Table of Contents Author Guidelines Submit a Manuscript
Abstract and Applied Analysis
Volume 2018, Article ID 8732457, 7 pages
https://doi.org/10.1155/2018/8732457
Research Article

A Version of Uncertainty Principle for Quaternion Linear Canonical Transform

1Department of Mathematics, Hasanuddin University, Makassar 90245, Indonesia
2Department of Mathematics, Tadulako University, Palu 94148, Indonesia

Correspondence should be addressed to Mawardi Bahri; moc.liamg@irhabidrawam

Received 2 October 2017; Revised 18 March 2018; Accepted 2 April 2018; Published 7 May 2018

Academic Editor: Feyzi Başar

Copyright © 2018 Mawardi Bahri et al. 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.

Abstract

In recent years, the two-dimensional (2D) quaternion Fourier and quaternion linear canonical transforms have been the focus of many research papers. In the present paper, based on the relationship between the quaternion Fourier transform (QFT) and the quaternion linear canonical transform (QLCT), we derive a version of the uncertainty principle associated with the QLCT. We also discuss the generalization of the Hausdorff-Young inequality in the QLCT domain.

1. Introduction

The quaternion Fourier transform (QFT) is an extension of the classical two-dimensional Fourier transform (FT) [14] in the framework of quaternion algebra. It plays an important role in the representation of the two-dimensional quaternion signals. A number of useful properties of the extended transform are generalizations of the corresponding properties of the FT with some modifications (see, e.g., [515]). The QFT has found many applications in color image processing and signal analysis; we refer the reader to [1619] and the references mentioned therein. An extension of the QFT in the framework of the classical linear canonical transform (LCT) (see [2022]), known as the quaternion linear canonical transform, has received much attention in recent years. Due to the several definitions of the QFT, there are basically three ways of obtaining the quaternion linear canonical transform (QLCT): the right-sided quaternion linear canonical transform, the left-sided quaternion linear canonical transform, and the two-sided quaternion linear canonical transform. The right-sided quaternion linear canonical transform is obtained by substituting the Fourier kernel with the right-sided QFT kernel in the LCT definition, and so on. Recent works related to some important properties of the QLCT such as Parseval’s theorem, reconstruction formula, and component-wise uncertainty principles were also published [18, 2325]. It was found that the properties of the QLCT are extensions of the corresponding version of the QFT with some modifications.

On the other hand, the uncertainty principle plays one important role in signal processing. It describes a function and its FT, which cannot both be simultaneously sharply localized. One example of this is the Heisenberg uncertainty principle concerning position and momentum wave functions in quantum physics. In signal processing, an uncertainty principle states that the product of the variances of the signal in the time and frequency domains has a lower bound. Up till now, several attempts have been made to extend the uncertainty principles associated with the QFT and QLCT domains. The component-wise and directional uncertainty principles associated with the QFT were proposed in [11]. In [26, 27], the authors established a component-wise uncertainty principle for the QLCT and proved that the equality is achieved for optimal quaternion Gaussian function. Recently, the authors [23] proposed the logarithmic uncertainty principle associated with the QLCT which is the generalization of the logarithmic uncertainty principle for the QFT.

Therefore, the main objective of the present paper is to establish full uncertainty principle for the two-sided QLCT, which is a new general form of component-wise uncertainty principle for the two-sided QLCT. This uncertainty principle is derived using the connection between the QFT and QLCT. We also obtain full uncertainty principle for the right-sided QLCT using a relation between the right-sided QLCT and two-sided QLCT. We also derive the Hausdorff-Young inequality associated with the two-sided QLCT.

This paper is organized as follows. In Section 2, we briefly review the basic knowledge of quaternion and its split used in the next section. In Section 3, we introduce the definition of the QFT and the QLCT. The relationship between the QFT and the QLCT is also discussed in this section. In Section 4, we derive a new version uncertainty principle associated with the QLCT, which shows that the spread of a quaternion-valued function and its QLCT are inversely proportional. Section 5 concludes this paper.

2. Preliminaries

In this section, let us briefly recall some basic definitions and properties of the quaternions (for more details, see [28]).

