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

Wang et al. EURASIP Journal on Advances in Signal EURASIP Journal on Advances Processing (2018) 2018:29 https://doi.org/10.1186/s13634-018-0550-z in Signal Processing

RESEARCH Open Access Discrete linear canonical wavelet transform and its applications Jiatong Wang, Yue Wang, Weijiang Wang and Shiwei Ren*

Abstract The continuous generalized wavelet transform (GWT) which is regarded as a kind of time-linear canonical domain (LCD)-frequency representation has recently been proposed. Its constant-Q property can rectify the limitations of the wavelet transform (WT) and the linear canonical transform (LCT). However, the GWT is highly redundant in signal reconstruction. The discrete linear canonical wavelet transform (DLCWT) is proposed in this paper to solve this problem. First, the continuous linear canonical wavelet transform (LCWT) is obtained with a modification of the GWT. Then, in order to eliminate the redundancy, two aspects of the DLCWT are considered: the multi-resolution approximation (MRA) associated with the LCT and the construction of orthogonal linear canonical wavelets. The necessary and sufficient conditions pertaining to LCD are derived, under which the integer shifts of a chirp-modulated function form a Riesz basis or an orthonormal basis for a multi-resolution subspace. A fast algorithm that computes the discrete orthogonal LCWT (DOLCWT) is proposed by exploiting two-channel conjugate orthogonal mirror filter banks associated with the LCT. Finally, three potential applications are discussed, including shift sampling in multi-resolution subspaces, denoising of non-stationary signals, and multi-focus image fusion. Simulations verify the validity of the proposed algorithms. Keywords: Discrete linear canonical wavelet transform, Multi-resolution approximation, Filter banks, Shift sampling, Denoising, Image fusion

1 Introduction original window function (i.e., mother chirplet) [13]. Due The linear canonical transform (LCT), the generalization to the chirping operation, the users are available to new of the Fourier transform (FT), the fractional Fourier trans- degrees of freedom in shaping the time-frequency cells form (FrFT), the Fresnel transform and the scaling opera- with respect to the other TFRs. However, as the non- tions, has been found useful in many applications such as orthogonality between the chirplet with different chirp optics [1, 2] and signal processing [3–11]. Higher concen- rates, the CT is very redundant which makes the compu- tration and lower sampling rate make the LCT more com- tational complexity too high. petent to resolve non-stationary signals. However, due to The short-time fractional fourier transform (STFrFT) theglobalkernelituses,theLCTcanonlyrevealtheover- introduced in [14] is regarded as a kind of time-fractional- all linear canonical domain (LCD)-frequency contents. Fourier-domain-frequency representation. It plays a pow- Therefore, the LCT is not competent in those scenarios erful role in the 2D analysis of the chirp signals because which require the signal processing tools to display the the short-time fractional Fourier domain support is com- time and LCD-frequency information jointly. pact when the matched order STFrFT is taken. However, Chirplet transform (CT) was first proposed in [12]to the continuous STFrFT is highly redundant on its 2D solve this problem. Like the other time-frequency repre- plane (t, u) in signal reconstruction, and its computational sentations (TFRs), the CT projects the input signal onto complexity is high. a set of functions that are all obtained by modifying an A novel fractional wavelet transform (NFrWT) based on the idea of the FrFT and the wavelet transform (WT) was *Correspondence: [email protected] proposed in [15, 16]. It takes the fractional convolution School of Information and Electronics, Beijing Institute of Technology, South between the signal and the conventional wavelets which ZhongGuanCun Street, Haidian District, 100081 Beijing, China

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makes the NFrWT available with tunable time and frac- 2 Methods tional Fourier domain frequency resolutions and constant The aim of this paper is to eliminate the redundancy of Q-factor. However, in the process of the multi-resolution the GWT. First, we modify the definition of the GWT analysis of the discrete NFrWT, the coefficients of each slightly without having any effect on the partition of time- layer need to be chirp modulated and de-modulated with LCD-frequency plane. Then, we discrete the continuous different chirp rates in different layers (see Fig. 1). Such dilation parameter and shift parameter to construct a kind of operation simply increases the complexity of the set of orthonormal linear canonical wavelets. Finally, we NFrWT which makes it hardly to use in practice. Similar exploit two-channel conjugate orthogonal mirror filter idea of taking the linear canonical convolution between banks to compute this novel discrete orthonormal trans- the signal and the conventional wavelets was introduced form with lower computational complexity. The following in [17]. However, the GWT is lack of reasonable physi- is the definition of the GWT. cal explanation. The continuous GWT is highly redundant as well. 2.1 The generalized wavelet transform In this paper, we propose the discrete linear canonical The GWT of x(t) with parameter M = (A, B, C, D) is wavelet transform (DLCWT) to solve these problems. In defined as [17] order to eliminate the redundancy, the multi-resolution +∞ approximation (MRA) associated with the LCT is M( ) = ( ) ∗ ( ) Wx a, b x t hM,a,b t dt (1) proposed, and the construction of a Riesz basis or an −∞ orthogonal basis is derived. Furthermore, to reduce the ( ) = −i A (t2−b2)ψ ( ) computational complexity, a fast algorithm of DOLCWT where hM,a,b t e 2B a,b t denotes generalized ψ ( ) = −(1/2)ψ t−b is proposed based on the relationship between the discrete wavelets and a,b t a a denotes the scaled orthogonal LCWT (DOLCWT) and the two-channel and shifted mother wavelet function ψ(t). It should be filter banks associated with the LCT. As a kind of time- noticed that the dilation parameter and the shift parame- LCD-frequency representation, the proposed DLCWT ter a, b ∈ R.Asaresult,(1) is highly redundant when it is allows multi-scale analysis and the signal reconstruction used in signal decomposition and reconstruction. without redundancy. Finally, three applications are dis- The signal analysis tool used in our paper is the LCT cussed to verify the effectiveness of our proposed method. which is introduced as follows: The rest of this paper is organized as follows. In Section 2, the goals and methodologies of our paper are 2.2 The linear canonical transform presented. The LCT is introduced as well. In Section 3,the The LCT of signal x(t) with parameter M = (A, B, C, D) is theories of the continuous LCWT are proposed, includ- defined as [18] ing the physical explanation and the reproducing kernel. X (u) = L (x(t))(u) In Section 4, the theories of the DLCWT are proposed, M M including the definition of multi-resolution approxima- +∞ ( ) ( ) = = √−∞ x t KM u, t dt, B 0 tion, the necessary and sufficient conditions to generate a i CD u2 (2) De 2 x(Du), B = 0 Riesz basis or an orthonormal basis, and the fast algorithm that computes the DOLCWT. In Section 5, three appli- where A, B, C, D ∈ R with AD − BC = 1, and the kernel cations are discussed, including shift sampling in multi- i A t2− 1 ut+ D u2 K (u, t) = √ 1 e 2B B 2B . The inverse LCT is resolution subspaces, denoising of non-stationary signals, M i2πB and multi-focus image fusion. Finally, the Conclusions is +∞ L ( ( ))( ) ∗ ( ) = presented in Section 6. ( ) = √−∞ M x t u KM u, t du, b 0 x t −i CA t2 .(3) Ae 2 LM(x(t))(At), b = 0

