3-D Data Denoising and Inpainting with the Low-Redundancy Fast Curvelet Transform

Abstract

In this paper, we first present a new implementation of the 3-D fast curvelet transform, which is nearly 2.5 less redundant than the Curvelab (wrapping-based) implementation as originally proposed in Ying et al. (Proceedings of wavelets XI conference, San Diego, 2005) and Candès et al. (SIAM Multiscale Model. Simul. 5(3):861–899, 2006), which makes it more suitable to applications including massive data sets. We report an extensive comparison in denoising with the Curvelab implementation as well as other 3-D multi-scale transforms with and without directional selectivity. The proposed implementation proves to be a very good compromise between redundancy, rapidity and performance. Secondly, we exemplify its usefulness on a variety of applications including denoising, inpainting, video de-interlacing and sparse component separation. The obtained results are good with very simple algorithms and virtually no parameter to tune.

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References

  1. 1.

    Ballester, C., Bertalmío, V.C.M., Garrido, L., Marques, A., Ranchin, F.: An inpainting-based deinterlacing method. IEEE Trans. Image Process. 16, 2476–2491 (2007)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Candès, E., Donoho, D.: Curvelets—a surprisingly effective nonadaptive representation for objects with edges. In: Cohen, A., Rabut, C., Schumaker, L. (eds.) Curve and Surface Fitting: Saint-Malo 1999. Vanderbilt University Press, Nashville (1999)

    Google Scholar 

  3. 3.

    Candès, E., Donoho, D.: New tight frames of curvelets and optimal representations of objects with C 2 singularities. Commun. Pure Appl. Math. 57(2), 219–266 (2003)

    Article  Google Scholar 

  4. 4.

    Candès, E., Demanet, L., Donoho, D., Ying, L.: Fast discrete curvelet transforms. SIAM Multiscale Model. Simul. 5(3), 861–899 (2006)

    MATH  Article  Google Scholar 

  5. 5.

    Chandrasekaran, V., Wakin, M., Baron, D., Baraniuk, R.: Surflets: a sparse representation for multidimensional functions containing smooth discontinuities. In: Proceedings. International Symposium on Information Theory. ISIT 2004. July 2004

  6. 6.

    Chandrasekaran, V., Wakin, M., Baron, D., Baraniuk, R.: Representation and compression of multidimensional piecewise functions using surflets. IEEE Trans. Inf. Theory 55, 374–400 (2009)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Demanet, L.: Curvelets, wave atoms, and wave equations. PhD thesis, California Institute of Technology, May 2006

  8. 8.

    Demanet, L., Ying, L.: Curvelets and wave atoms for mirror-extended images. In: SPIE Wavelets XII Conference, August 2007

  9. 9.

    Donoho, D.: Wedgelets: nearly minimax estimation of edges. Ann. Stat. 27(3), 859–897 (1999)

    MathSciNet  MATH  Article  Google Scholar 

  10. 10.

    Doyle, T.: Interlaced to sequential conversion for EDTV applications. In: Proc. 2nd Int. Workshop Signal Processing of HDTV, pp. 412–430 (1990)

    Google Scholar 

  11. 11.

    Elad, M., Starck, J.-L., Querre, P., Donoho, D.: Simultaneous cartoon and texture image inpainting using morphological component analysis. Appl. Comput. Harmon. Anal. 19, 340–358 (2005)

    MathSciNet  MATH  Article  Google Scholar 

  12. 12.

    Fadili, M.J., Starck, J.-L., Murtagh, F.: Inpainting and zooming using sparse representations. Comput. J. 52(1), 64–79 (2007)

    Article  Google Scholar 

  13. 13.

    Haan, G.D., Bellers, E.B.: De-interlacing of video data. IEEE Trans. Consum. Electron. 43, 819–825 (1997)

    Article  Google Scholar 

  14. 14.

    Haan, G.D., Bellers, E.B.: Deinterlacing: an overview. Proc. IEEE 86, 1839–1857 (1998)

    Article  Google Scholar 

  15. 15.

    Hennenfent, G., Herrmann, F.: Seismic denoising with nonuniformly sampled curvelets. IEEE Comput. Sci. Eng. 8, 16–25 (2006)

    Google Scholar 

  16. 16.

    Herrmann, F., Hennenfent, G.: Non-parametric seismic data recovery with curvelet frames. Geophys. J. Int. 173(1), 233–248 (2008)

    Article  Google Scholar 

  17. 17.

    Kingsbury, N.: Complex wavelets for shift invariant analysis and filtering of signals. Appl. Comput. Harmon. Anal. 10, 234–253 (2001)

    MathSciNet  MATH  Article  Google Scholar 

  18. 18.

    Lu, Y., Do, M.: Multidimensional directional filter banks and surfacelets. IEEE Trans. Image Process. 16(4), 918–931 (2007)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Lu, Y., Do, M.N.: 3-D directional filter banks and surfacelets. In: Proc. of SPIE Conference on Wavelet Applications in Signal and Image Processing XI, San Diego, USA, 2005

  20. 20.

