Hostname: page-component-76fb5796d-dfsvx Total loading time: 0 Render date: 2024-04-25T14:44:07.285Z Has data issue: false hasContentIssue false
  • English
  • Français

Continuous Inversion Formulas for Multi-Dimensional Stockwell Transforms

Published online by Cambridge University Press:  28 January 2013

L. Riba  and
M W. Wong
L. Riba
Affiliation:
Dipartimento di Matematica, Università di Torino, via Carlo Alberto 10, 10123 Torino, Italy
M W. Wong*
Affiliation:
Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, Ontario M3J 1P3, Canada
*
∗Corresponding author. E-mail: mwwong@mathstat.yorku.ca, This research has been supported by a discovery grant from the Natural Sciences and Engineering Research Council of Canada.
Get access

Abstract

Stockwell transforms as hybrids of Gabor transforms and wavelet transforms have been studied extensively. We introduce in this paper multi-dimensional Stockwell transforms that include multi-dimensional Gabor transforms as special cases. Continuous inversion formulas for multi-dimensional Stockwell transforms are proved.

Keywords

Gabor transformswavelet transformsStockwell transformscontinuous inversion formulas
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 8 , Issue 1: Harmonic analysis , 2013 , pp. 215 - 229
Copyright
© EDP Sciences, 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

J.-P. Antoine, R. Murenzi, P. Vandergheynst, S. T. Ali. Two-Dimensional Wavelets and their Relatives. Cambridge University Press, 2004.
L. Cohen. Time-Frequency Analysis. Prentice Hall, 1995.
Cordero, E., De Mari, F., Nowak, K., Tabacco, A.. Analytic features of reproducing groups for the metaplectic representation. J. Fourier Anal. Appl. 12 (2006), 157–180. CrossRefGoogle Scholar
Cordoba, A., Fefferman, C.. Wave packets and Fourier integral operators. Comm. Partial Differential Equations 3 (1978), 979–1005. CrossRefGoogle Scholar
I. Daubechies. Ten Lectures on Wavelets. SIAM, 1992.
Eramian, M. G., Schincariol, R. A., Mansinha, L., Stockwell, R. G.. Generation of aquifer heterogeneity maps using two dimensional spectral texture segmentation techniques. Math. Geology 31 (1999), 327–348. CrossRefGoogle Scholar
G. B. Folland. Harmonic Analysis in Phase Space. Princeton University Press, 1989.
Goodyear, B. G., Zhu, H., Brown, R. A., Mitchell, J. R.. Removal of phase artifacts from fMRI data using a Stockwell transform filter improves brain activity detection. Magn. Reson. Med. 51 (2004), 16–21. CrossRefGoogle ScholarPubMed
Grossmann, A., Morlet, J.. Decomposition of Hardy functions into square integrable wavelets of constant shape SIAM J. Math. Anal. 15 (1984), 723–736. CrossRefGoogle Scholar
Q. Guo, S. Molahajloo, M. W. Wong. Modified Stockwell transforms and time-frequency analysis in New Developments in Pseudo-Differential Operators. Operator Theory : Advances and Applications 189, Birkhäuser, 2009, 275–285.
Guo, Q., Wong, M. W.. Modified Stockwell transforms, Memorie della Accademia delle Scienze di Torino, Classe di Scienze, Fische. Matematiche e Naturali, Serie V, Vol. 32 (2008), 3–20. Google Scholar
Y. Liu and M. W. Wong. Inversion formulas for two-dimensional Stockwell transforms, in Pseudo-Differential Operators : Partial Differential Equations and Time-Frequency Analysis. Fields Institute Communications 52, American Mathematical Society, 2007, 323–330.
Stockwell, R. G., Mansinha, L., Lowe, R. P.. Localization of the complex spectrum : the S transform. IEEE Trans. Signal Processing 44 (1996), 998–1001. CrossRefGoogle Scholar
Bernier, D., Taylor, K. F.. Wavelets from square-integrable representations. SIAM J. Math. Anal. 27 (1996), 594–608. CrossRefGoogle Scholar
M. W. Wong. Weyl Transforms. Springer, 1998.
M. W. Wong. Wavelet Transforms and Localization Operators. Birkhäuser, 2002.
M. W. Wong, H. Zhu. A characterization of the Stockwell spectrum, in Modern Trends in Pseudo-Differential Operators. Birkhäuser, 2007, 251–257.
Zhu, H., Goodyear, B. G., Lauzon, M. L., Brown, R. A., Mayer, G. S., Mansinha, L., Law, A. G., Mitchell, J. R.. A new multiscale Fourier analysis for MRI. Med. Phys. 30 (2003), 1134–1141. CrossRefGoogle Scholar