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Continuous curvelet transform. I: Resolution of the wavefront set. (English) Zbl 1086.42022
The authors introduce the continuous curvelet transform and study its applications to the analysis of singularities. Under admissibility conditions for two window functions, they give a reproducing formula similar to that for the continuous wavelet transform. Then they show how the decay of the continuous curvelet transform can be used to characterize singularities of functions in various forms: point singularity, linear singularity, polygonal singularity, curvilinear singularity, and in general the singular support.

##### MSC:
 42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
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##### References:
 [1] J. Bros, D. Iagolnitzer, Support essentiel et structure analytique des distributions, Séminaire Goulaouic-Lions-Schwartz, exp. no. 19, 1975-1976 · Zbl 0333.46029 [2] Candès, E.J.; Guo, F., New multiscale transforms, minimum total variation synthesis: applications to edge-preserving image reconstruction, Signal process., 82, 1519-1543, (2002) · Zbl 1009.94510 [3] Candès, E.J.; Demanet, L., Curvelets and Fourier integral operators, C. R. acad. sci. Paris, Sér. I, 359-398, (2003) · Zbl 1056.42025 [4] Candès, E.J.; Donoho, D.L., Curvelets: a surprisingly effective nonadaptive representation of objects with edges, () [5] Candès, E.J.; Donoho, D.L., New tight frames of curvelets and optimal representations of objects with piecewise $$C^2$$ singularities, Comm. pure appl. math., 219-266, (2004) · Zbl 1038.94502 [6] E.J. Candès, D.L. Donoho, Continuous curvelet transform II. Discretization and frames, Manuscript, 2002 [7] Cordoba, A.; Fefferman, C., Wave packets and Fourier integral operators, Comm. partial differential equations, 3, 979-1005, (1978) · Zbl 0389.35046 [8] Delort, J.M., FBI transformation: second microlocalization and semilinear caustics, Lecture notes in mathematics, (1992) · Zbl 0760.35004 [9] Donoho, D.L., Sparse components of images and optimal atomic decomposition, Constr. approx., 17, 3, 353-382, (2001) · Zbl 0995.65150 [10] Duistermaat, J.J., Fourier integral operators, Progress in mathematics, (1996), Birkhäuser Basel · Zbl 0841.35137 [11] Fefferman, C., A note on spherical summation multipliers, Israel J. math., 15, 44-52, (1973) · Zbl 0262.42007 [12] Frazier, M.; Jawerth, B.; Weiss, G., Littlewood – paley theory and the study of function spaces, () · Zbl 0757.42006 [13] Gérard, P., Moyennisation et regularité deux-microlocale, Ann. sci. école norm. sup. (4), 23, 89-121, (1990) · Zbl 0725.35003 [14] Hörmander, L., The analysis of linear partial differential operators, (1983), Springer New York/Berlin [15] Meyer, Y., Ondelettes et operateurs: I, (1990), Hermann Paris · Zbl 0694.41037 [16] Seeger, A.; Sogge, C.D.; Stein, E.M., Regularity properties of Fourier integral operators, Ann. math., 134, 231-251, (1993) · Zbl 0754.58037 [17] Simoncelli, E.P.; Freeman, W.T.; Adelson, E.H.; Heeger, D.J., Shiftable multi-scale transforms, IEEE trans. inform. theory, special issue on wavelets, 38, 2, 587-607, (March 1992) [18] J. Sjostrand, Singularités Analytiques Microlocales Asterisque 95, 1992 [19] Smith, H.F., A Hardy space for Fourier integral operators, J. geom. anal., 8, 4, 629-653, (1998) · Zbl 1031.42020 [20] H.F. Smith, Wave equations with low regularity coefficients, Documenta Mathematica, Extra Volume ICM 1998, II, 723-730 · Zbl 0909.35081 [21] Stein, E.M., Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, (1993), Princeton Univ. Press Princeton, NJ · Zbl 0821.42001 [22] Sogge, C.D., Fourier integrals in classical analysis, (1993), Cambridge Univ. Press Cambridge · Zbl 0783.35001 [23] Watson, A.B., The cortex transform: rapid computation of simulated neural images, Computer vision graphics image process., 39, 3, 311-327, (September 1987)
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