Some Properties of Fractional Boas Transforms of Wavelets

Khanna, Nikhil; Zothansanga, A.; Kaushik, S. K.; Kumar, Dilip

doi:https://doi.org/10.1155/2021/6689779

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Volume 2021 | Article ID 6689779 | https://doi.org/10.1155/2021/6689779

Some Properties of Fractional Boas Transforms of Wavelets

Nikhil Khanna,¹A. Zothansanga,²S. K. Kaushik,³and Dilip Kumar²

Academic Editor: Nan-Jing Huang

Received04 Nov 2020

Revised23 Jan 2021

Accepted10 Feb 2021

Published15 Mar 2021

Abstract

In this paper, we introduce fractional Boas transforms and discuss some of their properties. We also introduce the notion of wavelets associated with fractional Boas transforms and give some results related to their vanishing moments. Finally, a comparative study of Hilbert transforms and fractional Boas transforms is done.

1. Introduction

The ordinary transformations have been replaced with the fractional ones, which play a significant role in information processing. This transition has occurred naturally due to its various applications in quantum mechanics and optics and purvey us a tool, to characterize a signal completely, in the form of the fractional order, which happens to be the new degree of freedom or an appended parameter for encoding. Among all fractional transforms, the fractional Fourier transform (FRFT), a generalization of the Fourier transform, has been widely studied. In the last three decades, the fractional Fourier transform (FRFT) has played a substantial role in signal processing, optical systems, and quantum physics [1–3]. Another important variation of FRFT is canonical fractional FT [4], which is very effective in optical information processing, since it is easily achievable using simple optical setups and it renders a mere rotation of the two important phase-space distributions: the Wigner distribution and the ambiguity function. The canonical fractional FT was first introduced in [5] more than 90 years ago, which was later improvised by various researchers for applications in quantum mechanics [3], optics [6], and signal processing [2]. Another fractional transform, the complex fractional FT, closely related to the canonical fractional FT has been introduced in [7]. The generalization of Legendre transformation to the fractional Legendre transformation was formulated on the lines of FRFT in [8]. Based on the approach of eigenfunction kernel decomposition similar to the one given in [9], some new fractional integral transforms, including the fractional Mellin transform, a fractional transform associated with the Jacobi polynomials, a Riemann-Liouville fractional derivative operator, and a fractional Riemann-Liouville integral, have been proposed in [10]. In the analogy with canonical fractional Fourier and Hankel transforms, the fractional Laplace and Barut-Girardello transforms have been introduced in [11]. The applications of these transforms in science and engineering are still subject of research.

In order to process one-sided signals, fractional cosine (CT) and sine transforms (ST) were employed. Their digital application along with that of fractional Hartley transforms (HaT) was discussed in [9, 12]. One may refer to [13] for image watermarking scheme classified on the basis of variant fractional transforms such as fractional discrete FT, HaT, CT, and ST.

Gabor [14] introduced the Hilbert transform (HT), an important tool in optics, by constructing an analytic signal from a one-dimensional signal. In 1950, its optical implementation was performed in two different approaches, when Kastler [15] employed it for image processing, primarily for edge enhancement, and Wolter [16] utilized it for spectroscopy. Further advancements in HT can be seen in [17]. The efficacy of HT was raised with the origination of fractional Hilbert transform (FRHT) by Lohmann et al. [18] in 1996, which proffered an additional degree of freedom in the form of a fractional order. Two ways of fractional HT were proposed, which resulted in increase of fractional order and provided improvements in image processing. The first method was based on a spatial filter with a fractional parameter and the other was based on the FRFT. For details, one may see [6, 19]. The FRHT for two-dimensional objects was presented in [20]. Later, Tseng and Pei [13] formulated an SSB modulation by considering the parameter of the fractional phase in FRHT as a secret key. Zayed [21] generalized the HT in a distinctive way and suppressed the negative frequency component of the signal in the FRF domain to obtain the analytic part of a signal. Using FRHT, Cusmariu [22] proposed three possible versions of fractional analytic signals. Tao [23] employed FRFT and FRHT and presented a secured SSB modulation system.

