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Discrete linear canonical wavelet transform and its applications
EURASIP Journal on Advances in Signal Processing volume 2018, Article number: 29 (2018)
Abstract
The continuous generalized wavelet transform (GWT) which is regarded as a kind of timelinear canonical domain (LCD)frequency representation has recently been proposed. Its constantQ 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 multiresolution 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 chirpmodulated function form a Riesz basis or an orthonormal basis for a multiresolution subspace. A fast algorithm that computes the discrete orthogonal LCWT (DOLCWT) is proposed by exploiting twochannel conjugate orthogonal mirror filter banks associated with the LCT. Finally, three potential applications are discussed, including shift sampling in multiresolution subspaces, denoising of nonstationary signals, and multifocus image fusion. Simulations verify the validity of the proposed algorithms.
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 [3–11]. Higher concentration and lower sampling rate make the LCT more competent to resolve nonstationary 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 LCDfrequency information jointly.
Chirplet transform (CT) was first proposed in [12] to solve this problem. Like the other timefrequency 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 timefrequency cells with respect to the other TFRs. However, as the nonorthogonality between the chirplet with different chirp rates, the CT is very redundant which makes the computational complexity too high.
The shorttime fractional fourier transform (STFrFT) introduced in [14] is regarded as a kind of timefractionalFourierdomainfrequency representation. It plays a powerful role in the 2D analysis of the chirp signals because the shorttime 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.
A novel fractional wavelet transform (NFrWT) based on the idea of the FrFT and the wavelet transform (WT) was proposed in [15, 16]. It takes the fractional convolution between the signal and the conventional wavelets which makes the NFrWT available with tunable time and fractional Fourier domain frequency resolutions and constant Qfactor. However, in the process of the multiresolution analysis of the discrete NFrWT, the coefficients of each layer need to be chirp modulated and demodulated with different chirp rates in different layers (see Fig. 1). Such kind of operation simply increases the complexity of the NFrWT which makes it hardly to use in practice. Similar idea of taking the linear canonical convolution between the signal and the conventional wavelets was introduced in [17]. However, the GWT is lack of reasonable physical explanation. The continuous GWT is highly redundant as well.
In this paper, we propose the discrete linear canonical wavelet transform (DLCWT) to solve these problems. In order to eliminate the redundancy, the multiresolution 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 twochannel filter banks associated with the LCT. As a kind of timeLCDfrequency representation, the proposed DLCWT allows multiscale 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 multiresolution 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 multiresolution subspaces, denoising of nonstationary signals, and multifocus 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 timeLCDfrequency plane. Then, we discrete the continuous dilation parameter and shift parameter to construct a set of orthonormal linear canonical wavelets. Finally, we exploit twochannel 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
The GWT of x(t) with parameter M=(A,B,C,D) is defined as [17]
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
The LCT of signal x(t) with parameter M=(A,B,C,D) is defined as [18]
where \(A,B,C,D \in \mathbb {R}\) with AD−BC=1, and the kernel \({K_{M}}(u,t) = \frac {1}{{\sqrt {i2\pi B} }}{e^{i\left (\frac {A}{{2B}}{t^{2}}  \frac {1}{B}ut + \frac {D}{{2B}}{u^{2}}\right)}}\). The inverse LCT is
The convolution theorem of LCT is [19]
and
