# An Intertwining of Curvelet and Linear Canonical Transforms

Authors: Azhar Y. Tantary; Firdous A. Shah
Journal: Journal of Mathematics
 (2020)

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## Abstract

In this article, we introduce a novel curvelet transform by combining the merits of the well-known curvelet and linear canonical transforms. The motivation towards the endeavour spurts from the fundamental question of whether it is possible to increase the flexibility of the curvelet transform to optimize the concentration of the curvelet spectrum. By invoking the fundamental relationship between the Fourier and linear canonical transforms, we formulate a novel family of curvelets, which is comparatively flexible and enjoys certain extra degrees of freedom. The preliminary analysis encompasses the study of fundamental properties including the formulation of reconstruction formula and Rayleigh’s energy theorem. Subsequently, we develop the Heisenberg-type uncertainty principle for the novel curvelet transform. Nevertheless, to extend the scope of the present study, we introduce the semidiscrete and discrete analogues of the novel curvelet transform. Finally, we present an example demonstrating the construction of novel curvelet waveforms in a lucid manner.


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## Body

## 1. Introduction

The wavelet transform is a multiscale integral transform, which serves as one of the corner stones of nonstationary signal processing. It can be used in time-frequency analysis, wherein the scale and frequency are inverse to each other. The wavelet transform decomposes a signal into components determined by the translations and dilations of a single function known as the mother wavelet. By applying these local decomposition filters, the wavelet transform has proved to be of substantial importance in capturing the local characteristics of nonstationary signals and has paved its way to a number of fields including signal and image processing, sampling theory, geophysics, astrophysics, and quantum mechanics [1–4]. However, the efficiency of the wavelet transform fades away in the realm of higher-dimensional signal processing due to the fact that the wavelet transform employs isotropic scalings in dimensions n≥2. Such isotropic scalings are incompetent to capture the edges and corners in higher-dimensional signals appearing due to the spatial occlusion between different objects; for instance, in medical imaging curves separate bones and different kinds of soft tissue. Therefore, the key problem in multidimensional signal analysis is to extract and characterize the relevant and directional information regarding the occurrence of curves and boundaries in signals. As a result, some off-shoots of the wavelet transform, such as the Stockwell transform [5, 6], ridgelet transform [7], curvelet transform [8, 9], contourlet transform [10], and the shearlet transform [11], have been introduced to address these shortcomings of the wavelet transform.The curvelet transform aims to deal with certain interesting phenomena occurring along curved edges in higher-dimensional signals. Unlike the wavelet transform, the curvelet transform provides time-frequency localization with a reasonable directionality and anisotropy by using angled polar wedges or angled trapezoid windows in frequency domain. The intrinsic multiscale and anisotropic nature of curvelet waveforms leads to optimally sparse representations of objects which display curve-punctuated smoothness, that is, smoothness except for discontinuity along a general curve with bounded curvature. Another remarkable property of curvelets is that they elegantly model the geometry of wave propagation; curvelets may be viewed as coherent waveforms with enough frequency localization to behave like waves but, at the same time, with sufficient spatial localization to behave like particles [12]. For more about curvelets and their applications, we refer to the monographs in [12–19]. Keeping in view the merits of the curvelet transform, in the present study, we aim to answer the fundamental question of whether it is possible to increase the flexibility of the curvelet transform to optimize the concentration of the curvelet spectrum. The answer to this question is affirmative and lies in intertwining the curvelet transform with the well-known linear canonical transform, an integral transform known for its flexibility and higher degrees of freedom in modelling physical phenomenon [20].The highlights of the article are given as follows:(i)
We introduce the notion of novel curvelet transform by combining the merits of the curvelet and linear canonical transforms(ii)
We study the fundamental properties of the proposed transform including the reconstruction and Rayleigh’s energy formulae(iii)
We formulate a Heisenberg-type uncertainty principle associated with the novel curvelet transform(iv)
To extend the scope of the study, we introduce both the semidiscrete and discrete analogues of the novel curvelet transform(v)
Finally, we present an example regarding the construction of novel curveletsThe rest of the article is structured as follows: In Section2, we recapitulate the linear canonical transform and the ordinary curvelet transform. In Section 3, we present the formal aspects of the study, which are continued to Section 4, and Section 5 is devoted to illustrating the construction of novel curvelets. Finally, in Section 6, we extract a conclusion and provide an impetus to the future research work in the realm of novel curvelet transform.


## 2. Linear Canonical and Curvelet Transforms

In this section, we shall present a gentle overview of the linear canonical and curvelet transforms, which facilitates the formulation of the proposed novel curvelet transform.


### 2.1. Two-Dimensional Linear Canonical Transform

The origin of the theory of linear canonical transforms dates back to early 1970s with the independent seminal works of Collins [21] in paraxial optics and Moshinsky and Quesne [22] in quantum mechanics to study the conservation of information and uncertainty under linear maps of phase space. It was only in 1990s that both these independent works began to be referred to jointly in the open literature. The linear canonical transform (LCT) encompasses several well-known signal processing transforms as special cases including the Fourier transform, the fractional Fourier transform, the Fresnel transform, and even simple multiplication by quadratic phase factors [20]. As of now, the theory of linear canonical transforms has expanded into an independent and broad field of research with numerous applications to optics, mathematical physics, and signal and image processing. For more about LCT and its applications, the reader is referred to the monographs in [20–27].Below, we shall present the formal definition of the two-dimensional LCT [25]. For notational convenience, we shall write a 2×2 matrix M=ABCD as M=A,B:C,D.Definition 1.
For anyf∈L2ℝ2, the two-dimensional LCT with respect to a real, unimodular matrix M=A,B:C,D is denoted by ℒMf and is defined as(1)ℒMfξ=∫ℝ2ftKMt,ξdt,B≠0,DexpiCDξ22fDξ,B=0,where KMt,ξ, with t=t1,t2T and ξ=ξ1,ξ2T, denotes the kernel of the two-dimensional LCT and is given by(2)KMt,ξ=12πBexpiAt2−2tTξ+Dξ22B,B≠0.
It is pertinent to mention that, for the caseB=0, the two-dimensional LCT (1) corresponds to a chirp multiplication operation. Moreover, the case B<0 is also of no particular interest to us. As such, in the rest of the article, we shall focus our attention on the case B>0. We also note that the phase-space transform (1) is lossless if and only if the matrix M is unimodular; that is, AD−BC=1. The inversion formula corresponding to the two-dimensional LCT (1) is given by(3)ft=ℒM−1ℒMfξt=∫ℝ2ℒMfξKMt,ξ¯dξ.
Also, Parseval’s formula associated with (1) reads(4)f,g2=ℒMf,ℒMg2,∀f,g∈L2ℝ2.
In the remaining part of this subsection, we shall present an analogue of the two-dimensional LCT using the polar coordinates. We emphasize that the polar LCT plays a key role in the development of the novel curvelet transform. Forξ1=rcosω,ξ2=rsinω and t1=ρcosη,t2=ρsinη, where r,ρ≥0 and ω,η∈0,2π, the polar LCT is given by(5)ℒMfr,ω=12πB∫02π∫0∞fρ,ηexpiAρ2+Dr2−2ρrcosη−ω2Bρdρdη.
Also, the inversion formula corresponding to (5) is given by(6)fρ,η=12πB∫02π∫0∞ℒMfr,ωexp−iAρ2+Dr2−2ρrcosη−ω2Brdrdω.Remarks 1.
The aforementioned definitions (1) and (5) embody several well-known integral transforms, some of which are listed below:(i)
As a special case whenM=0,1:−1,0, the LCT definitions (1) and (5) reduce to their respective counterparts of the Fourier transform(ii)
Plugging the matrixM=cosα,sinα:−sinα,cosα, α≠nπ,n∈ℤ (1) and (5), yields the respective counterparts of the fractional Fourier transform(iii)
For the matrixM=1,B:0,1, B≠0, the LCT definitions (1) and (5) boil down to their analogues for the Fresnel transform


