Balanced multiwavelets in
Authors:
Charles K. Chui and Qingtang Jiang
Journal:
Math. Comp. 74 (2005), 13231344
MSC (2000):
Primary 42C40, 65T60; Secondary 94A08
Published electronically:
June 7, 2004
MathSciNet review:
2137005
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: The notion of balancing was introduced a few years ago as a condition for the construction of orthonormal scaling function vectors and multiwavelets to ensure the property of preservation/annihilation of scalarvalued discrete polynomial data of order (or degree ), when decomposed by the corresponding matrixvalued lowpass/highpass filters. While this condition is indeed precise, to the best of our knowledge only the proof for is known. In addition, the formulation of the balancing condition for is so prohibitively difficult to satisfy that only a very few examples for and vector dimension 2 have been constructed in the open literature. The objective of this paper is to derive various characterizations of the balancing condition that include the polynomial preservation property as well as equivalent formulations that facilitate the development of methods for the construction purpose. These results are established in the general multivariate and biorthogonal settings for any .
 1.
K. Attakitmongcol, D. P. Hardin, D. M. Wilkes, Multiwavelet prefilters II: Optimal orthogonal prefilters, IEEE Tran. Image Proc., 10 (2001), 14761487.
 2.
Charles
K. Chui, Multivariate splines, CBMSNSF Regional Conference
Series in Applied Mathematics, vol. 54, Society for Industrial and
Applied Mathematics (SIAM), Philadelphia, PA, 1988. With an appendix by
Harvey Diamond. MR 1033490
(92e:41009)
 3.
C. K. Chui, Q. T. Jiang, Multivariate balanced vectorvalued refinable functions, In Mordern Development in Multivariate Approximation, ISNM 145, 71102, V.W. Haussmann, K. Jetter, M. Reimer, and J. Stöckler (eds.), Birhhäuser Verlag, Basel, 2003.
 4.
Charles
K. Chui and Jianao
Lian, A study of orthonormal multiwavelets, Appl. Numer.
Math. 20 (1996), no. 3, 273–298. Selected
keynote papers presented at 14th IMACS World Congress (Atlanta, GA, 1994).
MR
1402703 (98g:42051), http://dx.doi.org/10.1016/01689274(95)001115
 5.
George
C. Donovan, Jeffrey
S. Geronimo, Douglas
P. Hardin, and Peter
R. Massopust, Construction of orthogonal wavelets using fractal
interpolation functions, SIAM J. Math. Anal. 27
(1996), no. 4, 1158–1192. MR 1393432
(97f:42053), http://dx.doi.org/10.1137/S0036141093256526
 6.
Jeffrey
S. Geronimo, Douglas
P. Hardin, and Peter
R. Massopust, Fractal functions and wavelet expansions based on
several scaling functions, J. Approx. Theory 78
(1994), no. 3, 373–401. MR 1292968
(95h:42033), http://dx.doi.org/10.1006/jath.1994.1085
 7.
D. P. Hardin, D. W. Roach, Multiwavelet prefilters I: Orthogonal prefilters preserving approximation order , IEEE Tran. Circuits and SystemII 45 (1998), 11191125.
 8.
RongQing
Jia and QingTang
Jiang, Approximation power of refinable vectors of functions,
Wavelet analysis and applications (Guangzhou, 1999) AMS/IP Stud. Adv.
Math., vol. 25, Amer. Math. Soc., Providence, RI, 2002,
pp. 155–178. MR 1887509
(2003e:41030)
 9.
R. Q. Jia, Q. T. Jiang, Spectral analysis of transition operators and its applications to smoothness analysis of wavelets, SIAM J. Matrix Anal. Appl. 24 (2003), 10711109.
 10.
RongQing
Jia, Qingtang
Jiang, and Zuowei
Shen, Distributional solutions of nonhomogeneous discrete and
continuous refinement equations, SIAM J. Math. Anal.
32 (2000), no. 2, 420–434 (electronic). MR 1781224
(2001k:42044), http://dx.doi.org/10.1137/S0036141099350882
 11.
Qingtang
Jiang, Multivariate matrix refinable
functions with arbitrary matrix dilation, Trans. Amer. Math. Soc. 351 (1999), no. 6, 2407–2438. MR 1650101
(99i:42047), http://dx.doi.org/10.1090/S0002994799024496
 12.
J. Lebrun, Balancing Multiwavelets, PhD Thesis, Swiss Federal Institute of Technology, Lausanne, Switzerland, 2000.
 13.
J. Lebrun, M. Vetterli, Balanced multiwavelets: Theory and design, IEEE Trans. Signal Processing 46 (1998), 11191125.
 14.
J. Lebrun, M. Vetterli, High order balanced multiwavelets, In Proc. IEEE Int. Conf. Acoust., Speech, Signal Processing (ICASSP), Seattle, 1998.
 15.
J. A. Lian, C. K. Chui, Balanced multiwavelets with short filters, IEEE Signal Processing Letters, 11 (2004), 7578.
 16.
Ivan
W. Selesnick, Multiwavelet bases with extra approximation
properties, IEEE Trans. Signal Process. 46 (1998),
no. 11, 2898–2908. MR 1719939
(2000g:94009), http://dx.doi.org/10.1109/78.726804
 17.
I. W. Selesnick, Balanced multiwavelet bases based on symmetric FIR filters, IEEE Trans. Signal Proc. 48 (2000), 184191.
 18.
G. Strang, Eigenvalues of and convergence of cascade algorithm, IEEE Trans. Signal Proc. 44 (1996), 233238.
 19.