The convolution theorem of LCT is [19] 1 −i A t2 i A t2 i A t2 [xy] (t) = √ e 2B x(t)e 2B ∗ y(t)e 2B (t) i2πB (4)

and

−i D u2 LM x(t)y(t) (u) = e 2B XM(u)YM(u),(5) Fig. 1 The flow chart of coefficient decomposition of one layer DFrWT where denotes the convolution for the LCT and ∗ [15] denotes the conventional convolution for the FT. Wang et al. EURASIP Journal on Advances in Signal Processing (2018) 2018:29 Page 3 of 18

The WD of XM computed with arguments (u, v) is equal where M,a,b(u) is the LCT of the linear canonical wavelet to the WD of x computed with arguments (t, ω): ψM,a,b(t), (u) is the FT of the conventional mother +∞ wavelet ψ(t). ( ω) = 2iuv (ε) ∗ ( − ε) −2ivε ε X t, 2e XM XM 2u e d .(6) By the inner-product between the signal and the linear −∞ canonical wavelets, the LCWT of x(t) with parameter The equation shows that the LCT performs a homoge- M = (A, B, C, D), therefore can be defined as neous linear mapping in the Wigner domain [20]: M( ) = ( ) ψ ( ) Wx a, b x t , M,a,b t u AB t 2 +∞ = · −i A b ∗ A 2 .(7)= 2B a ( )ψ ( ) i 2B t v CD ω e x t a,b t e dt. (10) −∞ According to (7), the LCD-frequency u is rotated by an According to convolution theory of LCT, the definition θ (θ) = / angle with tan B A in the time-frequency plane of LCWT can be rewritten as (see Fig. 2). M −i A γ 2t2 − 1 ∗ t −i A t2 W (a, b) = e 2B x(t) a 2 ψ − e 2B 3 The proposed continuous LCWT and its x a reproducing kernel (11) 3.1 Definition of the continuous LCWT γ = 1−a With some modifications of the generalized wavelets with a . Substituting (11)into(3), we can obtain the defined in [17], we define the linear canonical wavelets as expression of LCWT in the LCD as 2 +∞ A 2 A b ∗ −i t i 2B a M( ) = L M( ) ( ) ( ) ψ (t) = ψ (t)e 2B e .(8)Wx a, b M Wx a, b u KM u, b du M,a,b a,b −∞ 2 +∞ i A b ∗ au ∗ 2B a = ( ) ( ) Duetothecomplexamplitudee , the DOLCWT XM u KM u, b du, −∞ B can be obtained by a fast filter banks algorithm which we (12) will explain in Section 4. Besides, the LCT of ψ (t) is M,a,b still band-pass in the LCD since ψ (t) is band-pass in the a,b where = a eiγ b, X (u) is the LCT of x(t),and(u) FD, i.e., i2πB M is the Fourier transform (FT) of the conventional mother ( ) = L (ψ ( ))( ) M,a,b u M M,a,b t u wavelet ψ(t). 2 A b What Wei et al. [17] did not point out is that the 1 i i D u2 = √ e 2B a e 2B π chirp multiplication in the definition of linear canonical i2 B +∞ − wavelets causes rotations of all cells on the time-frequency 1 t b −i 1 ut × √ ψ e B dt planes and shears them along the frequency axis −∞ a a [13, 21, 22]. Therefore, due to the chirping operation, each 2 a i A b + D u2− b u a time-frequency cell is rotated by a degree of arctan − A = e 2B a 2B B u (9) B i2πB B 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. +∞ |( )|2 If Cψ = −∞ d <∞,thenx(t) can be derived M( ) from Wx a, b , i.e., π +∞ +∞ ( ) = 2 −2 M( )ψ ( ) x t a Wx a, b M,a,b t dadb. Cψ −∞ −∞ (13)

Cψ < ∞ is called the admissibility condition of the Fig. 2 The time-LCD-frequency plane LCWT which coincides with the admissibility condition 2 defined in WT. It implies that not any ψM,a,b(t) ∈ L (R) Wang et al. EURASIP Journal on Advances in Signal Processing (2018) 2018:29 Page 4 of 18

ψ ( ) and M,a0,b0 . According to (14), the LCWT of x t at = = M( ) a a0 and b b0 i.e., Wx a0, b0 can always be repre- M( ) sented by other Wx a, b through the reproducing kernel ( ) M( ) KψM a0, b0; a, b .ThismeansallWx a, b son2-Dplane (a, b) are related to each other, and there always exists u 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 ( ) = δ( − − ) property: KψM a0, b0; a, b a a0, b b0 .How- t ever, it is difficult to find a set of orthonormal linear o t o ψ ( ) ( ) = δ canonical wavelets M,a,b t to make KψM a0, b0; a, b ab(a − a0, b − b0) when a and b are both continuous. There- Fig. 3 a The time-frequency plane divided by time-frequency cells. fore, we need to discrete the dilation and shift parameters b The time-LCD-frequency plane divided by time-LCD-frequency cells of the linear canonical wavelet in (8)bymakinga = 2j, j b = 2 kb0 and b0 = 1, i.e., −j/2 −j −i A (t2−k2) could be the linear canonical wavelet unless the admissi- ψM,j,k(t) = 2 ψ 2 t − k e 2B , k, j ∈ Z, bility condition of the LCWT is satisfied. (17) ψ( ) ∈ ⊂ 2(R) 3.2 Reproducing kernel and corresponding equation where t W0 L , and find a set of orthonormal ψ ( ) ψ ( Like the conventional wavelet transform, the LCWT is a linear canonical wavelets M,j,k t to make K M a0, b0; ) = δ( − − ) redundant representation with a redundancy character- a, b a a0, b b0 hold and eliminate the redundancy ized by reproducing kernel equation. induced from the continuous LCWT.

Theorem 1 Suppose (a0, b0) ∈ (a, b), then 2D function 4 The proposed DLCWT and its fast algorithm M( ) ( ) The DLCWT of x(t) with parameter M = (A, B, C, D) can Wx a, b is the LCWT of signal x t if and only if it satisfies the following reproducing kernel equation, i.e., be defined as M W (j, k) = x(t), ψM j k(t) M( ) = 1 M( ) ( ) x , , Wx a0, b0 Wx a, b KψM a0, b0; a, b dadb +∞ a2 − A 2 ∗ A 2 = i 2B k ( )ψ ( ) i 2B t e x t j,k t e dt (14) −∞ −j/2 −i A k2 where Kψ (a , b ; a, b) is the reproducing kernel with = 2 e 2B M 0 0 +∞ 2π ∗ − A 2 × ( )ψ j − i 2B t Kψ (a0, b0; a, b) = ψM,a,b(t), ψM,a ,b (t) (15) x t 2 t k e dt, (18) M Cψ 0 0 −∞ where j ∈ Z and k ∈ Z. Proof Inserting the reconstruction formula (13)intothe definition of the LCWT (10)yields 4.1 Multi-resolution approximation associated with LCT W M(a , b ) The theory of multi-resolution approximation associated x 0 0 π with LCT is first proposed here since it sets the ground = 2 1 M( )ψ ( ) Wx a, b M,a,b t dadb for the DLCWT and the construction of orthogonal linear Cψ a2 ∗ canonical wavelets. According to the definition of multi- × ψ (t)dt M,a0,b0 resolution approximation in [16], we give the following 1 definition. = W M(a b) 2 x , a π Definition 1 A sequence of closet subspaces V M , j ∈ × 2 ψ ( )ψ∗ ( ) j M,a,b t t dt dadb 2 ψ M,a0,b0 Z of L (R) is a multi-resolution approximation associated C with LCT if the following six properties are satisfied: = 1 M( ) ( ) Wx a, b KψM a0, b0; a, b dadb. (16) a2 ∀( ) ∈ Z2 1) j, k , − A j ( − j ) The theorem is proved. ( ) ∈ M ⇔ − j i B 2 k t 2 k ∈ M x t Vj x t 2 k e Vj ; M M 2) ∀j ∈ Z, V ⊃ V + ; The reproducing kernel Kψ (a , b ; a, b) measures the j j 1 M 0 0 − 3A 2 ∀ ∈ Z ( ) ∈ M ⇔ t i 8B t ∈ M correlation of the two linear canonical wavelets, ψM,a,b 3) j , x t Vj x 2 e Vj+1; Wang et al. EURASIP Journal on Advances in Signal Processing (2018) 2018:29 Page 5 of 18

