An Intertwining of Curvelet and Linear Canonical Transforms

In this article, we introduce a novel curvelet transform by combining the merits of the well-known curvelet and linear canonical transforms. ,e 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. ,e preliminary analysis encompasses the study of fundamental properties including the formulation of reconstruction formula and Rayleigh’s energy theorem. Subsequently, we develop the Heisenbergtype 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.


Introduction
e 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. e 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][2][3][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. erefore, 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. e curvelet transform aims to deal with certain interesting phenomena occurring along curved edges in higherdimensional 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. e 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][13][14][15][16][17][18][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. e 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]. e 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 curvelets e rest of the article is structured as follows: In Section 2, 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.

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.

Two-Dimensional Linear Canonical Transform.
e 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. e 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 LCTand its applications, the reader is referred to the monographs in [20][21][22][23][24][25][26][27].
Below, we shall present the formal definition of the twodimensional LCT [25]. For notational convenience, we shall .
and is defined as where K M (t, ξ), with t � (t 1 , t 2 ) T and ξ � (ξ 1 , ξ 2 ) T , denotes the kernel of the two-dimensional LCT and is given by It is pertinent to mention that, for the case B � 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, A D − BC � 1. e inversion formula corresponding to the two-dimensional LCT (1) is given by Also, Parseval's formula associated with (1) reads 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. For ξ 1 � r cos ω, ξ 2 � r sin ω and t 1 � ρ cos η, t 2 � ρ sin η, where r, ρ ≥ 0 and ω, η ∈ [0, 2π), the polar LCT is given by 2 Journal of Mathematics Also, the inversion formula corresponding to (5) is given by Remarks 1. e aforementioned definitions (1) and (5) embody several well-known integral transforms, some of which are listed below:

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 plane R 2 and let (r, ω), r ≥ 0, ω ∈ [0, 2π), denote the polar coordinates of an arbitrary point ξ ∈ R 2 . We choose a pair of window functions W: (0, ∞) ⟶ (0, ∞), called "radial window," and V: (− ∞, ∞) ⟶ (0, ∞), called "angular window," satisfying the following admissibility conditions: e window functions (7) and (8) are used to construct a family of complex-valued waveforms adopted to scale a > 0 location b ∈ R 2 and orientation θ ∈ [0, 2π) or (− π, π) according to convenience. For a fixed scale a ∈ (0, a 0 ) where a 0 < π 2 the basic curvelet Ψ a : R 2 ⟶ C is defined via the polar Fourier transform as where F denotes the well-known Fourier transform defined by which can be expressed via the polar coordinates as (11) Consequently, the family of analyzing waveforms Ψ a,b,θ (t) called curvelets is generated by translation and rotation of the basic element Ψ a (t); that is, where R θ � cos θ − sin θ sin θ cos θ denotes the 2 × 2 rotation matrix affecting the planar rotation by θ radians. From (9), we note that the support of the basic element Ψ a in the frequency domain is a polar wedge governed by the respective supports of the radial and angular windows. e scaling in the radial and angular windows is parabolic in nature with ω being the "thin" variable. e coarsest scale a 0 is fixed once for all and must obey a 0 < π 2 . ese elements become increasingly needle-like at fine scales. Formally, we have the following definition of the ordinary curvelet transform [8,9].

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 scale a ∈ (0, a 0 ) where a 0 < π 2 , consider a basic waveform Ψ a : R 2 ⟶ C defined via the polar LCT (5) as Journal of Mathematics where the radial and angular windows W(r) and V(ω) satisfy the slightly modified set of admissibility conditions given by Applying the inverse LCT (6) on both sides of the expression (14), we have and upon simplifying (17), we obtain a novel basic waveform Ψ M a (t) via the following expression: where Ψ M a (t) � exp iA|t| 2 /2B}Ψ a (t). Hence, the family of novel curvelets Ψ M a,b,θ (t) (or linear canonical curvelets) is obtained by translating the basic waveform Ψ M a (t) by b ∈ R 2 and then inducing a rotation of θ ∈ [0, 2π) radians; that is, Having formulated a new family of curvelets Ψ M a,b,θ (t) by invoking the two-dimensional linear canonical transform (1), we are ready to introduce the formal definition of the novel curvelet transform.
is given by (19).  (20) can be expressed as