where Θ denotes the convolution for the LCT and ∗ denotes the conventional convolution for the FT.
The WD of X_{ M } computed with arguments (u,v) is equal to the WD of x computed with arguments (t,ω):
The equation shows that the LCT performs a homogeneous linear mapping in the Wigner domain [20]:
According to (7), the LCDfrequency u is rotated by an angle θ with tan(θ)=B/A in the timefrequency plane (see Fig. 2).
3 The proposed continuous LCWT and its reproducing kernel
3.1 Definition of the continuous LCWT
With some modifications of the generalized wavelets defined in [17], we define the linear canonical wavelets as
Due to the complex amplitude \({e^{i\frac {A}{{2B}}{{\left ({\frac {b}{a}} \right)}^{2}}}}\), the DOLCWT can be obtained by a fast filter banks algorithm which we will explain in Section 4. Besides, the LCT of ψ_{ M,a,b }(t) is still bandpass in the LCD since ψ_{ a,b }(t) is bandpass in the FD, i.e.,
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).
By the innerproduct 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
According to convolution theory of LCT, the definition of LCWT can be rewritten as
with \(\gamma = \frac {{1  a}}{a}\). Substituting (11) into (3), we can obtain the expression of LCWT in the LCD as
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 timefrequency planes and shears them along the frequency axis [13, 21, 22]. Therefore, due to the chirping operation, each timefrequency cell is rotated by a degree of \(arctan\left (\frac {A}{B}\right)\) on the timefrequency plane and sheared along the frequency axes (see Fig. 3a). The timeLCDfrequency plane is therefore divided by the LCWT with the timeLCDfrequency atoms as shown in Fig. 3b.
The constantQ 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.
If \({C_{\psi }} = \int _{\infty }^{+ \infty } {\frac {{{{\left  {\Psi (\Omega)} \right }^{2}}}}{\Omega }d\Omega } < \infty \), then x(t) can be derived from \(W_{x}^{M}(a,b)\), i.e.,
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
Suppose (a_{0},b_{0})∈(a,b), then 2D function \(W_{x}^{M}(a,b)\)is the LCWT of signal x(t)if and only if it satisfies the following reproducing kernel equation, i.e.,
where \({K_{{\psi _{M}}}}({a_{0}},{b_{0}};a,b)\) is the reproducing kernel with
Proof
Inserting the reconstruction formula (13) into the definition of the LCWT (10) yields
The theorem is proved. □
The reproducing kernel \({K_{{\psi _{M}}}}({a_{0}},{b_{0}};a,b)\) measures the correlation of the two linear canonical wavelets, ψ_{ M,a,b } and \(\psi _{M,{a_{0}},{b_{0}}}\). According to (14), the LCWT of x(t) at a=a_{0} and \(b = b_{0} \left (\text {i.e}., W_{x}^{M}({a_{0}},{b_{0}})\right)\) can always be represented by other \({W_{x}^{M}(a,b)}\) through the reproducing kernel \({K_{{\psi _{M}}}}({a_{0}},{b_{0}};a,b)\). This means all \({W_{x}^{M}(a,b)}\)s on 2D plane (a,b) are related to each other, and there always exists redundancy when the continuous LCWT is used for signal reconstruction. In order to reduce the redundancy, we need the reproducing kernel to have the following property: \({K_{{\psi _{M}}}}({a_{0}},{b_{0}};a,b) = \delta (a  {a_{0}},b  {b_{0}})\). However, it is difficult to find a set of orthonormal linear canonical wavelets ψ_{ M,a,b }(t) to make \({K_{{\psi _{M}}}}({a_{0}},{b_{0}};a,b) = \delta (a  {a_{0}},b  {b_{0}})\) when a and b are both continuous. Therefore, we need to discrete the dilation and shift parameters of the linear canonical wavelet in (8) by making a=2^{j}, b=2^{j}kb_{0} and b_{0}=1, i.e.,
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
The DLCWT of x(t) with parameter M=(A,B,C,D) can be defined as
where \(j \in \mathbb {Z}\) and \(k \in \mathbb {Z}\).
4.1 Multiresolution approximation associated with LCT
The theory of multiresolution 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 multiresolution approximation in [16], we give the following definition.
Definition 1
A sequence of closet subspaces \(\left \{ V_{j}^{M}\right \}, j \in \mathbb {Z}\) of \({ L^{2}}(\mathbb {R})\) is a multiresolution approximation associated with LCT if the following six properties are satisfied:

