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Convolution theorems for quaternion Fourier transform: properties and applications. (English) Zbl 1297.42015

Summary: General convolution theorems for two-dimensional quaternion Fourier transforms (QFTs) are presented. It is shown that these theorems are valid not only for real-valued functions but also for quaternion-valued functions. We describe some useful properties of generalized convolutions and compare them with the convolution theorems of the classical Fourier transform. We finally apply the obtained results to study hypoellipticity and to solve the heat equation in quaternion algebra framework.

MSC:

42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
42B37 Harmonic analysis and PDEs
16H05 Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.)
35H10 Hypoelliptic equations
35K05 Heat equation
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[1] Ebling, J.; Scheuermann, G., Clifford Fourier transform on vector fields, IEEE Transactionson Visualization and Computer Graphics, 11, 4, 469-479 (2005) · doi:10.1109/TVCG.2005.54
[2] Hitzer, E.; Mawardi, B., Clifford Fourier transform on multivector fields and uncertainty principles for dimensions \(n = 2(\text{mod} 4)\) and \(n = 3(\text{mod} 4)\), Advances in Applied Clifford Algebras, 18, 3-4, 715-736 (2008) · Zbl 1177.15029 · doi:10.1007/s00006-008-0098-3
[3] Mawardi, B.; Hitzer, E., Clifford Fourier transformation and uncertainty principle for the Clifford geometric algebra \(C l_{3, 0}\), Advances in Applied Clifford Algebras, 16, 1, 41-61 (2006) · Zbl 1133.42305 · doi:10.1007/s00006-006-0003-x
[4] de Bie, H.; de Schepper, N., Fractional Fourier transform of hypercomplex signals, Signal, Image and Video Processing, 6, 3, 381-388 (2012) · doi:10.1007/s11760-012-0315-3
[5] Guanlei, X.; Xiaotong, W.; Xiagang, X., Fractional quaternion Fourier transform, convolutionand correlation, Signal Processing, 88, 10, 2511-2517 (2008) · Zbl 1151.94364 · doi:10.1016/j.sigpro.2008.04.012
[6] Bahri, M.; Hitzer, E.; Ashino, R.; Vaillancourt, R., Windowed Fourier transform of two-dimensional quaternionic signals, Applied Mathematics and Computation, 216, 8, 2366-2379 (2010) · Zbl 1196.42009 · doi:10.1016/j.amc.2010.03.082
[7] Bahri, M.; Ashino, R.; Vaillancourt, R., Two-dimensional quaternion wavelet transform, Applied Mathematics and Computation, 218, 1, 10-21 (2011) · Zbl 1232.65192 · doi:10.1016/j.amc.2011.05.030
[8] Bahri, M., Quaternion algebra-valued wavelet transform, Applied Mathematical Sciences, 5, 71, 3531-3540 (2011) · Zbl 1247.42032
[9] Sangwine, S. J., Biquaternion (complexified quaternion) roots of \(- 1\), Advances in Applied Clifford Algebras, 16, 1, 63-68 (2006) · Zbl 1107.30038 · doi:10.1007/s00006-006-0005-8
[10] Bülow, T., Hypercomplex spectral signal representations for the processing and analysis of images [Ph.D. thesis] (1999), Kiel, Germany: University of Kiel, Kiel, Germany
[11] Ell, T. A., Quaternion-Fourier transformations for analysis of two-dimensional lineartime-invariant partial differential systems, Proceedings of the 32nd IEEE Conference on Decision and Control
[12] Hitzer, E., Quaternion Fourier transform on quaternion fields and generalizations, Advances in Applied Clifford Algebras, 17, 3, 497-517 (2007) · Zbl 1143.42006 · doi:10.1007/s00006-007-0037-8
[13] Bahri, M.; Hitzer, E.; Hayashi, A.; Ashino, R., An uncertainty principle for quaternion Fourier transform, Computers & Mathematics with Applications, 56, 9, 2411-2417 (2008) · Zbl 1165.42310 · doi:10.1016/j.camwa.2008.05.032
[14] Assefa, D.; Mansinha, L.; Tiampo, K. F.; Rasmussen, H.; Abdella, K., Local quaternion Fourier transform and color images texture analysis, Signal Processing, 90, 6, 1825-1835 (2010) · Zbl 1197.94021 · doi:10.1016/j.sigpro.2009.11.031
[15] Bujack, R.; Scheuermann, G.; Hitzer, E., A general geometric fourier transform convolution theorem, Advances in Applied Clifford Algebras, 23, 1, 15-38 (2013) · Zbl 1263.15021 · doi:10.1007/s00006-012-0338-4
[16] Bujack, R.; Scheuermann, G.; Hitzer, E.; Guerlebeck, K., A general geometric Fourier transform, Proceedings of the 9th International Conference on Clifford Algebras and Their Applications in Mathematical Physics (ICCA ’11) · Zbl 1263.15021 · doi:10.1007/s00006-012-0338-4
[17] Hitzer, E., Directional uncertainty principle for quaternion Fourier transform, Advances in Applied Clifford Algebras, 20, 2, 271-284 (2010) · Zbl 1198.42006 · doi:10.1007/s00006-009-0175-2
[18] Folland, G. B., Real Analysis: Modern Techniques and Their Applications. Real Analysis: Modern Techniques and Their Applications, Pure and Applied Mathematics (1999), New York, NY, USA: John Wiley & Sons, New York, NY, USA · Zbl 0924.28001
[19] Ell, T. A.; Sangwine, S. J., Hypercomplex Fourier transforms of color images, IEEE Transactions on Image Processing, 16, 1, 22-35 (2007) · Zbl 1279.94014 · doi:10.1109/TIP.2006.884955
[20] Mallat, S., A Wavelet Tour of Signal Processing (1999), San Diego, Calif, USA: Academic Press, San Diego, Calif, USA · Zbl 0945.68537
[21] Petrou, M.; Petrou, C., Image Processing: The Fundamentals (2010), West Sussex, UK: John Wiley & Sons, West Sussex, UK · Zbl 1191.68792
[22] Hörmander, L., The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis (1983), Berlin, Germany: Springer, Berlin, Germany · Zbl 0521.35001 · doi:10.1007/978-3-642-96750-4
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