On construction of multivariate symmetric MRA-based wavelets

TL;DR

This paper provides a comprehensive method for constructing symmetric multivariate wavelet systems with specified approximation order, applicable to various matrix dilations and symmetry groups, enhancing wavelet design for multidimensional data.

Contribution

It explicitly characterizes all symmetric masks with a given sum rule and constructs symmetric/antisymmetric wavelet functions for arbitrary matrix dilations.

Findings

Explicit formulas for symmetric masks with sum rule of order n.

Construction of frame-like wavelet systems with symmetry and approximation order n.

Application to matrix dilations compatible with axial symmetry groups.

Abstract

For an arbitrary matrix dilation, any integer n and any integer/semi-integer c, we describe all masks that are symmetric with respect to the point c and have sum rule of order n. For each such mask, we give explicit formulas for wavelet functions that are point symmetric/antisymmetric and generate frame-like wavelet system providing approximation order n. For any matrix dilations (which are appropriate for axial symmetry group on R^2 in some natural sense) and given integer n, axial symmetric/antisymmetric frame-like wavelet systems providing approximation order n are constructed. Also, for several matrix dilations the explicit construction of highly symmetric frame-like wavelet systems providing approximation order n is presented.