An Intertwining of Curvelet and Linear Canonical Transforms

Tantary, Azhar Y.; Shah, Firdous A.

doi:https://doi.org/10.1155/2020/8814998

Journal of Mathematics

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Wavelets and Wavelet-Based Numerical Methods

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Research Article | Open Access

Volume 2020 | Article ID 8814998 | https://doi.org/10.1155/2020/8814998

An Intertwining of Curvelet and Linear Canonical Transforms

Azhar Y. Tantary¹and Firdous A. Shah¹

Academic Editor: Georgios Psihoyios

Received27 Aug 2020

Revised14 Oct 2020

Accepted20 Oct 2020

Published30 Nov 2020

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.

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 . 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 curvelets

The 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.

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 matrix as .

Definition 1. For any , the two-dimensional LCT with respect to a real, unimodular matrix is denoted by and is defined aswhere , with and , denotes the kernel of the two-dimensional LCT and is given byIt is pertinent to mention that, for the case , the two-dimensional LCT (1) corresponds to a chirp multiplication operation. Moreover, the case is also of no particular interest to us. As such, in the rest of the article, we shall focus our attention on the case . We also note that the phase-space transform (1) is lossless if and only if the matrix is unimodular; that is, . The inversion formula corresponding to the two-dimensional LCT (1) is given byAlso, Parseval’s formula associated with (1) readsIn 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 and , where and , the polar LCT is given byAlso, the inversion formula corresponding to (5) is given by

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 when , the LCT definitions (1) and (5) reduce to their respective counterparts of the Fourier transform(ii)Plugging the matrix , (1) and (5), yields the respective counterparts of the fractional Fourier transform(iii)For the matrix , , 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 plane and let , , denote the polar coordinates of an arbitrary point . We choose a pair of window functions , called “radial window,” and , called “angular window,” satisfying the following admissibility conditions:

The window functions (7) and (8) are used to construct a family of complex-valued waveforms adopted to scale location and orientation or according to convenience. For a fixed scale where the basic curvelet is defined via the polar Fourier transform aswhere denotes the well-known Fourier transform defined bywhich can be expressed via the polar coordinates as

Consequently, the family of analyzing waveforms called curvelets is generated by translation and rotation of the basic element ; that is,where denotes the rotation matrix affecting the planar rotation by radians. From (9), we note that the support of the basic element 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 being the “thin” variable. The coarsest scale is fixed once for all and must obey . 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 function , the ordinary curvelet transform is defined aswhere , and 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 scale where , consider a basic waveform defined via the polar LCT (5) aswhere the radial and angular windows and satisfy the slightly modified set of admissibility conditions given by

Applying the inverse LCT (6) on both sides of the expression (14), we haveand upon simplifying (17), we obtain a novel basic waveform via the following expression:where .

Hence, the family of novel curvelets (or linear canonical curvelets) is obtained by translating the basic waveform by and then inducing a rotation of radians; that is,

Having formulated a new family of curvelets 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 matrix with , for any square-integrable function on , the novel curvelet transform is defined aswhere , and is given by (19).
Definition 3 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 matrix , , Definition 3 yields a new curvelet transform combining the merits of the ordinary curvelet transform and the well-known fractional Fourier transform(ii)For , , Definition 3 intertwines the advantages of the ordinary curvelet and the well-known Fresnel transforms into a new curvelet transform(iii)Nevertheless, when , 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 transform as a function of the translation variable , with the respective Fourier transforms of the given function and the basic waveform .

Proposition 1. Given any , the novel curvelet transform defined in (20) can be expressed as

Proof. To accomplish the motive, we shall firstly compute the Fourier transform of the novel curvelet family defined in (19). We proceed asLet , , and denote the polar coordinates of the variables , , and , respectively. Then, we can rewrite (22) as follows:Finally, using Definition 3 and invoking the well-known Parseval’s formula in polar coordinates, we haveNext, translating the expression (24) into cartesian coordinates yields the following:Applying the Fourier transform on both sides of (25), we obtain the desired resultThis 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 . As such, the proposed transform is capable of optimizing the concentration of the curvelet spectrum.
Let be the polar coordinates of the translation variable . Then, as a consequence of Proposition 1, we can express the novel curvelet transform (20) asFrom (23), we observe that the support of the analyzing elements in the frequency domain is completely determined by the support of the radial window and the angular window . Moreover, we observe thatHence, we conclude that the support of the analyzing elements in the frequency domain depends upon the choice of the matrix parameter and is completely independent of the translation parameter . Therefore, an appropriate matrix parameter can be chosen to optimize the concentration of novel curvelet spectrum.
On the other hand, since the curvelet functions 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 , the decay of the novel curvelet functions depends upon the smoothness of the corresponding Fourier transform; the smoother is, the faster the decay is. Moreover, by definition, is supported away from the vertical axis but near the horizontal axis . Hence, for smaller values of , the basic waveform is less oscillatory in direction and more oscillatory in 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 functions , the convolution operation is denoted by and is defined asMoreover, the convolution theorem corresponding to (29) reads

Theorem 1 (Reconstruction Formula). For any satisfying , the reconstruction formula for the novel curvelet transform defined in (20) is given bywhere the radial and angular windows and satisfy their respective admissibility conditions (15) and (16).

