On Inversion of Continuous Wavelet Transform

Lintao Liu¹, Xiaoqing Su², Guocheng Wang¹

¹State Key Laboratory of Geodesy and Earth’s Dynamics, Institute of Geodesy and Geophysics, Chinese Academy of Sciences, Wuhan, China.
²Shandong University of Technology, Zibo, China.

DOI: 10.4236/ojs.2015.57071                

Abstract

This study deduces a general inversion of continuous wavelet transform (CWT) with timescale being real rather than positive. In conventional CWT inversion, wavelet’s dual is assumed to be a reconstruction wavelet or a localized function. This study finds that wavelet’s dual can be a harmonic which is not local. This finding leads to new CWT inversion formulas. It also justifies the concept of normal wavelet transform which is useful in time-frequency analysis and time-frequency filtering. This study also proves a law for CWT inversion: either wavelet or its dual must integrate to zero.

Keywords

Continuous Wavelet Transform, Wavelet’s Dual, Inversion, Normal Wavelet Transform, Time-Frequency Filtering

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	http://scienceworld.wolfram.com/biography/Zweig.html
[2]	Goupillaud, P., Grossman, A. and Morlet, J. (1984) Cycle-Octave and Related Transforms in Seismic Signal Analysis. Geoexploration, 23, 85-102. http://dx.doi.org/10.1016/0016-7142(84)90025-5
[3]	Daubechies, I. (1992) Ten Lectures on Wavelets. SIAM, Philadelphia. http://dx.doi.org/10.1137/1.9781611970104
[4]	Chui, C.K. (1992) An Introduction to Wavelets (Wavelet Analysis & Its Applications). Academic Press, Waltham.
[5]	Mallat, S. (1998) A Wavelet Tour of Signal Processing. The Sparse Way.
[6]	Holschneider, M. (1995) Wavelet: An Analysis Tool. Clarendon Press, Oxford.
[7]	Gross, R.S. (1992) Correspondence between Theory and Observations of Polar Motion. Geophysical Journal International, 109, 162-170. http://dx.doi.org/10.1111/j.1365-246X.1992.tb00086.x
[8]	Su, X.Q., Liu, L.T., Hsu, H. and, Wang, G.C. (2014) Long-Term Polar Motion Prediction Using Normal Time-Frequency Transform. Journal of Geodesy, 88, 145-155. http://dx.doi.org/10.1007/s00190-013-0675-7
[9]	Liu, L.T. and, Hsu, H. (2012) Inversion and Normalization of Time-Frequency Transform. Applied Mathematics & Information Sciences, 6, 67-74.
[10]	Stockwell, R.G., Mansinha, L. and Lowe, R.P. (1996) Localization of the Complex Spectrum: The S-Transform. IEEE Transactions on Signal Processing, 44, 998-1001. http://dx.doi.org/10.1109/78.492555
[11]	Pinnegar, C. and Mansinha, L. (2003) The S-Transform with Windows of Arbitrary and Varying Shape. Geophysics, 68, 381-385. http://dx.doi.org/10.1190/1.1543223