Computational implementation of the inverse continuous wavelet transform without a requirement of the admissibility condition
For researchers using wavelet transforms, this work provides a practical computational method to invert transforms with approximate wavelets, though it is an incremental extension of existing theory.
The paper presents a computational implementation of the inverse continuous wavelet transform using non-admissible (approximate) wavelets, avoiding the admissibility condition. It demonstrates the method on test functions and real neuroscience signals.
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.