V. Strela, P. Heller, G. Strang, P. Topiwala, C. Heil, The application of multiwavelet filter banks to image processing, IEEE Trans. Image Proc. 8 (1999), 548563.
 20.
X. G. Xia, D. P. Hardin, J. S. Geronimo, B. Suter, Design of prefilters for discrete multiwavelet transforms, IEEE Trans. Signal Proc. 44 (1996), 2535.
 1.
 K. Attakitmongcol, D. P. Hardin, D. M. Wilkes, Multiwavelet prefilters II: Optimal orthogonal prefilters, IEEE Tran. Image Proc., 10 (2001), 14761487.
 2.
 C. K. Chui, Multivariate Splines, NSFCBMS Series, vol. 54, SIAM Publ., Philadelphia, 1988. MR 92e:41009
 3.
 C. K. Chui, Q. T. Jiang, Multivariate balanced vectorvalued refinable functions, In Mordern Development in Multivariate Approximation, ISNM 145, 71102, V.W. Haussmann, K. Jetter, M. Reimer, and J. Stöckler (eds.), Birhhäuser Verlag, Basel, 2003.
 4.
 C. K. Chui, J. Lian, A study of orthonormal multiwavelets, J. Appl. Numer. Math. 20 (1996), 273298. MR 98g:42051
 5.
 G. Donovan, J. Geronimo, D. P. Hardin, P. Massopust, Construction of orthogonal wavelets using fractal interpolation functions, SIAM J. Math. Anal. 27 (1996), 11581192. MR 97f:42053
 6.
 J. Geronimo, D. P. Hardin, P. Massopust, Fractal functions and wavelet expansions based on several scaling functions, J. Approx. Theory 78 (1994), 373401. MR 95h:42033
 7.
 D. P. Hardin, D. W. Roach, Multiwavelet prefilters I: Orthogonal prefilters preserving approximation order , IEEE Tran. Circuits and SystemII 45 (1998), 11191125.
 8.
 R. Q. Jia, Q. T. Jiang, Approximation power of refinable vectors of functions, In Wavelet analysis and applications, Studies Adv. Math., vol. 25, 155178, Amer. Math. Soc., Providence, RI, 2002. MR 2003e:41030
 9.
 R. Q. Jia, Q. T. Jiang, Spectral analysis of transition operators and its applications to smoothness analysis of wavelets, SIAM J. Matrix Anal. Appl. 24 (2003), 10711109.
 10.
 R. Q. Jia, Q. T. Jiang, Z. W. Shen, Distributional solutions of nonhomogeneous discrete and continuous refinement equations, SIAM J. Math. Anal. 32 (2000), 420434. MR 2001k:42044
 11.
 Q. T. Jiang, Multivariate matrix refinable functions with arbitrary matrix dilation, Trans. Amer. Math. Soc. 351 (1999), 24072438.MR 99i:42047
 12.
 J. Lebrun, Balancing Multiwavelets, PhD Thesis, Swiss Federal Institute of Technology, Lausanne, Switzerland, 2000.
 13.
 J. Lebrun, M. Vetterli, Balanced multiwavelets: Theory and design, IEEE Trans. Signal Processing 46 (1998), 11191125.
 14.
 J. Lebrun, M. Vetterli, High order balanced multiwavelets, In Proc. IEEE Int. Conf. Acoust., Speech, Signal Processing (ICASSP), Seattle, 1998.
 15.
 J. A. Lian, C. K. Chui, Balanced multiwavelets with short filters, IEEE Signal Processing Letters, 11 (2004), 7578.
 16.
 I. W. Selesnick, Multiwavelets with extra approximation properties, IEEE Trans. Signal Proc. 46 (1998), 28982909. MR 2000g:94009
 17.
 I. W. Selesnick, Balanced multiwavelet bases based on symmetric FIR filters, IEEE Trans. Signal Proc. 48 (2000), 184191.
 18.
 G. Strang, Eigenvalues of and convergence of cascade algorithm, IEEE Trans. Signal Proc. 44 (1996), 233238.
 19.
 V. Strela, P. Heller, G. Strang, P. Topiwala, C. Heil, The application of multiwavelet filter banks to image processing, IEEE Trans. Image Proc. 8 (1999), 548563.
 20.
 X. G. Xia, D. P. Hardin, J. S. Geronimo, B. Suter, Design of prefilters for discrete multiwavelet transforms, IEEE Trans. Signal Proc. 44 (1996), 2535.
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Additional Information
Charles K. Chui
Affiliation:
Department of Mathematics & Computer Science, University of Missouri–St. Louis, St. Louis, Missouri 63121 and Department of Statistics, Stanford University, Stanford, California 94305
Email:
ckchui@stanford.edu
Qingtang Jiang
Affiliation:
Department of Mathematics & Computer Science, University of Missouri–St. Louis, St. Louis, Missouri 63121
Email:
jiang@math.umsl.edu
DOI:
http://dx.doi.org/10.1090/S0025571804016813
PII:
S 00255718(04)016813
Keywords:
Multiwavelets,
characterization of balancing condition,
polynomial preservation/annihilation
Received by editor(s):
May 18, 2003
Received by editor(s) in revised form:
January 9, 2004
Published electronically:
June 7, 2004
Additional Notes:
The first author was supported in part by NSF Grants #CCR9988289 and #CCR0098331, and ARO Grant #DAAD 190010512
The second author was supported in part by a University of Missouri–St.\ Louis research award
Article copyright:
© Copyright 2004
American Mathematical Society
