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.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have