Proposition 1. Given any
Proof. To accomplish the motive, we shall firstly compute the Fourier transform of the novel curvelet family Ψ M a,b,θ (t) defined in (19). We proceed as Let (σ, μ), (ρ, η), and (r, ω) denote the polar coordinates of the variables b, t, and ξ, respectively. en, we can rewrite (22) as follows: Finally, using Definition 3 and invoking the well-known Parseval's formula in polar coordinates, we have Next, translating the expression (24) into cartesian coordinates yields the following: Applying the Fourier transform on both sides of (25), we obtain the desired result is completes the proof of Proposition 1. Next, we shall analyze the support and oscillatory behaviour of the novel curvelet transform by invoking Proposition 1. 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. Let en, as a consequence of Proposition 1, we can express the novel curvelet transform (20) as From (23), we observe that the support of the analyzing elements Ψ M a,b,θ in the frequency domain is completely determined by the support of the radial window W(aBr) and the angular window ). Moreover, we observe that Hence, we conclude that the support of the analyzing elements 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. e analogue for low-frequency signals will be dealt with afterwards. To facilitate the narrative, we need the following definition. □ Definition 4. Given any two functions f, g ∈ L 2 (R 2 ), the convolution operation is denoted by ⊛ and is defined as Moreover, the convolution theorem corresponding to (29) reads Theorem 1 (Reconstruction Formula). For any f ∈ L 2 (R 2 ) satisfying F[f](ξ) � 0, ∀|ξ| < 2/a 0 B, a 0 < π 2 , the reconstruction formula for the novel curvelet transform (20) is given by where the radial and angular windows W and V satisfy their respective admissibility conditions (15) and (16).
Proof. We note that the novel curvelet transform (20) can be expressed via the convolution ⊛ as follows: Next, we define a function Invoking (32), we can express (33) as follows: Applying the convolution theorem (30), we can compute the Fourier transform of the function F a,θ (t) as Consequently, we have 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/a 0 B, a 0 < π 2 , we have Implementing (37) in (36), we obtain at is, is completes the proof of eorem 1.

Theorem 2 (Rayleigh's Energy Formula). For any
at is, the total energy of the signal is preserved from the natural domain L 2 (R 2 ) to transformed domain Proof. Invoking the well-known Parseval's formula and using (30), we have 6 Journal of Mathematics which evidently completes the proof. (20) is an isometry from the space of signals

Remark 2. From (40), we infer that the novel curvelet transform defined in
We note that the reconstruction formula (31) is concerned for those signals f ∈ L 2 (R 2 ) satisfying F[f](ξ) � 0, ∀|ξ| < 2/a 0 B, a 0 < π 2 . 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 R 2 and define Here, we note that where (F[Ω M ](ξ)) 2 � B 2 /(2π) 2 a 0 B|ξ| 0 |W(a)| 2 da/a. Furthermore, using additivity of the Fourier transform, we observe that Also, thanks to the convolution theorem (30), we infer from (44) and (45) that (46) Moreover, we note that Also, Finally, we define the father wavelet Consequently, (43) implies that erefore, we conclude that the complete reconstruction formula for the novel curvelet transform (20) is composed of both curvelet waveforms and isotropic father wavelets. e above discussion can be summarized into the following theorem: [20] is given by where the radial and angular windows W and V satisfy their respective admissibility conditions (15) and (16).
e 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]. at is, if we limit the behaviour of one, we lose control over the other. e 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. is classical uncertainty inequality has been extended in different settings and, as of now, many analogues have appeared in the literature [28][29][30][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 a, b, θ), regarded as a function of the translation variable b. a, b, θ) is the novel curvelet transform of any nontrivial function f ∈ L 2 (R 2 ), satisfying F[f](ξ) � 0, ∀|ξ| < 2/a 0 B, a 0 < π 2 , the following uncertainty inequality holds:

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.