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 \(V_{0}^{M}\).
Condition (1) means that \(V_{j}^{M}\) is invariant by any translation proportional to the scale 2^{j} together with modulation. Dilating operation and chirping operation in \(V_{j}^{M}\) enlarge the detail and condition (3) guarantees that it defines an approximation at a coarser resolution 2^{−j−1}. The existence of a Riesz basis of \(V_{j}^{M}\) provides a discretization theorem. The theorem below gives the existence condition of Riesz basis in \(V_{j}^{M}\). The following is detailed proof of condition (3). Let XM′(u) denote the LCT of \(x\left ({\frac {t}{2}} \right){e^{ i\frac {{3A}}{{8B}}{t^{2}}}}\). According to the definition of the LCT, one can obtain
Replacing \({\frac {t}{2}}\) with t^{′} in (19) results in
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 \(\{ \theta (tk),k \in \mathbb {Z}\}\) forms a Riesz basis of the subspace V_{0} with θ(t)∈V_{0} as the basis function.
Proof
if \(\left \{{\theta _{M,0,k}}(t) = \theta (t  k){e^{ i\frac {A}{{2B}}\left ({t^{2}}  {k^{2}}\right)}}, k \in \mathbb {Z}\right \}\) is a Riesz basis of the subspace \(V_{0}^{M}\), then for \(\forall x(t) \in V_{0}^{M}\), we have
After taking the LCT on both sides of (21), we have
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 \(\frac {1}{B}\).
According to the Parseval’s relation associated with LCT,
Since \(x(t) \in {L^{2}}(\mathbb {R})\), it can be easily obtained that
and
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 \(V_{0}^{M}\).
This is to say, \(\left \{{\theta _{M,0,k}}(t) = \theta (t  k){e^{ i\frac {A}{{2B}}\left ({t^{2}}  {k^{2}}\right)}},k \in \mathbb {Z}\right \}\) will be a Riesz basis of the subspace \(V_{0}^{M}\), if and only if there exist constants P>0 and Q>0 such that (25) holds. Considering \(\{ \theta (tk),k \in \mathbb {Z}\}\) as a Riesz basis of the subspace of V_{0}, we can deduce that (25) definitely holds due to the inequation
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 chirpmodulated shiftinvariant 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)∈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
First, it is easy to find that \({\phi _{M,j,k}} \in V_{j}^{M}\). From it, we have
Taking the LCT with M=(A,B,C,D) on both sides of (27), we have
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 \(\left \{ {\phi _{M,0,k}}(t) = \phi (t  k){e^{ i\frac {A}{{2B}}({t^{2}}  {k^{2}})}},k \in \mathbb {Z}\right \} \) forms an orthonormal basis of \(V_{0}^{M}\), according to Theorem 2, we have
Applying (28) and (29), we can obtain
As \(\sum \limits _{k =  \infty }^{\infty } {{{\left  {\hat \theta (u/B + 2k\pi)} \right }^{2}}}\) is limited, combining (28) and (30) yields
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 \(V_{0}^{M}\).
Moreover, it is easy to prove that for \(\forall j,{k_{1}},{k_{2}} \in \mathbb {Z}\),
The theorem is proved. □
Thus, according to Theorems 2 and 3, one can use the mother wavelet ψ(t)∈W_{0} to construct mother linear canonical wavelet \(\psi _{M}(t) \in W_{0}^{M}\) such that the dilated, translated, and chirpmodulated family
is an orthonormal basis of \(W_{j}^{M}\). As \(W_{j}^{M}\) is the orthogonal complement of \(V_{j}^{M}\) in \(V_{j1}^{M}\), i.e.,
and
the orthogonal projection of input signal x on \(V_{j1}^{M}\) 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 twochannel 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 twochannel filter banks associated with LCT