∞ A 2 2 M M −i (t −k ) 4) lim V = ∩ V ={0}; Proof if θM,0,k(t) = θ(t − k)e 2B , k ∈ Z is a j→∞ j j=−∞ j M ∀ ( ) ∈ M ∞ Riesz basis of the subspace V0 ,thenfor x t V0 ,we 5) lim V M = Closure ∪ V M = L2(R); →−∞ j =−∞ j have j j θ( ) ∈ ⊂ 2(R) − A ( 2− 2) 6) There exists a basic function t V0 L ( ) = θ( − ) i 2B t k − A ( 2− 2) x t ck t k e . (21) θ ( ) = θ( − ) i 2B t k ∈ Z such that M,0,k t t k e , k k∈Z is a Riesz basis of subspace V M. 0 After taking the LCT on both sides of (21), we have M Condition (1) means that V is invariant by any transla- − A ( 2− 2) j ( ) = L θ( − ) i 2B t k ( ) j XM u ck M ck t k e u tion proportional to the scale 2 together with modulation. ∈Z M k Dilating operation and chirping operation in Vj enlarge 1 i D u2 −i 1 uk i A k2 ˆ u the detail and condition (3) guarantees that it defines an = √ cke 2B e B e 2B θ −j−1 i2πB B approximation at a coarser resolution 2 .Theexis- k∈Z tence of a Riesz basis of V M provides a discretization ˜ ˆ u j = CM(u)θ , (22) theorem. The theorem below gives the existence con- B dition of Riesz basis in V M. The following is detailed j ˜ ( ) ( ) where CM u denotes the DTLCT of ck with a period of proof of condition (3). Let XM u denote the LCT of π θˆ u θ( ) − 3A 2 2 B,and B denotes the FT of t with its argument t i 8B t x 2 e . According to the definition of the LCT, one 1 scaled by B . can obtain According to the Parseval’s relation associated with +∞ LCT, ( ) = t −i 3A t2 ( ) XM u x e 8B KM u, t dt 2 2 −∞ 2 x(t) = XM(u) +∞ 1 2 u 2 = √ = ˜ ( ) θˆ π CM u du i2 B −∞ B +∞ 2 +∞ t i A ( t ) i − 1 ut+ D u2 2πB × x e 2B 2 e B 2B dt ˜ 2 ˆ u 2 −∞ 2 = CM(u) θ + 2kπ du. B 0 k=−∞ 1 −i D 3u2 = √ e 2B (23) i2πB +∞ 2 2 Since x(t) ∈ L (R), it can be easily obtained that t i A ( t ) −i 1 2u t i D (2u)2 × x e 2B 2 e B 2 e 2B dt (19) π 2 2 B 2 −∞ 2 ˜ Px(t) ≤ CM(u) du 0 +∞ Replacing t with t in (19) results in 2 2 2 = |ck| ≤ Qx(t) (24) k=−∞ 2 −i D 3u2 X (u) = √ e 2B M i2πB and +∞ i A t 2 −i 1 2ut i D (2u)2 +∞ × x t e 2B e B e 2B dt 1 ˆ u 2 1 −∞ ≤ θ + 2kπ ≤ . (25) Q B P −i D 3u2 k=−∞ = 2e 2B XM(2u) (20) On the other hand, if (25)holds,then(24)canbe ( ) ( ) M ⊂ 2(R) obtained. If x(t) = 0, then according to (24), for where XM u denotes the LCT of x t .SinceVj L −i A k t− k M ⊂ 2(R) ∀ = ( − ) B 2 ∈ Z and Vj+1 L denote the subspace of all func- k, ck 0. r t k e , k is therefore linear tions bandlimited to the interval −2−jπB, +2−jπB and independent with each other. θ (t) = θ(t − k) − −(j+1)π + −(j+1)π M,0,k 2 B, 2 B in the LCT domain separately, − A ( 2− 2) i 2B t k ∈ Z M − 3A 2 e , k is a Riesz basis of the subspace V0 . t i 8B t ∈ M therefore x 2 e Vj+1 according to (20). −i A (t2−k2) This is to say, θM,0,k(t) = θ(t − k)e 2B , k ∈ Z {θ ( ) ∈ Z} M Theorem 2 M,0,k t , k is a Riesz basis of the sub- will be a Riesz basis of the subspace V0 ,ifandonlyif M {θ( − ) ∈ Z} > > space V0 if and only if t k , k forms a Riesz basis there exist constants P 0andQ 0suchthat(25) of the subspace V0 with θ(t) ∈ V0 as the basis function. holds. Considering {θ(t − k), k ∈ Z} as a Riesz basis of the Wang et al. EURASIP Journal on Advances in Signal Processing (2018) 2018:29 Page 6 of 18