### 2.2. Ordinary Curvelet Transform

In this subsection, we shall recapitulate the mathematical frameworks of the classical curvelet transform, which serve as preliminaries for the development of the novel curvelet transform.Consider the frequency planeℝ2 and let r,ω, r≥0,ω∈0,2π, denote the polar coordinates of an arbitrary point ξ∈ℝ2. We choose a pair of window functions W:0,∞⟶0,∞, called “radial window,” and V:−∞,∞⟶0,∞, called “angular window,” satisfying the following admissibility conditions:(7)∫0∞Wr2drr=1,suppW⊆12,2,(8)2π2∫−11Vω2dω=1,suppV⊆−1,1.The window functions (7) and (8) are used to construct a family of complex-valued waveforms adopted to scale a>0 location b∈ℝ2 and orientation θ∈0,2π or −π,π according to convenience. For a fixed scale a∈0,a0 where a0<π2 the basic curvelet Ψa:ℝ2⟶ℂ is defined via the polar Fourier transform as(9)ℱΨar,ω=a3/4WarVωa,where ℱ denotes the well-known Fourier transform defined by(10)ℱfξ=12π∫ℝ2fte−itTξdt,which can be expressed via the polar coordinates as(11)ℱfr,ω=12π∫02π∫0∞fρ,ηexp−iρrcosη−ωρdρdη.Consequently, the family of analyzing waveformsΨa,b,θt called curvelets is generated by translation and rotation of the basic element Ψat; that is,(12)Ψa,b,θt=ΨaRθt−b,t∈ℝ2,where Rθ=cosθ−sinθsinθcosθ denotes the 2×2 rotation matrix affecting the planar rotation by θ radians. From (9), we note that the support of the basic element Ψa in the frequency domain is a polar wedge governed by the respective supports of the radial and angular windows. The scaling in the radial and angular windows is parabolic in nature with ω being the “thin” variable. The coarsest scale a0 is fixed once for all and must obey a0<π2. These elements become increasingly needle-like at fine scales. Formally, we have the following definition of the ordinary curvelet transform [8, 9].Definition 2.
Given a functionf∈L2ℝ2, the ordinary curvelet transform is defined as(13)ΓΨfa,b,θ=∫ℝ2ftΨa,b,θ¯tdt,where a<a0,b∈ℝ2,θ∈0,2π, and Ψa,b,θt is given by (12).


## 2.1. Two-Dimensional Linear Canonical Transform

The origin of the theory of linear canonical transforms dates back to early 1970s with the independent seminal works of Collins [21] in paraxial optics and Moshinsky and Quesne [22] in quantum mechanics to study the conservation of information and uncertainty under linear maps of phase space. It was only in 1990s that both these independent works began to be referred to jointly in the open literature. The linear canonical transform (LCT) encompasses several well-known signal processing transforms as special cases including the Fourier transform, the fractional Fourier transform, the Fresnel transform, and even simple multiplication by quadratic phase factors [20]. As of now, the theory of linear canonical transforms has expanded into an independent and broad field of research with numerous applications to optics, mathematical physics, and signal and image processing. For more about LCT and its applications, the reader is referred to the monographs in [20–27].Below, we shall present the formal definition of the two-dimensional LCT [25]. For notational convenience, we shall write a 2×2 matrix M=ABCD as M=A,B:C,D.Definition 1.
For anyf∈L2ℝ2, the two-dimensional LCT with respect to a real, unimodular matrix M=A,B:C,D is denoted by ℒMf and is defined as(1)ℒMfξ=∫ℝ2ftKMt,ξdt,B≠0,DexpiCDξ22fDξ,B=0,where KMt,ξ, with t=t1,t2T and ξ=ξ1,ξ2T, denotes the kernel of the two-dimensional LCT and is given by(2)KMt,ξ=12πBexpiAt2−2tTξ+Dξ22B,B≠0.
It is pertinent to mention that, for the caseB=0, the two-dimensional LCT (1) corresponds to a chirp multiplication operation. Moreover, the case B<0 is also of no particular interest to us. As such, in the rest of the article, we shall focus our attention on the case B>0. We also note that the phase-space transform (1) is lossless if and only if the matrix M is unimodular; that is, AD−BC=1. The inversion formula corresponding to the two-dimensional LCT (1) is given by(3)ft=ℒM−1ℒMfξt=∫ℝ2ℒMfξKMt,ξ¯dξ.
Also, Parseval’s formula associated with (1) reads(4)f,g2=ℒMf,ℒMg2,∀f,g∈L2ℝ2.
In the remaining part of this subsection, we shall present an analogue of the two-dimensional LCT using the polar coordinates. We emphasize that the polar LCT plays a key role in the development of the novel curvelet transform. Forξ1=rcosω,ξ2=rsinω and t1=ρcosη,t2=ρsinη, where r,ρ≥0 and ω,η∈0,2π, the polar LCT is given by(5)ℒMfr,ω=12πB∫02π∫0∞fρ,ηexpiAρ2+Dr2−2ρrcosη−ω2Bρdρdη.
Also, the inversion formula corresponding to (5) is given by(6)fρ,η=12πB∫02π∫0∞ℒMfr,ωexp−iAρ2+Dr2−2ρrcosη−ω2Brdrdω.Remarks 1.
The aforementioned definitions (1) and (5) embody several well-known integral transforms, some of which are listed below:(i)
As a special case whenM=0,1:−1,0, the LCT definitions (1) and (5) reduce to their respective counterparts of the Fourier transform(ii)
Plugging the matrixM=cosα,sinα:−sinα,cosα, α≠nπ,n∈ℤ (1) and (5), yields the respective counterparts of the fractional Fourier transform(iii)
For the matrixM=1,B:0,1, B≠0, the LCT definitions (1) and (5) boil down to their analogues for the Fresnel transform


