Computational implementation of the inverse continuous wavelet transform without a requirement of the admissibility condition

TL;DR

This paper explores a computational method for the inverse continuous wavelet transform using non-admissible kernels, enabling exact reconstruction without the traditional admissibility condition, with practical applications in signal processing.

Contribution

It introduces a simplified computational approach for inverse wavelet transform with non-admissible kernels, focusing on real-valued finite samples and Morlet wavelets.

Findings

Exact inverse transform possible with non-admissible wavelets

Simplified reconstruction formula for practical signals

Applications demonstrated on neuroscience signals

Abstract

Recently, it has been proven [R. Soc. Open Sci. 1 (2014) 140124] that the continuous wavelet transform with non-admissible kernels (approximate wavelets) allows for an existence of the exact inverse transform. Here we consider the computational possibility for the realization of this approach. We provide modified simpler explanation of the reconstruction formula, restricted on the practical case of real valued finite (or periodic/periodized) samples and the standard (restricted) Morlet wavelet as a practically important example of an approximate wavelet. The provided examples of applications includes the test function and the non-stationary electro-physical signals arising in the problem of neuroscience.