Fast Algorithms for Compressed Sensing MRI Reconstruction

  • Bhabesh Deka
  • Sumit Datta
Part of the Springer Series on Bio- and Neurosystems book series (SSBN, volume 9)


Extensive research work is being carried out in the area of fast convex optimization-based compressed sensing magnetic resonance (MR) image reconstruction algorithms. The main focus here is to achieve throughputs of clinical compressed sensing MR image reconstruction in terms of quality of reconstruction and computational time. In this chapter, we briefly review some of the recently developed convex optimization-based algorithms for compressed sensing MR image reconstruction. All these algorithms may be classified broadly into four categories based on their approaches of solving the reconstruction/recovery problem. We then detail algorithms of each category with sufficient mathematical details and report their relative advantages and disadvantages.


  1. 1.
    Afonso, M., Bioucas-Dias, J., Figueiredo, M.: Fast image recovery using variable splitting and constrained optimization. IEEE Trans. Image Process. 19(9), 2345–2356 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Barzilai, J., Borwein, J.M.: Two-point step size gradient methods. IMA J. Numer. Anal. 8, 141–148 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Beck, A., Teboulle, M.: Fast gradient-based algorithms for constrained total variation image denoising and deblurring problems. IEEE Trans. Image Process. 18(11), 2419–2434 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Beck, A., Teboulle, M.: A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J. Imaging Sci. 2(1), 183–202 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Becker, S., Bobin, J., Candes, E.J.: NESTA: a fast and accurate first-order method for sparse recovery. SIAM J. Imaging Sci. 4(1), 1–39 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    van den Berg, E., Friedlander, M.P.: Probing the pareto frontier for basis pursuit solutions. SIAM J. Sci. Comput. 31(2), 890–912 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Bertsekas, D.: Constrained Optimization and Lagrange Multiplier Methods. Athena scientific series in optimization and neural computation, 1st edn. Athena Scientific, Massachusetts (1996)Google Scholar
  8. 8.
    Bertsekas, D.: Nonlinear Programming. Athena Scientific, Massachusetts (1999)zbMATHGoogle Scholar
  9. 9.
    Bioucas-Dias, J.M.: Fast GEM wavelet-based image deconvolution algorithm. In: IEEE International Conference on Image Processing- ICIP 2003, vol. 2, pp. 961–964 (2003)Google Scholar
  10. 10.
    Bioucas-Dias, J.M.: Bayesian wavelet-based image deconvolution: a GEM algorithm exploiting a class of heavy-tailed priors. IEEE Trans. Image Process. 15(4), 937–951 (2006)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Bioucas-Dias, J.M., Figueiredo, M.A.T.: A new TwIST: two-step iterative shrinkage/thresholding algorithms for image restoration. IEEE Trans. Image Process. 16(12), 2992–3004 (2007)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Bioucas-Dias, J.M., Figueiredo, M.A.T.: Multiplicative noise removal using variable splitting and constrained optimization. IEEE Trans. Image Process. 19(7), 1720–1730 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, USA (2004)zbMATHCrossRefGoogle Scholar
  14. 14.
    Bregman, L.: The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming. USSR Comput. Math. Math. Phys. 7(3), 200–217 (1967)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Bregman, L.M.: The method of successive projection for finding a common point of convex sets. Sov. Math. Dokl. 6, 688–692 (1965)zbMATHGoogle Scholar
  16. 16.
    Candes, E.J., Romberg, J.K.: Signal recovery from random projections. In: Proceedings of SPIE Computational Imaging III, vol. 5674, pp. 76–86. San Jose (2005)Google Scholar
  17. 17.
    Combettes, P.L.: Iterative construction of the resolvent of a sum of maximal monotone operators. J. Convex Anal. 16, 727–748 (2009)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Combettes, P.L., Pesquet, J.C.: A proximal decomposition method for solving convex variational inverse problems. Inverse Probl. 24(6), 1–27 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Combettes, P.L., Pesquet, J.C.: Proximal splitting methods in signal processing. In: FixedPoint Algorithms for Inverse Problems in Science and Engineering, pp. 185–212. Springer, New York (2011)Google Scholar
