Comptes Rendus
no. 5
Mathematical Analysis
Curvelets and Fourier Integral Operators
[Curvelets et Opérateurs Intégraux de Fourier]

Présenté par : Yves Meyer

Emmanuel Candès1 ; Laurent Demanet1
1 Applied and Computational Mathematics, California Institute of Technology, Mail Code 217-50, Pasadena, CA 91125, USA
Comptes Rendus. Mathématique, Volume 336 (2003) no. 5, pp. 395-398.

Une série de récents articles ont introduit l'analyse en curvelets E. Candès, D. Donoho, in : (i) Curvelets – a surprisingly effective nonadaptive representation for objects with edges (A. Cohen, C. Rabut, L. Schumaker (Eds.)), Vanderbilt University Press, Nashville, 2000, pp. 105–120 ; (ii) http://www.acm.caltech.edu/~emmanuel/publications.html, 2002 : les curvelets offrent une représentation multi-échelle qui ouvre de nouvelles perspectives pour l'analyse de problèmes importants en théorie de l'approximation et en traitement de l'image. Cet article montre que les curvelets permettent une représentation optimale de la classe des opérateurs intégraux de Fourier. Par « optimale », nous entendons par exemple, la plus économe.

A recent body of work introduced new tight-frames of curvelets E. Candès, D. Donoho, in: (i) Curvelets – a suprisingly effective nonadaptive representation for objects with edges (A. Cohen, C. Rabut, L. Schumaker (Eds.)), Vanderbilt University Press, Nashville, 2000, pp. 105–120; (ii) http://www.acm.caltech.edu/~emmanuel/publications.html, 2002 to address key problems in approximation theory and image processing. This paper shows that curvelets essentially provide optimally sparse representations of Fourier Integral Operators.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/S1631-073X(03)00095-5
Emmanuel Candès 1 ; Laurent Demanet 1

1 Applied and Computational Mathematics, California Institute of Technology, Mail Code 217-50, Pasadena, CA 91125, USA 
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Emmanuel Candès; Laurent Demanet. Curvelets and Fourier Integral Operators. Comptes Rendus. Mathématique, Volume 336 (2003) no. 5, pp. 395-398. doi : 10.1016/S1631-073X(03)00095-5. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(03)00095-5/

[1] G. Beylkin; R. Coifman; V. Rokhlin Fast wavelet transforms and numerical algorithms, Comm. Pure Appl. Math., Volume 44 (1991), pp. 141-183

[2] E. Candès, L. Demanet, Curvelets, warpings and optimally sparse representations of Fourier Integral Operators, Manuscript, 2002

[3] E. Candès; D. Donoho Curvelets – a suprisingly effective nonadaptive representation for objects with edges (A. Cohen; C. Rabut; L. Schumaker, eds.), Curves and Surface Fitting: Saint-Malo 1999, Vanderbilt University Press, Nashville, 2000, pp. 105-120

[4] E. Candès, D. Donoho, New tight Frames of curvelets and optimal representations of objects with C2 singularities, submitted, http://www.acm.caltech.edu/~emmanuel/publications.html, 2002

[5] E. Candès; F. Guo New multiscale transforms, minimum total variation synthesis: applications to edge-preserving image reconstruction, Signal Processing, Volume 82 (2002), pp. 1519-1543

[6] C. Fefferman A note on spherical summation multipliers, Israel J. Math., Volume 15 (1973), pp. 44-52

[7] H. Smith A Hardy space for Fourier integral operators, J. Geom. Anal., Volume 7 (1997)

[8] H. Smith A parametrix construction for wave equations with C1,1 coefficients, Ann. Inst. Fourier (Grenoble), Volume 48 (1998) no. 3, pp. 797-835

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