Since both ψ_{M,j−1,k}(t) and ϕ_{M,j−1,k}(t) form an orthonormal basis for \(W^{M}_{j1}\) and \(V^{M}_{j1}\), we can decompose ϕ_{M,j,0}(t) and ψ_{M,j,0}(t) as
and
with
and
Equation (36) are the twoscale difference equations belonging to LCWT which reveal the relationship between linear canonical wavelets and linear canonical scale functions in multiresolution approximation analysis associated with LCT. Moreover, due to the orthogonality between {ϕ_{ M,j,k }(t)} and {ψ_{ M,j,k }(t)}, we have
and
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 twochannel filter banks in the LCT domain.
Assume that j=1. By taking the LCT of both sides of (36), we obtain
and
where H_{0}(u) and H_{1}(u) are the discrete time Fourier transform (DTFT) of h_{0}(k) and h_{1}(k), respectively.
According to the orthogonality of \(\{{\phi _{M,0,k}}(t),k \in \mathbb {Z}\}\), we have
Since H_{0}(u) is 2π periodic, splitting k into odd and even parts, i.e., substituting k=2p and k=2p+1, \(p \in \mathbb {Z}\) into (40) yields
Notice that \(\sum \limits _{p =  \infty }^{\infty } {{{\left  {\Phi (u/B + 2p\pi)} \right }^{2}}} = 1\) and \(\sum \limits _{p =  \infty }^{\infty } {{{\left  {\Phi (u/B + 2p\pi + \pi)} \right }^{2}}} = 1\), it is easy to find that
Similar with \(\{ {\phi _{M,0,k}}(t),k \in \mathbb {Z}\}\), the relationship
holds.
Moreover, because \(W_{0}^{M}\) and \(V_{0}^{M}\) are orthogonal with each other, \(\left \{ {{\psi _{M,0,k}}(t),k \in \mathbb {Z}} \right \}\) and \(\left \{ {{\phi _{M,0,k}}(t),k \in \mathbb {Z}} \right \}\) are orthogonal, i.e.,
for \(\forall {k_{1}},{k_{2}} \in \mathbb {Z}\), and it is easy to verify that
Therefore, substituting (39a) and (39b) into (44), we have
Similarly, since H_{0}(u) and H_{1}(u) are both 2π periodic, splitting k into odd and even parts, i.e., substituting k=2p and k=2p+1, \(p \in \mathbb {Z}\) into (45) gives
Equations (42a), (42b), and (46) together indicate that if \({\psi _{M,j,k}}(t) = {2^{ j/2}}\psi ({2^{ j}}t  k){e^{ i\frac {A}{{2B}}({t^{2}}  {k^{2}})}}\) is an orthonormal basis for \(W^{M}_{j}\), then
where † denotes conjugate transpose, I is identity matrix, and
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 twochannel 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 \(V_{j1}^{M}\). 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
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
and
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
With the change of variable t^{′}=2^{−j}t−2k, we obtain
Equation (50) implies that
Taking the inner product by x(t) on both sides of (52) yields
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
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
Combining (52), we obtain
and
Substituting (55) into (54) yields
Taking the inner product by x(t) on both sides of (56) yields
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 demodulated with the same chirp rate in different layers, in the process of the multiresolution 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
where k=2, f_{0}=0.1, and f_{1}=4.5. Figure 5 shows an example of twolayer 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 f_{0} and f_{1} 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).
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 multiresolution subspaces
