Abstract
This paper provides a new formulation of wavelet transforms in terms of generalized matrix products. After defining the generalized matrix product, a fast algorithm using parallelism for compactly supported wavelet transforms that satisfy m-scale scaling equations for m ≥ 2 is established. Several special examples, such as the Fourier-wavelet matrix expansion and wavelet decompositions and reconstructions, that demonstrate that the new formulation and algorithm offer unique advantages over existing wavelet algorithms are provided.
This research was supported in part by U.S. Air Force contract F08635-89-C-0134.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
G.X. Ritter, “Heterogeneous matrix products,” Proc. Soc. Photo-Opt. Instrum. Eng., vol. 1568, 1991, pp. 92–100.
G.X. Ritter, “Recent developments in image algebra,” in Advances in Electronics and Electron Physics, vol. 80, Academic Press, New York, 1991, pp. 243–308.
G.X. Ritter and H. Zhu, “The generalized matrix product and its applications,” J. Math. Imag. Vis., vol. 1, 1992, pp. 201–213.
G. Beylkin, R. Coifman, I. Daubechies, S. Mallat, Y. Meyer, L. Rophael, and B. Ruskai, eds., Wavelets and Their Applications, Joness and Barllett, Cambridge, MA, 1992.
C.K. Chui, ed., Wavelets-A tutorial in Theory and Applications, Academic Press, Boston, 1992.
R. Kronland-Martinet, J. Morlet, and A. Grossmann, “Analysis of sound patterns through wavelet transform,” Int. J. Patt. Recog. Artif. Intell., 1988.
S.G. Mallat, “A compact multiresolution representation: The wavelet model,” in Proc. IEEE Workshop Comput. Vision, Miami, FL, December 1987.
I. Daubechies, “Orthonormal bases of wavelets,” Comm. Pure Appl. Math., vol. 41, 1988, pp. 909–996.
S.G. Mallat, “A theory for multiresolution signal decomposition: The wavelet representation,” IEEE Trans. Patt. Anal. Mach. Intell., vol. 11, 1989, pp. 674–693.
C.K. Chui, ed., Introduction to Wavelets, Academic Press, Boston, 1992.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1993 Springer Science+Business Media New York
About this chapter
Cite this chapter
Zhu, H., Ritter, G.X. (1993). The Generalized Matrix Product and the Wavelet Transform. In: Laine, A. (eds) Wavelet Theory and Application. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-3260-6_6
Download citation
DOI: https://doi.org/10.1007/978-1-4615-3260-6_6
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4613-6434-4
Online ISBN: 978-1-4615-3260-6
eBook Packages: Springer Book Archive