## 2.2. Ordinary Curvelet Transform

In this subsection, we shall recapitulate the mathematical frameworks of the classical curvelet transform, which serve as preliminaries for the development of the novel curvelet transform.Consider the frequency planeℝ2 and let r,ω, r≥0,ω∈0,2π, denote the polar coordinates of an arbitrary point ξ∈ℝ2. We choose a pair of window functions W:0,∞⟶0,∞, called “radial window,” and V:−∞,∞⟶0,∞, called “angular window,” satisfying the following admissibility conditions:(7)∫0∞Wr2drr=1,suppW⊆12,2,(8)2π2∫−11Vω2dω=1,suppV⊆−1,1.The window functions (7) and (8) are used to construct a family of complex-valued waveforms adopted to scale a>0 location b∈ℝ2 and orientation θ∈0,2π or −π,π according to convenience. For a fixed scale a∈0,a0 where a0<π2 the basic curvelet Ψa:ℝ2⟶ℂ is defined via the polar Fourier transform as(9)ℱΨar,ω=a3/4WarVωa,where ℱ denotes the well-known Fourier transform defined by(10)ℱfξ=12π∫ℝ2fte−itTξdt,which can be expressed via the polar coordinates as(11)ℱfr,ω=12π∫02π∫0∞fρ,ηexp−iρrcosη−ωρdρdη.Consequently, the family of analyzing waveformsΨa,b,θt called curvelets is generated by translation and rotation of the basic element Ψat; that is,(12)Ψa,b,θt=ΨaRθt−b,t∈ℝ2,where Rθ=cosθ−sinθsinθcosθ denotes the 2×2 rotation matrix affecting the planar rotation by θ radians. From (9), we note that the support of the basic element Ψa in the frequency domain is a polar wedge governed by the respective supports of the radial and angular windows. The scaling in the radial and angular windows is parabolic in nature with ω being the “thin” variable. The coarsest scale a0 is fixed once for all and must obey a0<π2. These elements become increasingly needle-like at fine scales. Formally, we have the following definition of the ordinary curvelet transform [8, 9].Definition 2.
Given a functionf∈L2ℝ2, the ordinary curvelet transform is defined as(13)ΓΨfa,b,θ=∫ℝ2ftΨa,b,θ¯tdt,where a<a0,b∈ℝ2,θ∈0,2π, and Ψa,b,θt is given by (12).