Paley and Wiener [24] studied a class of square integrable functions whose Fourier transforms vanish outside the intervals in great details. This class denoted by was later named by Paley-Weiner class of entire functions and a member of this class is a function band limited to . Contrary to this study, Boas was curious in examining the properties of square integrable functions whose Fourier transforms vanish on , that is, the class . Boas noticed that these properties were not trivial and led to the introduction of Boas transforms (BT) in [25]. Later, BT was studied by Goldberg [26], Heywood [27], and Zaidi [28] who played a substantial role in outlining the properties and the results. For complete review of BT, one may read [29]. In a roundabout way, it was employed in the theory of filters in electrical engineering. Recall from [29] that any finite energy signal on passing through a high pass filter whose system transfer function is given by gives an output such that . Thus, vanishes on . Using Boas’ theorems, one can characterize the output of the high-pass filter in two ways: (i) A signal is the output of a high-pass filter if and only if . (ii) If is an output of high-pass filter, then . Not much research has been done about it until 2019, when Khanna et al. introduced the notion of BT of wavelets (in a preprint form), which was later published in [30]. During the same year, Khanna and Kathuria [31] studied convolution of Boas transforms of wavelets. The motivation behind this study was the relationship between Boas and Fourier transforms of wavelets and the observation that wavelets for which the Fourier transform vanishes almost everywhere on can be characterized by the Boas transform of wavelets . Since Boas transforms are closely related to Hilbert transforms, readers must be interested in reading Hilbert transforms of wavelets. For more details, see [30–40].

1.1. Plan of the Work

The paper is organized as follows: In Section 2, we introduce the notion of fractional Boas transforms (FRBT) and give some properties in the form of observations. Titchmarsh-type and Tricomi-type results are established and a relationship between FRHT and FRBT is given followed by an inversion formula of FRBT and fractional Boas transform product theorem. In Section 3, wavelets associated with FRBT are introduced and a relationship between two wavelets in terms of operator is given. We give a necessary and sufficient condition under which FRBT of a given wavelet is multiple of the first-order derivative of the given wavelet. We further give sufficient condition for the higher vanishing moments of FRBT of wavelets. Finally, we give a sufficient condition on two wavelets to obtain a two-dimensional wavelet and the number of vanishing moments of their convolution is given.

2. Fractional Boas Transforms

The Boas transform of a function , denoted by in terms of principal value integral, is defined asfor any for which the integral exists.

The relationship between the Boas transform and the Hilbert transform of a function is given bywhere

The FRHT of is defined as . It can be easily verified that the operator satisfies the properties of linearity, translation-invariance, dilation-invariance, orthogonality, unitary nature, and linear independence. The linear independence property endorses one to induce a novel base from a given set of linearly independent functions.

Now, we define an operator on bywhere is called a fractional Boas transform. For .

Observations. (i) is translation-invariant; that is, if , then for translation operator , we have(ii) is dilation-invariant. Let denote the dilation operator defined as . Then . Thus, fractional Boas transform operator commutes with .(iii)The transformation is a nonsurjective bounded linear transform on . In fact, we have Now, . Since , it follows from Parseval’s identity that . Thus,(iv)Let . Then . In particular, if and , then . Further, . Indeed, we have If , then which gives . Further, if , then(v)For , we have Indeed, we have Then, the iteration property of the fractional Boas transform (11) can be easily obtained by taking .(vi)Let be a function such that , for . Then, Indeed, we have(vii)Let denote the reflection operator, defined by . Then, . We have(viii)It is easy to verify that if , then(ix)The fractional Boas transform of a convolution of two functions and can be expressed as a convolution of one of the functions with the fractional Boas transform of the other function; that is,Next, we give a Titchmarsh-type result for the fractional Boas transform.

Proposition 1. If , then

Proof. We computeNext, we give a Tricomi-type result for the fractional Boas transform.

Proposition 2. Let be functions such that(i)(ii)(iii) vanishes for and vanishes for Then, .

Proof. We haveIn view of Boas transform product theorem (Theorem 3.9, [31]), we haveNext, we discuss fractional Boas transform of product of analytic functions (or signals).

Proposition 3. For analytic functions and , we have

Proof. We computeIn particular, if , thenThe generalization to arbitrary powers isIn the following result, we give a relationship between fractional Hilbert transform and fractional Boas transform.

Proposition 4. Let and let . Then,where and , for .

Proof. We haveTaking inverse Fourier transform, we haveNote that, for any , we haveSince , it follows by Lebesgue convergence theorem thatThus, the series in (26) converges in norm.
Next, we discuss the inversion of the fractional Boas transform.

Proposition 5. If , then can be retrieved from its fractional Boas transform by means of the formulawhere with sine integral .

Proof. Let , whereThen, we haveNow, observe thatTherefore, . Also, .
Taking the inverse Fourier transform, we obtainTowards the end, we give fractional Boas transform product theorem.

Proposition 6. Let be functions such that(i)(ii) vanishes for and vanishes for Then, .

Proof. In view of Theorem 3.1 in [31], we have

3. Fractional Boas Transforms of Wavelets

The wavelet theory operates with the general properties of the wavelets and the wavelet transform. A wavelet function is chosen according to the application; for example, for space-frequency analysis, a wavelet that is localized in terms of both spatial width and frequency bandwidth is preferred, whereas a smooth wavelet is more appropriate in dealing with smooth signals. In case of analysis of a signal with certain discontinuities, wavelets with good spatial localization to scrupulously track swift changes in the signal are required. For more details on wavelets, one may read [41–45].

