Construction of bivariate symmetric orthonormal wavelets with short support

TL;DR

This paper characterizes bivariate symmetric orthonormal wavelets with small support, introduces new tight frame functions, and proves limitations on vanishing moments for larger support functions.

Contribution

It provides a parameterization of symmetric orthonormal scaling functions with 6x6 support and explores the existence of tight frame functions and vanishing moments for larger supports.

Findings

Parameterization of 6x6 symmetric orthonormal scaling functions

Construction of tight frame refinable functions

Impossibility of multiple vanishing moments for 8x8 support

Abstract

In this paper, we give a parameterization of the class of bivariate symmetric orthonormal scaling functions with filter size $6\times 6$ using the standard dilation matrix 2I. In addition, we give two families of refinable functions which are not orthonormal but have associated tight frames. Finally, we show that the class of bivariate symmetric scaling functions with filter size $8\times 8$ can not have two or more vanishing moments.