On optimal wavelet bases for the realization of microcanonical cascade processes

TL;DR

This paper introduces an efficient formalism to identify optimal wavelet bases for representing multiplicative cascade processes, particularly in turbulent flows, making the cascade relation explicit at each point.

Contribution

The authors propose a new formalism that simplifies finding optimal wavelet bases for cascade processes, reducing data requirements and computational complexity.

Findings

Validated on synthetic signals with known optimal bases

Approach is effective within specific wavelet families

Potential for generalization to continuous search schemes

Abstract

Multiplicative cascades are often used to represent the structure of multiscaling variables in many physical systems, specially turbulent flows. In processes of this kind, these variables can be understood as the result of a successive tranfer in cascade from large to small scales. For a given signal, only its optimal wavelet basis can represent it in such a way that the cascade relation between scales becomes explicit, i.e., it is geometrically achieved at each point of the system. Finding such a basis is a data-demanding, highly-complex task. In this paper we propose a formalism that allows to find the optimal wavelet in an efficient, less data-demanding way. We confirm the appropriateness of this approach by analyzing the results on synthetic signals constructed with prescribed optimal bases. Although we show the validity of our approach constrainted to given families of wavelets, it can be generalized for a continuous unconstrainted search scheme.