∞ 2 subspace of V0, we can deduce that (25) definitely holds As θ(ˆ u/B + 2kπ) is limited, combining (28) due to the inequation k=−∞ and (30)yields ∞ 1 ˆ 2 1 ≤ θ(u + 2kπ) ≤ (26) θ(ˆ u/B) Q P (u/B) = . (31) k=−∞ 1/2 ∞ 2 θ(ˆ / + π) = / ∈ u B 2k holds (see Theorem 3.4 [23] for details), where u u B k=−∞ [ −π, π]. −j/2 −j The theorem is proved. If {2 φ(2 t − k), k ∈ Z} is an orthonormal basis of φ( ) the subspace Vj, then the FT of t definitely makes (31) {θ ( ) ∈ Z} −i A (t2−k2) In particular, the family M,0,k t , k is an orthonor- hold. Therefore, φM,0,k(t) = φ(t − k)e 2B , k ∈ Z M = = mal basis of the space Vj if and only if P Q 1. M forms an orthonormal basis of the subspace V0 . Theorem 2 implies that V M are actually the chirp- j Moreover, it is easy to prove that for ∀j, k1, k2 ∈ Z, modulated shift-invariant subspaces of L2(R),because φ (t), φ (t) = δ(k − k ). (32) they are spaces in which the generators are modulated by M,j,k1 M,j,k2 1 2 chirps and then translated by integers [24–26]. The theorem is proved. The following theorem provides the condition to con- M struct an orthogonal basis of each space Vj by dilating, Thus, according to Theorems 2 and 3,onecanusethe translating, and chirping the scaling function φ(t) ∈ V0. mother wavelet ψ(t) ∈ W0 to construct mother linear ψ ( ) ∈ M canonical wavelet M t W0 such that the dilated, Theorem 3 Define V M , j ∈ Z as a sequence of closet translated, and chirp-modulated family j subspaces, and {φM,j,k(t), j, k ∈ Z} as a set of scaling func- −j/2 −j −i A (t2−k2) ψM,j,k(t) = 2 ψ(2 t − k)e 2B , k ∈ Z tions. If {φj,k(t), j, k ∈ Z} is an orthonormal basis of the subspace Vj,thenforallj∈ Z, φM,j,k formsanorthonormal (33) M basis of subspace Vj . M M is an orthonormal basis of Wj .AsWj is the orthogonal complement of V M in V M , i.e., φ ∈ M j j−1 Proof First, it is easy to find that M,j,k Vj .Fromit, M ⊥ M we have Wj Vj (34) and φM,0,0 = ckθM,0,k(t). (27) k M = M ⊕ M Vj−1 Vj Wj , (35) Taking the LCT with M = (A, B, C, D) on both sides of M the orthogonal projection of input signal x on V − can be (27), we have j 1 V M decomposed as the sum of orthogonal projections on j i A k2 −i 1 ut ˆ M (u/B) = cke 2B e B θ(u/B) and Wj . k∈Z 4.2 Discrete orthogonal LCWT and its fast algorithm = E˜ (u/B)θ(ˆ u/B), (28) In this section, we will give the relationship between j A k2 ˜ iω the DOLCWT and the conjugate mirror filter banks where ek = cke 2B ,andE(e ) is the DTFT of ek. Notice that (u) and θ(ˆ u) are the FT of φ(t) and θ(t), associated with LCT, and the condition to construct the orthonormal linear canonical wavelets. These two- respectively. −i A (t2−k2) channel filter banks implement a fast computation of If φ (t) = φ(t − k)e 2B , k ∈ Z forms an M,0,k DOLCWT which only has O(N) computational complex- M orthonormal basis of V0 , according to Theorem 2,we ity for signals of length N. have ∞ 4.2.1 Relationship between DOLCWT and two-channel filter |(u/B + 2kπ)|2 = 1. (29) banks associated with LCT ψ − (t) φ − (t) k=−∞ Since both M,j 1,k and M,j 1,k form an orthonor- M M φ ( ) mal basis for Wj−1 and Vj−1,wecandecompose M,j,0 t Applying (28)and(29), we can obtain and ψM,j,0(t) as ∞ ∞ ˜ 2 ˆ 2 E(u/B) θ(u/B + 2kπ) = 1. (30) φM,j,0(t) = hM,0(k)φM,j−1,k(t) (36a) k=−∞ k=−∞ Wang et al. EURASIP Journal on Advances in Signal Processing (2018) 2018:29 Page 7 of 18

∞ and Notice that (u/B + 2pπ)2 = 1and ∞ p=−∞ ψ ( ) = ( )φ ( ) ∞ M,j,0 t hM,1 k M,j−1,k t (36b) ( / + π + π)2 = =−∞ u B 2p 1, it is easy to find that k p=−∞ with 2 2 |H0(u/B)| + |H0(u/B + π)| = 2. (42a) −i A k2 hM,0(k) = h0(k)e 2B (37a) Similar with {φM,0,k(t), k ∈ Z}, the relationship and | ( / )|2 + | ( / + π)|2 = −i A k2 H1 u B H1 u B 2 (42b) hM,1(k) = h1(k)e 2B . (37b) holds. Equation (36) are the two-scale difference equations M M belonging to LCWT which reveal the relationship Moreover, because W0 and V0 are orthogonal with ψ ( ) ∈ Z φ ( ) ∈ Z between linear canonical wavelets and linear canoni- each other, M,0,k t , k and M,0,k t , k are cal scale functions in multi-resolution approximation orthogonal, i.e., analysis associated with LCT. Moreover, due to the φ ( ) ψ ( ) = M,0,k1 t , M,0,k2 t 0 (43) orthogonality between {φM,j,k(t)} and {ψM,j,k(t)},wehave for ∀k1, k2 ∈ Z, and it is easy to verify that hM,0(k) = φM,j,0(·), φM,j−1,k(·) (38a) ∞ and ( / + π)∗( / + π) = u B 2k u B 2k 0. (44) =−∞ hM,1(k) = ψM,j,0(·), φM,j−1,k(·) . (38b) k

As can be seen from (38), hM,0(k) and hM,1(k) are irrel- Therefore, substituting (39a)and(39b)into(44), we have evant to j, because of the complex amplitude we multiply ∞ to the mother linear canonical wavelet (see (8)). Moreover, H0(u/B + kπ)(u/B + kπ) ( ) ( ) it should be noticed that the sequence h0 k and h1 k k=−∞ are the conjugate mirror filters in the FT domain. There- × ∗( / + π)∗( / + π) = H1 u B k u B k 0. (45) fore, according to Zhao [7], hM,0(k) and hM,1(k) actually represent the two-channel filter banks in the LCT domain. Similarly, since H0(u) and H1(u) are both 2π periodic, Assume that j = 1. By taking the LCT of both sides of splitting k into odd and even parts, i.e., substituting k = 2p (36), we obtain and k = 2p + 1, p ∈ Z into (45)gives 1 ( / ) ∗( / )+ ( / +π) ∗( / +π) = (u/B) = √ H0(u/2B)(u/2B) (39a) H0 u B H1 u B H0 u B H1 u B 0. (46) 2 and Equations (42a), (42b), and (46) together indicate that if −j/2 −j −i A (t2−k2) ψM,j,k(t) = 2 ψ(2 t−k)e 2B is an orthonormal ( / ) = √1 ( / )( / ) u B H1 u 2B u 2B , (39b) basis for W M,then 2 j ( ) ( ) † where H0 u and H1 u are the discrete time Fourier M · M = 2I, (47) transform (DTFT) of h0(k) and h1(k),respectively. According to the orthogonality of {φM,0,k(t), k ∈ Z},we where † denotes conjugate transpose, I is identity matrix, have and

∞ ( / ) ( / + π) = H0 u B H0 u B |H (u/2B + kπ)|2|(u/2B + kπ)|2 = 2. (40) M . (48) 0 H1(u/B) H1(u/B + π) k=−∞ Equation (47) indicates that when {ψ (t), k ∈ Z} Since H (u) is 2π periodic, splitting k into odd and even M,j,k 0 M ( ) ( ) parts, i.e., substituting k = 2p and k = 2p + 1, p ∈ Z into forms an orthonormal basis for Wj , hM,0 k and hM,1 k (40)yields are actually the two-channel conjugate orthogonal mirror filter banks associated with the LCT. ∞ Overall, the construction of the orthonormal linear |H (u/B)|2 (u/B + 2pπ)2+ 0 canonical wavelets can be summarized in the following p=−∞ theorem. ∞ |H (u/B + π)|2 (u/B + 2pπ + π)2 = 2. 0 Theorem 4 Define V M , j ∈ Z as a sequence of closet p=−∞ j M M (41) subspaces. Wj is the orthogonal complement of Vj in Wang et al. EURASIP Journal on Advances in Signal Processing (2018) 2018:29 Page 8 of 18

