (39b) into (
Similarly, since H
Equations (42a), (42b), and (
where † denotes conjugate transpose, I is identity matrix, and M=H0(u/B)H0(u/B+π)H1(u/B)H1(u/B+π).
Equation (
Overall, the construction of the orthonormal linear canonical wavelets can be summarized in the following theorem.
Define VjM,j∈ℤ as a sequence of closet subspaces. WjM is the orthogonal complement of VjM in Vj−1M. If {ϕM,j,k(t),j,k∈ℤ} is a set of orthonormal basis of VjM, then {ψM,j,k(t),j,k∈ℤ} is a set of orthonormal basis of WjM if and only if M satisfy (
Since {ϕM,j,k(t),j,k∈ℤ} and {ψM,j,k(t),j,k∈ℤ} are orthonormal bases of V
and dM,j(k)=x(t),ψM,j,k(t).
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.
From the orthogonormal functions ϕM,j+1,k∈Vj+1M, ϕM,j,k∈VjM, and Vj+1M⊂VjM, we get ϕM,j+1,k(t)=∑n=−∞∞cnϕM,j,n(t).
With the change of variable t
Equation (
Taking the inner product by x(t) on both sides of (
From the orthogonal functions ψM,j+1,k∈Wj+1M, ϕM,j,k∈VjM, and Wj+1M⊂VjM, we have dM,j+1(k)=aM,j(k)Θh̄M,1(2k),
where h̄(k)=h(−k).
Since VjM=Vj+1M⊕Wj+1M, ϕM,j+1,k(t)∈Vj+1M, ψM,j+1,k(t)∈Wj+1M, and ϕ
Combining (
and ϕM,j,k(t),ψM,j+1,n(t)=hM,1(k−2n)×eiA2B(5n2−4nk).
Substituting (
Taking the inner product by x(t) on both sides of (
Equations (53a) and (53b) prove that a
The following is an example showing decompositions and reconstructions of 1D signal utilizing the DLCWT. We observe a chirp signal given by x(t)=sin(2πf0t)+sin(2πf1t)e−ik2t2
where k=2, f
Direct computation of (
In this section, we provide simulation results of three applications to illustrate the performance of the proposed DLCWT.
First, we consider shift sampling [[
The idea of sampling in multi-resolution subspaces is to find an invertible map T between c
If the sampling times are t
where D−uf(n)=f(n+u)eiA2B(2nu+u2). Substitute ck=i2πBe−iA2Bk2∫−πBπBC~M(ω)e−iD2Bω2ei1Bωkdω
in (
Then, interchanging the order of integration and summation, and replacing n−k with k
where Φ~uωB=∑k′∈ℤϕ(k′+u)e−i1Bk′ω
denotes the DTFT (with its argument scaled by 1B) of ϕ(k
Therefore, we can obtain D−uf(n)=∫−πBπBe−iD2Bω2C~M(ω)Φ~Mu(ω)KM∗(n,ω)dω
by substituting (
According to (
Second, Let us now construct synthesis functions. G~M−u(ω)=∑k∈ℤDugM,0,0(k)KM(k,ω) are the synthesis filters in the Fig. 6, i.e., G~−uωB=∑k∈ℤg(k−u)e−i1Bkω=1/Φ~uωB.
The perfect reconstruction property of the filter bank associated with LCT implies that ck=e−iA2Bk2∑n∈Zf(tn)eiA2Btn2gk−n−u.
Then, for any f(t)∈VjM, we have f(t)=∑n∈ℤf(tn)eiA2Btn2∑k∈ℤgk−n−uϕ(t−k)e−iA2Bt2=∑n∈ℤf(tn)Sun(t)e−iA2B(t2−tn2).
If we define Su(t)=∑k∈ℤgk−uϕ(t−k), then Sun(t)=∑k∈ℤgk−n−uϕt−n−(k−n)=Su(t−n)
Therefore, all the synthesizing functions are obtained as shifts of the L basic functions S
Finally, we give simulations to verify the proposed algorithm. We choose scaling function ϕ(t)=N
where k=1, ρ
It is plotted in Fig. 8. As can be seen from Fig. 8, the derived synthesizing function S(t) is compactly supported. It decay (drop to zero) much faster than the synthesizing function Sinc(t−u) used in [[
Now, we try to reconstruct f(t), t∈[−12,10.5] according to (
The simulations illustrate that the proposed sampling and reconstruction algorithm outperforms the conventional algorithm in [[
Besides, when using the algorithm in [[
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.
Consider the following model x(t)=s(t)+wn1(t)+wn2(t),
where w
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.
An example is given here to demonstrate the performance of the LCWT-based denoising. The source chirp signal is given by s(t)=exp−(t−t0)22σ2expjπk0t2+j2πω0t.
The interference is a cubic polynomial phase function wn2(t)=a∗expjπvt3+jπut2+j2πω1t.
