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

Research output: Journal Publications and Reviews (RGC: 21, 22, 62)21_Publication in refereed journalpeer-review

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## Detail(s)

Original language English 204-225 Applied and Computational Harmonic Analysis 33 2 Published - Sep 2012 Yes

## 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 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.

## Research Area(s)

• Biorthogonal wavelets, Bottom-up wavelet filter construction, CBC algorithm, Dual-chain method, Integer dilation, Lazy wavelets, Polyphase matrices, Symmetry property

## Citation Format(s)

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