Fast discrete curvelet transforms

Multiscale Modeling and Simulation

Volumn 5, Issue 3, 2006, Pages 861-899

  Candès, Emmanuel ^a   Demanet, Laurent ^a   Donoho, David ^b   Ying, Lexing ^a  

^a CALIFORNIA INSTITUTE OF TECHNOLOGY   (United States)

^b STANFORD UNIVERSITY   (United States)

Author keywords

Digital shear; Fast Fourier transforms; Filtering; Interpolation; Smooth partitioning; Two dimensional and three dimensional curvelet transforms; Unequally spaced fast Fourier transforms; Wrapping

Indexed keywords

ALGORITHMS; INTERPOLATION; THREE DIMENSIONAL; TWO DIMENSIONAL; VECTORS;

CARTESIAN ARRAY; CURVELET TRANSFORM; SMOOTH PARTITIONING;

DISCRETE FOURIER TRANSFORMS;

EID: 33845602456     PISSN: 15403459     EISSN: 15403467     Source Type: Journal    
DOI: 10.1137/05064182X     Document Type: Article

References (42)

1. C. ANDERSON AND M. D. DAHLEH, Rapid computation of the discrete Fourier transform, SIAM J. Sci. Comput., 17 (1996), pp. 913-919.
2. G. BEYLKIN, R. COIFMAN, AND V. ROKHLIN, Fast wavelet transforms and numerical algorithms, Comm. Pure Appl. Math., 44 (1991), pp. 141-183.
3. G. BEYLKIN, On the fast Fourier transform of functions with singularities, Appl. Comput. Harmon. Anal., 2 (1995), pp. 363-381.
4. E. J. CANDÈS, Harmonic analysis of neural networks, Appl. Comput. Harmon. Anal., 6 (1999), pp. 197-218.
5. E. J. CANDÈS AND L. DEMANET, The curvelet representation of wave propagators is optimally sparse, Comm. Pure Appl. Math., 58 (2005), pp. 1472-1528.
6. L. DEMANET, Curvelets, Wave Atoms, and Wave Equations, Ph.D. thesis, California Institute of Technology, Pasadena, CA, 2006.
7. E. J. CANDES AND D. L. DONOHO, Ridgelets: The key to higher-dimensional intermittency?, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci., 357 (1999), pp. 2495-2509.
8. E. J. CANDÈS AND D. L. DONOHO, Curvelets - a surprisingly effective nonadaptive representation for objects with edges, in Curves and Surfaces, C. Rabut, A. Cohen, and L. L. Schumaker, eds., Vanderbilt University Press, Nashville, TN, 2000, pp. 105-120.
9. E. J. CANDÈS AND D. L. DONOHO, Recovering edges in ill-posed inverse problems: Optimality of curvelet frames, Ann. Statist., 30 (2002), pp. 784-842.
10. 2 singularities, Comm. Pure Appl. Math., 57 (2004), pp. 219-266.
11. E. J. CANDÈS AND D. L. DONOHO, Curvelets: New Tools for Limited-Angle Tomography, manuscript, 2004.
12. E. J. CANDÈS AND F. GUO, New multiscale transforms, minimum total variation synthesis: Application to edge-preserving image reconstruction, Sig. Process., 82 (2002), pp. 1519-1543.
13. R. H. CHAN AND M. K. NG, Conjugate gradient methods for Toeplitz systems, SIAM Rev., 38 (1996), pp. 427-482.
14. M. N. DO, Directional Multiresolution Image Representations, Ph.D. thesis, Swiss Federal Institute of Technology, Lausanne, Switzerland, 2001.
15. M. N. DO AND M. VETTERLI, Contourlets, in Beyond Wavelets, G. V. Welland, ed., Academic Press, San Diego, CA, 2003.
16. M. N. DO AND M. VETTERLI, The contourlet transform: An efficient directional multiresolution image representation, IEEE Trans. Image Process., 14 (2005), pp. 2091-2106.
17. D. L. DONOHO, Wedgelets: Nearly-minimax estimation of edges, Ann. Statist., 27 (1999), pp. 859-897.
18. D. L. DONOHO AND M. R. DUNCAN, Digital Curvelet Transform: Strategy, Implementation, Experiments, Technical report, Stanford University, Stanford, CA, 1999.
19. D. L. DONOHO AND X. HUO, Beamlets and multiscale image analysis, in Multiscale and Multiresolution Methods, Lect. Notes Comput. Sci. Eng. 20, Springer, Berlin, 2001, pp. 149-196.
