Design of Wavelet Filter Banks for Any Dilation Using Extended Laplacian Pyramid Matrices

TL;DR

This paper introduces a novel method for designing wavelet filter banks adaptable to any dilation matrix and dimension, leveraging extended Laplacian pyramid matrices and a new sum of vanishing products condition.

Contribution

It generalizes wavelet frame construction by establishing the equivalence between the SVP condition and the mixed unitary extension principle, enabling flexible filter bank design.

Findings

Introduces the SVP condition for wavelet filter design

Establishes equivalence between SVP and MUEP conditions

Provides illustrative examples demonstrating the method

Abstract

In this paper, we present a new method for designing wavelet filter banks for any dilation matrices and in any dimension. Our approach utilizes extended Laplacian pyramid matrices to achieve this flexibility. By generalizing recent tight wavelet frame construction methods based on the sum of squares representation, we introduce the sum of vanishing products (SVP) condition, which is significantly easier to satisfy. These flexible design methods rely on our main results, which establish the equivalence between the SVP and mixed unitary extension principle conditions. Additionally, we provide illustrative examples to showcase our main findings.