Homogeneous Wavelets and Framelets with the Refinable Structure

TL;DR

This paper explores the relationship between homogeneous and nonhomogeneous wavelets and framelets, emphasizing their connection to refinable structures and multiresolution analysis to deepen understanding of their properties.

Contribution

It provides a comprehensive study linking homogeneous wavelets and framelets to nonhomogeneous ones with refinable structures, enhancing theoretical understanding.

Findings

Clarifies the connection between homogeneous and nonhomogeneous wavelets

Shows how refinable structures underpin wavelet and framelet properties

Improves understanding of multiresolution analysis in wavelet theory

Abstract

Homogeneous wavelets and framelets have been extensively investigated in the classical theory of wavelets and they are often constructed from refinable functions via the multiresolution analysis. On the other hand, nonhomogeneous wavelets and framelets enjoy many desirable theoretical properties and are often intrinsically linked to the refinable structure and multiresolution analysis. In this paper we shall provide a comprehensive study on connecting homogeneous wavelets and framelets to nonhomogeneous ones with the refinable structure. This allows us to understand better the structure of homogeneous wavelets and framelets as well as their connections to the refinable structure and multiresolution analysis.