A variation on uncertainty principle and logarithmic uncertainty principle for continuous quaternion wavelet transforms. (English) Zbl 1470.42062

Bahri, Mawardi; Ashino, Ryuichi

Abstr. Appl. Anal. 2017, Article ID 3795120, 11 p. (2017).

Summary: The continuous quaternion wavelet transform (CQWT) is a generalization of the classical continuous wavelet transform within the context of quaternion algebra. First of all, we show that the directional quaternion Fourier transform (QFT) uncertainty principle can be obtained using the component-wise QFT uncertainty principle. Based on this method, the directional QFT uncertainty principle using representation of polar coordinate form is easily derived. We derive a variation on uncertainty principle related to the QFT. We state that the CQWT of a quaternion function can be written in terms of the QFT and obtain a variation on uncertainty principle related to the CQWT. Finally, we apply the extended uncertainty principles and properties of the CQWT to establish logarithmic uncertainty principles related to generalized transform.

MSC:

42C40	Nontrigonometric harmonic analysis involving wavelets and other special systems
43A30	Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc.

References:

[1]	Chui, C. K., An Introduction to Wavelets, (1992), New York, NY, USA: Academic Press, New York, NY, USA · Zbl 0925.42016
[2]	Gröchenig, K., Foundation of Time-Frequency Analysis, (2001), Boston, Mass, USA: Birkhäuser, Boston, Mass, USA
[3]	He, J. X.; Yu, B., Continuous wavelet transforms on the space \(L^2\) (R, \(\mathbb{H}; d x\)), Applied Mathematics Letters, 17, 1, 111-121, (2001) · Zbl 1040.42031
[4]	Pathak, R. S., The Wavelet Transform, (2009), Amsterdam, The Netherlands: Atlantis Press, Amsterdam, The Netherlands · Zbl 1173.42328
[5]	Ali, S. T.; Thirulogasanthar, K., The quaternionic affine group and related continuous wavelet transforms on complex and quaternionic Hilbert spaces, Journal of Mathematical Physics, 55, 6, (2014) · Zbl 1305.42035
[6]	Bahri, M.; Ashino, R.; Vaillancourt, R., Continuous quaternion Fourier and wavelet transforms, International Journal of Wavelets, Multiresolution and Information Processing, 12, 4, (2014) · Zbl 1301.42015
[7]	Bahri, M.; Ashino, R.; Vaillancourt, R., Two-dimensional quaternion wavelet transform, Applied Mathematics and Computation, 218, 1, 10-21, (2011) · Zbl 1232.65192
[8]	Bayro-Corrochano, E., The theory and use of the quaternion wavelet transform, Journal of Mathematical Imaging and Vision, 24, 1, 19-35, (2006)
[9]	Xi, Y.; Yang, X.; Song, L.; Traversoni, L.; Lu, W.; Bayro-Corrochano, E.; Scheuermann, G., QWT: retrospective and new application, Applied Geometric Algebras in Computer Science and Engineering, 249-273, (2010), London, UK: Springer, London, UK · Zbl 1202.94086
[10]	Gai, S.; Yang, G.; Zhang, S., Multiscale texture classification using reduced quaternion wavelet transform, International Journal of Electronics and Communications, 67, 3, 233-241, (2013)
[11]	Akila, L.; Roopkumar, R., Ridgelet transform for quarternion-valued functions, International Journal of Wavelets, Multiresolution and Information Processing, 14, 1, (2016) · Zbl 1334.44004
[12]	Akila, L.; Roopkumar, R., Multidimensional quaternionic Gabor transforms, Advances in Applied Clifford Algebras, 26, 3, 985-1011, (2016) · Zbl 1350.42015
[13]	Hitzer, E. M., Quaternion Fourier transform on quaternion fields and generalizations, Advances in Applied Clifford Algebras, 17, 3, 497-517, (2007) · Zbl 1143.42006
[14]	Bahri, M.; Hitzer, E. S.; Hayashi, A.; Ashino, R., An uncertainty principle for quaternion Fourier transform, Computers and Mathematics with Applications, 56, 9, 2398-2410, (2008) · Zbl 1165.42310
[15]	Chen, L.-P.; Kou, K. I.; Liu, M.-S., Pitt’s inequality and the uncertainty principle associated with the quaternion Fourier transform, Journal of Mathematical Analysis and Applications, 423, 1, 681-700, (2015) · Zbl 1425.42012
[16]	Hitzer, E. M., Directional uncertainty principle for quaternion Fourier transform, Advances in Applied Clifford Algebras, 20, 2, 271-284, (2010) · Zbl 1198.42006
[17]	Bahri, M., A modified uncertainty principle for two-sided quaternion Fourier transform, Advances in Applied Clifford Algebras, 26, 2, 513-527, (2016) · Zbl 1342.42009
[18]	Yang, Y.; Dang, P.; Qian, T., Tighter uncertainty principles based on quaternion Fourier transform, Advances in Applied Clifford Algebras, 26, 1, 479-497, (2016) · Zbl 1338.42013
[19]	Bülow, T., Hypercomplex spectral signal representations for the processing and analysis of images [Ph.D. thesis], (1999), Kiel, Germany: University of Kiel, Kiel, Germany
[20]	Morais, J. P.; Georgiev, S.; Sprößig, W., Real Quaternionic Calculus Handbook, (2014), New York, NY, USA: Birkhäuser, New York, NY, USA · Zbl 1297.30002
[21]	Kou, K. I.; Morais, J., Asymptotic behaviour of the quaternion linear canonical transform and the Bochner-Minlos theorem, Applied Mathematics and Computation, 247, 15, 675-688, (2014) · Zbl 1338.43005
[22]	Weyl, H., The Theory of Groups and Quantum Mechanics, (1950), New York, NY, USA: Dover, New York, NY, USA · Zbl 0041.25401