    Ma, J., Hussaini, M.: Three-dimensional curvelets for coherent vortex analysis of turbulence. Appl. Phys. Lett. 91, 184101 (2007)

    Article  Google Scholar 

  21. 21.

    Mallat, S.: A wavelet tour of signal processing. Academic Press, San Diego (1998)

    MATH  Google Scholar 

  22. 22.

    Oh, H., Kim, Y., Jung, Y., Ko, S., Morales, A.: Spatio-temporal edge-based median filtering for deinterlacing. In: IEEE International Conference on Consumer Electronics, pp. 52–53. IEEE, New York (1999/2000)

    Google Scholar 

  23. 23.

    Remi, K., Evans, A., Pike, G.: MRI simulation-based evaluation of image-processing and classification methods. IEEE Trans. Med. Imaging 18(11), 1085 (1999)

    Article  Google Scholar 

  24. 24.

    Romberg, J., Wakin, M., Baraniuk, R.: Multiscale wedgelet image analysis: fast decompositions and modeling. In: IEEE Int. Conf. on Image Proc. 2002, vol. 3, pp. 585–588 (2002)

    Google Scholar 

  25. 25.

    Selesnick, I.: The double-density dual-tree DWT. IEEE Trans. Image Process. 52, 1304–1314 (2004)

    MathSciNet  Google Scholar 

  26. 26.

    Starck, J., Bijaoui, A., Lopez, B., Perrier, C.: Image reconstruction by the wavelet transform applied to aperture synthesis. Astron. Astrophys. 283, 349–360 (1999)

    Google Scholar 

  27. 27.

    Starck, J., Donoho, D., Candès, E.: Very high quality image restoration by combining wavelets and curvelets. In: Laine, A., Unser, M., Aldroubi, A. (eds.) SPIE Conference on Signal and Image Processing: Wavelet Applications in Signal and Image Processing IX, San Diego, 1–4 August. SPIE, Bellingham (2001)

    Google Scholar 

  28. 28.

    Starck, J.-L., Candès, E., Donoho, D.: The curvelet transform for image denoising. IEEE Trans. Image Process. 11, 670–684 (2002)

    MathSciNet  Article  Google Scholar 

  29. 29.

    Starck, J.-L., Murtagh, F., Candès, E., Donoho, D.: Gray and color image contrast enhancement by the curvelet transform. IEEE Trans. Image Process. 12(6), 706–717 (2003)

    MathSciNet  Article  Google Scholar 

  30. 30.

    Starck, J., Nguyen, M., Murtagh, F.: Wavelets and curvelets for image deconvolution: a combined approach. Signal Process. 83, 2279–2283 (2003)

    MATH  Article  Google Scholar 

  31. 31.

    Starck, J., Nguyen, M., Murtagh, F.: Deconvolution based on the curvelet transform. In: International Conference on Image Processing, pp. 993–996 (2003)

    Google Scholar 

  32. 32.

    Starck, J., Elad, M., Donoho, D.: Redundant multiscale transforms and their application for morphological component analysis. Adv. Imaging Electron Phys. 132, 287–348 (2004)

    Article  Google Scholar 

  33. 33.

    Starck, J., Elad, M., Donoho, D.: Image decomposition via the combination of sparse representations and a variational approach. IEEE Trans. Image Process. 14, 1570–1582 (2005)

    MathSciNet  Article  Google Scholar 

  34. 34.

    Starck, J.-L., Murtagh, F., Fadili, M.: Sparse Signal and Image Processing: Wavelets, Curvelets and Morphological Diversity. Cambridge University Press, Cambridge (2010)

    MATH  Book  Google Scholar 

  35. 35.

    Tekalp, A.M.: Digital Video Processing. Prentice Hall, New York (1995)

    Google Scholar 

  36. 36.

    Wang, F., Anastassiou, D., Netravali, A.: Time-recursive deinterlacing for IDTV and pyramid coding. Signal Process. Image Commun. 2(3), 365–374 (1990)

    Article  Google Scholar 

  37. 37.

    Yang, S., Jung, Y., Young, H., Park, R.: Motion compensation assisted motion adaptive interlaced-to-progressive conversion. IEEE Trans. Circ. Syst. Video Technol. 14(9), 1138–1148 (2004)

    Article  Google Scholar 

  38. 38.

    Ying, L., Demanet, L., Candès, E.: 3D discrete curvelet transform. In: Proceedings of Wavelets XI Conference, San Diego, July 2005

  39. 39.

    Yoo, H., Jeong, J.: Direction-oriented interpolation and its application to de-interlacing. IEEE Trans. Consum. Electron. 48, 954–962 (2002)

    Google Scholar 

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Correspondence to A. Woiselle.

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Woiselle, A., Starck, JL. & Fadili, J. 3-D Data Denoising and Inpainting with the Low-Redundancy Fast Curvelet Transform. J Math Imaging Vis 39, 121–139 (2011). https://doi.org/10.1007/s10851-010-0231-5

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Keywords

  • 3-D curvelets
  • Sparsity
  • Denoising
  • Inpainting
  • Morphological component analysis
  • Video deinterlacing