Now, we give a sufficient condition under which fractional Boas transform of a wavelet is again a wavelet.

Theorem 1. Let be a wavelet such that and . Then, is again a wavelet.

Proof. In view of Theorem 2.1 in [30], . To verify the admissibility condition, we haveNow, sincewe deduce that

Remark 1. The condition that such that is not necessary for to be a wavelet. Indeed, let be a Haar wavelet defined bySince , where and , we conclude that is a wavelet. However, .
In the next result, we give a relationship between two wavelets in terms of operator.

Theorem 2. Let be a wavelet such that and let be a function such that . If and , then is a wavelet such that

Proof. Note that . Thus, in view of Theorem 2.3 in [30], is a wavelet such that .
The following result gives a necessary and sufficient condition under which fractional Boas transform of a given wavelet is multiple of the first-order derivative of the given wavelet.

Theorem 3. Let be a wavelet such that and . Then if and only if for every .

Proof. Let . Then, taking Fourier transform on both sides, we haveIf , then , whereSince , we get .
The proof of the converse part is straightforward.
A wavelet is said to have vanishing moments if . This property actually represents the regularity of the wavelet function and ability of wavelet transform to capture the localized information. If a wavelet with large number of vanishing moments is employed, then the corresponding wavelet series of a smooth function will converge very rapidly to the function. Thus, only few wavelet coefficients are required in order to obtain a good approximation. During image compression, it requires only to keep a few wavelet coefficients, where the image is smooth and, in contrary to this, more coefficients are needed at the edges. For more details, see [33–40].
Next, we define the notion of -function of order .

Definition 1. Let be a function such that . Then, is said to be a -function of order if , for , where .
Recall from [44] that a function is said to have fast decay with decay rate , if there exists a constant such that , for all .
In the following result, we give a sufficient condition for the higher vanishing moments of fractional Boas transform of wavelets.

Theorem 4. If is an orthonormal system on , thenwhere such that is a -function of order , and has fast decay with decay exponent such that , .

Proof. We havewhere .
In view of Theorem 3.1 in [30], we have , for all .
Also, since and for , using the moment formula for the Hilbert transform, we haveNow , for . Hence, it follows that for . □
Recall from [46] that a two-dimensional function is called an admissible wavelet if it satisfies the admissibility conditionwhere .
In the following result, we give a sufficient condition on two wavelets and such that the product forms a two-dimensional wavelet.

Theorem 5. Let be wavelets such that(i)(ii) and for Then, are admissible wavelets in .

Proof. Note thatSince , for , it follows that is bounded. Also, is bounded and so it must be in . Thus, clearly for .
Also, we haveThen, for , is an admissible wavelet in .
Next, we give a sufficient condition on two wavelets and such that the convolution of two-dimensional wavelets again forms a wavelet with vanishing moments.

Theorem 6. Let be wavelets as defined in Theorem 5 such that(i), and , where is Dirac delta function and is given by (3)(ii) are -function of order (iii), and have vanishing momentsThen, have vanishing moments. Further, if , then is admissible wavelet with vanishing moments.

Proof. We havewhere and .
Since have vanishing moments, using the arguments given in Theorem 3.1 in [30], we conclude that have vanishing moments.
Assume that . If , then ; otherwise, , which gives . Thus, the number of vanishing moments of is .
Also, we computeFurther, note thatNow, since and , it follows that .
Again, since for i = 1, 2, we haveTherefore, we obtainSince and , it follows that are bounded. Thus, we havewhere and are constants. A similar argument works for and . Therefore, we have . Now, observe thatwhere and .
Taking , we obtainThus, in view of Theorem 4.3 in [33], and have, respectively, and vanishing moments.
If , then and . If not, then . Thus, .
Therefore, , and all vanish if .
Hence, the number of vanishing moments of is .

4. Conclusion

In this paper, we define and study the notion of fractional Boas transforms (FRBT) with the aim of obtaining better comparative results. Various properties of FRBT are discussed, and several results are obtained. A comparative study is done to show that the FRBT of wavelets gives better results as compared to the usual wavelets of the classical Boas transform. We illustrate this study by considering the HT and the FRBTs of Daubechies wavelet and Mexican Hat wavelet, respectively, through Figures 1 and 2 and show that the FRBT of wavelet at is the wavelet itself, and the FRBT of wavelet at is the BT of wavelet. It is easy to conclude that, after applying FRBT on a wavelet, the resulting wavelet is approximately equal to the original wavelet. Hence, it is better to employ FRBT of the wavelet instead of using BT of the wavelet, or HT of the wavelet.

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Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors equally contributed to this paper and read and approved the final manuscript.

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