Algorithm for constructing symmetric dual framelet filter banks

Abstract

Abstract. Dual wavelet frames and their associated dual framelet filter banks are often constructed using the oblique extension principle. In comparison with the construction of tight wavelet frames and tight framelet filter banks, it is indeed quite easy to obtain some particular examples of dual framelet filter banks with or without symmetry from any given pair of low-pass filters. However, such constructed dual framelet filter banks are often too particular to have some desirable properties such as balanced filter supports between primal and dual filters. From the point of view of both theory and application, it is important and interesting to have an algorithm which is capable of finding all possible dual framelet filter banks with symmetry and with the shortest possible filter supports from any given pair of low-pass filters with symmetry. However, to our best knowledge, this issue has not been resolved yet in the literature and one often has to solve systems of nonlinear equations to obtain nontrivial dual framelet filter banks. Given the fact that the construction of dual framelet filter banks is widely believed to be very flexible, the lack of a systematic algorithm for constructing all dual framelet filter banks in the literature is a little bit surprising to us. In this paper, by solving only small systems of linear equations, we shall completely settle this problem by introducing a step-by-step efficient algorithm to construct all possible dual framelet filter banks with or without symmetry and with the shortest possible filter supports. As a byproduct, our algorithm leads to a simple algorithm for constructing all symmetric tight framelet filter banks with two high-pass filters from a given low-pass filter with symmetry. Examples will be provided to illustrate our algorithm. To explain and to understand better our algorithm and dual framelet filter banks, we shall also discuss some properties of our algorithms and dual framelet filter banks in this paper.

Duality for Frames

“…Moreover, biorthogonal dual wavelets are easier to construct than orthonormal wavelet bases, since one "only" has to factor a polynomial rather than finding the square root of a polynomial. Several one dimensional dual wavelet frames have been constructed in [31,37,40,41,66,74]. The construction of multivariate dual frames, similar to the multivariate tight frame construction, becomes increasingly difficult, since it involves the completion of two matrices with polynomial entries.…”

Quasi-tight framelets with high vanishing moments derived from arbitrary refinable functions

“…and φ(ξ) := ∞ j=1 a(2 −j ξ), ψ 1 (ξ) := b 1 (ξ/2) φ(ξ/2), and ψ 2 (ξ) := b 2 (ξ/2) φ(ξ/2) with φ, ψ 1 , ψ 2 ∈ L 2 (R). The above example in [20, Example 3.2.2] was accidentally obtained by applying the general algorithm developed in [19] for constructing dual framelet filter banks to the above low-pass filter a.…”

Generalized matrix spectral factorization and quasi-tight framelets with a minimum number of generators

“…(1.16) That is, the highest possible order of vanishing moments achieved by a quasi-tight framelet filter bank derived from given filters a, Θ ∈ l 0 (Z) is min(sr(a), 1 2 vm(Θ(z) − Θ(z 2 )a(z)a ⋆ (z))). As demonstrated in [17,Theorem 7] and [18,Theorem 1.4.7], for general filters a, Θ ∈ l 0 (Z), det(M a,Θ (z)) is often not identically zero and the minimum number s of high-pass filters in a quasitight framelet filter bank is at least 2. Given a Laurent polynomial p(z), for simplicity, we use p(z) ≡ 0 (p(z) ≡ 0) to indicate that p(z) is (is not) identically zero.…”