Proof. We note that the novel curvelet transform defined in (20) can be expressed via the convolution as follows:Next, we define a functionInvoking (32), we can express (33) as follows:Applying the convolution theorem (30), we can compute the Fourier transform of the function asConsequently, we haveNext, we shall evaluate the integral on the right-hand side of (36). To do so, we shall use the polar coordinates of and invoke the admissibility conditions (15) and (16). For , we haveImplementing (37) in (36), we obtainThat is,This completes the proof of Theorem 1.

Theorem 2 (Rayleigh’s Energy Formula). For any satisfying , we have

That is, the total energy of the signal is preserved from the natural domain to transformed domain , where .

Proof. Invoking the well-known Parseval’s formula and using (30), we havewhich 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 to the space of transforms , where .
We note that the reconstruction formula (31) is concerned for those signals satisfying . 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 on and defineHere, we note thatwhere .
Furthermore, using additivity of the Fourier transform, we observe thatwhere .
Also, thanks to the convolution theorem (30), we infer from (44) and (45) thatMoreover, we note thatAlso,Finally, we define the father wavelet , so thatConsequently, (43) implies thatTherefore, 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 any , the reproducing formula for the novel curvelet transform defined in [20] is given bywhere the radial and angular windows and 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 and its Fourier transform, a different class of uncertainty principles comparing the localization of 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 with the corresponding novel curvelet transform , regarded as a function of the translation variable .

Theorem 4 (Heisenberg-Type Uncertainty Principle). If is the novel curvelet transform of any nontrivial function , satisfying , the following uncertainty inequality holds:

Proof. The classical Heisenberg-Pauli-Weyl inequality is given by [29]Identifying as a function of the translation variable and invoking (53), we haveIntegrating (54) with respect to the measure , we obtainAs a consequence of the Cauchy-Schwartz’s inequality, Fubini’s theorem, and (40), the above inequality can be expressed asInvoking (21) and noting that satisfies , we havePlugging (57) in (56), we obtain the desired Heisenberg-type uncertainty inequality asThis completes the proof of Theorem 4.

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 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 .

4.1. Novel Semidiscrete Curvelet Transform

To formulate the semidiscrete analogue of the proposed transform (20), we shall discretize the scaling parameter and the rotation parameter in the following manner:(i)For , we choose the scale as , and.(ii)For a fixed , we sample the rotation parameter into 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 , where denotes the integer part of . Consequently, the scale-dependent angular discretization is given below:

Now, for a given unimodular matrix , with , the radial and angular windows and are chosen to satisfy the discrete admissibility conditions:

Having discretized the scale and angular parameters, we define a semidiscrete family of linear canonical curvelets aswhere the novel basic waveform is defined in the polar coordinate setting aswith .

With the semidiscrete family of novel curvelets at hand, we have the following definition.

Definition 5. Given a real, unimodular matrix , with , the novel semidiscrete curvelet transform corresponding to any is defined aswhere the novel semidiscrete family 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 any satisfying , we havewhere the radial and angular windows and satisfy their respective admissibility conditions (61) and (62).

Proof. For , we define the functionThen, we observe thatNoting that , and invoking the admissibility condition (61), we haveInvoking the admissibility condition (62) yieldswhere is proportional to the distance from to the nearest . Thus, we haveHence,From (72), we obtain the desired reconstruction formula asThis completes the proof of Theorem 5.

Corollary 1. Invoking (69) and (70), we observe that

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 by taking both the previous discretizations of the scale and angular parameters into consideration. For and , we sample the spatial variable as

Consequently, the novel discrete family of curvelets takes the following form:where the basic waveform is given by (64). Moreover, an easy computation yields that

The formal definition of the novel discrete curvelet transform is given below.

Definition 6. Given a real, unimodular matrix , with , the novel discrete curvelet transform corresponding to any is defined aswhere the novel discrete family of curvelets 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 asIn analogy to the continuous case, we need to take care of the low-frequency signals. We introduce another radial window satisfyingAnd, for , the father wavelet is defined byThese father wavelets are nondirectional in nature. Therefore, the complete family of novel discrete curvelets takes the following form:

5. Construction of Novel Curvelets: An Example

In this section, we shall present a lucid construction of the radial and angular window functions and 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 and ; 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 a real, unimodular matrix , with , we consider the following window functions:where is a smooth function, such thatCertain choices of the function include or even smoother polynomials like and . We note that the smoothness of the window functions and is governed by the function . The 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 below:Next, we show that the aforementioned window functions obey the admissibility conditions (61) and (62). By definition, we have . Firstly, we shall compute the sum , where . For a fixed , the aforementioned sum contains only two nonvanishing terms, and for we haveIn order to show that the admissibility condition (61) holds for the window function we choose the scale . Since , it follows that . Consequently, the sum on the left-hand side of (61) has only two nonvanishing terms corresponding to , namely, and . Thus, for , we haveMoreover, we observe thatPlugging equation (88) in equation (87), we obtainFinally, if we choose , then we shall demonstrate that the window functions and satisfy the admissibility conditions (15) and (16). We proceed withFinally, for , we take so that we have

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.

Data Availability

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

Conflicts of Interest

The authors declare that there are no conflicts of interest.

Acknowledgments

The second author was supported by SERB (DST), Government of India, under Grant no. EMR/2016/007951

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Copyright

Copyright © 2020 Azhar Y. Tantary and Firdous A. Shah. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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