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 (a,̃a) of d-dual lowpass filters, with (a ₀,a ₁) = (a,̃a), a top-down chain of filters a ₀ → a ₁ →.→ar = δ is constructed with consecutive d-dual pairs (a _j ,a _j+1), j = 1, . , r-1, and #(a1) > #(a ₂) > . > #(a _r ) = 1, where δ(0) = 1 and δ(k) = 0 for all k ε Z\{0}, and #(a _j ) denotes the number of filter taps of a _j . This enables the formulation of the filter system (a _r ; b _r,1, . , b _r,δ-1) =: (a _r ; b _r ), with b _r = [δ(.-1), . , δ(.-d + 1)], to be used as the second component of the initial filter system ((a _r-1; b _r-1), (a _r ; b _r )) of the bottom-up ddual chain: ((a _r-1; b _r-1), (a _r ; b _r ))→((a _r-2; b _r-2), (a _r-1; b _r-1))→.→((a ₀; b ₀), (a ₁; b ₁)), constructed bottom-up iteratively, from j =r to j = 0, by using both the d-duality property of (a _j ,a _j+1), j = 0, . , r-1 and the unimodular property of the polyphase Laurent polynomial matrix associated with the filter system (a _j ; b _j ). Then the desired dual wavelet filter systems, associated with a and ̃a, are given by (b ₁, . , b _d-1) := (b _0,1, . , b _0,d-1) and ( ̃b 1, . , ̃b _d-1) := (b 1,1, . , b 1, _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. © 2011 Elsevier Inc.