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3-D data denoising and inpainting with the low-redundancy fast curvelet transform. (English) Zbl 1255.68273
Summary: 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 by Ying et al. and Candès et al., 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.

68U10 Computing methodologies for image processing
94A08 Image processing (compression, reconstruction, etc.) in information and communication theory
Full Text: DOI
[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) · Zbl 05453877 · doi:10.1109/TIP.2007.903844
[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)
[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) · Zbl 1038.94502 · doi:10.1002/cpa.10116
[4] Candès, E., Demanet, L., Donoho, D., Ying, L.: Fast discrete curvelet transforms. SIAM Multiscale Model. Simul. 5(3), 861–899 (2006) · Zbl 1122.65134 · doi:10.1137/05064182X
[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 · Zbl 1367.94077
[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) · Zbl 1367.94077 · doi:10.1109/TIT.2008.2008153
[7] Demanet, L.: Curvelets, wave atoms, and wave equations. PhD thesis, California Institute of Technology, May 2006
[8] Demanet, L., Ying, L.: Curvelets and wave atoms for mirror-extended images. In: SPIE Wavelets XII Conference, August 2007
[9] Donoho, D.: Wedgelets: nearly minimax estimation of edges. Ann. Stat. 27(3), 859–897 (1999) · Zbl 0957.62029 · doi:10.1214/aos/1018031261
[10] Doyle, T.: Interlaced to sequential conversion for EDTV applications. In: Proc. 2nd Int. Workshop Signal Processing of HDTV, pp. 412–430 (1990)
[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) · Zbl 1081.68732 · doi:10.1016/j.acha.2005.03.005
[12] Fadili, M.J., Starck, J.-L., Murtagh, F.: Inpainting and zooming using sparse representations. Comput. J. 52(1), 64–79 (2007) · Zbl 05534259 · doi:10.1093/comjnl/bxm055
[13] Haan, G.D., Bellers, E.B.: De-interlacing of video data. IEEE Trans. Consum. Electron. 43, 819–825 (1997) · doi:10.1109/30.628721
[14] Haan, G.D., Bellers, E.B.: Deinterlacing: an overview. Proc. IEEE 86, 1839–1857 (1998) · doi:10.1109/5.705528
[15] Hennenfent, G., Herrmann, F.: Seismic denoising with nonuniformly sampled curvelets. IEEE Comput. Sci. Eng. 8, 16–25 (2006) · Zbl 05092100 · doi:10.1109/MCSE.2006.49
[16] Herrmann, F., Hennenfent, G.: Non-parametric seismic data recovery with curvelet frames. Geophys. J. Int. 173(1), 233–248 (2008) · doi:10.1111/j.1365-246X.2007.03698.x
[17] Kingsbury, N.: Complex wavelets for shift invariant analysis and filtering of signals. Appl. Comput. Harmon. Anal. 10, 234–253 (2001) · Zbl 0990.94005 · doi:10.1006/acha.2000.0343
[18] Lu, Y., Do, M.: Multidimensional directional filter banks and surfacelets. IEEE Trans. Image Process. 16(4), 918–931 (2007) · Zbl 05453802 · doi:10.1109/TIP.2007.891785
[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] Ma, J., Hussaini, M.: Three-dimensional curvelets for coherent vortex analysis of turbulence. Appl. Phys. Lett. 91, 184101 (2007)
[21] Mallat, S.: A wavelet tour of signal processing. Academic Press, San Diego (1998) · Zbl 0937.94001
[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)
[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) · doi:10.1109/42.816072
[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)
[25] Selesnick, I.: The double-density dual-tree DWT. IEEE Trans. Image Process. 52, 1304–1314 (2004) · Zbl 1370.94234 · doi:10.1109/TSP.2004.826174
[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)
[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)
[28] Starck, J.-L., Candès, E., Donoho, D.: The curvelet transform for image denoising. IEEE Trans. Image Process. 11, 670–684 (2002) · Zbl 1288.94011 · doi:10.1109/TIP.2002.1014998
[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) · Zbl 1288.94013 · doi:10.1109/TIP.2003.813140
[30] Starck, J., Nguyen, M., Murtagh, F.: Wavelets and curvelets for image deconvolution: a combined approach. Signal Process. 83, 2279–2283 (2003) · Zbl 1145.94329 · doi:10.1016/S0165-1684(03)00150-6
[31] Starck, J., Nguyen, M., Murtagh, F.: Deconvolution based on the curvelet transform. In: International Conference on Image Processing, pp. 993–996 (2003)
[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) · doi:10.1016/S1076-5670(04)32006-9
[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) · Zbl 1288.94012 · doi:10.1109/TIP.2005.852206
[34] Starck, J.-L., Murtagh, F., Fadili, M.: Sparse Signal and Image Processing: Wavelets, Curvelets and Morphological Diversity. Cambridge University Press, Cambridge (2010) · Zbl 1196.94008
[35] Tekalp, A.M.: Digital Video Processing. Prentice Hall, New York (1995)
[36] Wang, F., Anastassiou, D., Netravali, A.: Time-recursive deinterlacing for IDTV and pyramid coding. Signal Process. Image Commun. 2(3), 365–374 (1990) · doi:10.1016/0923-5965(90)90012-7
[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) · Zbl 05451650 · doi:10.1109/TCSVT.2004.833163
[38] Ying, L., Demanet, L., Candès, E.: 3D discrete curvelet transform. In: Proceedings of Wavelets XI Conference, San Diego, July 2005
[39] Yoo, H., Jeong, J.: Direction-oriented interpolation and its application to de-interlacing. IEEE Trans. Consum. Electron. 48, 954–962 (2002)
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