2.1. Quaternions

The quaternions, a generalization of complex numbers, are members of a noncommutative division algebra. The set of quaternions is denoted by . Every element of can be written in the following form: with the units , , , which obey the following: For a quaternion , is simply called the scalar part of denoted by and is called the vector part of denoted by .

Let and , be their vector parts, respectively. Equation (2) yields the quaternionic multiplication aswhere Analogously to the complex case, a quaternionic conjugation is given bywhich leads to the anti-involution; that is, With the help of (5), we get the norm or modulus of asOne can easily verify thatUsing conjugate (5) and the modulus of q, we can define the inverse of as which shows that is a normed division algebra.

In quaternionic notation, we may define an inner product for quaternion-valued functions as follows:with symmetric real scalar partIn particular, for , we obtain the -normA quaternion module is then defined as

2.2. Split Quaternion and Properties

In this section, we study some of the basic formulas of split quaternion (see [9]), which will be used to prove the fundamental results in the sequel.

Definition 1. For two quaternion square roots such that , one may express a quaternion as

For the special case of , any quaternion may be split up into commuting and anticommuting parts with respect to ; that is,

It easily seems that the commuting and anticommuting parts satisfy the interesting properties:We learn from the above equation thatwhere In particular, taking and , (14) becomesThe above givesThis leads to the following modulus identity:Furthermore, one can obtain

3. Relationship between Quaternion Fourier Transform (QFT) and Quaternion Linear Canonical Transform (QLCT)

In this section, we introduce the QFT and its relationship to the QLCT. We begin by introducing different types of the QFT.

Definition 2 (two-sided, right-sided, and left-sided QFTs). The two-sided, right-sided, and left-sided quaternion Fourier transforms (QFTs) of are given by, respectively,where , , and the quaternion exponential product is the quaternion Fourier kernel. From (24), we get the partial right-sided QFT:

It is not difficult to check that the relationship between the two-sided QFT and the right-sided QFT takes the form

Definition 3 (left-sided, right-sided, and two-sided QLCTs). Suppose that and are real matrix parameters satisfying . The left-sided, right-sided, and two-sided QLCTs of a quaternion signal are defined by, respectively,where the kernel functions of the QLCT above are given by

From the definition of the QLCT, we can easily see that when and , the QLCT of a signal is essentially a quaternion chirp multiplication. Therefore, in this work, we always assume . As a special case, when , for , the QLCT definition (30) will lead to the QFT definition; that is,

The following lemma describes the general relationship between the two-sided QFT and the two-sided QLCT of 2D quaternion-valued signals.

Lemma 4. The QLCT of a signal with matrix parameters and can be seen as the QFT of the signal in the following form:

We introduceThis implies that (33) can be rewritten in the form

Further, we have the following lemma which describes a relation between the right-sided QLCT and the two-sided QLCT of 2D quaternion-valued signals.

Lemma 5. For , one haswhere the matrix parameters and .

The following lemma allows us to represent the right-sided QLCT to the single right-sided QFT of 2D quaternion-valued signals.

Lemma 6. For , one has

Proof. Proofs of Lemmas 4, 5, and 6 are straightforward and are therefore omitted for brevity.

4. A Version of Uncertainty Principle Associated with QLCT

The classical uncertainty principle of harmonic analysis describes that a nontrivial function and its Fourier transform cannot be sharply localized simultaneously. In quantum mechanics, the uncertainty principle asserts that one cannot at the same time be certain of the position and of the velocity of an electron (or any particle). Let us now establish a version of the uncertainty principle associated with the QLCT. However, before proceeding the statement of this main result, we need to introduce a modified uncertainty principle for the QFT as follows (see [29] for more details).

Theorem 7 (the two-sided QFT uncertainty principle). Let be the quaternion Schwartz space. If the quaternion-valued function , then the following inequality holds:

It is shown that the quaternionic Gabor filters minimize the above uncertainty. Analogously, we get the uncertainty principle associated with the right-sided QFT in the following lemma.

Lemma 8 (the right-sided QFT uncertainty principle). Under the hypotheses of Theorem 7, one has the following inequality:

Theorem 7 has been recently generalized in the context of the QLCT by the authors of [30]. Our generalization is given by the following theorem.

