Abstract
Background: In order to overcome the limitation of long scanning time, compressive sensing (CS) technology exploits the sparsity of image in some transform domain to reduce the amount of acquired data. Therefore, CS has been widely used in magnetic resonance imaging (MRI) reconstruction.
Discussion: Blind compressed sensing enables to recover the image successfully from highly under- sampled measurements, because of the data-driven adaption of the unknown transform basis priori. Moreover, analysis-based blind compressed sensing often leads to more efficient signal reconstruction with less time than synthesis-based blind compressed sensing. Recently, some experiments have shown that nonlocal low-rank property has the ability to preserve the details of the image for MRI reconstruction.
Methods: Here, we focus on analysis-based blind compressed sensing, and combine it with additional nonlocal low-rank constraint to achieve better MR images from fewer measurements. Instead of nuclear norm, we exploit non-convex Schatten p-functionals for the rank approximation.
Results & Conclusion: Simulation results indicate that the proposed approach performs better than the previous state-of-the-art algorithms.
Keywords: Blind compressed sensing, nonlocal low-rank, nonconvex optimization, low-rank approximation, Schatten p-functionals, Magnetic Resonance Imaging (MRI).
Graphical Abstract
[1]
Lustig M, Donoho DL, Santos JM, Pauly JM. Compressed sensing MRI. IEEE Signal Process Mag 2008; 25(2): 72-82.
[2]
Weizman L, Eldar YC, Ben Bashat D. Compressed sensing for longitudinal MRI: An adaptive-weighted approach. Med Phys 2015; 42(9): 5195-208.
[4]
Ravishankar S, Bresler Y. MR image reconstruction from highly undersampled k-space data by dictionary learning. IEEE Trans Med Imaging 2011; 30(5): 1028-41.
[6]
Aharon M, Elad M, Bruckstein A. The K-SVD: An algorithm for designing overcomplete dictionaries for sparse representation. IEEE Trans Signal Process 2006; 54(11): 4311-22.
[7]
Huang Y, Paisley J, Lin Q, Ding X, Fu X, Zhang XP. Bayesian nonparametric dictionary learning for com- pressed sensing MRI. IEEE Trans Image Process 2014; 23(12): 5007-19.
[8]
Lingala SG, Jacob M. Blind compressive sensing dynamic MRI. IEEE Trans Med Imaging 2013; 32(6): 1132-45.
[9]
Ravishankar S, Bresler Y. Data-driven learning of a union of sparsifying transforms model for blind compressed sensing. IEEE Trans Comput Imaging 2016; 2(3): 294-309.
[10]
Ravishankar S, Bresler Y. Blind compressed sensing using sparsifying transforms. In: International conference on Sampling Theory and Applications (SampTA) 2 0 1 5 . IEEE: Washington, DC, USA; pp. 513-7.
[11]
Ravishankar S, Bresler Y. Efficient blind compressed sensing using sparsifying transforms with convergence guarantees and application to magnetic resonance imaging. SIAM J Imaging Sci 2015; 8(4): 2519-57.
[12]
Zhao B, Haldar JP, Brinegar C, Liang ZP. Low rank matrix recovery for real-time cardiac MRI. In: IEEE International symposium on biomedical imaging: from nano to macro 2010. IEEE: Rotterdam, Netherlands; pp. 996-9.
[13]
Otazo R, Cand’es E, Sodickson DK. Low-rank plus sparse matrix decomposition for accelerated dynamic MRI with separation of background and dynamic components. Magn Reson Med 2015; 73(3): 1125-36.
[14]
Weizman L, Miller KL, Eldar YC, Chiew M. PEAR: PEriodic And fixed Rank separation for fast fMRI. Med Phys 2017; arXiv:1706.04772.
[15]
Majumdar A, Ward RK. An algorithm for sparse MRI reconstruction by Schatten p-norm minimization. Magn Reson Imaging 2011; 29(3): 408-17.
[16]
Haldar JP. Low-rank modeling of local-space neighborhoods (LORAKS) for constrained MRI. IEEE Trans Med Imaging 2014; 33(3): 668-81.
[17]
Buades A, Coll B, Morel JM. A non-local algorithm for image denoising.In: IEEE Computer society conference on Computer Vision and Pattern Recognition (CVPR'05) 2005. 60-5.
[18]
Dabov K, Foi A, Katkovnik V, Egiazarian K. Image denoising by sparse 3-D transform-domain collaborative filtering. IEEE Trans Image Process 2007; 16(8): 2080-95.
