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Two-dimensional quaternion wavelet transform. (English) Zbl 1232.65192

Summary: We introduce the continuous quaternion wavelet transform (CQWT). We express the admissibility condition in terms of the (right-sided) quaternion Fourier transform. We show that its fundamental properties, such as inner product, norm relation, and inversion formula, can be established whenever the quaternion wavelets satisfy a particular admissibility condition. We present several examples of the CQWT. As an application we derive a Heisenberg type uncertainty principle for these extended wavelets.

MSC:

65T50 Numerical methods for discrete and fast Fourier transforms
65T60 Numerical methods for wavelets
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[1] T.A Ell, Quaternionic-Fourier transforms for analysis of two-dimensional linear time-invariant partial differential systems, in: Proceedings of 32nd IEEE Conference on Decision and Control, pp. 148-1841, San Antonio, TX, 1993.; T.A Ell, Quaternionic-Fourier transforms for analysis of two-dimensional linear time-invariant partial differential systems, in: Proceedings of 32nd IEEE Conference on Decision and Control, pp. 148-1841, San Antonio, TX, 1993.
[2] Pei, S. C.; Ding, J. J.; Chang, J. H., Efficient implementation of quaternion Fourier transform, convolution, and correlation by 2-D complex FFT, IEEE Trans. Signal Process, 49, 11, 2783-2797 (2001) · Zbl 1369.65190
[3] Antoine, J. P.; Murenzi, R., Two-dimensional directional wavelet and the scale-angle representation, Signal Process., 52, 3, 259-281 (1996) · Zbl 0875.94074
[4] Antoine, J. P.; Vandergheynst, P., Two-dimensional directional wavelet in imaging processing, Int. J. Imag. Syst. Technol., 7, 3, 152-165 (1996)
[5] Gröchenig, K., Foundation of Time-Frequency Analysis (2001), Birkhäuser: Birkhäuser Boston
[6] Traversoni, L., Imaging analysis using quaternion wavelet, in geometric algebra with applications, (Corrochano, E. B.; Sobczyk, G., Science and Engineering (2001), Birkhäuser: Birkhäuser Boston)
[7] Hitzer, E., Quaternion Fourier transform on quaternion fields and generalizations, Adv. Appl. Clifford Algebr., 17, 3, 497-517 (2007) · Zbl 1143.42006
[8] Hitzer, E.; Mawardi, B., Clifford Fourier transform on multivector fields and uncertainty principle for dimensions \(n=2\) (mod 4) and \(n=3\) (mod 4), Adv. Appl. Clifford Algebr., 18, 3-4, 715-736 (2008) · Zbl 1177.15029
[9] Brackx, F.; Delange, R.; Sommen, F., Clifford-Hermite wavelets in Euclidean space, J. Fourier Anal. Appl., 8, 3, 299-310 (2000) · Zbl 0961.42021
[10] Brackx, F.; Sommen, F., Benchmarking of three-dimensional Clifford wavelet functions, Complex Variables: Theory and Applications, 47, 7, 577-588 (2002) · Zbl 1041.42026
[11] Bayro-Corrochano, E., The theory and use of the quaternion wavelet transform, J. Math. Imag. Vision, 24, 1, 19-35 (2006) · Zbl 1478.94010
[12] Zhou, J.; Xu, Y.; Yang, X., Quaternion wavelet phase based stereo matching for uncalibrated images, Pattern Recogn. Lett., 28, 12, 1509-1522 (2007)
[13] Mallat, S., A Wavelet Tour of Signal Processing (1999), Academic Press: Academic Press San Diego, CA · Zbl 0998.94510
[14] Mawardi, B.; Hitzer, E.; Hayashi, A.; Ashino, R., An uncertainty principle for quaternion Fourier transform, Comput. Math. Appl., 56, 9, 2411-2417 (2008)
[15] Mawardi, B.; Hitzer, E.; Ashino, R.; Vaillancourt, R., Windowed Fourier transform of two-dimensional quaternionic signals, App. Math. Comput., 216, 8, 2366-2379 (2010) · Zbl 1196.42009
[16] Mawardi, B.; Adji, S.; Zhao, J., Clifford algebra-valued wavelet transform on multivector fields, Adv. Appl. Clifford Algebr., 21, 1, 13-30 (2011) · Zbl 1217.15032
[17] Mawardi, B.; Hitzer, E., Clifford algebra \(Cl}_{3,0} \)-valued wavelet transformation, Clifford wavelet uncertainty inequality and Clifford Gabor wavelets, Int. J. Wavelets Multiresolut. Inf. Process., 5, 6, 997-1019 (2007) · Zbl 1197.42019
[18] Debnath, L., Wavelet Transforms and Their Applications (2002), Birkhäuser: Birkhäuser Boston · Zbl 1019.94003
[19] He, J. X., Continuous wavelet transform on the space \(L^2(R, H \text{;} dx)\), Appl. Math. Lett., 17, 1, 111-121 (2001)
[20] Weyl, H., The Theory of Groups and Quantum Mechanics (1950), Dover: Dover New York · Zbl 0041.25401
[21] Zhao, J.; Peng, L., Quaternion-valued admissible wavelets associated with the 2-dimensional Euclidean group with dilations, J. Nat. Geom., 20, 1, 21-32 (2001) · Zbl 1020.42025
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