A dual-chain approach for bottom–up construction of wavelet filters with any integer dilation

Abstract

A dual-chain approach is introduced in this paper to construct dual wavelet filter systems with an arbitrary integer dilation d⩾2. Starting from a pair (a,a˜) of d-dual low-pass filters, with (a0,a1)=(a,a˜), a top–down chain of filters a0→a1→⋯→ar=δ is constructed with consecutive d-dual pairs (aj,aj+1), j=1,…,r−1, and #(a1)>#(a2)>⋯>#(ar)=1, where δ(0)=1 and δ(k)=0 for all k∈Z\\{0}, and #(aj) denotes the number of filter taps of aj. This enables the formulation of the filter system (ar;br,1,…,br,d−1)=:(ar;b→r), with b→r=[δ(⋅−1),…,δ(⋅−d+1)], to be used as the second component of the initial filter system ((ar−1;b→r−1),(ar;b→r)) of the bottom–up d-dual chain: ((ar−1;b→r−1),(ar;b→r))→((ar−2;b→r−2),(ar−1;b→r−1♯))→⋯→((a0;b→0),(a1;b→1♯)), constructed bottom–up iteratively, from j=r to j=0, by using both the d-duality property of (aj,aj+1), j=0,…,r−1 and the unimodular property of the polyphase Laurent polynomial matrix associated with the filter system (aj;b→j). Then the desired dual wavelet filter systems, associated with a and a˜, are given by (b1,…,bd−1):=(b0,1,…,b0,d−1) and (b˜1,…,b˜d−1):=(b1,1♯,…,b1,d−1♯). More importantly, the constructive algorithm for this dual-chain approach can be appropriately modified to preserve the symmetry property of the initial d-dual pair (a,a˜). For any dilation factor d, the dual-chain algorithms developed in this paper provide two systematic methods for the construction of both biorthogonal wavelets and bottom–up wavelets with or without the symmetry property.