Symmetric wavelet tight frames with two generators. (English) Zbl 1066.42027

Summary: This paper uses the UEP approach for the construction of wavelet tight frames with two (anti-) symmetric wavelets, and provides some results and examples that complement recent results by [Q. Jiang, Adv. Comput. Math. 18, No. 2–4, 247–268 (2003; Zbl 1020.42020)]. A description of a family of solutions when the lowpass scaling filter is of even-length is provided. When one wavelet is symmetric and the other is antisymmetric, the wavelet filters can be obtained by a simple procedure based on matching the roots of associated polynomials. The design examples in this paper begin with the construction of a lowpass filter \(h_0(n)\) that is designed so as to ensure that both wavelets have at least a specified number of vanishing moments.


42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
42C15 General harmonic expansions, frames


Zbl 1020.42020
Full Text: DOI


[1] A.F. Abdelnour, I.W. Selesnick, Symmetric nearly shift invariant tight frame wavelets, IEEE Trans. Signal Process., 2003, in press · Zbl 1370.42026
[2] Chui, C.; He, W., Compactly supported tight frames associated with refinable functions, Appl. comput. harmon. anal., 8, 3, 293-319, (May 2000)
[3] Chui, C.K.; He, W.; Stöckler, J., Compactly supported tight and sibling frames with maximum vanishing moments, Appl. comput. harmon. anal., 13, 3, 177-283, (November 2003)
[4] Coifman, R.R.; Donoho, D.L., Translation-invariant de-noising, () · Zbl 0866.94008
[5] Cooklev, T.; Nishihara, A., Maximally flat FIR filters, (), vol. 1, pp. 96-99
[6] Daubechies, I., Orthonormal bases of compactly supported wavelets, Comm. pure appl. math., 41, 909-996, (1988) · Zbl 0644.42026
[7] Daubechies, I.; Han, B., Pairs of dual wavelet frames from any two refinable functions, Constr. approx., 20, 3, 325-352, (2004) · Zbl 1055.42025
[8] Daubechies, I.; Han, B.; Ron, A.; Shen, Z., Framelets: MRA-based constructions of wavelet frames, Appl. comput. harmon. anal., 14, 1, 1-46, (January 2003)
[9] Eirola, T., Sobolev characterization of solutions of dilation equations, SIAM J. math. anal., 23, 4, 1015-1030, (1992) · Zbl 0761.42014
[10] Herrmann, O., Design of nonrecursive filters with linear phase, Electron. lett., 6, 11, 328-329, (28 May 1970), also in [15]
[11] Jiang, Q., Parameterizations of masks for tight affine frames with two symmetric/antisymmetric generators, Adv. comput. math., 18, 247-268, (February 2003)
[12] Kingsbury, N.G., Complex wavelets for shift invariant analysis and filtering of signals, Appl. comput. harmon. anal., 10, 3, 234-253, (May 2001)
[13] Petukhov, A., Explicit construction of framelets, Appl. comput. harmon. anal., 11, 2, 313-327, (September 2001)
[14] Petukhov, A., Symmetric framelets, Constr. approx., 19, 2, 309-328, (January 2003)
[15] ()
[16] Ron, A.; Shen, Z., Compactly supported tight affine spline frames in L2(rd), Math. comput., 67, 191-207, (1998) · Zbl 0892.42018
[17] Ron, A.; Shen, Z., Construction of compactly supported affine frames in L2(rd), ()
[18] Selesnick, I.W., The double density DWT, () · Zbl 1370.94234
[19] Selesnick, I.W., Smooth wavelet tight frames with zero moments, Appl. comput. harmon. anal., 10, 2, 163-181, (March 2001)
[20] Unser, M.; Blu, T., Mathematical properties of the JPEG2000 wavelet filters, IEEE trans. signal process., 12, 9, 1080-1090, (September 2003)
[21] Villemoes, L., Energy moments in time and frequency for two-scale difference equation solutions and wavelets, SIAM J. math. anal., 23, 1519-1543, (1992) · Zbl 0759.39005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.