×
Chyzak, Frédéric; Paule, Peter; Scherzer, Otmar; Schoisswohl, Armin; Zimmermann, Burkhard

The construction of orthonormal wavelets using symbolic methods and a matrix analytical approach for wavelets on the interval. (English) Zbl 0974.42027

Exp. Math. 10, No. 1, 67-86 (2001).
Using computer algebraic methods, the authors discuss closed form representations of filter coefficients of orthonormal/biorthogonal wavelets on the real line, \([0,\infty)\) and \([0,1]\), respectively. Gröbner basis method is used to solve systems of polynomial equations for the filter coefficients. A big advantage of using computer algebra is that one can avoid instabilities which occur in numerical calculations of filter coefficients. Further, the authors present a matrix analytical approach that unifies constructions of orthonormal/biorthogonal wavelets on \([0,1]\).
Reviewer: Manfred Tasche (Rostock)

Cited in 7 Documents

MSC:

42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
68W30 Symbolic computation and algebraic computation

Keywords:

orthonormal wavelets; biorthogonal wavelets; symbolic methods; computer algebraic methods; Gröbner basis method; filter coefficients
Cite Review PDF
Full Text: DOI Euclid EuDML EMIS

References:

[1] Bradley J., ”The FBI wavelet/scalar quantisation standard for gray-scale fingerprint image compression” (1993)
[2] Buchberger B., Ph.D. thesis, in: Ein Algorithmus zum Auffinden der Basiselemente des Restklassen-ringes nach einem nulldimensionalen Polynomideal (1965) · Zbl 1245.13020
[3] DOI: 10.1007/BF01817776 · doi:10.1007/BF01817776
[4] DOI: 10.1017/CBO9780511565847 · doi:10.1017/CBO9780511565847
[5] DOI: 10.1137/0524032 · Zbl 0792.42019 · doi:10.1137/0524032
[6] DOI: 10.1002/cpa.3160450502 · Zbl 0776.42020 · doi:10.1002/cpa.3160450502
[7] Cohen A., C. R. Acad. Sci. Paris, ser. 1 316 (5) pp 417– (1993)
[8] DOI: 10.1006/acha.1993.1005 · Zbl 0795.42018 · doi:10.1006/acha.1993.1005
[9] Cox D., Using algebraic geometry (1998) · Zbl 0920.13026
[10] Dahmen W., Results in Mathematics 34 pp 255– (1998) · Zbl 0942.41003
[11] DOI: 10.1006/acha.1998.0247 · Zbl 0922.42021 · doi:10.1006/acha.1998.0247
[12] DOI: 10.1002/cpa.3160410705 · Zbl 0644.42026 · doi:10.1002/cpa.3160410705
[13] DOI: 10.1137/1.9781611970104 · Zbl 0776.42018 · doi:10.1137/1.9781611970104
[14] DOI: 10.1137/0524031 · Zbl 0792.42018 · doi:10.1137/0524031
[15] DOI: 10.1109/18.119733 · Zbl 0754.68118 · doi:10.1109/18.119733
[16] Faugère J., Ph.D. thesis, in: Résolution des systémes dèquations polynomiales (1994)
[17] von zur Ga-then J., Modem Computer Algebra (1999)
[18] Graham R., Concrete Mathematics,, 2. ed. (1994)
[19] DOI: 10.1137/0729059 · Zbl 0761.65083 · doi:10.1137/0729059
[20] Klappenecker, A. ”On algebraic properties of selfreciprocal polynomials and of Daubechies filters of low order”. Proc. of 1997 IEEE Int. Symp. on Inform. Theory. 1997, Ulm, Germany. pp.80New York: IEEE. [Klappenecker 1997]
[21] DOI: 10.1109/34.192463 · Zbl 0709.94650 · doi:10.1109/34.192463
[22] DOI: 10.4171/RMI/107 · Zbl 0753.42015 · doi:10.4171/RMI/107
[23] DOI: 10.1006/jsco.1995.1071 · Zbl 0851.68052 · doi:10.1006/jsco.1995.1071
[24] Petkovšek M., A = B (1996)
[25] Scherzer O., ”Wavelet Compression of 3D Ultrasound Data” (1998)
[26] Strang G., Z. Angew. Math. Mech. 76 (2) pp 37– (1996)
[27] DOI: 10.1002/nme.1620371403 · Zbl 0812.65144 · doi:10.1002/nme.1620371403
[28] DOI: 10.1007/978-3-7091-6571-3 · Zbl 0853.12003 · doi:10.1007/978-3-7091-6571-3
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.