V M .If{φ (t), j, k ∈ Z} is a set of orthonormal basis of c = φ + (t), φ (t) j−1 M,j,k n M,j 1,k M ,j,n M {ψ ( ) ∈ Z} +∞ Vj ,then M,j,k t , j, k is a set of orthonormal basis −j− 1 t i A k2 = 2 2 φ − k e 2B of W M if and only if M satisfy (47), i.e., {ψ (t), j, k ∈ Z} is j+1 j j,k −∞ 2 a set of orthonormal basis of Wj. ∗ t −i A n2 × φ − n e 2B dt 2j 4.2.2 Fast algorithm +∞ {φ ( ) ∈ Z} {ψ ( ) ∈ Z} − 1 t i A k2 Since M,j,k t , j, k and M,j,k t , j, k are = 2 2 φ e 2B −∞ 2 orthonormal bases of VM,j and WM,j, the projection in ∗ −i A n2 these spaces can be characterized by × φ (t − n + 2k)e 2B dt i A (5k2−4nk) = e 2B φM,1,0(t), φM,0,n−2k(t) aM,j(k) = x(t), φ (t) (49a) M,j,k i A (5k2−4nk) = e 2B hM,0(n − 2k). (51) and Equation (50)impliesthat ( ) = ( ) ψ ( ) dM,j k x t , M,j,k t . (49b) ∞ i A (5k2−4nk) φM,j+1,k(t) = e 2B =−∞ An actual implementation of the MAR of LCWT n × ( − )φ ( ) requires computation of the inner products shown above, hM,0 n 2k M,j,n t . (52) which is computationally rather involved. Therefore, in this section, we develop a fast filter bank algorithm asso- Taking the inner product by x(t) on both sides of (52) ciated with the LCT that computes the orthogonal linear yields canonical wavelet coefficients of a signal measured at a finite resolution. ¯ a + (k) = a (k)h (2k). (53a) φ ∈ M M,j 1 M,j M,0 From the orthogonormal functions M,j+1,k Vj+1, φ ∈ V M,andV M ⊂ V M,weget M,j,k j j+1 j ψ ∈ M From the orthogonal functions M,j+1,k Wj+1, M M M φM,j,k ∈ V ,andW + ⊂ V ,wehave ∞ j j 1 j φ + (t) = c φ (t). (50) M,j 1,k n M,j,n ¯ n=−∞ dM,j+1(k) = aM,j(k)hM,1(2k), (53b)

− ¯ With the change of variable t = 2 jt − 2k,weobtain where h(k) = h(−k).

b Fig. 4 Filter-bank interpretation of the fast algorithm we proposed. a A fast linear canonical wavelet transform is computed with a cascade of filtering ¯ ¯ with hM,0(k) and hM,1(k) in the LCT domain followed by a factor 2 subsampling. Note that the linear canonical convolution is used here according to (53a)and(53b). b A fast inverse linear canonical wavelet transform reconstructs progressively each aM,j by inserting zeroes between samples of aM,j+1 and dM,j+1, filtering in the LCT domain and adding the outputs. Note that the linear canonical convolution is used here according to (57) Wang et al. EURASIP Journal on Advances in Signal Processing (2018) 2018:29 Page 9 of 18

2 0 −2 Re{x(k)} 20 40 60 80 100 120 a

(k)} 2

M,0 0 −2

Re{d 10 20 30 40 50 60 b

(k)} 2

M,1 0 −2

Re{a 5 10 15 20 25 30 c

(k)} 2

M,1 0 −2

Re{d 5 10 15 20 25 30 d 2 0 −2 Re{x'(k)} 20 40 60 80 100 120 e

Fig. 5 Two-layer DLCWT of a signal x(n) computed using db3 wavelets. a Real part of original signal x(k). b Real part of coefficients dM,0(k). c Real part of coefficients aM,1(k). d Real part of coefficients dM,1(k). e Reconstructed signal x (k)

M = M ⊕ M φ ( ) ∈ M ψ Since Vj Vj+1 Wj+1, M,j+1,k t Vj+1, M,j+1,k Equations (53a)and(53b)provethataM,j+1 and dM,j+1 ( ) ∈ M φ ( ) can be obtained by taking every other sample of the linear t Wj+1,and M,j,k t can be decomposed as ¯ ¯ canonical convolution of aM,j with hM,0(k) and hM,1(k), φ ( ) M,j,k t respectively, as illustrated by Fig. 4a. The reconstruc- ∞ tion (57) is an interpolation that inserts zeroes to expand = φ ( ) φ ( ) φ ( ) M,j,k t , M,j+1,n t M,j+1,n t aM,j+1 and dM,j+1 and filters these signals in the LCT n=−∞ domain, as shown in Fig. 4b.Comparedtothestructure ∞ shown in Fig. 1, the coefficients of each layer can be chirp + φ ( ) ψ ( ) ψ ( ) M,j,k t , M,j+1,n t M,j+1,n t . (54) modulated and de-modulated with the same chirp rate =−∞ n in different layers, in the process of the multi-resolution Combining (52), we obtain analysis of the DLCWT. The following is an example showing decompositions φ (t), φ + (t) = h (k − 2n) M,j,k M,j 1,n M,0 and reconstructions of 1D signal utilizing the DLCWT. i A (5n2−4nk) × e 2B (55a) We observe a chirp signal given by and ( ) = ( π ) + ( π ) −i k t2 x t sin 2 f0t sin 2 f1t e 2 (58) φM,j,k(t), ψM,j+1,n(t) = hM,1(k − 2n) where k = 2, f0 = 0.1, and f1 = 4.5. Figure 5 shows i A (5n2−4nk) × e 2B . (55b) an example of two-layer DLCWT of this signal x(t) computed using db3 wavelets with M = (2, 1, 1, 1).Notethat Substituting (55)into(54)yields the initial data aM,−1(k) = x(k) where x(k) denote sam- φM,j,k(t) ples of continuous signal x(t) with sampling rate t = 0.1. ∞ As shown in Fig. 5, the coefficients dM,1(k) are basically i A (5n2−4nk) = φM,j+1,n(t)hM,0(k − 2n)e 2B n=−∞ ∞ i A (5n2−4nk) + ψM,j+1,n(t)hM,1(k − 2n)e 2B . (56) n=−∞ Taking the inner product by x(t) on both sides of (56) yields

aM,j(k) = aM,j+1(k)hM,0(2k) + dM,j+1(k)hM,1(2k). Fig. 6 Interpretation of the uniform sampling result in terms of digital filtering associated with LCT (57) Wang et al. EURASIP Journal on Advances in Signal Processing (2018) 2018:29 Page 10 of 18

sampling and discretization mapping reconstruction

Fig. 7 Block diagram representation of the shift sampling and reconstruction procedure in multi-resolution subspaces associated with the LCWT. −1 a The analog input signal f(t) is shift sampled and discretized (the box C/D stands for continuous-to-discrete transformation). b The operator T ( ) −u ( ) ( ) ( ) = δ( − ) maps samples f tn (or D [ f] n ) to sequences ck. c f t can be reconstructed through (59), where cδ t k∈Z c[ k] t k

