Symmetric Framelets

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].…”

“…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. Later on, a general result on matrix splitting with symmetry has been established in [13,Theorem 2.3].…”

“…Therefore, the technical GCD condition in [13,Theorem 2.4] cannot be dropped and it is impossible to obtain a real-valued tight framelet filter bank {a; b 1 , b 2 } Θ with symmetry from [13,Example 3.1]. We are largely motivated by the above two negative results in [13,17] on real-valued symmetric tight framelet filter banks to further explore this topic.…”

“…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.…”

“…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.…”