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) For λ > 1, we choose the j th scale as a j � λ − j , j ≥ 0, andj ∈ Z. (ii) For a fixed L 0 ∈ Z, we sample the rotation parameter θ into L 0 equispaced pieces as 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 choose L 0 � λ ⌊j/2⌋ , where j/2 denotes the integer part of j/2 . Consequently, the scale-dependent angular discretization is given below: Now, for a given unimodular matrix M � (A, B: C, D), with B > 0, the radial and angular windows W and V are chosen to satisfy the discrete admissibility conditions: Having discretized the scale and angular parameters, we define a semidiscrete family of linear canonical curvelets as where the novel basic waveform Ψ M j (t) is defined in the polar coordinate setting as with Ψ M j (t) � exp iA|t| 2 /2B}Ψ j (t). With the semidiscrete family of novel curvelets Ψ M j,b,ℓ (t) at hand, we have the following definition.
Definition 5. Given a real, unimodular matrix M � (A, B: C, D), with B > 0, the novel semidiscrete curvelet transform corresponding to any f ∈ L 2 (R 2 ) is defined as where the novel semidiscrete family Ψ M j,b,ℓ (t) is given by (63). We now intend to establish a reconstruction formula associated with the novel semidiscrete curvelet transform defined in (65).

Journal of Mathematics
where the radial and angular windows W and V satisfy their respective admissibility conditions (61) and (62).
Proof. For λ > 1, we define the function en, we observe that Noting that F[f](ξ) � 0, ∀|ξ| < 2/a 0 B, a 0 < π 2 , and invoking the admissibility condition (61), we have Invoking the admissibility condition (62) yields where y is proportional to the distance from ω to the nearest θ ℓ j . us, we have Hence, From (72), we obtain the desired reconstruction formula as is completes the proof of eorem 5.

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 � (m 1 , m 2 ) ∈ Z 2 and β 1 , β 2 > 0, we sample the spatial variable b as Consequently, the novel discrete family of curvelets takes the following form: where the basic waveform Ψ M j (t) is given by (64). Moreover, an easy computation yields that e formal definition of the novel discrete curvelet transform is given below. Definition 6. Given a real, unimodular matrix M � (A, B: C, D), with B > 0, the novel discrete curvelet transform corresponding to any f ∈ L 2 (R 2 ) is defined as where the novel discrete family of curvelets Ψ M j,m,ℓ (t) 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 In analogy to the continuous case, we need to take care of the low-frequency signals. We introduce another radial window W 0 (r) satisfying And, for m ∈ Z 2 , the father wavelet Φ M m is defined by ese father wavelets are nondirectional in nature. erefore, the complete family of novel discrete curvelets F Φ,Ψ takes the following form:

Construction of Novel Curvelets: An Example
In this section, we shall present a lucid construction of the radial and angular window functions W 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]. (84) Certain choices of the function ] include ](y) � y or even smoother polynomials like ](y) � 3y 2 − 2y 3 and ](y) � y 5 − 5y 4 + 5y 3 . We note that the smoothness of the window functions W and V is governed by the function ]. e smoother ] 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 Moreover, we observe that cos 2 π 2 (](6r − 4)) + cos 2 π 2 (](5 − 6r)) � cos 2 π 2 (](z)) + cos 2 π 2 (](1 − z)) � cos 2 π 2 (](z)) + cos 2 π 2 (1 − ](z)) � cos 2 π 2 (](z)) + sin 2 π 2 (](z)) � 1, Plugging equation (88) in equation (87), we obtain Finally, if we choose ln 2 W ′ (r) � W(r), then we shall demonstrate that the window functions W ′ and V satisfy the admissibility conditions (15) and (16). We proceed with Finally, for r ∈ (0, ∞), we take r � 2 x , x ∈ (− ∞, ∞) so that we have

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. e 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.

Data Availability
No data were used to support the findings of this study.

Conflicts of Interest
e authors declare that there are no conflicts of interest.