equal to zero, and two frequency components f0 and f1 of The idea of sampling in multi-resolution subspaces is ( ) M M ( ) T x t lie in subspaces V1 and W0 , separately. Signal x k is to find an invertible map between ck and samples perfectly reconstructed from coefficients dM,0(k), aM,1(k) {f (tn)} where tn denotes the sampling times. To simplify and dM,1(k), denoted as x (k). the problem, in the rest of the section, we normalize the sampling interval as t = 1. 4.2.3 Computational complexity If the sampling times are tn = n + u, n ∈ Z,0≤ u < 1, 2 Direct computation of (11) would involve O(N ) opera- then the samples f (tn) can be written as follows tions per scale with N as the length of the input sequence. However, when using the fast algorithm shown in Fig. 4, −u −i A (n2−k2) D f (n) = c φ(n − k + u)e 2B , (60) the DOLCWT’s computational complexity depends on k ∈ that of the linear canonical convolution. According to k Z (4), (53a), and (53b), each takes O(N) time at the first − i A (2nu+u2) level. Then, the downsampling operation splits the signal where D u f (n) = f (n + u)e 2B . Substitute into two branches of size N/2. But the filter bank only ( ) recursively splits one branch convolved with hM,0 n .This πB i −i A k2 ˜ −i D ω2 i 1 ωk leads to a recurrence relation which conduces to an O(N) = 2B (ω) 2B B ω ck π e CM e e d (61) time for the entire operation. Furthermore, because the 2 B −πB proposed fast filter bank algorithm can inherit the conventional lifting scheme, the computational complexity in (60), we have could be halved for long filters [27]. πB −u ( ) = 1 ˜ (ω) ∗ ( ω) 5 Simulations results and discussion D f n CM KM k, 2πB −πB In this section, we provide simulation results of three k∈Z −i A (n2−k2) applications to illustrate the performance of the proposed × φ(n − k + u)e 2B dω DLCWT. πB = 1 ˜ (ω) ∗ ( ω) CM KM n, 2πB −π 5.1 Shift sampling in multi-resolution subspaces k∈Z B − 1 ω( − ) First, we consider shift sampling [26, 28] in multi- × φ( − + ) i B n k ω M n k u e d . (62) resolution subspace V0 . The shift sampling instants is defined as tn = n + u with n ∈ Z and fractional shift u ∈ ) M [0, 1 .WeworkinV0 only since all the relevant proper- φ( ) ∈ 2(R) ties are independent of the scale. Let t L be the 2 Synthesizing function S (t) used in the proposed algorithm linear canonical scaling function of a MRA V M asso- u j j∈Z 1.5 Synthesizing function Sinc(t−u) used in [4] ciated with the LCWT such that the sampling sequence φ(n + u) of φ(t) belongs to 2(Z) for some u ∈[0,1). 1 According to Theorem 2,since{φ } is a Riesz basis 0.5

M,0,k Amplitude M ( ) ∈ M for V0 , then for any f t V0 , there exists a unique 0 2 sequence {ck}∈ (Z) such that −0.5 −8 −6 −4 −2 0 2 4 6 8 Time ( ) Fig. 8 Comparison of synthesizing functions: Su t is the synthesizing −i A (t2−k2) function used in the proposed algorithm, while Sinc(t − u) is the f (t) = c φ(t − k)e 2B . (59) k synthesizing function used in [4] k∈Z Wang et al. EURASIP Journal on Advances in Signal Processing (2018) 2018:29 Page 11 of 18

Second, Let us now construct synthesis functions. Original signal ˜ −u 4 Reconstructed signal using the proposed method G (ω) = Du g (k)K (k, ω) are the synthesis Reconstructed signal using Stern’s method [4] M M,0,0 M Reconstructed signal using Janssen’s method [28] k∈Z 2 Sampling points filters in the Fig. 6, i.e., ω 1 ω 0 ˜ −i kω ˜ Amplitude = ( − ) B = / −0.8 G− g k u e 1 . (67) u B u B −1 k∈Z −2 −10.8 −10.7 −10.6 −10.5 −10 −5 0 5 10 The perfect reconstruction property of the filter bank Time associated with LCT implies that Fig. 9 The original, sampling points and the reconstructed signals − A 2 A 2 − = i 2B k ( ) i 2B tn u ck e f tn e gk−n. (68) n∈Z Then, interchanging the order of integration and sum- Then, for any f (t) ∈ V M,wehave j mation, and replacing n − k with k in (62)yields i A t2 −u −i A t2 f (t) = f (tn)e 2B n g − φ(t − k)e 2B πB k n ω n∈Z k∈Z −u ( ) = 1 ˜ (ω) ∗ ( ω)˜ ω D f n CM KM n, u d , −i A (t2−t2) 2πB B = f (tn)Sun(t)e 2B n . (69) −π B n∈Z (63) ( ) = −uφ( − ) If we define Su t gk t k ,then where k∈Z ω − ˜ −i 1 k ω ( ) = u φ − − ( − ) u = φ(k + u)e B (64) Sun t gk−n [t n k n ] k∈Z B k∈Z 1 = S (t − n) (70) denotes the DTFT (with its argument scaled by B )of u φ(k + u). Notice that Therefore, all the synthesizing functions are obtained as − ˜ u (ω) = u φ ( ) ( ω) shifts of the L basic functions Su(t). The corresponding M D M,0,0 k KM k, k∈Z sampling and reconstruction procedure is shown in Fig. 7. ω i D ω2 ˜ Finally, we give simulations to verify the proposed = e 2B u . (65) φ( ) = ( ) B algorithm. We choose scaling function t N1 t , which is the B-spline of order 1 [29]. It is easy to verify Therefore, we can obtain −i A (t2−k2) that φM,0,k(t) N1(t − k)e 2B forms a Riesz k∈Z πB − − D ω2 ∗ M(φ) ⊂ 2(R) u ( ) = i 2B ˜ (ω)˜ u (ω) ( ω) ω basis for V0 L . We observe a signal given by D f n e CM M KM n, d −πB ( ) = ρ ( π ) −i k t2 (66) f t 0 sin 2 f0t e 2 (71) where k = 1, ρ = 2, and f = 0.03. It is band-limited by substituting (65)into(63). 0 0 −u in the LCT domain with M = (1, 1, 0.5, 1.5). According According to (66)and( 60 ), D f (n) is equal to the convolution of D−u φ (k) and c . The interpretation to the uniform sampling theory of the LCT domain [4], M,0,0 k = of the uniform sampling result in terms of digital filter- the uniform sampling period can be chosen as T 1. We = ing associated with LCT is shown in Fig. 6. Therefore, choose u 0.5. Hence, the synthesizing function can be −1 derived as the operator T and its inverse T can be represented ω ω by ˜ and 1/˜ respectively, under the condition ≤ < u B u B ( ) = 1, 0 t 1, ˜ ω = Su t (72) that u B 0. 0, others.

Table 1 Reconstruction performances comparison Metrics Definition Reconstructed Proposed Stern’s method [4] Janssen’s method [28] signal method

2 fˆ(t)−f(t) Real part − 27.1518 dB − 24.7675 dB 2.4739 dB NMSE L2 ( )2 f t L2 Imaginary part − 27.0527 dB − 26.0184 dB 3.0201 dB

ˆ( )− ( ) Real part − 26.4773 dB − 5.9511 dB 5.1289 dB ∞ f t f t ∞ Normalized L error L ( ) ∞ f t L Imaginary part − 26.9728 dB − 8.2247 dB 6.0701 dB Wang et al. EURASIP Journal on Advances in Signal Processing (2018) 2018:29 Page 12 of 18

