Abstract
In this paper, we investigate compactly supported symmetric orthonormal dyadic complex wavelets such that the symmetric orthonormal refinable functions have high linearphase moments and the antisymmetric wavelets have high vanishing moments. Such wavelets naturally lead to realvalued symmetric tight wavelet frames with some desirable moment properties, and are related to coiflets which are realvalued and are of interest in numerical algorithms. For any positive integer m, employing only the Riesz lemma without solving any nonlinear equations, we obtain a 2πperiodic trigonometric polynomial \(\hat a\) with complex coefficients such that

(i)
\(\hat a\) is an orthogonal mask: \(\hat a(\xi)^2+\hat a(\xi+\pi)^2=1\).

(ii)
\(\hat a\) has m + 1 − odd_{ m } sum rules: \(\hat a(\xi+\pi)=O(\xi^{m+1odd_m})\) as ξ→0, where \(odd_m:=\frac{1(1)^m}{2}\).

(iii)
\(\hat a\) has m + odd_{ m } linearphase moments: \(\hat a(\xi)=e^{{{\mathrm{i}}} c\xi}+O(\xi^{m+odd_m})\) as ξ→0 with phase c = − 1/2.

(iv)
\(\hat a\) has symmetry and coefficient support [2 − 2m,2m − 1]: \(\hat a(\xi)=\sum_{k=22m}^{2m1} h_k e^{{{\mathrm{i}}} k\xi}\) with h_{1 − k } = h_{ k }.

(v)
\(\hat a(\xi)\ne 0\) for all ξ ∈ ( − π,π).
Define \(\hat \phi(\xi):=\prod_{j=1}^\infty \hat a(2^{j}\xi)\) and \(\hat \psi(2\xi)=e^{{{\mathrm{i}}} \xi} {\overline{\hat a(\xi+\pi)}}\hat \phi(\xi)\). Then ψ is a compactly supported antisymmetric orthonormal wavelet with m + 1 − odd _{ m } vanishing moments, and ϕ is a compactly supported symmetric orthonormal refinable function with the special linearphase moments: \(\int_{{{\mathbb R}}} \phi(x)dx=1\) and \(\int_{{{\mathbb R}}} (x1/2)^j \phi(x) dx=0\) for all j = 1,...,m + odd _{ m } − 1. Both functions ϕ and ψ are supported on [2 − 2m,2m − 1].
The mask of a coiflet has real coefficients and satisfies (i), (ii), and (iii), often with a general phase c and the additional condition that the order of the linearphase moments is equal (or close) to the order of the sum rules. On the one hand, as Daubechies showed in [3, 5] that except the Haar wavelet, any compactly supported dyadic orthonormal realvalued wavelets including coiflets cannot have symmetry. On the other hand, solving nonlinear equations, [4, 12] constructed many interesting realvalued dyadic coiflets without symmetry. But it remains open whether there is a family of realvalued orthonormal wavelets such as coiflets whose masks can have arbitrarily high linearphase moments. This partially motivates this paper to study the complex wavelet case with symmetry property. Though symmetry can be achieved by considering complex wavelets, the symmetric Daubechies complex orthogonal masks in [11] generally have no more than 2 linearphase moments. In this paper, we shall study and construct orthonormal dyadic complex wavelets and masks with symmetry, linearphase moments, and sum rules. Examples and two general construction procedures for symmetric orthogonal masks with high linearphase moments and sum rules are given to illustrate the results in this paper. We also answer an open question on construction of symmetric Daubechies complex orthogonal masks in the literature.
Similar content being viewed by others
References
Clonda, D., Lina, J.M., Goulard, B.: Complex Daubechies wavelets: properties and statistical image modeling. Signal Process. 84, 1–23 (2004)
Beylkin, G., Coifman, R., Rokhlin, V.: Fast wavelet transforms and numerical algorithms. I. Comm. Pure Appl. Math. 44, 141–183 (1991)
Daubechies, I.: Orthonormal bases of compactly supported wavelets. Comm. Pure Appl. Math. 41, 674–996 (1988)
Daubechies, I.: Orthonormal bases of compactly supported wavelets, II. Variations on a theme. SIAM J. Math. Anal. 24, 499–519 (1993)
Daubechies, I.: Ten Lectures on Wavelets. SIAM, CBMS Series. SIAM, Philadelphia (1992)
Daubechies, I., Han, B., Ron, A., Shen, Z.: Framelets: MRAbased constructions of wavelet frames. Appl. Comput. Harmon. Anal. 14, 1–46 (2003)
Han, B.: Symmetric orthonormal scaling functions and wavelets with dilation factor 4. Adv. Comput. Math. 8, 221–247 (1998)
Han, B.: Vector cascade algorithms and refinable function vectors in Sobolev spaces. J. Approx. Theory 124, 44–88 (2003)
Han, B.: Computing the smoothness exponent of a symmetric multivariate refinable function. SIAM J. Matrix Anal. Appl. 24, 693–714 (2003)
Han, B.: Refinable functions and cascade algorithms in weighted spaces with Hölder continuous masks. SIAM J. Math. Anal. 41, 70–102 (2008)
Lawton, W.: Applications of complex valued wavelet transforms to subband decomposition. IEEE Trans. Signal Process. 41, 3566–3568 (1993)
Monzón, L., Beylkin, G., Hereman, W.: Compactly supported wavelets based on almost interpolating and nearly linear phase filters (coiflets). Appl. Comput. Harmon. Anal. 7, 184–210 (1999)
Sayed, A.H., Kailath, T.: A survey of spectral factorization methods. Numer. Linear Algebra Appl. 8, 467–496 (2001)
Zhang, X.P., Desai, M.D., Peng, Y.N.: Orthonormal complex filter banks and wavelets: some properties and design. IEEE Trans. Signal Process. 47, 1039–1048 (1999)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Qiyu Sun.
Research supported in part by NSERC Canada under Grant RGP 228051.
Rights and permissions
About this article
Cite this article
Han, B. Symmetric orthonormal complex wavelets with masks of arbitrarily high linearphase moments and sum rules. Adv Comput Math 32, 209–237 (2010). https://doi.org/10.1007/s1044400891027
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s1044400891027