Wavelets and Functions with Bounded Variation from Image Processing to Pure Mathematics
Meyer, Yves
Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni, Tome 11 (2000), p. 77-105 / Harvested from Biblioteca Digitale Italiana di Matematica
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JPEG-2000 has the potential to be the most significant advance in still image compression since the introduction of JPEG over a decade ago. JPEG-2000 is a wavelet based algorithm and it relies on new estimates on wavelet coefficients of functions of bounded variation. These new estimates have far reaching implications in pure mathematics.

Publié le : 2000-12-01
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     author = {Yves Meyer},
     title = {Wavelets and Functions with Bounded Variation from Image Processing to Pure Mathematics},
     journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
     volume = {11},
     year = {2000},
     pages = {77-105},
     zbl = {1149.42309},
     mrnumber = {1845666},
     language = {en},
     url = {http://dml.mathdoc.fr/item/RLIN_2000_9_11_S1_77_0}
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Meyer, Yves. Wavelets and Functions with Bounded Variation from Image Processing to Pure Mathematics. Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni, Tome 11 (2000) pp. 77-105. http://gdmltest.u-ga.fr/item/RLIN_2000_9_11_S1_77_0/

[1] Bonami, A. - Poornima, S., Non-multipliers of Sobolev spaces. Journ. Funct. Anal., 71, 1987, 175-181. | MR 879706 | Zbl 0627.46034

[2] Brislaw, Ch. M., Fingerprints go digital. Notices of the AMS, vol. 42, no. 11, nov. 1995, 1278-1283. http://www.c3. lanl. gov/ brislawn

[3] Cohen, A. - Devore, R. - Petrushev, P. - Xu, H., Nonlinear approximation and the space B⁢V⁢(ℝ2). American Journal of Mathematics, 121, 1999, 587-628. | MR 1738406 | Zbl 0931.41019

[4] Daubechies, I., Ten lectures on wavelets. SIAM, Philadelphia1992. | MR 1162107 | Zbl 0776.42018

[5] Devore, R. - Jawerth, B. - Popov, V., Compression of wavelet decompositions. American Journal of Mathematics, 114, 1992, 737-785. | MR 1175690 | Zbl 0764.41024

[6] Devore, R. - Lucier, B. J., Fast wavelet techniques for near optimal image compression. IEEE Military Communications Conference (October 11-14, 1992).

[7] Donoho, D., Nonlinear Solution of Linear Inverse Problems by Wavelet-Vaguelette Decomposition. Applied and Computational Harmonic Analysis, 2, 1995, 101-126. | MR 1325535 | Zbl 0826.65117

[8] Donoho, D. - Johnstone, I., Wavelet shrinkage: Asymptopia? J. R. Statist. Soc., B, 57 (2), 1995, 301-369. | MR 1323344 | Zbl 0827.62035

[9] Kahane, J.-P. - Lemarié-Rieusset, P.-G., Fourier Series and Wavelets. Gordon and Breach Science Publishers, 1996. | MR 3585499

[10] Mallat, S., A Wavelet Tour of Signal Processing. Academic Press, 1998. | MR 1614527 | Zbl 0937.94001

[11] Meyer, Y., Wavelets, Algorithms & Applications. SIAM, Philadelphia1993. | MR 1219953

[12] Meyer, Y., Wavelets and Operators. Cambridge studies in advanced mathematics, 37 CUP, 1992. | MR 1228209

[13] Meyer, Y., Wavelets, vibrations and scalings. CRM Monograph Series, vol. 9, 1998. | MR 1483896 | Zbl 0893.42015

[14] Morel, J.-M., Filtres itératifs des images et équations aux dérivées partielles. Notes de cours du centre Emile Borel, 1998.

[15] Morel, J.-M. - Solimini, S., Variational methods in image segmentation. Birkhäuser, Boston 1995.

[16] Mumford, D., Book reviews on [12]. Bulletin of the American Mathematical Society, vol. 33, n. 2, April 1996.

[17] Mumford, D. - Shah, J., Boundary detection by minimizing functionals. Proc. IEEE Conf. Comp. Vis. Pattern Recognition, 1985.

[18] Oru, F., Le rôle des oscillations dans quelques problèmes d'analyse non-linéaire. Thèse, CMLA, Ecole Norm. Sup. de Cachan, 9 Juin 1998.

[19] Rudin, L. - Osher, S., Total variation based image restoration with free local constraints. Proceedings Intern. Conf. on Image Processing (Austin, 1994). IEEE Press, Piscataway, N.J., 1994, 31-35.