It is plotted in Fig. 8. As can be seen from Fig. 8, 1 the derived synthesizing function S(t) is compactly sup- 0.8 ported. It decay (drop to zero) much faster than the synthesizing function Sinc(t − u) used in [4]. 0.6 (u)| ( ) ∈ − 1 Now, we try to reconstruct f t , t [ 12, 10.5] accord- |X 0.4 ing to (69) under the condition that the number of sam- 0.2 pling points is constrained to 24. The real parts of the original, sampling points and the reconstructed signals are −40 −30 −20 −10 0 10 20 30 40 50 shown in Fig. 9. We use two different quantitative metrics: u the normalized mean-square error (NMSE) and the nor- a ∞ malized L error [30] to show the comparisons between 1 our proposed method and other classic methods. Com- ˆ 0.8 parison results are presented in Table 1 where f (t) and f (t) 0.6 denote the original signal and the reconstructed signal, (u)| 2 respectively. |X 0.4 The simulations illustrate that the proposed sampling 0.2 and reconstruction algorithm outperforms the conventional algorithm in [4] when we are given only finite −40 −30 −20 −10 0 10 20 30 40 50 u numbers of samples. This is because the synthesis func- b tion Su(t) is compactly supported while the Sinc function used in [4] is slowly decayed. The Haar scaling function Fig. 11 Comparing the LCTs of x(t) under different parameters. a The ( ) = (− π − / π) = used here is rather simple (a rectangle in time domain) LCT of x t with parameter M 6 ,1,0, 1 6 at SNR 5dB. b The LCT of x(t) with parameter M = (−6.4π,1,0,− 1/6.4π)at which causes some distortions to the signal in LCD. SNR = 5dB Therefore, some other scaling functions or wavelets with proper frequency shapes can be considered. The synthe- /˜ ω sis filters 1 u B may be found by using their Laurent Besides, when using the algorithm in [28], the real series [31]. 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 0 NMSE of LCWT−denoising sampling theorem to signals non-bandlimited in the FT NMSE of WT−denoising [32] −5 domain may lead to wrong (or at least suboptimal) conclu- −10 sions [5]. Therefore, our proposed algorithm can be found more applicable for non-stationary signal processing, such −15 NMSE/dB as radar chirp signals. −20

−25 5.2 Denoising of non-stationary signals 1 2 3 4 5 6 7 8 J The LCWT enjoys both high concentrations and tunable a resolutions when dealing with chirp signals. The DOL- 0 CWT and its fast algorithm we propose eliminate the NMSE of LCWT−denoising at SNR=20dB NMSE of LCWT−denoising at SNR=5dB −5 0 −10 NMSE of LCWT−denoising NMSE of WT−denoising [32] −5 NMSE of LCD−filtering [19] −15 MSE/dB

−10 −20

−15

−25 NMSE/dB −3.8 −3.6 −3.4 −3.2 −3 −2.8 −2.6 −2.4 −2.2 −2 A/B/2π −20

b −25 −5 0 5 10 15 20 SNR/dB Fig. 10 Choosing the level and matched-parameter of the LCWT. a Performance of LCWT-based denoising and WT-based denoising Fig. 12 Performance of the LCWT-based denoising, WT-based [32] in different decomposition level J with SNR = 20 dB and denoising [32], and LCD-filtering [19]. WT-based denoising’s M = (−6π,1,0,− 1/6π). b NMSE of LCWT-based denoising at decomposition level is 2. The LCD filter is a 128 order low-pass FIR SNR = 20 dB and SNR = 5 dB with level J = 5 filter with 8 Hz pass frequency and 10 Hz stop frequency Wang et al. EURASIP Journal on Advances in Signal Processing (2018) 2018:29 Page 13 of 18

1 0 Re(s) −1 1 2 3 4 5 6 7 8 9 10 a 1 0 Re(x) −1 1 2 3 4 5 6 7 8 9 10

) b 1

LCWT 0 −1 Re(s 1 2 3 4 5 6 7 8 9 10 c ) 1 WT 0

Re(s −1 1 2 3 4 5 6 7 8 9 10 d ) 1 LCD 0 −1 Re(s 1 2 3 4 5 6 7 8 9 10 e Fig. 13 Comparing different reconstructed signals in the time domain. a The received chirp signal s(t). b The contaminated signal x(t) at SNR = 5dB.c The reconstructed signal of LCWT-based denoising. d The reconstructed signal of WT-based denoising. e The retrieved signal of LCD-filtering redundancy and imply that it is a potent signal process- Step 3: Compute the LCWT decomposition of the ing tool. The LCWT-based denoising of chirp signals is signal at N level and apply threshold rule to the investigated here to validate the theory proposed above. detail coefficients. Consider the following model Step 4: Compute the inverse LCWT to reconstruct the signal. ( ) = ( ) + ( ) + ( ) x t s t wn1 t wn2 t , (73) An example is given here to demonstrate the performance of the LCWT-based denoising. The source chirp where w (t) isthewhiteGaussiannoiseandw (t) is signal is given by n1 n2 ! " the interference. An LCWT-based denoising algorithm is (t − t )2 proposed with the steps summarized below. s(t) = exp − 0 exp jπk t2 + j2πω t . (74) 2σ 2 0 0 Step 1: Choose a linear canonical wavelet, a level N and The interference is a cubic polynomial phase function the threshold rule. 3 2 Step 2: Decide the matched-parameter M of LCWT. wn2(t) = a ∗ exp jπvt + jπut + j2πω1t . (75)

Fig. 14 Thumbnails of all five multi-focus image pairs used for evaluation purposes Wang et al. EURASIP Journal on Advances in Signal Processing (2018) 2018:29 Page 14 of 18

Fig. 15 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’

After digitalization, the length of the sequence is denoising step, the energy of the interference in detail N = 1024 and the sampling frequency is Fs = 100Hz.The coefficients can be eliminated by the heursure threshold phase parameters of the signal are k0 = 3and√ω0 = 1. selection rule. Furthermore, because the initial frequency The envelope parameters are t0 = 5andσ = 0.5. The ω0 = 1 Hz, the chirp signal is nearly centralized at the base interference’s phase parameters are v =−0.3, u = 6, and LCD-frequency which makes the LCWT-based denoising ω1 = 10. The amplitude is a = 0.1. perform better at higher decomposition level. Therefore, Firstly, we select the db4 wavelet as the mother linear we choose canonical wavelet and use the heursure threshold selec- ⎧ tion rule with soft thresholding. As for the selection of ⎨ (− 6.4π,1,0,− 1/6.4π),SNR=−5 ∼ 17 dB = (− π − / π) = decomposition level J,Fig.10a shows the different NMSE M ⎩ 6.6 ,1,0, 1 6.6 ,SNR 18 dB of the reconstructed signal in different decomposition (− 6π,1,0,− 1/6π),SNR= 19 ∼ 20 dB. level J with SNR = 20 dB and M = (−6π,1,0,− 1/6π).As (76) shown in Fig. 10b, the LCWT-based denoising achieves its best performance at levels 4 or 5. Therefore, we choose Then, execute steps 3 and 4. At last, the LCWT-based five levels of LCWT decomposition while the WT-based denoising is compared with the WT-based denoising [32] denoising performs best at levels 1 or 2. and LCD-filtering [19] in the aspect of NMSE of the Secondly, the major task of the LCWT-based denois- reconstructed signal. A two-hundred-time Monte Carlo ing is to decide the matched-parameter of LCWT. Sup- experiment is taken at a range of SNR from − 5to20dB pose that chirp rate is known or has been estimated. (see Fig. 12). During the selection of decomposition level, we choose The reconstruction signals in time domain de-noised M = (−6π,1,0,− 1/6π)because the chirp signal is highly by three different methods are shown in Fig. 13 as well. concentrated in this parameter. However, as the existence The simulations illustrate that the LCWT-based denois- of the interference signal and initial frequency, this param- ing outperforms the WT-based denoising [32]andLCD- eter might not be the best choice for LCWT. Figure 10b filtering [19]atawiderangeofSNRs.Ascanbeseenfrom shows that the LCWT-based denoising achieves its best Figs. 12 and 13, the chirp signal cannot concentrate in the performance with M = (−6.4π,1,0,− 1/6.4π) at a lower FD which makes the performances of WT-based denois- signal-to-noise ratio (SNR). This is because the interfer- ing method poorly. Though the chirp signal is highly ence signal can be hardly eliminated using the heursure concentrated in the matched-parameter LCD, the LCD- threshold selection rule since it is almost concentrated in filtering method still fails to eliminate both the white the LCT domain with parameter M = (−6π,1,0,− 1/6π) Gaussian noise and the interference which lie in the pass (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 Table 2 Transform setting for the LP, the DWT, the CVT, and the energy of the interference after applying the heursure CT (according to [37]) threshold selection rule. While the interference signal is Transform Filter Levels = (− π − / π) less concentrated at M 6.4 ,1,0, 1 6.4 (see LP LeGall 5/3 4 Fig. 11b). It is nearly submerged in the white Gaussian DWT bior6.8 4 noise, and it is well-known that the heursure threshold selection rule performs better when denoising signals CVT 4 corrupted by white Gaussian noise. Therefore, during the CT CDF 9/7 CDF 9/7 [4 8 8 16] Wang et al. EURASIP Journal on Advances in Signal Processing (2018) 2018:29 Page 15 of 18

