Mathematics of Computation

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Help
ISSN 1088-6842(e) ISSN 0025-5718(p)

Matricial filters and crystallographic composite dilation wavelets

Authors: Jeffrey D. Blanchard and Ilya A. Krishtal
Journal: Math. Comp. 81 (2012), 905-922
MSC (2010): Primary 42C40
Posted: July 12, 2011
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In 2006 Guo, Labate, Lim, Weiss, and Wilson introduced the theory of MRA composite dilation wavelets. We continue their work by studying the filter properties of such wavelets and present several important examples.

References

Bibliography

1. Damir Bakić, Ilya Krishtal, and Edward N. Wilson, Parseval frame wavelets with 𝐸_{𝑛}⁽²⁾-dilations, Appl. Comput. Harmon. Anal. 19 (2005), no. 3, 386–431. MR 2186451 (2006g:42056), http://dx.doi.org/10.1016/j.acha.2004.12.006
2. Jeffrey David Blanchard, Existence and accuracy results for composite dilation wavelets, ProQuest LLC, Ann Arbor, MI, 2007. Thesis (Ph.D.)–Washington University in St. Louis. MR 2710862
3. J. D. Blanchard and K. R. Steffen.
Crystallographic Haar-type composite dilation wavelets.
In Wavelets and Multi-scale Analysis: theory and applications. Birkhäuser Boston, Inc., March 2011.
4. Carlos Cabrelli, Christopher Heil, and Ursula Molter, Accuracy of lattice translates of several multidimensional refinable functions, J. Approx. Theory 95 (1998), no. 1, 5–52. MR 1645975 (99g:42038), http://dx.doi.org/10.1006/jath.1997.3211
5. Flavia Colonna, Glenn Easley, Kanghui Guo, and Demetrio Labate, Radon transform inversion using the shearlet representation, Appl. Comput. Harmon. Anal. 29 (2010), no. 2, 232–250. MR 2652460 (2011d:44005), http://dx.doi.org/10.1016/j.acha.2009.10.005
6. Kanghui Guo, Gitta Kutyniok, and Demetrio Labate, Sparse multidimensional representations using anisotropic dilation and shear operators, Wavelets and splines: Athens 2005, Mod. Methods Math., Nashboro Press, Brentwood, TN, 2006, pp. 189–201. MR 2233452 (2007c:42050)
7. Kanghui Guo and Demetrio Labate, Optimally sparse multidimensional representation using shearlets, SIAM J. Math. Anal. 39 (2007), no. 1, 298–318. MR 2318387 (2008k:42097), http://dx.doi.org/10.1137/060649781
8. Kanghui Guo and Demetrio Labate, Characterization and analysis of edges using the continuous shearlet transform, SIAM J. Imaging Sci. 2 (2009), no. 3, 959–986. MR 2551249 (2011b:94004), http://dx.doi.org/10.1137/080741537
9. Kanghui Guo, Demetrio Labate, Wang-Q Lim, Guido Weiss, and Edward Wilson, The theory of wavelets with composite dilations, Harmonic analysis and applications, Appl. Numer. Harmon. Anal., Birkhäuser Boston, Boston, MA, 2006, pp. 231–250. MR 2249312 (2007d:42072), http://dx.doi.org/10.1007/0-8176-4504-7_11
10. Kanghui Guo, Demetrio Labate, Wang-Q Lim, Guido Weiss, and Edward Wilson, Wavelets with composite dilations and their MRA properties, Appl. Comput. Harmon. Anal. 20 (2006), no. 2, 202–236. MR 2207836 (2006j:42056), http://dx.doi.org/10.1016/j.acha.2005.07.002
11. Eugenio Hernández and Guido Weiss, A first course on wavelets, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1996. With a foreword by Yves Meyer. MR 1408902 (97i:42015)
12. Robert Timothy Houska, The nonexistence of shearlet-like scaling multifunctions that satisfy certain minimally desirable properties and characterizations of the reproducing properties of the integer lattice translations of a countable collection of square integrable functions, ProQuest LLC, Ann Arbor, MI, 2009. Thesis (Ph.D.)–Washington University in St. Louis. MR 2713543
13. Ilya A. Krishtal, Benjamin D. Robinson, Guido L. Weiss, and Edward N. Wilson, Some simple Haar-type wavelets in higher dimensions, J. Geom. Anal. 17 (2007), no. 1, 87–96. MR 2302875 (2008a:42018), http://dx.doi.org/10.1007/BF02922084
14. G. Kutyniok, J. Lemvig, and W.-Q Lim.
Compactly supported shearlets.
submitted, 2010.
15. Wang-Q Lim, The discrete shearlet transform: a new directional transform and compactly supported shearlet frames, IEEE Trans. Image Process. 19 (2010), no. 5, 1166–1180. MR 2723739 (2011f:94017), http://dx.doi.org/10.1109/TIP.2010.2041410
16. J. MacArthur.
Compatible dilations and wavelets for the wallpaper groups.
preprint, 2009.
17. J. MacArthur and K. Taylor.
Wavelets with crystal symmetry shifts.
submitted, 2009.
18. Amos Ron and Zuowei Shen, Affine systems in 𝐿₂(𝐑^{𝐝}): the analysis of the analysis operator, J. Funct. Anal. 148 (1997), no. 2, 408–447. MR 1469348 (99g:42043), http://dx.doi.org/10.1006/jfan.1996.3079

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2010): 42C40

Retrieve articles in all journals with MSC (2010): 42C40

Additional Information

Jeffrey D. Blanchard
Affiliation: Department of Mathematics and Statistics, Grinnell College, Grinnell, Iowa 50112
Email: jeff@math.grinnell.edu

Ilya A. Krishtal
Affiliation: Department of Mathematics, Northern Illinois University, Dekalb, Illinois 60115
Email: krishtal@niu.edu

DOI: http://dx.doi.org/10.1090/S0025-5718-2011-02518-4
PII: S 0025-5718(2011)02518-4
Keywords: Filters, wavelets, composite dilation wavelets, bracket product, unitary extension principle
Received by editor(s): November 22, 2009 and in revised form, January 16, 2011
Posted: July 12, 2011
Additional Notes: The first author was partially supported by NSF DMS (VIGRE) Grant number 0602219.
The second author was partially supported by NSF DMS Grant number 0908239.
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|