Discrete linear canonical wavelet transform and its applications

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.

According to (7), the LCD-frequency u is rotated by an angle θ with tan(θ)=B/A in the time-frequency plane (see Fig. 2).

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).

What Wei et al. [17] did not point out is that the chirp multiplication in the definition of linear canonical wavelets causes rotations of all cells on the time-frequency planes and shears them along the frequency axis [13, 21, 22]. Therefore, due to the chirping operation, each time-frequency cell is rotated by a degree of \(arctan\left (-\frac {A}{B}\right)\) on the time-frequency plane and sheared along the frequency axes (see Fig. 3a). The time-LCD-frequency plane is therefore divided by the LCWT with the time-LCD-frequency atoms as shown in Fig. 3b.

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.

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.

4.1 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.

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^−jπB,+2^−jπB] 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).

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 [24–26].

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)∈V₀.

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.

4.2.2 Fast algorithm

Since \(\{ {\phi _{M,j,k}}(t),j,k \in \mathbb {Z}\}\) and \(\{ {\psi _{M,j,k}}(t),j,k \in \mathbb {Z}\}\) are orthonormal bases of V _M,j and W _M,j , the projection in these spaces can be characterized by

$$\begin{array}{*{20}l} {a_{M,j}}(k) = \left\langle {x(t),{\phi_{M,j,k}}(t)} \right\rangle \end{array} $$

(49a)

and

$$\begin{array}{*{20}l} {d_{M,j}}(k) = \left\langle {x(t),{\psi_{M,j,k}}(t)} \right\rangle. \end{array} $$

(49b)

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 \({\phi _{M,j + 1,k}} \in V_{j + 1}^{M}\), \({\phi _{M,j,k}} \in V_{j}^{M}\), and \(V_{j + 1}^{M} \subset V_{j}^{M}\), we get

$$ {\phi_{M,j + 1,k}}(t) = \sum\limits_{n = - \infty}^{\infty} {{c_{n}}{\phi_{M,j,n}}(t)}. $$

(50)

With the change of variable t^′=2^−jt−2k, we obtain

$$\begin{array}{*{20}l} {c_{n}} &= \left\langle {{\phi_{M,j + 1,k}}(t),{\phi_{M,j,n}}(t)} \right\rangle \\ &= {2^{- j - \frac{1}{2}}}\int_{- \infty}^{+ \infty} {\phi\left(\frac{t}{{{2^{j + 1}}}} - k\right){e^{i\frac{A}{{2B}}{k^{2}}}}} \\ & \quad \times {\phi^{*}}\left(\frac{t}{{{2^{j}}}} - n\right){e^{- i\frac{A}{{2B}}{n^{2}}}}dt \\ &= {2^{- \frac{1}{2}}}\int_{- \infty}^{+ \infty} {\phi\left(\frac{{t}}{2}\right){e^{i\frac{A}{{2B}}{k^{2}}}}} \\ & \quad \times {\phi^{*}}(t - n + 2k){e^{- i\frac{A}{{2B}}{n^{2}}}}dt \\ &= {e^{i\frac{A}{{2B}}(5{k^{2}} - 4nk)}}\left\langle {{\phi_{M,1,0}}(t),{\phi_{M,0,n - 2k}}(t)} \right\rangle \\ &= {e^{i\frac{A}{{2B}}(5{k^{2}} - 4nk)}}{h_{M,0}}(n - 2k). \end{array} $$

(51)

Equation (50) implies that

$$\begin{array}{*{20}l} {\phi_{M,j + 1,k}}(t) &= \sum\limits_{n = - \infty}^{\infty} {{e^{i\frac{A}{{2B}}(5{k^{2}} - 4nk)}}} \\ & \qquad \qquad \times {h_{M,0}}(n - 2k){\phi_{M,j,n}}(t). \end{array} $$

(52)

Taking the inner product by x(t) on both sides of (52) yields

$$\begin{array}{*{20}l} {a_{M,j + 1}}(k) = {a_{M,j}}(k)\Theta {\bar h_{M,0}}(2k). \end{array} $$

(53a)

From the orthogonal functions \({\psi _{M,j + 1,k}} \in W_{j + 1}^{M}\), \({\phi _{M,j,k}} \in V_{j}^{M}\), and \(W_{j + 1}^{M} \subset V_{j}^{M}\), we have

$$\begin{array}{*{20}l} {d_{M,j + 1}}(k) = {a_{M,j}}(k)\Theta {\bar h_{M,1}}(2k), \end{array} $$

(53b)

where \(\bar h(k) = h(- k)\).

