From Frame-like Wavelets to Wavelet Frames keeping approximation properties and symmetry

TL;DR

This paper presents explicit methods for constructing symmetric wavelet frames with high approximation order, using lifting schemes and direct construction techniques, applicable to various symmetry groups.

Contribution

It introduces new explicit methods for constructing symmetric wavelet frames that preserve approximation order and symmetry properties, enhancing wavelet system design.

Findings

Constructed symmetric almost frame-like wavelet systems with approximation order n.

Developed a lifting scheme to convert almost frame-like wavelets into dual wavelet frames.

Provided techniques to ensure H-symmetry for wavelet functions across different methods.

Abstract

For a given symmetric refinable mask obeying the sum rule of order $n$, an explicit method is suggested for the construction of mutually symmetric almost frame-like wavelet system providing approximation order $n$. A transformation based on the lifting scheme is described that allows to improve almost frame-like wavelets to dual wavelet frames and preserve other properties. A direct method for the construction of dual wavelet frames providing approximation order $n$ and mutual symmetry properties is also discussed. For an abelian symmetry group $H$, a technique providing the $H$-symmetry property for each wavelet function is given for the above three methods.