Remote Access Electronic Research Announcements

Electronic Research Announcements

ISSN 1079-6762



Wavelets with composite dilations

Authors: Kanghui Guo, Demetrio Labate, Wang-Q Lim, Guido Weiss and Edward Wilson
Journal: Electron. Res. Announc. Amer. Math. Soc. 10 (2004), 78-87
MSC (2000): Primary 42C15, 42C40
Published electronically: August 3, 2004
MathSciNet review: 2075899
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A wavelet with composite dilations is a function generating an orthonormal basis or a Parseval frame for $L^2({\mathbb R}^n)$ under the action of lattice translations and dilations by products of elements drawn from non-commuting matrix sets $A$ and $B$. Typically, the members of $B$ are shear matrices (all eigenvalues are one), while the members of $A$ are matrices expanding or contracting on a proper subspace of ${\mathbb R}^n$. These wavelets are of interest in applications because of their tendency to produce ``long, narrow'' window functions well suited to edge detection. In this paper, we discuss the remarkable extent to which the theory of wavelets with composite dilations parallels the theory of classical wavelets, and present several examples of such systems.

References [Enhancements On Off] (What's this?)

  • [1] E.J. Candès and D.L. Donoho, Ridgelets: a key to higher-dimensional intermittency?, Phil. Trans. Royal Soc. London A 357 (1999), 2495-2509. MR 1721227 (2000g:42047)
  • [2] E.J. Candès and D.L. Donoho, New tight frames of curvelets and optimal representations of objects with $C^2$ singularities, Comm. Pure Appl. Math. 57 (2004), 219-266. MR 2012649
  • [3] R.R. Coifman, and F.G. Meyer, Brushlets: a tool for directional image analysis and image compression, Appl. Comp. Harmonic Anal. 5 (1997), 147-187. MR 1448220 (99c:42069)
  • [4] M.N. Do and M. Vetterli, Contourlets, in: Beyond Wavelets, G.V. Welland (ed.), 2003.
  • [5] D.L. Donoho and X. Huo, Beamlets and multiscale image analysis, Lecture notes in computational science and engineering, Springer, 2002. MR 1928566 (2003m:94009)
  • [6] K. Guo, D. Labate, W. Lim, G. Weiss, and E. Wilson, The theory of wavelets with composite dilations, preprint 2004.
  • [7] E. Hernández, D. Labate, and G. Weiss, A unified characterization of reproducing systems generated by a finite family, II, J. Geom. Anal. 12(4) (2002), 615-662. MR 1916862 (2003j:42036)
  • [8] E. Hernández and G. Weiss, A First Course on Wavelets, CRC Press, Boca Raton, FL, 1996. MR 1408902 (97i:42015)
  • [9] L. Hörmander, The analysis of linear partial differential operators. I. Distribution theory and Fourier analysis. Springer-Verlag, Berlin, 2003. MR 1996773
  • [10] E.M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, Princeton, NJ, 1970. MR 0304972 (46:4102)
  • [11] G.V. Welland (ed.), Beyond Wavelets, Academic Press, San Diego, CA, 2003.
  • [12] G. Weiss, and E. Wilson, The mathematical theory of wavelets, Proceedings of the NATO-ASI Meeting. Harmonic Analysis 2000--A Celebration. Kluwer, 2001. MR 1858791 (2002h:42078)

Similar Articles

Retrieve articles in Electronic Research Announcements of the American Mathematical Society with MSC (2000): 42C15, 42C40

Retrieve articles in all journals with MSC (2000): 42C15, 42C40

Additional Information

Kanghui Guo
Affiliation: Department of Mathematics, Southwest Missouri State University, Springfield, Missouri 65804

Demetrio Labate
Affiliation: Department of Mathematics, North Carolina State University, Raleigh, North Carolina 27695

Wang-Q Lim
Affiliation: Department of Mathematics, Washington University, St. Louis, Missouri 63130

Guido Weiss
Affiliation: Department of Mathematics, Washington University, St. Louis, Missouri 63130

Edward Wilson
Affiliation: Department of Mathematics, Washington University, St. Louis, Missouri 63130

Keywords: Affine systems, frames, multiresolution analysis (MRA), multiwavelets, wavelets
Received by editor(s): February 23, 2004
Received by editor(s) in revised form: April 13, 2004
Published electronically: August 3, 2004
Additional Notes: The fourth author was supported in part by a SW Bell Grant.
Communicated by: Boris Hasselblatt
Article copyright: © Copyright 2004 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society