First, we consider shift sampling [26, 28] in multiresolution subspace \(V_{0}^{M}\). The shift sampling instants is defined as t_{ n }=n+u with \(n \in \mathbb {Z}\) and fractional shift u∈[0,1). We work in \(V_{0}^{M}\) only since all the relevant properties are independent of the scale. Let \(\phi (t) \in {L^{2}}(\mathbb {R})\) be the linear canonical scaling function of a MRA \({\big \{ V_{j}^{M} \big \}_{j \in \mathbb {Z}}}\) associated with the LCWT such that the sampling sequence ϕ(n+u) of ϕ(t) belongs to \({\ell ^{2}}(\mathbb {Z})\) for some u∈[0,1). According to Theorem 2, since {ϕ_{M,0,k}} is a Riesz basis for \(V_{0}^{M}\), then for any \(f(t) \in V_{0}^{M}\), there exists a unique sequence \(\{c_{k}\} \in {\ell ^{2}}(\mathbb {Z})\) such that
The idea of sampling in multiresolution 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.
If the sampling times are t_{ n }=n+u, \(n \in \mathbb {Z}\), 0≤u<1, then the samples f(t_{ n }) can be written as follows
where \({D^{ {u}}}\left [f\right ](n) = f(n + {u}){e^{i\frac {A}{{2B}}(2n{u} + u^{2})}}\). Substitute
in (60), we have
Then, interchanging the order of integration and summation, and replacing n−k with k^{′} in (62) yields
where
denotes the DTFT (with its argument scaled by \(\frac {1}{B}\)) of ϕ(k^{′}+u). Notice that
Therefore, we can obtain
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 \({\tilde \Phi _{{u}}}\left ({\frac {\omega }{B}} \right) \neq 0\).
Second, Let us now construct synthesis functions. \(\tilde G_{M}^{ {u}}(\omega) = \sum \limits _{k \in \mathbb {Z}} {{D^{{u}}}\left [ {{g_{M,0,0}}} \right ](k){K_{M}}(k,\omega)}\) are the synthesis filters in the Fig. 6, i.e.,
The perfect reconstruction property of the filter bank associated with LCT implies that
Then, for any \(f(t) \in {V_{j}^{M}}\), we have
If we define \({S_{u}}(t) = \sum \limits _{k \in \mathbb {Z}} {g_{k}^{ {u}}\phi (t  k)}\), then
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.
Finally, we give simulations to verify the proposed algorithm. We choose scaling function ϕ(t)=N_{1}(t), which is the Bspline of order 1 [29]. It is easy to verify that \({\left \{{\phi _{M,0,k}}(t) \triangleq {N_{1}}(tk){e^{ i\frac {A}{{2B}}({t^{2}}  {k^{2}})}}\right \}_{k \in \mathbb {Z}}}\) forms a Riesz basis for \({V_{0}^{M}}(\phi) \subset {L^{2}}(\mathbb {R})\). We observe a signal given by
where k=1, ρ_{0}=2, and f_{0}=0.03. It is bandlimited in the LCT domain with M=(1,1,0.5,1.5). According to the uniform sampling theory of the LCT domain [4], the uniform sampling period can be chosen as T=1. We choose u=0.5. Hence, the synthesizing function can be derived as
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 meansquare 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.
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 nonbandlimited in the FT domain but bandlimited in LCT domain. When applying the common sampling theorem to signals nonbandlimited in the FT domain may lead to wrong (or at least suboptimal) conclusions [5]. Therefore, our proposed algorithm can be found more applicable for nonstationary signal processing, such as radar chirp signals.
5.2 Denoising of nonstationary 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 LCWTbased denoising of chirp signals is investigated here to validate the theory proposed above.
Consider the following model
where w_{n1}(t) is the white Gaussian noise and w_{n2}(t) is the interference. An LCWTbased denoising algorithm is proposed with the steps summarized below.