## 3. Novel Curvelet Transform

In this section, our aim is to introduce the notion of the novel curvelet transform and formulate the associated reconstruction formula and Rayleigh’s energy theorem. Subsequently, we shall also study the support and oscillation properties of the proposed novel curvelet transform.For a fixed scalea∈0,a0 where a0<π2, consider a basic waveform Ψa:ℝ2⟶ℂ defined via the polar LCT (5) as(14)ℒMΨar,ω=a3/4WarVωa,where the radial and angular windows Wr and Vω satisfy the slightly modified set of admissibility conditions given by(15)B2∫0∞Wr2drr=1,suppW⊆12,2,(16)2π2∫−11Vω2dω=1,suppV⊆−1,1.Applying the inverse LCT (6) on both sides of the expression (14), we have(17)Ψaρ,η=a3/42πB∫02π∫0∞WarVωaexp−iAρ2+Dr2−2ρrcosη−ω2Brdrdω,=a3/42πBexp−iAρ22B∫02π∫0∞exp−iDr22BWarVωa×expiρrcosη−ωBrdrdω.and upon simplifying (17), we obtain a novel basic waveform ΨaMt via the following expression:(18)ℱΨaMr,ω=a3/4Bexp−iDBr22WaBrVωa,where ΨaMt=expiAt2/2BΨat.Hence, the family of novel curveletsΨa,b,θMt (or linear canonical curvelets) is obtained by translating the basic waveform ΨaMt by b∈ℝ2 and then inducing a rotation of θ∈0,2π radians; that is,(19)Ψa,b,θMt=ΨaMRθt−b,t∈ℝ2.Having formulated a new family of curveletsΨa,b,θMt by invoking the two-dimensional linear canonical transform (1), we are ready to introduce the formal definition of the novel curvelet transform.Definition 3.
Given a real, unimodular matrixM=A,B:C,D with B>0, for any square-integrable function f on ℝ2, the novel curvelet transform is defined as(20)ΓΨMfa,b,θ=f,Ψa,b,θM2=∫ℝ2ftΨa,b,θM¯tdt,where a<a0,b∈ℝ2,θ∈0,2π, and Ψa,b,θMt is given by (19).
Definition3 embodies many new integral transforms that are yet to be reported in the open literature. Below we point out some important deductions.(i)
Choosing the matrixM=cosα,sinα:−sinα,cosα, α≠nπ,n∈ℤ, Definition 3 yields a new curvelet transform combining the merits of the ordinary curvelet transform and the well-known fractional Fourier transform(ii)
ForM=1,B:0,1, B≠0, Definition 3 intertwines the advantages of the ordinary curvelet and the well-known Fresnel transforms into a new curvelet transform(iii)
Nevertheless, whenM=0,1:−1,0, Definition 3 boils down to the ordinary curvelet transform (13)
Next, we shall present a proposition that interlinks the Fourier transform of the novel curvelet transformΓΨMfa,b,θ as a function of the translation variable b, with the respective Fourier transforms of the given function f and the basic waveform ΨaM.Proposition 1.
Given anyf∈L2ℝ2, the novel curvelet transform ΓΨMfa,b,θ defined in (20) can be expressed as(21)ℱΓΨMfa,b,θξ=2πℱfξℱΨaMRθξ.Proof.
To accomplish the motive, we shall firstly compute the Fourier transform of the novel curvelet familyΨa,b,θMt defined in (19). We proceed as(22)ℱΨa,b,θMξ=12π∫ℝ2Ψa,b,θMte−itTξdt=12π∫ℝ2ΨaMRθt−be−itTξdt=12π∫ℝ2ΨaMze−ib+R−θzTξdz=e−ibTξ2π∫ℝ2ΨaMze−izTRθξdz=e−ibTξℱΨaMRθξ.
Letσ,μ, ρ,η, and r,ω denote the polar coordinates of the variables b, t, and ξ, respectively. Then, we can rewrite (22) as follows:(23)ℱΨa,b,θMr,ω=e−irσcosμ−ωℱΨaMr,ω−θ=e−irσcosμ−ωa3/4Bexp−iDBr22WaBrVω−θa.
Finally, using Definition3 and invoking the well-known Parseval’s formula in polar coordinates, we have(24)ΓΨMfa,b,θ=f,Ψa,b,θM2=a3/4B∫02π∫0∞eirσcosμ−ωℱfr,ωexpiDBr22WaBrVω−θardrdω,
Next, translating the expression (24) into cartesian coordinates yields the following:(25)ΓΨMfa,b,θ=∫ℝ2ℱfξℱΨaM¯RθξeibTξdξ=2πℱ−1ℱfξℱΨaM¯Rθξb,
Applying the Fourier transform on both sides of (25), we obtain the desired result(26)ℱΓΨMfa,b,θξ=2πℱfξℱΨaMRθξ.
This completes the proof of Proposition1.
Next, we shall analyze the support and oscillatory behaviour of the novel curvelet transform by invoking Proposition1. We shall demonstrate that the proposed transform enjoys a certain degree of freedom as the radial window is comparatively more flexible with the degree of flexibility governed by the matrix parameter B. As such, the proposed transform is capable of optimizing the concentration of the curvelet spectrum.
Letσ,μ,σ≥0,μ∈0,2π be the polar coordinates of the translation variable b. Then, as a consequence of Proposition 1, we can express the novel curvelet transform (20) as(27)ΓΨMfa,b,θ=ΓΨMfa,σ,μ,θ=a3/4B∫02π∫0∞eirσcosμ−ωℱfr,ωexpiDBr22WaBrVω−θardrdω.
From (23), we observe that the support of the analyzing elements Ψa,b,θM in the frequency domain is completely determined by the support of the radial window WaBr and the angular window Vω−θ/a. Moreover, we observe that(28)suppWaBr⊆12aB,2aBand suppVω−θa⊆−a+θ,a+θ.
Hence, we conclude that the support of the analyzing elementsΨa,b,θM in the frequency domain depends upon the choice of the matrix parameter B and is completely independent of the translation parameter b. Therefore, an appropriate matrix parameter B can be chosen to optimize the concentration of novel curvelet spectrum.
On the other hand, since the curvelet functionsΨa,b,θM have compact support in the frequency domain, the well-known Heisenberg’s uncertainty principle implies that the novel curvelet functions cannot have compact support in the time domain. We note that, for large t, the decay of the novel curvelet functions Ψa,b,θMt depends upon the smoothness of the corresponding Fourier transform; the smoother ℱΨa,b,θMξ is, the faster the decay is. Moreover, by definition, ℱΨaMξ is supported away from the vertical axis ξ1=0 but near the horizontal axis ξ2=0. Hence, for smaller values of a<a0, the basic waveform ΨaMt is less oscillatory in t2 direction and more oscillatory in t1 direction.
Below, we shall present the formal reconstruction formula associated with the novel curvelet transform. We note that the said reconstruction formula is valid for high-frequency signals. The analogue for low-frequency signals will be dealt with afterwards. To facilitate the narrative, we need the following definition.Definition 4.
Given any two functionsf,g∈L2ℝ2, the convolution operation is denoted by ⊛ and is defined as(29)f⊛gz=∫ℝ2ftgz−tdt,Moreover, the convolution theorem corresponding to (29) reads(30)ℱf⊛gξ=2πℱfξℱgξ.Theorem 1 (Reconstruction Formula).
