Empirical wavelet frames

TL;DR

This paper investigates the theoretical foundations for constructing empirical wavelet frames, which are adaptable tools used in various fields, by establishing conditions for their existence based on recent developments.

Contribution

It provides the first theoretical conditions for building empirical wavelet frames from classic mother wavelets, expanding their applicability.

Findings

Established conditions for the existence of empirical wavelet frames

Extended the theoretical framework to multi-dimensional cases

Demonstrated the adaptability of empirical wavelets in practical applications

Abstract

Due to their adaptive nature, empirical wavelets had several successes in many fields from engineering, science, medical signal/image processing. Recently, a general theoretical framework has been developed in the one-dimensional case, showing the possibility to build empirical wavelets from any classic mother wavelets. Given extensive literature both in theory and applications of classic wavelet frames, it is legitimate to ask about the feasibility of building empirical wavelet frames. We address this question in this paper. We prove several results which provide conditions on the existence of empirical wavelet frames taking into account the above mentioned adaptability.