Convex optimization and greedy iterative algorithms for dictionary learning in the presence of Rician noise
Keywords:Compressive Sensing, Convex Techniques, Greedy Iterative Algorithms, Magnetic Resonance Imaging.
Compressive Sensing (CS) is the emerging trend of recovering signal/image accurately from samples acquired at a rate far below the Nyquist rate. MR imaging provides a natural fit for applying CS as they can be sparsely represented in the transform domain. Magnitude MR images are corrupted by noise which follows Rician distribution and is difficult to remove as it is image dependent. Sparse coding is an important stage in dictionary learning. In this paper an attempt is being made to bring out a sparse coding technique which can provide better reconstruction in the presence of Rician noise. Greedy iterative algorithms and convex solutions are widely used for sparse coding. In the present work, performance of greedy algorithms, namely, Orthogonal Matching Pursuit (OMP) and Compressive Sampling Matching Pursuit (CoSaMP) have been evaluated and compared with convex techniques viz. Basis Pursuit (BP) and Least Absolute Shrinkage and Selection Operator (LASSO) in the sparse coding stage of an adaptive patch-based dictionary learning. Experiments have been carried out by varying Rician noise level from 0 to 30 and sparsity threshold per patch on MR images, acquired by employing various sampling schemes. Results show that greedy algorithms achieve higher PSNR and have very high computational speed compared to convex techniques when the MR images are corrupted with Rician noise.
 Saiprasad Ravishankar, and Yoram Bresler, â€œMR Image Reconstruction from Highly Undersampled k-Space Data by Dictionary Learningâ€, IEEE Transactions on Medical Imaging, Vol. 30, No. 5, pp. 1028-1041, May 2011. https://doi.org/10.1109/TMI.2010.2090538.
 M. Lustig, D. L. Donoho, J. M. Santos, and J. M. Pauly, â€œCompressed sensing MRIâ€, IEEE Signal Processing Mag., Mar. 2008, vol. 25, no. 2, pp. 72â€“82. https://doi.org/10.1109/MSP.2007.914728.
 Saad Qaisar, Rana Muhammad Bilal, Wafa Iqbal, Muqaddas Naureen, and Sungyoung Lee, â€œCompressive Sensing: From Theory to Applications, a Surveyâ€, Journal of Communications and Networks, October 2013, Vol. 15, No. 5. https://doi.org/10.1109/JCN.2013.000083.
 J. Tropp, and A. C. Gilbert, â€œSignal Recovery from Random Measurements via Orthogonal Matching Pursuitâ€, IEEE Transactions on Information Theory, 2007, vol. 53, no.12, pp: 4655-4666. https://doi.org/10.1109/TIT.2007.909108.
 D. Needell and J.A. Tropp., "CoSaMP: Iterative signal recovery from incomplete and inaccurate samples". Communications of the ACM, vol. 53, no. 12, December 2010, pp: 93-100. https://doi.org/10.1145/1859204.1859229.
 Scott Shaobing Cheny, David L. Donohoz, And Michael A. Saunders, â€œAtomic Decomposition by Basis Pursuitâ€, Society for Industrial and Applied Mathematics, 1998, Vol. 20, No. 1, pp. 33-61.
 Emmanuel Cand`es and Justin Romberg, â€œ -magic: Recovery of Sparse Signals via Convex Programmingâ€, Caltech, October 2005.
 R. Tibshirani, â€œRegression shrinkage and selection via the lassoâ€, J. R. Statist. Soc. B, 1996, vol. 58, pp. 267â€“288. https://doi.org/10.1111/j.2517-6161.1996.tb02080.x.
 David Donoho, Victoria Stodden and Yaakov Tsaig, About SparseLab, Stanford University, Version .100, May 2006.
 M. Aharon, M. Elad, and A. Bruckstein, â€œK-SVD: An algorithm for designing overcomplete dictionaries for sparse representation,â€ IEEE Trans. Signal Process., Nov. 2006, vol. 54, no. 11, pp. 4311â€“4322. https://doi.org/10.1109/TSP.2006.881199.
 M.V.R. Manimala, C.D. Naidu and M. N. Giriprasad, â€œSparse Recovery Algorithms Based on Dictionary Learning for MR Image Reconstructionâ€, IEEE 2016 International Conference on Wireless Networks, Signal Processing and Networking, March 2016, pp. 1354-1360. https://doi.org/10.1109/WiSPNET.2016.7566358.
 Gudbjartsson H, Patz S., â€œThe Rician Distribution on Noisy MRI Dataâ€, Magn Reson Med 1995, vol. 34, pp: 910-914. https://doi.org/10.1002/mrm.1910340618.
 Gilles Hennenfent and Felix J. Herrmann, â€œSimply denoise: Wavefield reconstruction via jittered undersamplingâ€, Geophysics, May-June 2008, Vol. 73, No. 3, P. V19â€“V28. https://doi.org/10.1190/1.2841038.
 Yang et al., â€œBrain MR image denoising for Rician noise using pre-smooth non-local means filterâ€, BioMedical Engineering OnLine 2015. https://doi.org/10.1186/1475-925X-14-2.
 Javier Portilla, Antonio Tristan-Vega, Ivan W. Selesnick, â€œEfficient and Robust Image Restoration using Multiple-Feature L2-relaxed Sparse Analysis Priorsâ€, IEEE Transactions on Image Processing, December 2015, Vol. 24, No. 12, pp. 5046-5059. https://doi.org/10.1109/TIP.2015.2478405.
 Suyash P. Awate and Ross T. Whitaker, â€œFeature-Preserving MRI Denoising: A Nonparametric Empirical Bayes Approachâ€, IEEE Transactions on Medical Imaging, September 2007, Vol. 26, No. 9, pp. 1242-1255. https://doi.org/10.1109/TMI.2007.900319.
 Yue Huangy, John Paisleyy, Qin Lin, Xinghao Dingz, Xueyang Fu and Xiao-ping Zhang, â€œBayesian Nonparametric Dictionary Learning for Compressed Sensing MRIâ€, IEEE Transactions on Image Processing, Dec. 2014, Vol. 23, No. 12, pp:5007-5019. https://doi.org/10.1109/TIP.2014.2360122.