Since \(V_{j}^{M} = V_{j + 1}^{M} \oplus W_{j + 1}^{M}\), \({\phi _{M,j + 1,k}}(t) \in V_{j + 1}^{M}\), \({\psi _{M,j + 1,k}} (t)\in W_{j + 1}^{M}\), and ϕ _M,j,k (t) can be decomposed as

$$\begin{array}{*{20}l} & {\phi_{M,j,k}}(t) \\ & = \sum\limits_{n = - \infty }^{\infty} {\left\langle {{\phi_{M,j,k}}(t),{\phi_{M,j + 1,n}}(t)} \right\rangle {\phi_{M,j + 1,n}}(t)} \\ &\quad + \sum\limits_{n = - \infty }^{\infty} {\left\langle {{\phi_{M,j,k}}(t),{\psi_{M,j + 1,n}}(t)} \right\rangle {\psi_{M,j + 1,n}}(t)}. \end{array} $$

(54)

Combining (52), we obtain

$$\begin{array}{*{20}l} \left\langle {{\phi_{M,j,k}}(t),{\phi_{M,j + 1,n}}(t)} \right\rangle & = {h_{M,0}}(k - 2n) \\ & \quad \times {e^{i\frac{A}{{2B}}(5{n^{2}} - 4nk)}} \end{array} $$

(55a)

and

$$\begin{array}{*{20}l} \left\langle {{\phi_{M,j,k}}(t),{\psi_{M,j + 1,n}}(t)} \right\rangle & = {h_{M,1}}(k - 2n) \\ & \quad \times {e^{i\frac{A}{{2B}}(5{n^{2}} - 4nk)}}. \end{array} $$

(55b)

Substituting (55) into (54) yields

$$\begin{array}{*{20}l} & {\phi_{M,j,k}}(t) \\ & = \sum\limits_{n = - \infty}^{\infty} {{\phi_{M,j + 1,n}}(t){h_{M,0}}(k - 2n){e^{i\frac{A}{{2B}}(5{n^{2}} - 4nk)}}} \\ &\quad + \sum\limits_{n = - \infty}^{\infty} {{\psi_{M,j + 1,n}}(t){h_{M,1}}(k - 2n){e^{i\frac{A}{{2B}}(5{n^{2}} - 4nk)}}}. \end{array} $$

(56)

Taking the inner product by x(t) on both sides of (56) yields

$$ {a_{M,j}}(k) = {a_{M,j + 1}}(k)\Theta {h_{M,0}}(2k) + {d_{M,j + 1}}(k)\Theta {h_{M,1}}(2k). $$

(57)

Equations (53a) and (53b) prove that a_M,j+1 and d_M,j+1 can be obtained by taking every other sample of the linear canonical convolution of a _M,j with \({\bar h_{M,0}}(k)\) and \({\bar h_{M,1}}(k)\), respectively, as illustrated by Fig. 4a. The reconstruction (57) is an interpolation that inserts zeroes to expand a_M,j+1 and d_M,j+1 and filters these signals in the LCT domain, as shown in Fig. 4b. Compared to the structure shown in Fig. 1, the coefficients of each layer can be chirp modulated and de-modulated with the same chirp rate in different layers, in the process of the multi-resolution analysis of the DLCWT.

The following is an example showing decompositions and reconstructions of 1D signal utilizing the DLCWT. We observe a chirp signal given by

$$ x(t) = \left({\sin (2\pi {f_{0}}t) + \sin (2\pi {f_{1}}t)} \right){e^{- i\frac{k}{2}{t^{2}}}} $$

(58)

where k=2, f₀=0.1, and f₁=4.5. Figure 5 shows an example of two-layer DLCWT of this signal x(t) computed using db3 wavelets with M=(2,1,1,1). Note that the initial data a_M,−1(k)=x(k) where x(k) denote samples of continuous signal x(t) with sampling rate Δt=0.1. As shown in Fig. 5, the coefficients d_M,1(k) are basically equal to zero, and two frequency components f₀ and f₁ of x(t) lie in subspaces \(V_{1}^{M}\) and \(W_{0}^{M}\), separately. Signal x(k) is perfectly reconstructed from coefficients d_M,0(k), a_M,1(k) and d_M,1(k), denoted as x^′(k).

