New tight frames of curvelets and optimal representations of objects with piecewise C2 singularities

Communications on Pure and Applied Mathematics

Volumn 57, Issue 2, 2004, Pages 0219-0266

  Candès, Emmanuel J ^a   Donoho, David L ^b  

^a CALIFORNIA INSTITUTE OF TECHNOLOGY   (United States)

^b STANFORD UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 2442458882     PISSN: 00103640     EISSN: None     Source Type: Journal    
DOI: 10.1002/cpa.10116     Document Type: Article

References (37)

1. Antoine J. P.; Murenzi, R. Two-dimensional directional wavelets and the scale-angle representation. IEEE Signal Processing 52 (1996), 259-281, 1996.
2. Candès, E. J. Harmonic analysis of neural networks. Appl. Comput. Harmon. Anal. 6 (1999), 197-218.
3. Candès, E. J. Ridgelets and the representation of mutilated Sobolev functions. SIAM J. Math. Anal. 33 (2001), 347-368.
4. Candès, E. J.; Demanet, L. Curvelets, warpings, and optimally sparse representations of Fourier integral operators. Technical report, 2002. Available online at: http://www.acm.caltech.edu/~demanet/.
5. Candes, E. J.; Demanet, L. Curvelets and Fourier integral operators. C. R. Acad. Sci. Paris, Sér. I Math. 336 (2003), 395-398.
6. Candès, E. J.; Donoho, D. L. Ridgelets: a key to higher-dimensional intermittency? R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci. 357 (1999), no. 1760, 2495-2509.
7. Candès, E. J.; Donoho, D. L. Curvelets-a surprisingly effective nonadaptive representation for objects with edges. Curves and surfaces, 105-120. C. Rabut, A. Cohen, and L. L. Schumaker, eds. Vanderbilt University Press, Nashville, TN, 2000.
8. Candès, E. J.; Donoho, D. L. Recovering edges in ill-posed inverse problems: optimality of curvelet frames. Ann. Statist. 30 (2002), no. 3, 784-842.
9. Candès, E. J.; Donoho, D. L. Directional wavelet transform via parabolic scaling. I. Resolution of the wavefront set. objects with edges. Technical Report, Stanford University, 2003.
10. Candès, E. J.; Guo, F. New multiscale transforms, minimum total variation synthesis: applications to edge-preserving image reconstruction. IEEE Signal Processing 82 (2002), 1519-1543.
11. Daubechies, I. Ten lectures on wavelets. CBMS-NSF Regional Conference Series in Applied Mathematics, 61. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 1992.
12. Daubechies, I.; Grossmann, A.; Meyer, Y. Painless nonorthogonal expansions. J. Math. Phys. 27 (1986), no. 5, 1271-1283.
13. Do, M. N. Directional multiresolution image representations. Thesis, Swiss Federal Institute of Technology, Zürich, 2001.
14. Do, M. N.; Vetterli, M. Contourlets. Beyond wavelets. J. Stoeckler and G. V. Welland, eds. Academic, New York, 2002.
15. Donoho, D. L. Unconditional bases are optimal bases for data compression and for statistical estimation. Appl. Comput. Harmon. Anal. 1 (1993), no. 1, 100-115.
16. Donoho, D. L. De-noising by soft-thresholding. IEEE Trans. Inform. Theory 41 (1995), no. 3, 613-627.
17. Donoho, D. L. Unconditional bases and bit-level compression. Appl. Comput. Harmon. Anal. 3 (1996), no. 4, 388-392.
18. Donoho, D. L. Wedgelets: nearly-minimax estimation of edges. Ann. Statist. 27 (1999), no. 3, 859-897.
19. Donoho, D. L. Sparse components of images and optimal atomic decompositions. Constr. Approx. 17 (2001), no. 3, 353-382.
20. Donoho, D. L.; Duncan, M. R. Digital curvelet transform: strategy, implementation, experiments. Technical report, Stanford University, 1999.
21. Donoho, D. L.; Johnstone, I. M. Empirical atomic decomposition. Unpublished manuscript, 1995.
22. Donoho, D. L.; Vetterli, M.; DeVore, R. A.; Daubechies, I. Data compression and harmonic analysis. IEEE Trans. Inform. Theory 44 (1998), no. 6, 2435-2476.
23. Fefferman, C. A note on spherical summation multipliers. Israel J. Math. 15 (1973), 44-52.
24. Flesia, A. G.; Hel-Or, H.; Averbuch, A.; Candès, E. J.; Coifman, R. R.; Donoho, D. L. Digital implementation of ridgelet packets. Beyond wavelets. J. Stoeckler and G. V. Welland, eds. Academic, New York, 2002.
25. Frazier, M.; Jawerth, B.; Weiss, G. Littlewood-Paley theory and the study of function spaces. CBMS Regional Conference Series in Mathematics, 79. American Mathematical Society, Providence, R.I., 1991.
26. Gressman, P.; Labate, D.; Weiss, G.; Wilson, E. Affine, quasi-affine and co-affine wavelets. Beyond wavelets. J. Stoeckler and G. V. Welland, eds. Academic, New York, 2002.
27. Kashin, B. S. Approximation properties of complete orthonormal systems. Trudy Mat. Inst. Steklov. 172 (1985), 187-191. English translation in Proc. Steklov Inst. Math. (1987).
28. Kashin, B. S. Approximation properties of complete orthonormal systems. Trudy Mat. Inst. Steklov. 172 (1985), 187-191. English translation in Proc. Steklov Inst. Math. (1987).
29. Lemarié, P. G.; Meyer, Y. Ondelettes et bases Hilbertiennes. Rev. Mat. Iberoamericana 2 (1986), 1-18.
30. Mallat, S. A wavelet tour of signal processing. Academic, San Diego, 1998.
31. Meyer, Y. Wavelets: algorithms and applications. SIAM, Philadelphia, 1993.
32. Simoncelli, E. P.; Freeman, W. T.; Adelson, E. H.; Heeger, D. J. Shiftable multi-scale transforms. IEEE Trans. Inform. Theory 38 (1992), no. 2, 587-607.
33. 1,1 coefficients. Ann. Inst. Fourier (Grenoble) 48 (1998), no. 3, 797-835.
34. Starck, J.-L.; Candès, E. J.; Donoho, D. L. Very high quality image restoration by combining wavelets and curvelets. Wavelets: applications in signal and image processing IX, 9-19. M. A. Unser, A. Aldroubi, and A. F. Laine, eds. Proceedings of SPIE, 4478. The International Society for Optical Engineering, Bellingham, Wash., 2001.
35. Starck, J.-L.; Candès, E. J.; Donoho, D. L. The curvelet transform for image denoising. IEEE Trans. Image Process. 11 (2002), 670-684.
36. Stein, E. M. Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals. Princeton Mathematical Series, 43. Monographs in Harmonic Analysis, III. Princeton University, Princeton, N.J., 1993.
37. Taubman, D. High performance scalable image compression with EBCOT. IEEE Trans. Image Process. 9 (2000), no. 7. 1151-1170.