Design of Wavelet Filter Banks for Any Dilation Using Extended Laplacian Pyramid Matrices
Youngmi Hur, Sungjoo Kim
TL;DR
The paper tackles the problem of designing multidimensional wavelet filter banks for arbitrary dilation matrices by introducing extended Laplacian pyramid matrices and the sum of vanishing products (SVP) condition. It establishes a deep link between SVP and the mixed unitary extension principle (MUEP), showing that SVP is equivalent to MUEP and enables flexible, constructive filter bank design beyond SOS-based factorization. Key contributions include the SVP framework, extended Laplacian pyramid matrices, and the equivalence theorems that unify SOS, UEP, and LP-based approaches, together with concrete examples in multiple dimensions. This design paradigm broadens the practical toolkit for building tight or quasi-tight multi-dimensional wavelet frames with arbitrary dilation, reducing reliance on hard SOS factorization.
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.