20. H. DOUMA AND M. V. DE HOOP, Wave-character preserving prestack map migration using curvelets, presentation at the Society of Exploration Geophysicists, Denver, CO, 2004.
21. A. J. W. DUIJNDAM AND M. A. SCHONEWILLE, Nonuniform fast Fourier transform, Geophys., 64 (1999), pp. 539-551.
22. A. DUTT AND V. ROKHLIN, Fast Fourier transforms for nonequispaced data, SIAM J. Sci. Comput., 14 (1993), pp. 1368-1393.
23. A. DUTT AND V. ROKHLIN, Fast Fourier transforms for nonequispaced data II, Appl. Comput. Harmon. Anal., 2 (1995), pp. 85-100.
24. A. G. FLESIA, H. HEL-OR, A. AVERBUCH, E. J. CANDÈS, R. R. COIFMAN, AND D. L. DONOHO, Digital implementation of ridgelet packets, in Beyond Wavelets, J. Stoeckler and G. V. Welland, eds., Academic Press, San Diego, CA, 2003.
25. G. HENNENFENT AND F. J. HERRMANN, Seismic denoising with unstructured curvelets, Comput. in Sci. Eng., to appear.
26. F. J. HERRMANN AND E. VERSCHUUR, Separation of primaries and multiples by non-linear estimation in the curvelet domain, in Proceedings of the EAGE 66th Annual Conference & Exhibition, Paris, France, 2004.
27. F. J. HERRMANN, P. P. MOGHADDAM, AND C. C. STOLK, Sparsity- and continuity-promoting seismic image recovery with curvelet frames, submitted.
28. K. GUO, D. LABATE, W. LIM, G. WEISS, AND E. WILSON, Wavelets with composite dilations, Electron. Res. Announc. Amer. Math. Soc., 10 (2004), pp. 78-87.
29. K. GUO, D. LABATE, W. LIM, G. WEISS, AND E. WILSON, Wavelets with composite dilations and their MRA properties, Appl. Comput. Harmon. Anal., 20 (2006), pp. 202-236.
30. D. LABATE, W. LIM, G. KUTYNIOK, AND G. WEISS, Sparse multidimensional representation using shearlets, in Proceedings of the SPIE Conference on Wavelets XI, San Diego, CA, 2005.
31. E. LE PENNEC AND S. MALLAT, Sparse geometric image representations with bandelets, IEEE Trans. Image Process., 14 (2005), pp. 423-438.
32. R. MURENZI, Ondelettes multidimensionelles et applications a l'analyse d'images, Thèse, Université catholique de Louvain, Louvain-la-Neuve, Belgium, 1990.
33. F. NATTERER, The Mathematics of Computerized Tomography, B. G. Teubner, Stuttgart, John Wiley & Sons, Chichester, UK, 1986; reprinted, SIAM, Philadelphia, 2001.
34. E. P. SIMONCELLI AND W. T. FREEMAN, The steerable pyramid: A flexible architecture for multi-scale derivative computation, in Proceedings of the IEEE Second International Conference on Image Processing, Washington, DC, 1995.
35. E. P. SIMONCELLI, W. T. FREEMAN, E. H. ADELSON, AND D. J. HEEGER , Shiftable multi-scale transforms [or what's wrong with orthonormal wavelets], IEEE Trans. Inform. Theory, 38 (1992), pp. 587-607.
36. 1,1 coefficients, Ann. Inst. Fourier (Grenoble), 48 (1998), pp. 797-835.
37. J. L. STARCK, E. J. CANDÈS, AND D. L. DONOHO, The curvelet transform for image denoising, IEEE Trans. Image Process., 11 (2002), pp. 670-684.
38. J.-L. STARCK, N. AGHANIM, AND O. FORNI, Detecting cosmological non-Gaussian signatures by multi-scale methods, Astronomy and Astrophysics, 416 (2004), pp, 9-17.
39. J. L. STARCK, M. ELAD, AND D. L. DONOHO, Redundant multiscale transforms and their application for morphological component analysis, Advances in Imaging and Electron Physics, 132 (2004), pp. 132-195.
40. G. STRANG, A proposal for Toeplitz matrix calculations, Stud. Appl. Math., 74 (1986), pp. 171-176.
41. P. VANDERGHEYNST AND J. F. GOBBERS, Directional dyadic wavelet transforms: Design and algorithms, IEEE Trans. Image Process., 11 (2002), pp. 363-372.
42. A. ZYGMUND, Trigonometric Series, Cambridge University Press, Cambridge, UK, 1964.