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Nonconvex compressed sampling of natural images and applications to compressed MR imaging. (English) Zbl 1251.94009

Summary: There have been proposed several compressed imaging reconstruction algorithms for natural and MR images. In essence, however, most of them aim at the good reconstruction of edges in the images. In this paper, a nonconvex compressed sampling approach is proposed for structure-preserving image reconstruction, through imposing sparseness regularization on strong edges and also oscillating textures in images. The proposed approach can yield high-quality reconstruction as images are sampled at sampling ratios far below the Nyquist rate, due to the exploitation of a kind of approximate \(\ell_0\) seminorms. Numerous experiments are performed on the natural images and MR images. Compared with several existing algorithms, the proposed approach is more efficient and robust, not only yielding higher signal-to-noise ratios but also reconstructing images of better visual effects.

MSC:

94A08 Image processing (compression, reconstruction, etc.) in information and communication theory
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[1] A. Skodras, C. Christopoulos, and T. Ebrahimi, “The JPEG 2000 still image compression standard,” IEEE Signal Processing Magazine, vol. 18, no. 5, pp. 36-58, 2001. · Zbl 0990.68071 · doi:10.1109/79.952804
[2] T. Acharya and P. S. Tsai, JPEG 2000 Standard for Image Compression, John Wiley & Sons, Hoboken, NJ, USA, 2005.
[3] A. K. Katsaggelos, R. Molina, and J. Mateos, Super-Resolution of Images and Videos, Morgan and Claypool, 2007.
[4] S. C. Park, M. K. Park, and M. G. Kang, “Super-resolution image reconstruction: a technical overview,” IEEE Signal Processing Magazine, vol. 20, no. 3, pp. 21-36, 2003. · doi:10.1109/MSP.2003.1203207
[5] E. J. Candes and T. Tao, “Decoding by linear programming,” IEEE Transactions on Information Theory, vol. 51, no. 12, pp. 4203-4215, 2005. · Zbl 1264.94121 · doi:10.1109/TIT.2005.858979
[6] E. J. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Transactions on Information Theory, vol. 52, no. 2, pp. 489-509, 2006. · Zbl 1231.94017 · doi:10.1109/TIT.2005.862083
[7] E. J. Candès and T. Tao, “Near-optimal signal recovery from random projections: universal encoding strategies?” IEEE Transactions on Information Theory, vol. 52, no. 12, pp. 5406-5425, 2006. · Zbl 1309.94033 · doi:10.1109/TIT.2006.885507
[8] E. J. Candès, J. K. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Communications on Pure and Applied Mathematics, vol. 59, no. 8, pp. 1207-1223, 2006. · Zbl 1098.94009 · doi:10.1002/cpa.20124
[9] E. Candès and J. Romberg, “Sparsity and incoherence in compressive sampling,” Inverse Problems, vol. 23, no. 3, article no. 008, pp. 969-985, 2007. · Zbl 1120.94005 · doi:10.1088/0266-5611/23/3/008
[10] E. Hale, W. Yin, and Y. Zhang, “A fixed-point continuation method for L1-regularized minimization with applications to compressed sensing,” Tech. Rep., Rice University, 2007.
[11] S. Ma, W. Yin, Y. Zhang, and A. Chakraborty, “An efficient algorithm for compressed MR imaging using total variation and wavelets,” in Proceedings of the 26th IEEE Conference on Computer Vision and Pattern Recognition (CVPR ’08), pp. 1-8, June 2008. · doi:10.1109/CVPR.2008.4587391
[12] K. T. Block, M. Uecker, and J. Frahm, “Undersampled radial MRI with multiple coils. Iterative image reconstruction using a total variation constraint,” Magnetic Resonance in Medicine, vol. 57, no. 6, pp. 1086-1098, 2007. · doi:10.1002/mrm.21236
[13] M. Lustig, D. Donoho, and J. M. Pauly, “Sparse MRI: the application of compressed sensing for rapid MR imaging,” Magnetic Resonance in Medicine, vol. 58, no. 6, pp. 1182-1195, 2007. · doi:10.1002/mrm.21391
[14] C. Y. Jong, S. Tak, Y. Han, and W. P. Hyun, “Projection reconstruction MR imaging using FOCUSS,” Magnetic Resonance in Medicine, vol. 57, no. 4, pp. 764-775, 2007. · doi:10.1002/mrm.21202