For anyf∈L2ℝ2 satisfying ℱfξ=0,∀ξ<2/a0B,a0<π2, the reconstruction formula for the novel curvelet transform ΓΨMfa,b,θ defined in (20) is given by(31)ft=∫02π∫ℝ2∫0a0ΓΨMfa,b,θΨa,b,θMtdadbdθa3,where the radial and angular windows W and V satisfy their respective admissibility conditions (15) and (16).Proof.
We note that the novel curvelet transformΓΨMfa,b,θ defined in (20) can be expressed via the convolution ⊛ as follows:(32)ΓΨMfa,b,θ=∫ℝ2ftΨaM¯Rθt−bdt=∫ℝ2ftΨa,0,θM¯−b−tdt=f⊛Ψ˜a,0,θMb,Ψ˜Mt=ΨM¯−t.
Next, we define a function(33)Fa,θMt=∫ℝ2ΓΨMfa,b,θΨa,b,θMtdb,Invoking (32), we can express (33) as follows:(34)Fa,θMt=f⊛Ψ˜a,0,θMb⊛Ψa,0,θMbt.
Applying the convolution theorem (30), we can compute the Fourier transform of the function Fa,θt as(35)ℱFa,θMξ=2πℱf⊛Ψ˜a,0,θMξℱΨa,0,θMξ=2π2ℱfξℱΨa,0,θMξ2.
Consequently, we have(36)∫0a0∫02πℱFa,θMξdθdaa3=2π2ℱfξ∫0a0∫02πℱΨa,0,θMξ2dθdaa3.
Next, we shall evaluate the integral on the right-hand side of (36). To do so, we shall use the polar coordinates of ξ and invoke the admissibility conditions (15) and (16). For r≥2/a0B,a0<π2, we have(37)2π2∫0a0∫02πℱΨa,0,θMr,ω2dθdaa3=2πB2∫0a0∫02πWaBr2Vω−θa2dθdaa3/2=B2∫0a0WaBr22π2∫02πVω−θa2dθdaa3/2=B2∫0a0WaBr2daa=B2∫0a0BrWr′2dr′r′=1.Implementing (37) in (36), we obtain(38)ℱfξ=∫0a0∫02πℱFa,θMξdθdaa3.
That is,(39)ft=∫02π∫ℝ2∫0a0ΓΨMfa,b,θΨa,b,θMtdadbdθa3.
This completes the proof of Theorem1.Theorem 2 (Rayleigh’s Energy Formula).
For anyf∈L2ℝ2 satisfying ℱfξ=0,∀ξ<2/a0B,a0<π2, we have(40)ΓΨMfa,b,θ22=f22.That is, the total energy of the signal is preserved from the natural domainL2ℝ2 to transformed domain L20,a0×ℝ2×0,2π, where a0<π2.Proof.
Invoking the well-known Parseval’s formula and using (30), we have(41)ΓΨMfa,b,θ22=∫02π∫ℝ2∫0a0ΓΨMfa,b,θ2dadbdθa3=∫02π∫ℝ2∫0a0f⊛Ψ˜a,0,θMb2dadbdθa3=∫02π∫ℝ2∫0a0ℱf⊛Ψ˜a,0,θMξ2dadξdθa3=2π2∫02π∫ℝ2∫0a0ℱfξ2ℱΨa,0,θMξ2dadξdθa3=∫ℝ2ℱfξ22π2∫0a0∫02πℱΨa,0,θMξ2dθdaa3dξ=∫ℝ2ℱfξ2dξ=f22,which evidently completes the proof.Remark 2.
From (40), we infer that the novel curvelet transform defined in (20) is an isometry from the space of signals L2ℝ2 to the space of transforms L20,a0×ℝ2×0,2π, where a0<π2.
We note that the reconstruction formula (31) is concerned for those signals f∈L2ℝ2 satisfying ℱfξ=0,∀ξ<2/a0B,a0<π2. In order to have a complete reconstruction formula, we need to take care of the other frequency components as well. To facilitate the narrative, we consider an arbitrary square integrable function f on ℝ2 and define(42)T1ft=∫02π∫ℝ2∫0a0ΓΨMfa,b,θΨa,b,θMtdadbdθa3=∫02π∫0a0f⊛Ψ˜a,0,θMb⊛Ψa,0,θMbtdadθa3,(43)T0ft=ft−T1ft.
Here, we note that(44)ℱT1fξ=2π2∫02π∫0a0ℱfξℱΨa,0,θMξ2dadθa3=B2ℱfξ∫0a0BξWa2daa=2π2ℱfξℱΩMξ2,where ℱΩMξ2=B2/2π2∫0a0BξWa2da/a.
Furthermore, using additivity of the Fourier transform, we observe that(45)ℱT0fξ=ℱfξ−ℱT1fξ=ℱfξ1−2π2ℱΩMξ2=2π2ℱfξ12π2−ℱΩMξ2=2π2ℱfξℱΦMξ2,where ℱΦMξ2=1/2π2−ℱΩMξ2.
Also, thanks to the convolution theorem (30), we infer from (44) and (45) that(46)T0ft=f⊛ΦM⊛ΦMt,T1ft=f⊛ΩM⊛ΩMt.
Moreover, we note that(47)2π2ℱΩMξ2+ℱΦMξ2=1.
Also,(48)ℱΦMξ=0,ξ>2a0B,ℱΦMξ=12π,ξ<12a0B.
Finally, we define the father waveletΦbMt=ΦMt−b, so that(49)T0ft=∫ℝ2f,ΦbM2ΦbMtdb.
Consequently, (43) implies that(50)ft=∫ℝ2f,ΦbM2ΦbMtdb+∫02π∫ℝ2∫0a0ΓΨMfa,b,θΨa,b,θMtdadbdθa3.
Therefore, we conclude that the complete reconstruction formula for the novel curvelet transform (20) is composed of both curvelet waveforms and isotropic father wavelets. The above discussion can be summarized into the following theorem:Theorem 3 (Complete Reconstruction Formula).
For anyf∈L2ℝ2, the reproducing formula for the novel curvelet transform ΓΨMfa,b,θ defined in [20] is given by(51)ft=∫ℝ2f,ΦbM2ΦbMtdb+∫02π∫ℝ2∫0a0ΓΨMfa,b,θΨa,b,θMtdadbdθa3,where the radial and angular windows W and V satisfy their respective admissibility conditions (15) and (16).The classical Heisenberg’s uncertainty principle in harmonic analysis gives information about the spread of a signal and its Fourier transform by asserting that a signal cannot be sharply localized in both the time and frequency domains [29]. That is, if we limit the behaviour of one, we lose control over the other. The essence of the uncertainty principle is that it provides a lower bound for optimal resolution of a signal in both the time and frequency domains. This classical uncertainty inequality has been extended in different settings and, as of now, many analogues have appeared in the literature [28–31]. In analogy to the uncertainty principles governing the simultaneous localization of a function f and its Fourier transform, a different class of uncertainty principles comparing the localization of f with the localization of its Gabor or wavelet transform were studied by Wilczok [28]. Motivated by this fact, we shall also obtain an uncertainty inequality comparing the localization of the Fourier transform of a function f with the corresponding novel curvelet transform ΓΨMfa,b,θ, regarded as a function of the translation variable b.Theorem 4 (Heisenberg-Type Uncertainty Principle).
IfΓΨMfa,b,θ is the novel curvelet transform of any nontrivial function f∈L2ℝ2, satisfying ℱfξ=0,∀ξ<2/a0B,a0<π2, the following uncertainty inequality holds:(52)∫02π∫ℝ2∫0a0b2ΓΨMfa,b,θ|2dadbdθa3|1/2∫ℝ2ξ2ℱfξ2dξ1/2≥12f22.Proof.
The classical Heisenberg-Pauli-Weyl inequality is given by [29](53)∫ℝ2t2ft2dt1/2∫ℝ2ξ2ℱfξ2dξ1/2≥12∫ℝ2ft2dt.
IdentifyingΓΨMfa,b,θ as a function of the translation variable b and invoking (53), we have(54)∫ℝ2b2ΓΨMfa,b,θ|2db1/2∫ℝ2ξ2ℱΓΨMfa,b,θξ2dξ1/2≥12∫ℝ2ΓΨMfa,b,θ2db.
Integrating (54) with respect to the measure dadθ/a3, we obtain(55)∫02π∫0a0∫ℝ2b2ΓΨMfa,b,θ|2db1/2∫ℝ2ξ2ℱΓΨMfa,b,θξ2dξ1/2dadθa3≥12∫02π∫0a0∫ℝ2ΓΨMfa,b,θ2dbdadθa3.
As a consequence of the Cauchy-Schwartz’s inequality, Fubini’s theorem, and (40), the above inequality can be expressed as(56)∫02π∫ℝ2∫0a0b2ΓΨMfa,b,θ2dadbdθa31/2×∫02π∫ℝ2∫0a0ξ2ℱΓΨMfa,b,θξ2dadξdθa31/2≥12f22.
Invoking (21) and noting that f∈L2ℝ2 satisfies ℱfξ=0,∀r<2/a0B,a0<π2, we have(57)∫02π∫ℝ2∫0a0ξ2ℱΓΨMfa,b,θξ2dadξdθa3=2π2∫02π∫ℝ2∫0a0ξ2ℱfξ2ℱΨaMRθξ2dadξdθa3=2π2∫ℝ2ξ2ℱfξ2∫02π∫0a0ℱΨaMRθξ2dadθa3dξ=∫ℝ2ξ2ℱfξ2B2∫0a0WaBr2daadξ=∫ℝ2ξ2ℱfξ2B2∫0a0BrWr′|2dr′r′dξ=∫ℝ2ξ2ℱfξ2dξ.
Plugging (57) in (56), we obtain the desired Heisenberg-type uncertainty inequality as(58)∫02π∫ℝ2∫0a0b2ΓΨMfa,b,θ2dadbdθa31/2∫ℝ2ξ2ℱfξ2dξ1/2≥12f22.
This completes the proof of Theorem4.