Step 1:
Choose a linear canonical wavelet, a level N and the threshold rule.

Step 2:
Decide the matchedparameter 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 LCWTbased denoising. The source chirp signal is given by
The interference is a cubic polynomial phase function
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.
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 LCWTbased denoising achieves its best performance at levels 4 or 5. Therefore, we choose five levels of LCWT decomposition while the WTbased denoising performs best at levels 1 or 2.
Secondly, the major task of the LCWTbased denoising is to decide the matchedparameter 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 LCWTbased denoising achieves its best performance with M=(−6.4π,1,0,− 1/6.4π) at a lower signaltonoise 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 wellknown 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_{0}=1 Hz, the chirp signal is nearly centralized at the base LCDfrequency which makes the LCWTbased denoising perform better at higher decomposition level. Therefore, we choose
Then, execute steps 3 and 4. At last, the LCWTbased denoising is compared with the WTbased denoising [32] and LCDfiltering [19] in the aspect of NMSE of the reconstructed signal. A twohundredtime Monte Carlo experiment is taken at a range of SNR from − 5 to 20 dB (see Fig. 12).
The reconstruction signals in time domain denoised by three different methods are shown in Fig. 13 as well. The simulations illustrate that the LCWTbased denoising outperforms the WTbased denoising [32] and LCDfiltering [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 WTbased denoising method poorly. Though the chirp signal is highly concentrated in the matchedparameter LCD, the LCDfiltering 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 LCDfiltering method unsatisfactory as well. The LCWTbased denoising enjoys both the abilities of multiresolution analysis and high signal concentration which makes the LCWTbased 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 uneliminated. The combination of the LCWTbased denoising method and the LCDfiltering method can be utilized to solve this problem. A better performance is, therefore, promising. Potential applications of the LCWTbased denoising algorithm include speech recovery [33], estimations of the timeofarrival and pulse width of chirp signals [14].
5.3 Multifocus image fusion
In this section the performance of multifocus image fusion using the proposed 2D LCWT will be investigated. The corresponding thumbnails of all used imagepairs are shown in Fig. 14.
The performance of the 2D LCWTbased 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.
First, we give the definition of 2D LCWT. According to the definition of 1D linear canonical wavelet in (8), we introduce the 2D linear canonical wavelet to be the 2D wavelet elementary function
Then the onedimensional LCWT can be extended to 2D LCWT, i.e., the 2D LCWT of \(f(x,y) \in {L^{2}}\left ({\mathbb {R}^{2}}\right)\) with parameters M_{1} = (A_{1},B_{1},C_{1},D_{1}) and M_{2}=(A_{2},B_{2},C_{2},D_{2}) is defined as
In particular, the filterbank 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 twotimes expansive.
Figure 15 shows the magnitudes, real parts and imaginary parts of a example of two layers 2DLCWT decomposition of 512×512 ’Barbara.’ Note that parameters M = (A,B,C,D) in rows and columns are different with each other.
The fusion rule we applied here is the maximum selection fusion rule. By this rule, the fused approximation coeffients \(X_{F}^{J}\) are obtained by a averaging operation
whereas for each decomposition level j, orientation band p and location n, the fused detail coefficients \(y_{F}^{j}\) are defined as
As for the filter choices, number of decomposition levels and directions, we refer to the best results of each multiresolution transform published in [37]. Table 2 lists the used settings for each transform. Particularly, for the CT, the symbols ’CDF 9/7’ and ’CDF 9/7’ denote the pyramid and orientation filter, respectively. The ’levels’ represents decomposition levels and the corresponding number of orientation for each level.
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.
Tables 3 and 4 list the average results as well as the corresponding standard deviations for multifocus 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_{0}, and Q_{ W } fusion metrics. Besides, the complexity and memory requirement of the 2D LCWT is much smaller than the CVT because of the fast algorithm we proposed here.
The fusion results for a multifocus image pair can be seen from Fig. 16.
6 Conclusions
In this paper, the theories of DLCWT and multiresolution 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 multiresolution subspaces, denoising of nonstationary signals, and multifocus image fusion.
Further improvements of our proposed methods include the lifting scheme [42] to accelerate the fast filter banks algorithm, the periodic nonuniform sampling of signals in multiresolution subspaces associated with the DLCWT, etc. Potential applications include singleimage superresolution reconstruction [43], blind reconstruction of multiband signal in LCT domain [44, 45], multichannel SAR imaging [11, 46], speech recovery [33], estimations of the timeofarrival, and pulse width of chirp signals [14], etc.
Abbreviations
 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:

Multiresolution approximation
 MI:

Mutual information NFrWT: Novel fractional wavelet transform
 NMSE:

Normalized meansquare error
 STFrFT:

Shorttime FrFT
 SNR:

Signaltonoise ratio
 TFR:

Timfrequency representation
 WT:

Wavelet transform
 WD:

Wigner distribution
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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).
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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.
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Wang, J., Wang, Y., Wang, W. et al. Discrete linear canonical wavelet transform and its applications. EURASIP J. Adv. Signal Process. 2018, 29 (2018). https://doi.org/10.1186/s136340180550z
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DOI: https://doi.org/10.1186/s136340180550z