[15] H. Jung, J. C. Ye, and E. Kim, “Improved k-t BLAST and k-t SENSE using FOCUSS,” Physics in Medicine and Biology, vol. 52, no. 11, article 018, pp. 3201-3226, 2007. · doi:10.1088/0031-9155/52/11/018
[16] M. Seeger and H. Nickisch, “Compressed sensing and Bayesian experimental design,” in Proceedings of the 25th International Conference on Machine Learning, pp. 912-919, July 2008.
[17] J. Yang, Y. Zhang, and W. Yin, “A fast TVL1-L2 minimization algorithm for signal reconstruction from partial Fourier data,” Tech. Rep., Rice University, 2009.
[18] R. Chartrand, “Exact reconstruction of sparse signals via nonconvex minimization,” IEEE Signal Processing Letters, vol. 14, no. 10, pp. 707-710, 2007. · doi:10.1109/LSP.2007.898300
[19] E. J. Candès, “The restricted isometry property and its implications for compressed sensing,” Comptes Rendus Mathematique, vol. 346, no. 9-10, pp. 589-592, 2008. · Zbl 1153.94002 · doi:10.1016/j.crma.2008.03.014
[20] S. Kim, K. Koh, M. Lustig, S. Boyd, and D. Gorinevsky, “A method for large-scale L1-regularized least squares problems with applications in signal processing and statistics,” Tech. Rep., Stanford University, 2007.
[21] I. Daubechies, M. Defrise, and C. De Mol, “An iterative thresholding algorithm for linear inverse problems with a sparsity constraint,” Communications on Pure and Applied Mathematics, vol. 57, no. 11, pp. 1413-1457, 2004. · Zbl 1077.65055 · doi:10.1002/cpa.20042
[22] M. A. T. Figueiredo, R. D. Nowak, and S. J. Wright, “Gradient projection for sparse reconstruction: application to compressed sensing and other inverse problems,” IEEE Journal on Selected Topics in Signal Processing, vol. 1, no. 4, pp. 586-597, 2007. · doi:10.1109/JSTSP.2007.910281
[23] I. Daubechies, R. DeVore, M. Fornasier, and S. Güntürk, “Iteratively Re-weighted Least Squares minimization: proof of faster than linear rate for sparse recovery,” In Proceedings of the 42nd Annual Conference on Information Sciences and Systems, pp. 26-29, 2008.
[24] E. J. Candès, M. B. Wakin, and S. P. Boyd, “Enhancing sparsity by reweighted\ell 1 minimization,” Journal of Fourier Analysis and Applications, vol. 14, no. 5-6, pp. 877-905, 2008. · Zbl 1176.94014 · doi:10.1007/s00041-008-9045-x
[25] S. Ji, Y. Xue, and L. Carin, “Bayesian compressive sensing,” IEEE Transactions on Signal Processing, vol. 56, no. 6, pp. 2346-2356, 2008. · Zbl 1390.94231 · doi:10.1109/TSP.2007.914345
[26] S. D. Babacan, R. Molina, and A. K. Katsaggelos, “Bayesian compressive sensing using laplace priors,” IEEE Transactions on Image Processing, vol. 19, no. 1, Article ID 5256324, pp. 53-63, 2010. · Zbl 1371.94480 · doi:10.1109/TIP.2009.2032894
[27] J. L. Starck, M. Elad, and D. L. Donoho, “Image decomposition via the combination of sparse representations and a variational approach,” IEEE Transactions on Image Processing, vol. 14, no. 10, pp. 1570-1582, 2005. · Zbl 1288.94012 · doi:10.1109/TIP.2005.852206
[28] Y. Meyer, Oscillating Patterns in IImage Processing and Nonlinear Evolution Equation, vol. 22 of University Lecture Series, American Mathematical Society, 2001.
[29] R. Chartrand and W. Yin, “Iteratively reweighted algorithms for compressive sensing,” in Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing, pp. 3869-3872, April 2008. · doi:10.1109/ICASSP.2008.4518498
[30] G. Aubert and P. Kornprobst, Mathematical Problems in Image Processing, Springer, New York, NY, USA, 2000. · Zbl 1110.35001
[31] E. Esser, X. Zhang, and T. Chan, “A general framework for a class of first order primal-dual algorithms for TV minimization,” Tech. Rep., UCLA, 2009.
[32] E. J. Candès and M. B. Wakin, “An introduction to compressive sampling: a sensing/sampling paradigm that goes against the common knowledge in data acquisition,” IEEE Signal Processing Magazine, vol. 25, no. 2, pp. 21-30, 2008. · doi:10.1109/MSP.2007.914731
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