## 4. Novel Semidiscrete and Discrete Curvelet Transforms

In this section, our main aim is to study both the semidiscrete and discrete analogues of the proposed novel curvelet transform defined in [20]. In the beginning of the section, we formulate the definition of the novel semidiscrete curvelet transform, wherein the spatial variable b is continuous, whereas the scalings and orientations vary over a discrete grid. In the sequel, we obtain a reconstruction formula associated with the novel semidiscrete curvelet transform. Towards the culmination, we introduce the notion of the novel discrete curvelet transform by extending the aforementioned discretization to the spatial variable b.


### 4.1. Novel Semidiscrete Curvelet Transform

To formulate the semidiscrete analogue of the proposed transform (20), we shall discretize the scaling parameter a and the rotation parameter θ in the following manner:(i)
Forλ>1, we choose the jth scale as aj=λ−j,j≥0, andj∈ℤ.(ii)
For a fixedL0∈ℤ, we sample the rotation parameter θ into L0 equispaced pieces as(59)θℓ=2πℓL0,whereℓ∈ℤL0=0,1,2,…,L0−1.To prevent the expansion of the angular part as the radial parameter moves away from origin, it is desirable to make the spacing between the consecutive angles scale-dependent. As such, we chooseL0=λ⌊j/2⌋, where ⌊j/2⌋ denotes the integer part of ⌊j/2⌋. Consequently, the scale-dependent angular discretization is given below:(60)θℓj=2πℓλj/2,whereℓ∈ℤλj/2=0,1,2,…,λj/2−1.Now, for a given unimodular matrixM=A,B:C,D, with B>0, the radial and angular windows W and V are chosen to satisfy the discrete admissibility conditions:(61)B2∑j=−∞∞Wλjr2=1,λ>1,r>0,(62)2π2∑ℓ=−∞∞Vy−ℓ2=1,y∈ℝ.Having discretized the scale and angular parameters, we define a semidiscrete family of linear canonical curvelets as(63)Ψj,b,ℓMt=ΨjMRθℓjt−b,t∈ℝ2,where the novel basic waveform ΨjMt is defined in the polar coordinate setting as(64)ℱΨjMr,ω≔λ−3j/4Bexp−iDBr22WBrλjVωθ1j=λ−3j/4Bexp−iDBr22WBrλjVλj/2ω2π,with ΨjMt=expiAt2/2BΨjt.With the semidiscrete family of novel curveletsΨj,b,ℓMt at hand, we have the following definition.Definition 5.
Given a real, unimodular matrixM=A,B:C,D, with B>0, the novel semidiscrete curvelet transform corresponding to any f∈L2ℝ2 is defined as(65)ΓΨMfj,b,ℓ=f,Ψj,b,ℓM2=∫ℝ2ftΨj,b,ℓM¯tdt,where the novel semidiscrete family Ψj,b,ℓMt is given by (63).
We now intend to establish a reconstruction formula associated with the novel semidiscrete curvelet transform defined in (65).Theorem 5 (Reconstruction Formula).
For anyf∈L2ℝ2 satisfying ℱfξ=0,∀ξ<2/a0B,a0<π2, we have(66)ft=∑j≥0∑ℓ=0λj/2−1∫ℝ2ΓΨMfj,b,ℓΨj,b,ℓMtdbλ−3j/2,where the radial and angular windows W and V satisfy their respective admissibility conditions (61) and (62).Proof.
Forλ>1, we define the function(67)Fj,ℓMt=∫ℝ2ΓΨMfj,b,ℓΨj,b,ℓMtdbλ−3j/2.
Then, we observe that(68)ℱFj,ℓMξ=2π2ℱfξℱΨj,0,ℓMξ2=2πB2ℱfξWBrλj2Vλj/2ω−θℓj2π2,Noting that ℱfξ=0,∀ξ<2/a0B,a0<π2, and invoking the admissibility condition (61), we have(69)B2∑j≥0WBrλj2=B2∑j=−∞−1WBrλj2+B2∑j=0∞WBrλj2=B2∑j=−∞∞WBrλj2=B2∑j=−∞∞WλjBr2=1.
Invoking the admissibility condition (62) yields(70)2π2∑ℓ=0λj/2−1Vλj/2ω−θℓj2π2,=2π2∑ℓ=0λj/2−1Vλj/2ω2π−ℓ2,=2π2∑ℓ=−λj/2/2λj/2/2−1Vy−ℓ2=1,where y is proportional to the distance from ω to the nearest θℓj. Thus, we have(71)∑ℓ=0λj/2−1ℱFj,ℓMξ,=B2ℱfξWBrλj22π2∑ℓ=0λj/2−1Vλj/2ω−θℓj2π2,=B2ℱfξWBrλj2.
Hence,(72)∑j≥0∑ℓ=0λj/2−1ℱFj,ℓMξ=ℱfξB2∑j≥0WBrλj2=ℱfξ.
From (72), we obtain the desired reconstruction formula as(73)ft=∑j≥0∑ℓ=0λj/2−1∫ℝ2ΓΨMfj,b,ℓΨj,b,ℓMtdbλ−3j/2.
This completes the proof of Theorem5.Corollary 1.
Invoking (69) and (70), we observe that(74)∑j≥0∑ℓ=0λj/2−1λ3j/2ΓΨMfj,b,ℓ22,=∑j≥0∑ℓ=0λj/2−1λ3j/2ℱΓΨMfj,b,ℓξ22,=∑j≥0∑ℓ=0λj/2−1∫ℝ2ℱΓΨMfj,b,ℓξ2dξλ−3j/2,=∑j≥0∑ℓ=0λj/2−1λλj/2−1∫ℝ22πB2λ−3j/2ℱfξ2WBrλj2Vλj/2ω−θℓj2π2dξλ−3j/2,j/2=∫R2|ℱfξ2B2∑j≥0WBrλj22π2∑ℓ=0λj/2−1Vλj/2ω−θℓj2π2dξ,=∫R2Ffξ2dξ=f22.


### 4.2. Novel Discrete Curvelet Transform

In this subsection, we shall present a complete discrete analogue of the proposed novel curvelet transform defined in (20). Having formulated the semidiscrete analogue, we need to discretize the spatial variable b by taking both the previous discretizations of the scale and angular parameters into consideration. For m=m1,m2∈ℤ2 and β1,β2>0, we sample the spatial variable b as(75)bmjℓ≔R−θℓjKm,β1,β2,j=R−θℓjβ1m1λj,β2m2λj/2T.Consequently, the novel discrete family of curvelets takes the following form:(76)Ψj,m,ℓMt=ΨjMRθℓjt−Km,β1,β2,j,t∈ℝ2,where the basic waveform ΨjMt is given by (64). Moreover, an easy computation yields that(77)ℱΨj,m,ℓMξ=exp−ibmjℓTξℱΨjMRθℓjξ,=λ−3j/4Bexp−ibmjℓTξexp−iDBr22WBrλjVλj/2ω−θℓj2π.The formal definition of the novel discrete curvelet transform is given below.Definition 6.
Given a real, unimodular matrixM=A,B:C,D, with B>0, the novel discrete curvelet transform corresponding to any f∈L2ℝ2 is defined as(78)ΓΨMfj,m,ℓ=f,Ψj,m,ℓM2=∫ℝ2ftΨj,m,ℓM¯tdt,where the novel discrete family of curvelets Ψj,m,ℓMt is given by (76).
By implementing Parseval’s formula for the Fourier transform and taking the benefit of (77), we can express the above definition as(79)ΓΨMfj,m,ℓ=f,Ψj,m,ℓM2=ℱf,ℱΨj,m,ℓM2=∫ℝ2expibmjℓTξℱfξℱΨjM¯Rθℓjξdξ.
In analogy to the continuous case, we need to take care of the low-frequency signals. We introduce another radial windowW0r satisfying(80)W0Br2+∑j≥0WBrλj=12π2.
And, form∈ℤ2, the father wavelet ΦmM is defined by(81)ΦmMt=ΦMt−m,whereℱΦMξ=W0Bξ.
These father wavelets are nondirectional in nature. Therefore, the complete family of novel discrete curveletsℱΦ,Ψ takes the following form:(82)ℱΦ,Ψ≔ΦmMt:m∈ℤ2∪Ψj,m,ℓMt:j≥0,m∈ℤ2,ℓ∈ℤλj/2.