Table 3 Fusion results for multi-focus image pairs: LP, DWT, CVT AB/F Transform MI(mean/std) Q (mean/std) Q0(mean/std) QW (mean/std) QE(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

band of the filter. This makes the performance of LCD- Then the one-dimensional LCWT can be extended to filtering method unsatisfactory as well. The LCWT-based 2-D LCWT, i.e., the 2-D LCWT of f (x, y) ∈ L2 R2 denoising enjoys both the abilities of multi-resolution with parameters M1 = (A1, B1, C1, D1) and M2 = (A2, B2, analysis and high signal concentration which makes the C , D ) is defined as 2 2 LCWT-based denoising method performs better than the M1,M2 ( ) = ( ) other two. However, it should be noticed that there is Wf a, b1, b2 f x, y still a part of the interference (which lies around 9 s in × ψ (x, y)dxdy. (78) the time axis) left un-eliminated. The combination of the M1,M2;a,b1,b2 LCWT-based denoising method and the LCD-filtering In particular, the filter-bank structure illustrated in method can be utilized to solve this problem. A better per- Fig. 4 can be used to implement the orthogonal 2D LCWT. formance is, therefore, promising. Potential applications Note that both the linear canonical wavelet and the filter of the LCWT-based denoising algorithm include speech shownin(8)and(37) are complex. Hence, the coefficients recovery [33], estimations of the time-of-arrival and pulse of 2D LCWT are complex which makes the 2D LCWT width of chirp signals [14]. two-times expansive. Figure 15 shows the magnitudes, real parts and imagi- 5.3 Multi-focus image fusion nary parts of a example of two layers 2D-LCWT decom- In this section the performance of multi-focus image position of 512 × 512 ’Barbara.’ Note that parameters fusion using the proposed 2-D LCWT will be investigated. M = (A, B, C, D) in rows and columns are different with The corresponding thumbnails of all used image-pairs are each other. shown in Fig. 14. The fusion rule we applied here is the maximum selec- The performance of the 2D LCWT-based fusion scheme tion fusion rule. By this rule, the fused approximation J is compared to the results obtained by applying the coeffients XF are obtained by a averaging operation Laplacian pyramid (LP) [34],thediscretewavelettrans- xJ [n] + xJ [n] form (DWT) [23], the Curvelet (CVT) [35], and the Con- xJ [n] = A B , (79) tourlet (CT) [36] which are frequently used to perform F 2 image fusion task. whereas for each decomposition level j, orientation band First, we give the definition of 2D LCWT. According to j p and location n, the fused detail coefficients yF are the definition of 1D linear canonical wavelet in (8), we defined as introduce the 2D linear canonical wavelet to be the 2D j j j wavelet elementary function y n, p , if y n, p > y n, p j n = A A B yF , p j . yB n, p ,otherwise ψ ( ) = ψ ( )ψ ( ) M1,M2;a,b1,b2 x, y M1,a,b1 x M2,a,b2 y . (77) (80)

Table 4 Fusion results for multi-focus image pairs: the proposed LCWT AB/F Filters A/B MI(mean/std) Q (mean/std) Q0(mean/std) QW (mean/std) QE(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 Wang et al. EURASIP Journal on Advances in Signal Processing (2018) 2018:29 Page 16 of 18

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

As for the filter choices, number of decomposition levels than the CVT because of the fast algorithm we proposed and directions, we refer to the best results of each multi- here. resolution transform published in [37]. Table 2 lists the The fusion results for a multi-focus image pair can be used settings for each transform. Particularly, for the CT, seen from Fig. 16. the symbols ’CDF 9/7’ and ’CDF 9/7’ denote the pyramid and orientation filter, respectively. The ’levels’ represents 6Conclusions decomposition levels and the corresponding number of In this paper, the theories of DLCWT and multi- orientation for each level. resolution approximation associated with LCT are We choose five metrics recommended in [37]toquan- proposed to eliminate the redundancy of the continuous titatively evaluate the fusion performance. They are LCWT. In order to reduce the computational complexity AB/F mutual information (MI) [38], Q [39], Q0, QW ,and of DOLCWT, a fast filter banks algorithm associated QE [40, 41]. The scores of all five evaluation metrics with LCT is derived. Three potential applications are closer to 1 indicate a higher quality of the composite discussed as well, including shift sampling in multi- image. resolution subspaces, denoising of non-stationary signals, Tables 3 and 4 list the average results as well as the and multi-focus image fusion. corresponding standard deviations for multi-focus image Further improvements of our proposed methods pairs of each type of transform. From these two tables, it include the lifting scheme [42] to accelerate the fast fil- canbeobservedthatoveralltheCVTshowsbetterper- ter banks algorithm, the periodic non-uniform sampling formance than the LP, the DWT, and the CT, because of signals in multi-resolution subspaces associated with the CVT is good at capturing edge and line features. the DLCWT, etc. Potential applications include single- However, the complexity and memory requirement of image super-resolution reconstruction [43], blind recon- the CVT is much larger than the others. The proposed struction of multi-band signal in LCT domain [44, 45], LCWT can achieve better results with different filter than multi-channel SAR imaging [11, 46], speech recovery [33], the conventional fusion method. Especially, when choos- estimations of the time-of-arrival, and pulse width of chirp ing filter to be rbio1.3 and A/B = 53, the proposed signals [14], etc. LCWT yields better results than the CVT for the MI, Abbreviations Q0,andQW fusion metrics. Besides, the complexity and CT: Chirplet transform; CT: Contourlet; CVT: Curvelet; DLCWT: Discrete LCWT; memory requirement of the 2D LCWT is much smaller DOLCWT: Discrete orthogonal LCWT; DTFT: Discrete time Fourier transform; Wang et al. EURASIP Journal on Advances in Signal Processing (2018) 2018:29 Page 17 of 18

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