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A fast algorithm for constructing orthogonal multiwavelets

Published online by Cambridge University Press:  17 February 2009

Yang Shouzhi
Yang Shouzhi
Affiliation:
Department of Mathematics, Shantou University, Shantou 515063, P.R. China; e-mail: [email protected].
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Abstract

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Multiwavelets possess some nice features that uniwavelets do not. A consequence of this is that multiwavelets provide interesting applications in signal processing as well as in other fields. As is well known, there are perfect construction formulas for the orthogonal uniwavelet. However, a good formula with a similar structure for multiwavelets does not exist. In particular, there are no effective methods for the construction of multiwavelets with a dilation factor a (a ≥ 2, a ∈ Z). In this paper, a procedure for constructing compactly supported orthonormal multiscaling functions is first given. Based on the constructed multiscaling functions, we then propose a method of constructing multiwavelets, which is similar to that for constructing uniwavelets. In addition, a fast numerical algorithm for computing multiwavelets is given. Compared with traditional approaches, the algorithm is not only faster, but also computationally more efficient. In particular, the function values of several points are obtained simultaneously by using our algorithm once. Finally, we give three examples illustrating how to use our method to construct multiwavelets.

Type
Research Article
Information
The ANZIAM Journal , Volume 46 , Issue 2 , October 2004 , pp. 185 - 200
Copyright
Copyright © Australian Mathematical Society 2004

References

[1]Cabrelli, A. C. and Gordillo, L. M., “Existence of multiwavelets in Rn”, Proc. Amer. Math. Soc. 130 (2002) 1413–1424.Google Scholar
[2]Chui, C. K. and Lian, J., “A study on orthonormal multiwavelets”, Appl. Numer Math. 20 (1996) 273–298.CrossRefGoogle Scholar
[3]Chui, C. K. and Wang, J. Z., “A cardinal spline approach to wavelets”, Proc. Amer Math. Soc. 113 (1991) 785–793.Google Scholar
[4]Daubechies, I., Ten lectures on wavelets (SIAM, Philadelphia, PA, 1992).CrossRefGoogle Scholar
[5]Daubechies, I., “Orthonormal bases of compactly supported wavelets”, Comm. Pure Appl. Math. 41 (1998) 909–996.Google Scholar
[6]Daubechies, I. and Lagarias, J. C., “Two-scale difference equations. I Existence and global regularity of solutions”, SIAM J. Math. Anal. 22 (1991) 1388–1410.CrossRefGoogle Scholar
[7]Donovan, G. C., Geronimo, J. and Hardin, D. P., “Construction of orthogonal wavelets using fractal interpolation functions”, SIAM J. Math. Anal. 27 (1996) 1158–1192.Google Scholar
[8]Donovan, G. C., Geronimo, J. and Hardin, D. P., “Intertwining multiresolution analysis and construction of piecewise polynomial wavelets”, SIAM J. Math. Anal. 27 (1996) 1791–1815.Google Scholar
[9]Van Fleet, J. P., “Multiwavelets and integer transforms”, J. Comput. Anal. Appl. 3 (2003) 161–178.Google Scholar
[10]Geronimo, J., Hardin, D. P. and Massopust, P., “Fractal functions and wavelet expansions based on several scaling functions”, J. Approx. Theory 78 (1998) 373–401.Google Scholar
[11]Goodman, T. N. T., Lee, S. L. and Tang, W. S., “Wavelets in wandering subspaces”, Trans. Amer. Math. Soc. 338 (1993) 639–654.Google Scholar
[12]Kessler, B., “A construction of compactly supported biorthogonal scaling vectors and multiwavelets on R2”, J. Approx. Theory 117 (2002) 229–254.Google Scholar
[13]Massopust, P., Ruch, D. and Van Fleet, P., “On the support properties of scaling vectors”, Appl. Comput. Harmon. Anal. 3 (1996) 229–238.Google Scholar
[14]Selesnick, I., “Interpolating multiwavelet bases and the sampling theorem”, IEEE Trans. Signal Process. 47 (1999) 1615–1621.CrossRefGoogle Scholar
[15]So, W. and Wang, J., “Estimating the support of a scaling vector”, SIAM J. Matrix Anal. Appl. 1 (1997) 66–73.Google Scholar
[16]Strang, G. and Strela, V., “Orthogonal multiwavelets with vanishing moments”, J. Opt. Eng. 33 (1994) 2104–2107.Google Scholar
[17]Strela, V., Heller, P. N., Strang, G., Topiwala, P. and Heil, C., “The application of multiwavelet filter banks to image processing”, IEEE Trans. Image Process. 8 (1999) 548–563.Google Scholar
[18]Tymczak, C. J., Niklasson, A. M. N. and Röder, H., “Separable and nonseparable multiwavelets in multiple dimensions”, J. Comput. Phys. 175 (2002) 363–397.Google Scholar
[19]Zhou, D., “Interpolatory orthogonal multiwavelets and refinable functions”, IEEE Trans. Signal Process. 50 (2002) 520–527.Google Scholar