## 4.1. Novel Semidiscrete Curvelet Transform

To formulate the semidiscrete analogue of the proposed transform (20), we shall discretize the scaling parameter a and the rotation parameter θ in the following manner:(i)
Forλ>1, we choose the jth scale as aj=λ−j,j≥0, andj∈ℤ.(ii)
For a fixedL0∈ℤ, we sample the rotation parameter θ into L0 equispaced pieces as(59)θℓ=2πℓL0,whereℓ∈ℤL0=0,1,2,…,L0−1.To prevent the expansion of the angular part as the radial parameter moves away from origin, it is desirable to make the spacing between the consecutive angles scale-dependent. As such, we chooseL0=λ⌊j/2⌋, where ⌊j/2⌋ denotes the integer part of ⌊j/2⌋. Consequently, the scale-dependent angular discretization is given below:(60)θℓj=2πℓλj/2,whereℓ∈ℤλj/2=0,1,2,…,λj/2−1.Now, for a given unimodular matrixM=A,B:C,D, with B>0, the radial and angular windows W and V are chosen to satisfy the discrete admissibility conditions:(61)B2∑j=−∞∞Wλjr2=1,λ>1,r>0,(62)2π2∑ℓ=−∞∞Vy−ℓ2=1,y∈ℝ.Having discretized the scale and angular parameters, we define a semidiscrete family of linear canonical curvelets as(63)Ψj,b,ℓMt=ΨjMRθℓjt−b,t∈ℝ2,where the novel basic waveform ΨjMt is defined in the polar coordinate setting as(64)ℱΨjMr,ω≔λ−3j/4Bexp−iDBr22WBrλjVωθ1j=λ−3j/4Bexp−iDBr22WBrλjVλj/2ω2π,with ΨjMt=expiAt2/2BΨjt.With the semidiscrete family of novel curveletsΨj,b,ℓMt at hand, we have the following definition.Definition 5.
Given a real, unimodular matrixM=A,B:C,D, with B>0, the novel semidiscrete curvelet transform corresponding to any f∈L2ℝ2 is defined as(65)ΓΨMfj,b,ℓ=f,Ψj,b,ℓM2=∫ℝ2ftΨj,b,ℓM¯tdt,where the novel semidiscrete family Ψj,b,ℓMt is given by (63).
We now intend to establish a reconstruction formula associated with the novel semidiscrete curvelet transform defined in (65).Theorem 5 (Reconstruction Formula).
For anyf∈L2ℝ2 satisfying ℱfξ=0,∀ξ<2/a0B,a0<π2, we have(66)ft=∑j≥0∑ℓ=0λj/2−1∫ℝ2ΓΨMfj,b,ℓΨj,b,ℓMtdbλ−3j/2,where the radial and angular windows W and V satisfy their respective admissibility conditions (61) and (62).Proof.
Forλ>1, we define the function(67)Fj,ℓMt=∫ℝ2ΓΨMfj,b,ℓΨj,b,ℓMtdbλ−3j/2.
Then, we observe that(68)ℱFj,ℓMξ=2π2ℱfξℱΨj,0,ℓMξ2=2πB2ℱfξWBrλj2Vλj/2ω−θℓj2π2,Noting that ℱfξ=0,∀ξ<2/a0B,a0<π2, and invoking the admissibility condition (61), we have(69)B2∑j≥0WBrλj2=B2∑j=−∞−1WBrλj2+B2∑j=0∞WBrλj2=B2∑j=−∞∞WBrλj2=B2∑j=−∞∞WλjBr2=1.
Invoking the admissibility condition (62) yields(70)2π2∑ℓ=0λj/2−1Vλj/2ω−θℓj2π2,=2π2∑ℓ=0λj/2−1Vλj/2ω2π−ℓ2,=2π2∑ℓ=−λj/2/2λj/2/2−1Vy−ℓ2=1,where y is proportional to the distance from ω to the nearest θℓj. Thus, we have(71)∑ℓ=0λj/2−1ℱFj,ℓMξ,=B2ℱfξWBrλj22π2∑ℓ=0λj/2−1Vλj/2ω−θℓj2π2,=B2ℱfξWBrλj2.
Hence,(72)∑j≥0∑ℓ=0λj/2−1ℱFj,ℓMξ=ℱfξB2∑j≥0WBrλj2=ℱfξ.
From (72), we obtain the desired reconstruction formula as(73)ft=∑j≥0∑ℓ=0λj/2−1∫ℝ2ΓΨMfj,b,ℓΨj,b,ℓMtdbλ−3j/2.
This completes the proof of Theorem5.Corollary 1.
Invoking (69) and (70), we observe that(74)∑j≥0∑ℓ=0λj/2−1λ3j/2ΓΨMfj,b,ℓ22,=∑j≥0∑ℓ=0λj/2−1λ3j/2ℱΓΨMfj,b,ℓξ22,=∑j≥0∑ℓ=0λj/2−1∫ℝ2ℱΓΨMfj,b,ℓξ2dξλ−3j/2,=∑j≥0∑ℓ=0λj/2−1λλj/2−1∫ℝ22πB2λ−3j/2ℱfξ2WBrλj2Vλj/2ω−θℓj2π2dξλ−3j/2,j/2=∫R2|ℱfξ2B2∑j≥0WBrλj22π2∑ℓ=0λj/2−1Vλj/2ω−θℓj2π2dξ,=∫R2Ffξ2dξ=f22.