[19]
Elad M, Aharon M. Image denoising via sparse and redundant representations over learned dictionaries. IEEE Trans Image Process 2006; 15(12): 3736-45.
[20]
Dong W, Shi G, Li X, Ma Y, Huang F. Compressive sensing via nonlocal low-rank regularization. IEEE Trans Image Process 2014; 23(8): 3618-32.
[21]
Qu X, Hou Y, Lam F, Guo D, Zhong J, Chen Z. Magnetic resonance image reconstruction from undersampled measurements using a patch-based nonlocal operator. Med Image Anal 2014; 18(6): 843-56.
[22]
Lingala SG, Hu Y, DiBella E, Jacob M. Accelerated dynamic MRI exploiting sparsity and low-rank structure: kt SLR. IEEE Trans Med Imaging 2011; 30(5): 1042-54.
[23]
Donoho DL, Elad M. Maximal sparsity representation via l1 minimization. Submitt to IEEE Trans Inf Theory 2002. Available from: https://pdfs.semanticscholar.org/a906/07d10cf9468dff6fe22571e472910b2fa0cc.pdf
[24]
Donoho DL. For most large underdetermined systems of linear equations the minimal l1-norm solution is also the sparsest solution. Commun Pure Appl Math 2006; 59(6): 797-829.
[25]
Ravishankar S, Bresler Y. l0 Sparsifying transform learning with efficient optimal updates and convergence guarantees. IEEE Trans Signal Process 2015; 63(9): 2389-404.
[26]
Ravishankar S, Bresler Y. Online sparsifying transform learning part II: convergence analysis. IEEE J Sel Top Signal Process 2015; 9(4): 637-46.
[27]
Ravishankar S, Wen B, Bresler Y. Online sparsifying transform learningPart I: Algorithms. IEEE J Sel Top Signal Process 2015; 9(4): 625-36.
[28]
Wen B, Ravishankar S, Bresler Y. Structured overcomplete sparsifying transform learning with convergence guarantees and applications. Int J Comput Vis 2015; 114(2-3): 137-67.
[29]
Cai JF, Cand’es EJ, Shen Z. A singular value thresholding algorithm for matrix completion*. SIAM J Optim 1956; 20(4): 2010.
[30]
Cand’es EJ, Wakin MB, Boyd SP. Enhancing sparsity by reweighted l1 minimization. J Fourier Anal Appl 2008; 14(5-6): 877-905.
[31]
Chartrand R. Exact reconstruction of sparse signals via nonconvex minimization. IEEE Signal Process Lett 2007; 14(10): 707-10.
[32]
Gorodnitsky IF, Rao BD. Sparse signal reconstruction from limited data using FOCUSS: A re-weighted minimum norm algorithm. IEEE Trans Signal Process 1997; 45(3): 600-16.
[33]
Trzasko J, Manduca A. Relaxed conditions for sparse signal recovery with general concave priors. IEEE Trans Signal Process 2009; 57(11): 4347-54.
[34]
Wipf D, Nagarajan S. Iterative Reweighted l1 and l2 Methods for Finding Sparse Solutions. IEEE J Sel Top Signal Process 2010; 4(2): 317-29.
[35]
Cand’es EJ, Romberg J, Tao T. Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information. IEEE Trans Inf Theory 2006; 52(2): 489-509.
[36]
Cand’es EJ, Romberg J. l1- MAGIC: Recovery of sparse signals via convex programming. Available from: http://citeseerx.ist.psu.edu/ viewdoc/summary?doi=10.1.1.212.9120
[37]
Scho¨nemann PH. A generalized solution of the orthogonal Procrustes problem. Psychometrika 1966; 31(1): 1-10.
[38]
Ravishankar S, Bresler Y. Closed-form solutions within sparsifying transform learning. In: IEEE International conference on acoustics, speech and signal processing 2013 5378-82.
[39]
Ravishankar S, Bresler Y. Learning sparsifying transforms. IEEE Trans Signal Process 2013; 61(5): 1072-86.
[40]
Gu S, Zhang L, Zuo W, Feng X. Weighted nuclear norm minimization with application to image denoising. In: IEEE Conference on Computer Vision and Pattern Recognition 2014 2862-9.
[41]
Ga¨ıffas S, Lecu´e G. Weighted algorithms for compressed sensing and matrix completion 2011. arXiv Prepr arX- iv11071638.
[42]
Boyd S, Parikh N, Chu E, Peleato B, Eckstein J. Distributed optimization and statistical learning via the alternating direction method of multipliers. Found Trends Mach Learn 2011; 3(1): 1-122.
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