Theorem 9 (the two-sided QLCT uncertainty principle). Under the assumptions of Theorem 7, one haswhere is defined byIt is worth noting here that if , (41) can be reduced to

Proof. By replacing by defined by (34) on both sides of (39), we immediately getLet ; then, (44) becomesIn view of (34), we obtainThis leads toInserting (35) into (47), we easily obtain The above identity can be further simplified to where is defined in (42). Therefore, the proof is complete.

The above theorem is also valid for the right-sided QLCT as described in the following lemma.

Lemma 10 (the right-sided QLCT uncertainty principle). Under the assumptions of Theorem 7, one has

Proof. The proof follows directly from (37).

Now, we define a module of as where Furthermore, we obtain the -norm

It should be noticed that if , is real-valued function, then (53) reduces towhere

By Riesz’s interpolation theorem, one can get the Hausdorff-Young inequality related to the QFT (see [10]):holds for with . This gives the following important theorem.

Theorem 11 (Hausdorff-Young inequality). If and letting be such that , then, for all , it holds thatwhere

Proof. From the Hausdorff-Young inequality for the QFT, we have Based on the arguments used in the proof of Theorem 9, we immediately getHence, Or, equivalently, Thus, the theorem is completely proved.

5. Conclusion

In this paper, we derived a version of the uncertainty principle for the QLCT using a relation between the QFT and the QLCT. We presented Hausdorff-Young inequality associated with the QLCT. This inequality is very useful for establishing a variation on Heisenberg’s uncertainty principle related to the QLCT.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work is supported by Research Collaboration Between Universities (PEKERTI) (no. 622.48/UN28.2/PL/2017) of Indonesia.