## 4.2. Novel Discrete Curvelet Transform

In this subsection, we shall present a complete discrete analogue of the proposed novel curvelet transform defined in (20). Having formulated the semidiscrete analogue, we need to discretize the spatial variable b by taking both the previous discretizations of the scale and angular parameters into consideration. For m=m1,m2∈ℤ2 and β1,β2>0, we sample the spatial variable b as(75)bmjℓ≔R−θℓjKm,β1,β2,j=R−θℓjβ1m1λj,β2m2λj/2T.Consequently, the novel discrete family of curvelets takes the following form:(76)Ψj,m,ℓMt=ΨjMRθℓjt−Km,β1,β2,j,t∈ℝ2,where the basic waveform ΨjMt is given by (64). Moreover, an easy computation yields that(77)ℱΨj,m,ℓMξ=exp−ibmjℓTξℱΨjMRθℓjξ,=λ−3j/4Bexp−ibmjℓTξexp−iDBr22WBrλjVλj/2ω−θℓj2π.The formal definition of the novel discrete curvelet transform is given below.Definition 6.
Given a real, unimodular matrixM=A,B:C,D, with B>0, the novel discrete curvelet transform corresponding to any f∈L2ℝ2 is defined as(78)ΓΨMfj,m,ℓ=f,Ψj,m,ℓM2=∫ℝ2ftΨj,m,ℓM¯tdt,where the novel discrete family of curvelets Ψj,m,ℓMt is given by (76).
By implementing Parseval’s formula for the Fourier transform and taking the benefit of (77), we can express the above definition as(79)ΓΨMfj,m,ℓ=f,Ψj,m,ℓM2=ℱf,ℱΨj,m,ℓM2=∫ℝ2expibmjℓTξℱfξℱΨjM¯Rθℓjξdξ.
In analogy to the continuous case, we need to take care of the low-frequency signals. We introduce another radial windowW0r satisfying(80)W0Br2+∑j≥0WBrλj=12π2.
And, form∈ℤ2, the father wavelet ΦmM is defined by(81)ΦmMt=ΦMt−m,whereℱΦMξ=W0Bξ.
These father wavelets are nondirectional in nature. Therefore, the complete family of novel discrete curveletsℱΦ,Ψ takes the following form:(82)ℱΦ,Ψ≔ΦmMt:m∈ℤ2∪Ψj,m,ℓMt:j≥0,m∈ℤ2,ℓ∈ℤλj/2.


## 5. Construction of Novel Curvelets: An Example

In this section, we shall present a lucid construction of the radial and angular window functionsW and V satisfying the prescribed admissibility conditions. As is evident from (18) and (64), the construction of basic curvelet waveforms is governed by the admissible radial and angular window functions W and V; therefore, the upcoming example also guides the construction of novel basic curvelet waveforms. Consequently, the family of novel curvelets can be obtained by appropriately translating and rotating the basic waveform. It is pertinent to mention that our approach is motivated by [19].Example 1.
Given a2×2 real, unimodular matrix M=A,B:C,D, with B>0, we consider the following window functions:(83)Wr=1Bcosπ2ν5−6r,2/3≤r≤5/6,1B,5/6≤r≤4/3,1Bcosπ2ν3r−4,4/3≤r≤5/3,0,elsewhere,Vω=12πcosπ2ν−3ω−1,−2/3≤ω≤−1/3,12π,−1/3≤ω≤1/3,12πcosπ2ν3ω−1,−1/3≤ω≤2/3,0,elsewhere,where ν is a smooth function, such that(84)νy=0,y≤0,1,y≥1,νy+ν1−y=1,y∈ℝ.Certain choices of the function ν include νy=y or even smoother polynomials like νy=3y2−2y3 and νy=y5−5y4+5y3. We note that the smoothness of the window functions W and V is governed by the function ν. The smoother ν is, the smoother the window functions are and consequently the faster the decay of curvelets is. As an example, one of the sufficiently smooth functions is given below:(85)νy=−0,y≤0,αy−1αy−1+αy,0<y<1−1,y≥1.,where,αy=exp−11+y2−11−y2.
Next, we show that the aforementioned window functions obey the admissibility conditions (61) and (62). By definition, we have suppV⊆−2/3,2/3. Firstly, we shall compute the sum ∑ℓ=−∞∞Vω−ℓ2, where ω∈ℝ. For a fixed ω∈ℝ, the aforementioned sum contains only two nonvanishing terms, and for t∈1/3,2/3 we have(86)2π2∑ℓ=−∞∞Vω−ℓ2=2π2Vω2+Vω−12,=cos2π2ν3ω−1+cos2π2ν−3ω+2=cos2π2νx+cos2π2ν1−x=cos2π2νx+cos2π21−νx=cos2π2νx+sin2π2νx=1.
In order to show that the admissibility condition (61) holds for the window function W we choose the scale λ=2. Since suppW⊂1/2,2, it follows that SuppW2jr⊆2−j−1,21−j. Consequently, the sum on the left-hand side of (61) has only two nonvanishing terms corresponding to r∈1/2,1, namely, Wr2 and W2r2. Thus, for r∈1/2,1, we have(87)B2∑j=−∞∞W2jr2=B2Wr2+W2r2,=1,1/2≤r≤2/3,cos2π2ν6r−4+cos2π2ν5−6r,2/3≤r≤5/6,1,5/6≤r≤1,0,elsewhere.
Moreover, we observe that(88)cos2π2ν6r−4+cos2π2ν5−6r=cos2π2νz+cos2π2ν1−z=cos2π2νz+cos2π21−νz=cos2π2νz+sin2π2νz=1,
Plugging equation (88) in equation (87), we obtain(89)B2∑j=−∞∞W2jr2=1.
Finally, if we chooseln2W′r=Wr, then we shall demonstrate that the window functions W′and V satisfy the admissibility conditions (15) and (16). We proceed with(90)1=2π2∑ℓ=−∞∞Vω−ℓ2=2π2∫01∑ℓ=−∞∞Vω−ℓ2dω=2π2∫−∞∞Vω2dω.
Finally, forr∈0,∞, we take r=2x,x∈−∞,∞ so that we have(91)1=B2∑j=−∞∞W2jr2=B2∑j=−∞∞W2j+x2=B2∫01∑j=−∞∞W2j+x2dx=B2∑j=−∞∞1ln2∫2j2j+1Wy2dyy=B2∑j=−∞∞∫2j2j+1W′y2dyy=B2∫0∞W′y2dyy.


## 6. Conclusion and Future Work

In the present study, we intertwined the advantages of the curvelet and linear canonical transforms and introduced the notion of the novel curvelet transform. The prime advantage of this intertwining lies in the fact that the novel curvelet transform enjoys certain degrees of freedom and the new radial window achieves higher flexibility, which in turn can be employed in optimizing the concentration of the curvelet spectrum. As such, the proposed transform serves as a significant addition to the contemporary tools of signal and image processing. Nevertheless, the present study, in itself, appeals several ramifications and developments thereon. An immediate concern is to study the frame theory associated with the novel discrete curvelet transform.



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*Source: 8814998-2020-11-30.xml*