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

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 of d-dual low-pass filters, with , 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 , with , to be used as the second component of the initial filter system of the bottom–up d-dual chain: , 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 . Then the desired dual wavelet filter systems, associated with a and , are given by (b1,…,bd−1):=(b0,1,…,b0,d−1) and . 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 . 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.