Eugene B. Postnikov, Kursk State University, Russia
The talk presents a minireview of modern approaches to the data processing focused on the continuous wavelet transform with the Morlet wavelet. As a principal approach, which allows the effective and fast processing of signals and images, the diffusion smoothing will be considered. Since the Morlet wavelet consists of the product of the Gaussian and the imaginary exponential function, the corresponding integral transform can be decomposed into a set of periodically modulated patterns diffusionally smoothed at the next step.
One of the promising methods is the replacement of the Gaussians by cubic B-splines resembling their shape with a high accuracy. The fast method of their hierarchical generation has advantages, which can be listed as follows: i) the number of coefficients, which need to be stored growth practically linearly as a function of the processed sample’s length (instead of quadratic for conventional methods); ii) a size of decomposition matrices is simultaneously reduced in geometric progression for growing scales that result in the speed prevalence, which reaches 3÷6 times for the sample’s lengths of 512−2048.
Another possible approach utilizes the Gaussians (or even the Morlet wavelet as a whole) as the Green functions for a partial differential equation of the diffusion type. Since the diffusion operator is local within the finite difference representation, it allows on-the-fly processing of streaming data.
Finally, the properties of the directed 2D wavelets argues that image processing in this case can be reduced to the processing of appropriate 1D samples obtained via an appropriate rotation. In addition, it is shown that the usage of the differential operator allows an exact inversion of the transforms obtained with small central frequencies. This provides opportunities for extracting of sharp localized patterns. All these features will be illustrated by biophysical examples among which there are vascular blood flow pulsations, snail’s optical neuronal system, spatial patterning of neuronal oscillations.
Prof. Eugene Postnikov
Kursk State University
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