Bell-shaped nonstationary refinable ripplets

TL;DR

This paper investigates nonstationary refinable ripplets, demonstrating their bell-shaped nature, constructing associated prewavelets and biorthogonal bases, and showcasing their effectiveness in analyzing spike-like signals.

Contribution

It introduces a comprehensive framework for nonstationary refinable ripplets, including their properties, construction of prewavelets, biorthogonal bases, and efficient algorithms for signal analysis.

Findings

Ripplets satisfy Strang-Fix conditions and are bell-shaped.

Constructed nonstationary prewavelets and biorthogonal bases.

Demonstrated good performance in spike-like signal analysis.

Abstract

We study the approximation properties of the class of nonstationary refinable ripplets introduced in \cite{GP08}. These functions are solution of an infinite set of nonstationary refinable equations and are defined through sequences of scaling masks that have an explicit expression. Moreover, they are variation-diminishing and highly localized in the scale-time plane, properties that make them particularly attractive in applications. Here, we prove that they enjoy Strang-Fix conditions and convolution and differentiation rules and that they are bell-shaped. Then, we construct the corresponding minimally supported nonstationary prewavelets and give an iterative algorithm to evaluate the prewavelet masks. Finally, we give a procedure to construct the associated nonstationary biorthogonal bases and filters to be used in efficient decomposition and reconstruction algorithms. As an example, we calculate the prewavelet masks and the nonstationary biorthogonal filter pairs corresponding to the $C^2$ nonstationary scaling functions in the class and construct the corresponding prewavelets and biorthogonal bases. A simple test showing their good performances in the analysis of a spike-like signal is also presented. Keywords: total positivity, variation-dimishing, refinable ripplet, bell-shaped function, nonstationary prewavelet, nonstationary biorthogonal basis