This Article 
   
 Share 
   
 Bibliographic References 
   
 Add to: 
 
Digg
Furl
Spurl
Blink
Simpy
Google
Del.icio.us
Y!MyWeb
 
 Search 
   
Computing with Curvelets: From Image Processing to Turbulent Flows
March/April 2009 (vol. 11 no. 2)
pp. 72-80
Jianwei Ma, Tsinghua University
Gerlind Plonka, University of Duisburg-Essen
The curvelet transform allows an almost optimal nonadaptive sparse representation for curve-like features and edges. The authors describe some recent applications involving image processing, seismic data exploration, turbulent flows, and compressed sensing.

1. E. Candes et al., "Fast Discrete Curvelet Transforms," Multiscale Modeling and Simulation, vol. 5, no. 3, 2006, pp. 861–899.
2. E. Candes and D. Donoho, "Curvelets: A Surprisingly Effective Nonadaptive Representation for Objects with Edges," Curves and Surface Fitting: Saint-Malo 1999, A. Cohen, C. Rabut, and L. Schumaker eds., , Vanderbilt Univ. Press, 2000, pp. 105–120.
3. E. Candes and D. Donoho, "New Tight Frames of Curvelets and Optimal Representations of Objects with Piecewise Singularities," Comm. Pure and Applied Mathematics, vol. 57, no. 2, 2004, pp. 219–266.
4. J. Ma, "Curvelets for Surface Characterization," Applied Physics Letters, vol. 90, no. 5, 2007, pp. 054109:1–3.
5. J. Ma, A. Antoniadis, and F.-X. Le Dimet, "Curvelets-Based Multiscale Detection and Tracking for Geophysical Fluids," IEEE Trans. Geoscience and Remote Sensing, vol. 44, no. 12, 2006, pp. 3626–3637.
6. J. Ma and G. Plonka, "Combined Curvelet Shrinkage and Nonlinear Anisotropic Diffusion," IEEE Trans. Image Processing, vol. 16, no. 9, 2007, pp. 2198–2206.
7. J. Starck, E. Candes, and D. Donoho, "The Curvelet Transform for Image Denoising," IEEE Trans. Image Processing, vol. 11, no. 6, 2002, pp. 670–684.
8. L. Ying, L. Demanet, and E. Candes, "3D Discrete Curvelet Transform," Proc. SPIE Wavelets XI, vol. 5914, SPIE, 2005, p. 413.
9. N. Kingsbury, "Image Processing with Complex Wavelets," Philosophical Trans. Royal Soc. A., vol. 357, no. 1760, 1999, pp. 2543–2560.
10. E. Candes and D. Donoho, "Ridgelets: A Key to Higher-Dimensional Intermittency?," Philosophical Trans. Royal Soc. A., vol. 357, no. 1760, 1999, pp. 2495–2509.
11. J. Ma and M. Fenn, "Combined Complex Ridgelet Shrinkage and Total Variation Minimization," SIAM J. Scientific Computing, vol. 28, no. 3, 2006, pp. 984–1000.
12. D. Donoho, "Wedgelets: Nearly Minimax Estimation of Edges," Ann. Statistics, vol. 27, no. 3, 1999, pp. 859–897.
13. E. Le Pennec and S. Mallat, "Sparse Geometrical Image Approximation with Bandlets," IEEE Trans. Image Processing, vol. 14, no. 4, 2005, pp. 423–438.
14. M. Do and M. Vetterli, "The Contourlet Transform: An Efficient Directional Multiresolution Image Representation," IEEE Trans. Image Processing, vol. 14, no. 12, 2005, pp. 2091–2106.
15. K. Guo and D. Labate, "Optimally Sparse Multidimensional Representation Using Shearlets," SIAM J. Mathematical Analysis, vol. 39, no. 1, 2007, pp. 298–318.
16. R. Willett and K. Nowak, "Platelets: A Multiscale Approach for Recovering Edges and Surfaces in Photon-Limited Medical Imaging," IEEE Trans. Medical Imaging, vol. 22, no. 3, 2003, pp. 332–350.
17. J. Starck et al., "Gray and Color Image Contrast Enhancement by the Curvelet Transform," IEEE Trans. Image Processing, vol. 12, no. 6, 2003, pp. 706–717.
18. M. Choi et al., "Fusion of Multispectral and Panchromatic Satellite Images Using the Curvelet Transform," IEEE Geoscience Remote Sensing Letters, vol. 2, no. 2, 2005, pp. 136–140.
19. S. Dekel and A. Sherman, "Curvelets: A Low-Level Framework for Computer Vision," preprint, GE Healthcare, 2008.
20. G. Plonka and J. Ma, "Nonlinear Regularized Reaction-Diffusion Filters for Denoising of Images with Textures," IEEE Trans. Image Processing, vol. 17, no. 8, 2008, pp. 1283–1294.
21. G. Hennenfent and F. Herrmann, "Seismic Denoising with Nonuniformly Sampled Curvelets," Computing in Science &Eng., vol. 8, no. 3, 2006, pp. 16–25.
22. R. Neelamani et al., "Coherent and Random Noise Attenuation Using the Curvelet Transform," The Leading Edge, vol. 27, no. 2, 2008, pp. 240–248.
23. H. Douma and M. de Hoop, "Leading-Order Seismic Imaging Using Curvelets," Geophysics, vol. 72, no. 6, 2007, pp. S231–S248.
24. H. Chauris and T. Nguyen, "Seismic Demigration/Migration in the Curvelet Domain," Geophysics, vol. 73, no. 2, 2008, pp. S35–S46.
25. F. Herrmann, D. Wang, and D. Verschuur, "Adaptive Curvelet-Domain Primary-Multiple Separation," Geophysics, vol. 73, no. 3, 2008, pp. A17–A21.
26. F. Herrmann, P. Moghaddam, and C. Stolk, "Sparsity- and Continuity-Promoting Seismic Image Recovery with Curvelet Frames," Applied and Computational Harmonic Analysis, vol. 24, no. 2, 2008, pp. 150–173.
27. I. Bermejo-Moreno and D. Pullin, "On the Non-Local Geometry of Turbulence," J. Fluid Mechanics, vol. 603, 2008, pp. 101–135.
28. J. Ma and M. Hussaini, "Three-Dimensional Curvelets for Coherent Vortex Analysis of Turbulence," Applied Physics Letters, vol. 91, no. 18, 2007, pp. 184101:1–3.
29. E. Candes, J. Romberg, and T. Tao, "Stable Signal Recovery from Incomplete and Inaccurate Measurements," Comm. Pure Applied Mathematics, vol. 59, no. 8, 2005, pp. 1207–1233.
30. E. Candes and T. Tao, "Decoding by Linear Programming," IEEE Trans. Information Theory, vol. 51, no. 12, 2005, pp. 4203–4215.
31. D. Donoho, "Compressed Sensing," IEEE Trans. Information Theory, vol. 52, no. 4, 2006, pp. 1289–1306.
32. J. Ma and F.-X. Le Dimet, "Deblurring from Highly Incomplete Measurements for Remote Sensing," to be published in IEEE Trans. Geoscience Remote Sensing, 2008.

Index Terms:
Computer simulations, computational science, engineering, curvelets, turbulent flow
Citation:
Jianwei Ma, Gerlind Plonka, "Computing with Curvelets: From Image Processing to Turbulent Flows," Computing in Science and Engineering, vol. 11, no. 2, pp. 72-80, March-April 2009, doi:10.1109/MCSE.2009.26
Usage of this product signifies your acceptance of the Terms of Use.