Splitting a Matrix of Laurent Polynomials with Symmetry and itsApplication to Symmetric Framelet Filter Banks

Abstract

Let M be a 2 × 2 matrix of Laurent polynomials with real coefficients and symmetry. In this paper, we obtain a necessary and sufficient condition for the existence of four Laurent polynomials (or FIR filters) u 1 , u 2 , v 1 , v 2 with real coefficients and symmetry such that u 1 (z) v 1 (z) u 2 (z) v 2 (z) u 1 (1/z) u 2 (1/z) v 1 (1/z) v 2 (1/z) = M (z) ∀ z ∈ C\{0} and [Su 1 ](z)[Sv 2 ](z) = [Su 2 ](z)[Sv 1 ](z), where [Sp](z) = p(z)/p(1/z) for a nonzero Laurent polynomial p. Our criterion can be easily checked and a step-by-step algorithm will be given to construct the symmetric filters u 1 , u 2 , v 1 , v 2. As an application of this result to symmetric framelet filter banks, we present a necessary and sufficient condition for the construction of a symmetric MRA tight wavelet frame with two compactly supported generators derived from a given symmetric refinable function. Once such a necessary and sufficient condition is satisfied, an algorithm will be used to construct a symmetric framelet filter bank with two high-pass filters which is of interest in applications such as signal denoising and image processing. As an illustration of our results and algorithms in this paper, we give several examples of symmetric framelet filter banks with two high-pass filters which have good vanishing moments and are derived from various symmetric low-pass filters including some B-spline filters.

Matrix splitting with symmetry and symmetric tight framelet filter banks with two high-pass filters

“…Tight wavelet frames and tight framelet filter banks without symmetry have been extensively studied in a lot of papers, to mention only a few here, see [2,3,4,5,8,12,18,19,20] and many papers therein. Interesting examples of real-valued tight framelet filter banks with symmetry have been obtained in [3,5,6,13,14,15,16,17,18,21].…”

“…, b s } δ . This paper is largely motivated by two negative results presented in [13,17] on symmetric tight framelet filter banks and a positive result in [11]. [17,Corollary 2] shows that if a is a real-valued interpolatory filter (that is, a(0) = 1/2 and a(2k) = 0 for all k ∈ Z\{0}), except the trivial cases a(z) = 1/2 + (z 1−2m + z 2m−1 )/4 (which have no more than 2 sum rules) for some m ∈ N, it is impossible to obtain a real-valued tight framelet filter bank {a; b 1 , b 2 } with symmetry.…”

“…[17,Corollary 2] shows that if a is a real-valued interpolatory filter (that is, a(0) = 1/2 and a(2k) = 0 for all k ∈ Z\{0}), except the trivial cases a(z) = 1/2 + (z 1−2m + z 2m−1 )/4 (which have no more than 2 sum rules) for some m ∈ N, it is impossible to obtain a real-valued tight framelet filter bank {a; b 1 , b 2 } with symmetry. Later on, a general result on matrix splitting with symmetry has been established in [13,Theorem 2.3]. As an application, a necessary and sufficient condition for a real-valued tight framelet filter bank {a; b 1 , b 2 } Θ with symmetry has been obtained in [13,Theorem 2.4] and a detailed algorithm is given in [13,Algorithm 2.5] to derive the high-pass filters b 1 , b 2 with symmetry from the given filters a and Θ.…”

“…The oblique extension principle introduced in [3,5] is a general procedure to construct tight wavelet frames and their associated filter banks. Symmetric tight framelet filter banks with two high-pass filters have been studied in [13,16,17]. Tight framelet filter banks with or without symmetry have been constructed in [1]- [21] and references therein.…”

“…Tight framelet filter banks with or without symmetry have been constructed in [1]- [21] and references therein. This paper is largely motivated by several results in [11,13,17] to further study tight wavelet frames and their associated filter banks with symmetry and two high-pass filters. Our study not only leads to a simpler algorithm for the construction of tight framelet filter banks with symmetry and two high-pass filters, but also allows us to obtain a wider class of tight wavelet frames with symmetry which are not available in the current literature.…”