Eugene B. Postnikov, Kursk State University, Russia
Anastasia I. Lavrova, Immanuel Kant Baltic Federal University, Russia
Elena A. Lebedeva, Saint-Petersburg State University & Saint-Petersburg State Polytechnic University, Russia
The work presents a review of recently emerging approaches to the continuous wavelet transform (CWT) with the Morlet wavelet, its inversion and the extraction and reconstruction of non-stationary oscillating patterns from 1D (time series) and 2D (series of images) signals. The principal approach is based on the application of differential operators instead of integrals with non-localized kernels and allows for overriding difficulties connected with a violation of the admissibility condition, which occurs for the Morlet wavelet.
As an area of application, the data of neurophysiological records are considered. Among them, there are the sequences of images displaying a firing of hippocampal grid cells and very fast non-stationary electrocorticographic oscillations. The developed methods provide the tool for a more accurate spatiotemporal localization of periodic structures in comparison with conventional methods.
 A.I. Lavrova, E.B. Postnikov Wavelet analysis of location and intensity of spatial rhythms in hippocampus. AIP Conf. Proc. 1558 (2013) 715-718.
 E.B. Postnikov, V.K. Singh Local spectral analysis of images via the wavelet transform based on partial differential equations. Multidim. Syst. Sign. P., 25 (2014) 145-155.
 E.A. Lebedeva, E.B. Postnikov, On alternative wavelet reconstruction formula: a case study of approximate wavelets. R. Soc. Open Sci., 1 (2014), 140124.
Prof. Eugene Postnikov
Kursk State University
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