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).

According to (66) and (60), D^−u[f](n) is equal to the convolution of D^−u[ϕ_M,0,0](k) and c _k . The interpretation of the uniform sampling result in terms of digital filtering associated with LCT is shown in Fig. 6. Therefore, the operator \(\mathcal {T}\) and its inverse \(\mathcal {T}^{-1}\) can be represented by \({\tilde \Phi _{{u}}}\left ({\frac {\omega }{B}} \right)\) and \({1 / {{{\tilde \Phi }_{{u}}}\left ({\frac {\omega }{B}} \right)}}\) respectively, under the condition that \({\tilde \Phi _{{u}}}\left ({\frac {\omega }{B}} \right) \neq 0\).

Therefore, all the synthesizing functions are obtained as shifts of the L basic functions S _u (t). The corresponding sampling and reconstruction procedure is shown in Fig. 7.

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 [4].

Now, we try to reconstruct f(t), t∈[−12,10.5] according to (69) under the condition that the number of sampling points is constrained to 24. The real parts of the original, sampling points and the reconstructed signals are shown in Fig. 9. We use two different quantitative metrics: the normalized mean-square error (NMSE) and the normalized L ^∞ error [30] to show the comparisons between our proposed method and other classic methods. Comparison results are presented in Table 1 where f(t) and \(\hat f(t)\) denote the original signal and the reconstructed signal, respectively.

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.

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 k₀=3 and ω₀=1. The envelope parameters are t₀=5 and \(\sigma = \sqrt {0.5} \). The interference’s phase parameters are v=−0.3,u=6, and ω₁=10. The amplitude is a=0.1.

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.

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 ω₀=1 Hz, the chirp signal is nearly centralized at the base LCD-frequency which makes the LCWT-based denoising perform better at higher decomposition level. Therefore, we choose

$${} M = \left\{ \begin{array}{l} (-~6.4\pi,1,0, -~1/6.4\pi), \quad {\text{SNR}} = -~ 5 \sim 17~{\text{dB}} \\ (-~6.6\pi,1,0, -~1/6.6\pi), \quad {\text{SNR}} = 18~{\text{dB}} \\ (-~6\pi,1,0, -~1/6\pi), \qquad \ \ {\text{SNR}} = 19 \sim 20~{\text{dB}}. \end{array} \right. $$

(76)

Then, execute steps 3 and 4. At last, the LCWT-based denoising is compared with the WT-based denoising [32] and LCD-filtering [19] in the aspect of NMSE of the reconstructed signal. A two-hundred-time Monte Carlo experiment is taken at a range of SNR from − 5 to 20 dB (see Fig. 12).

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 [32] and LCD-filtering [19] at a wide range of SNRs. As can be seen from Figs. 12 and 13, the chirp signal cannot concentrate in the FD which makes the performances of WT-based denoising method poorly. Though the chirp signal is highly concentrated in the matched-parameter LCD, the LCD-filtering method still fails to eliminate both the white Gaussian noise and the interference which lie in the pass band of the filter. This makes the performance of LCD-filtering method unsatisfactory as well. The LCWT-based denoising enjoys both the abilities of multi-resolution analysis and high signal concentration which makes the LCWT-based denoising method performs better than the other two. However, it should be noticed that there is still a part of the interference (which lies around 9 s in the time axis) left un-eliminated. The combination of the LCWT-based denoising method and the LCD-filtering method can be utilized to solve this problem. A better performance is, therefore, promising. Potential applications of the LCWT-based denoising algorithm include speech recovery [33], estimations of the time-of-arrival and pulse width of chirp signals [14].

5.3 Multi-focus image fusion

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.

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.

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.

We choose five metrics recommended in [37] to quantitatively evaluate the fusion performance. They are mutual information (MI) [38], Q^AB/F [39], Q₀,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.

The fusion results for a multi-focus image pair can be seen from Fig. 16.