References

  1. L. Debnath and F. A. Shah, Wavelet Transforms and Their Applications, Birkhäuser, USA, 2010. View at Scopus
  2. S. Mallat, A Wavelet Tour of Signal Processing, Academic Press, San Diego, Calif, USA, 2nd edition, 1998. View at MathSciNet
  3. T. G. Feeman, The Mathematics of Medical Imaging, Springer Science+Business Media, New York, NY, USA, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  4. K. Grochenig, Foundations of Time–Frequency Analysis, Birkhauser, Boston, Mass, USA, 2000. View at Publisher · View at Google Scholar · View at MathSciNet
  5. M. Bahri, R. Ashino, and R. Vaillancourt, “Continuous quaternion Fourier and wavelet transforms,” International Journal of Wavelets, Multiresolution and Information Processing, vol. 12, no. 4, Article ID 1460003, 21 pages, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  6. M. Bahri, A. Lawi, N. Aris, A. F. Saleh, and M. Nur, “Relationships between convolution and correlation for fourier transform and quaternion fourier transform,” International Journal of Mathematical Analysis, vol. 7, no. 41-44, pp. 2101–2109, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  7. M. Bahri, “Quaternion algebra-valued wavelet transform,” Applied Mathematical Sciences, vol. 5, no. 69-72, pp. 3531–3540, 2011. View at Google Scholar · View at MathSciNet · View at Scopus
  8. M. Bahri, “Construction of quaternion-valued wavelets,” Matematika (Johor Bahru), vol. 26, no. 1, pp. 107–114, 2010. View at Google Scholar · View at MathSciNet
  9. H. De Bie, N. De Schepper, T. A. Ell, K. Rubrecht, and S. J. Sangwine, “Connecting spatial and frequency domains for the quaternion Fourier transform,” Applied Mathematics and Computation, vol. 271, pp. 581–593, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  10. L.-P. Chen, K. I. Kou, and M.-S. Liu, “Pitt's inequality and the uncertainty principle associated with the quaternion fourier transform,” Journal of Mathematical Analysis and Applications, vol. 423, no. 1, pp. 681–700, 2015. View at Publisher · View at Google Scholar · View at MathSciNet
  11. E. M. Hitzer, “Directional uncertainty principle for quaternion fourier transform,” Advances in Applied Clifford Algebras (AACA), vol. 20, no. 2, pp. 271–284, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  12. E. M. Hitzer, “Quaternion Fourier transform on quaternion fields and generalizations,” Advances in Applied Clifford Algebras (AACA), vol. 20, no. 3, pp. 497–517, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  13. E. Hitzer, “General two-sided quaternion fourier transform, convolution and mustard convolution,” Advances in Applied Clifford Algebras (AACA), vol. 27, no. 1, pp. 381–395, 2017. View at Publisher · View at Google Scholar · View at MathSciNet
  14. M. Fei, Y. Xu, and J. Yan, “Real Paley-Wiener theorem for the quaternion Fourier transform,” Complex Variables and Elliptic Equations. An International Journal, vol. 62, no. 8, pp. 1072–1080, 2017. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  15. Y. Yang, P. Dang, and T. Qian, “Tighter uncertainty principles based on quaternion Fourier transform,” Advances in Applied Clifford Algebras (AACA), vol. 26, no. 1, pp. 479–497, 2016. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  16. T. A. Ell, “Quaternion-fourier transforms for analysis of two-dimensional linear time-invariant partial differential systems,” in Proceeding of the 32nd Conference on Decision and Control, IEEE Control Systems Society, pp. 1830–1841, December 1993. View at Scopus
  17. A. M. Grigoryan and S. S. Agaian, “Tensor transform-based quaternion fourier transform algorithm,” Information Sciences, vol. 320, pp. 62–74, 2015. View at Publisher · View at Google Scholar · View at MathSciNet
  18. V. R. Dubey, “Quaternion Fourier transform for colour images,” International Journal of Computer Science and Information Technologies, vol. 5, no. 3, pp. 4411–4416, 2014. View at Google Scholar
  19. S. Sangwine and T. Ell, “Hypercomplex Fourier transforms of color images,” IEEE Transactions on Image Processing, vol. 16, no. 1, pp. 22–35, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  20. M. Bahri, Zulfajar, and R. Ashino, “Convolution and correlation theorem for linear canonical transform and properties,” Information, vol. 17, no. 6B, pp. 2509–2521, 2014. View at Google Scholar
  21. D. Wei, Q. Ran, and Y. Li, “A convolution and correlation theorem for the linear canonical transform and its application,” Circuits, Systems and Signal Processing, vol. 31, no. 1, pp. 301–312, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  22. S. Xu, L. Feng, Y. Chai, Y. Hu, and L. Huang, “The properties of generalized offset linear canonical Hilbert transform and its applications,” International Journal of Wavelets, Multiresolution and Information Processing, vol. 15, no. 4, Article ID 1750031, 21 pages, 2017. View at Publisher · View at Google Scholar · View at MathSciNet
  23. M. Bahri and R. Ashino, “Logarithmic uncertainty principle for quaternion linear canonical transform,” in Proceedings of the 2016 International Conference on Wavelet Analysis and Pattern Recognition, ICWAPR 2016, pp. 140–145, South Korea, July 2016. View at Publisher · View at Google Scholar · View at Scopus
  24. K. I. Kou, J.-Y. Ou, and J. Morais, “On uncertainty principle for quaternionic linear canonical transform,” Abstract and Applied Analysis, vol. 2013, Article ID 725952, 14 pages, 2013. View at Publisher · View at Google Scholar · View at Scopus
  25. X.-X. Hu and K. I. Kou, “Quaternion Fourier and linear canonical inversion theorems,” Mathematical Methods in the Applied Sciences, vol. 40, no. 7, pp. 2421–2440, 2017. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  26. M. Bahri and R. Ashino, “A simplified proof of uncertainty principle for quaternion linear canonical transform,” Abstract and Applied Analysis, Article ID 5874930, Art. ID 5874930, 11 pages, 2016. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  27. K. I. Kou, J. Ou, and J. Morais, “Uncertainty principles associated with quaternionic linear canonical transforms,” Mathematical Methods in the Applied Sciences, vol. 39, no. 10, pp. 2722–2736, 2016. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  28. T. Bülow, Hypercomplex Spectral Signal Representations for the Processing and Analysis of Images [PhD Thesis], University of Kiel, Germany, 1999.
  29. M. Bahri, “A modified uncertainty principle for two-sided quaternion Fourier transform,” Advances in Applied Clifford Algebras (AACA), vol. 26, no. 2, pp. 513–527, 2016. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  30. Y.-N. Zhang and B.-Z. Li, “Novel uncertainty principles for two-sided quaternion linear canonical transform,” Advances in Applied Clifford Algebras (AACA), vol. 28, no. 15, 2018. View at Publisher · View at Google Scholar · View at MathSciNet