After digitalization, the length of the sequence is N=1024 and the sampling frequency is F
Firstly, we select the db4 wavelet as the mother linear canonical wavelet and use the heursure threshold selection rule with soft thresholding. As for the selection of decomposition level J, Fig. 10a shows the different NMSE of the reconstructed signal in different decomposition level J with SNR =20 dB and M=(−6π,1,0,− 1/6π). As shown in Fig. 10b, the LCWT-based denoising achieves its best performance at levels 4 or 5. Therefore, we choose five levels of LCWT decomposition while the WT-based denoising performs best at levels 1 or 2.Choosing the level and matched-parameter of the LCWT. a Performance of LCWT-based denoising and WT-based denoising [[
Secondly, the major task of the LCWT-based denoising is to decide the matched-parameter of LCWT. Suppose that chirp rate is known or has been estimated. During the selection of decomposition level, we choose M=(−6π,1,0,− 1/6π) because the chirp signal is highly concentrated in this parameter. However, as the existence of the interference signal and initial frequency, this parameter might not be the best choice for LCWT. Figure 10b shows that the LCWT-based denoising achieves its best performance with M=(−6.4π,1,0,− 1/6.4π) at a lower signal-to-noise ratio (SNR). This is because the interference signal can be hardly eliminated using the heursure threshold selection rule since it is almost concentrated in the LCT domain with parameter M=(−6π,1,0,− 1/6π) (which lies around 15 Hz in the frequency axis, see Fig. 11a). As a result, detail coefficients which contain most of the energy of the interference are left with some energy of the interference after applying the heursure threshold selection rule. While the interference signal is less concentrated at M=(−6.4π,1,0,−1/6.4π) (see Fig. 11b). It is nearly submerged in the white Gaussian noise, and it is well-known that the heursure threshold selection rule performs better when denoising signals corrupted by white Gaussian noise. Therefore, during the denoising step, the energy of the interference in detail coefficients can be eliminated by the heursure threshold selection rule. Furthermore, because the initial frequency ω
Then, execute steps 3 and 4. At last, the LCWT-based denoising is compared with the WT-based denoising [[
The reconstruction signals in time domain de-noised by three different methods are shown in Fig. 13 as well. The simulations illustrate that the LCWT-based denoising outperforms the WT-based denoising [[
In this section the performance of multi-focus image fusion using the proposed 2-D LCWT will be investigated. The corresponding thumbnails of all used image-pairs are shown in Fig. 14.Thumbnails of all five multi-focus image pairs used for evaluation purposes
The performance of the 2D LCWT-based fusion scheme is compared to the results obtained by applying the Laplacian pyramid (LP) [[
First, we give the definition of 2D LCWT. According to the definition of 1D linear canonical wavelet in (
Then the one-dimensional LCWT can be extended to 2-D LCWT, i.e., the 2-D LCWT of f(x,y)∈L2ℝ2 with parameters M
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 (
Figure 15 shows the magnitudes, real parts and imaginary parts of a example of two layers 2D-LCWT decomposition of 512×512 ’Barbara.’ Note that parameters M = (A,B,C,D) in rows and columns are different with each other.Example of 2D DLCWT decomposition using db3 linear canonical wavelet with A/B=200 in rows and A/B=− 100 in columns. a Original 512×512 ’Barbara.’ b Magnitude of two layers 2D DLCWT decomposition of ’Barbara.’ c Real part of two layers 2D DLCWT decomposition of ’Barbara.’ d Imaginary part of two layers 2D DLCWT decomposition of ’Barbara.’ e Reconstructed ’Barbara’
The fusion rule we applied here is the maximum selection fusion rule. By this rule, the fused approximation coeffients XFJ are obtained by a averaging operation xFJn=xAJn+xBJn2,
whereas for each decomposition level j, orientation band p and location n, the fused detail coefficients yFj are defined as yFjn,p=yAjn,p,ifyAjn,p>yBjn,pyBjn,p,otherwise.
As for the filter choices, number of decomposition levels and directions, we refer to the best results of each multi-resolution transform published in [[
We choose five metrics recommended in [[
Tables 3 and 4 list the average results as well as the corresponding standard deviations for multi-focus image pairs of each type of transform. From these two tables, it can be observed that overall the CVT shows better performance than the LP, the DWT, and the CT, because the CVT is good at capturing edge and line features. However, the complexity and memory requirement of the CVT is much larger than the others. The proposed LCWT can achieve better results with different filter than the conventional fusion method. Especially, when choosing filter to be rbio1.3 and A/B = 53, the proposed LCWT yields better results than the CVT for the MI, Q
The fusion results for a multi-focus image pair can be seen from Fig. 16.Fusion results for a multi-focus image pair. a The LP. b The DWT. c The CVT. d The CT. e The LCWT
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 [[
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.
This work was supported by the National Natural Science Foundation of China (Grant No. 61271113).
JW is the first author of this paper. His main contributions include (
The authors declare that they have no competing interests.
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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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
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By Jiatong Wang; Yue Wang; Weijiang Wang and Shiwei Ren