  20. 20.
    Combettes, P.L., Wajs, V.R.: Signal recovery by proximal forward-backward splitting. Multiscale Model. Simul. 4(4), 1168–1200 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Daubechies, I., Defrise, M., De Mol, C.: An iterative thresholding algorithm for linear inverse problems with a sparsity constraint. Commun. Pure Appl. Math. 57(11), 1413–1457 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Deka, B., Datta, S.: High throughput MR image reconstruction using compressed sensing. ICVGIP 14, 89:1–89: 6 (2014). ACM, Bangalore, IndiaGoogle Scholar
  23. 23.
    Demmel, J.W.: Applied Numerical Linear Algebra. Society for industrial and applied mathematics, USA (1997)Google Scholar
  24. 24.
    Dolui, S.: Variable splitting as a key to efficient cient image reconstruction. Ph.D. thesis, Electrical and Computer Engineering, University of Waterloo (2012)Google Scholar
  25. 25.
    Eckstein, J.: Splitting methods for monotone operators with applications to parallel optimization. Ph.D. thesis, Department of Civil Engineering, Massachusetts Institute of Technology (1989)Google Scholar
  26. 26.
    Eckstein, J., Bertsekas, D.P.: On the douglas-rachford splitting method and the proximal point algorithm for maximal monotone operators. Math. Program. 55, 293–318 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Figueiredo, M., Bioucas-Dias, J., Nowak, R.: Majorization minimization algorithms for wavelet-based image restoration. IEEE Trans. Image Process. 16(12), 2980–2991 (2007)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Figueiredo, M., Nowak, R.: An EM algorithm for wavelet-based image restoration. IEEE Trans. Image Process. 12(8), 906–916 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Figueiredo, M., Nowak, R., Wright, S.: Gradient projection for sparse reconstruction: Application to compressed sensing and other inverse problems. IEEE J. Sel. Top. Signal Process. 1(4), 586–597 (2008)CrossRefGoogle Scholar
  30. 30.
    Gabay, D., Mercier, B.: A dual algorithm for the solution of nonlinear variational problems via finite element approximation. Comput. Math. Appl. 2(1), 17–40 (1976)zbMATHCrossRefGoogle Scholar
  31. 31.
    Glowinski, R., Le Tallec, P.: Augmented Lagrangian and Operator-splitting Methods in Nonlinear Mechanics. SIAM studies in applied mathematics. Society for Industrial and Applied Mathematics, Philadelphia (1989)Google Scholar
  32. 32.
    Glowinski, R., Marrocco, A.: Sur lapproximation, par elements finis dordre un, et la resolution, parpenalisation-dualite, dune classe de problems de dirichlet non lineares. Revue Francaise dAutomatique, Informatique, et Recherche Op erationelle 9, 41–76 (1975)Google Scholar
  33. 33.
    Goldstein, T., Osher, S.: The split bregman method for \(L_1\)-regularized problems. SIAM J. Imaging Sci. 2(2), 323–343 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Grippo, L., Sciandrone, M.: Nonmonotone globalization techniques for the Barzilai-Borwein gradient method. Comput. Optim. Appl. 23(2), 143–169 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Hale, E., Yin, W., Zhang, Y.: A fixed-point continuation method for \(L_1\)-regularized minimization with applications to compressed sensing. Technical report. Rice University, CAAM (2007)Google Scholar
  36. 36.
    Hestenes, M.: Multiplier and gradient methods. J. Optim. Theory Appl. 4(5), 303–320 (1969)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Hestenes, M.R., Stiefel, E.: Methods of conjugate gradients for solving linear systems. J. Res. Natl. Bur. Stand. 49, 409–436 (1952)MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Huang, J., Zhang, S., Li, H., Metaxas, D.N.: Composite splitting algorithms for convex optimization. Comput. Vis. Image Underst. 115(12), 1610–1622 (2011)CrossRefGoogle Scholar
  39. 39.
    Huang, J., Zhang, S., Metaxas, D.N.: Efficient MR image reconstruction for compressed MR imaging. Med. Image Anal. 15(5), 670–679 (2011)CrossRefGoogle Scholar
  40. 40.
    Kim, S., Koh, K., Lustig, M., Boyd, S., Gorinevsky, D.: An interior-point method for largescale \(L_1\)-regularized least squares. IEEE J. Sel. Top. Signal Process. 1(4), 606–617 (2008)Google Scholar
  41. 41.
    Lustig, M.: Sparse MRI. Ph.D. thesis, Electrical Engineering, Stanford University (2008)Google Scholar
  42. 42.
    Lustig, M., Donoho, D., Pauly, J.M.: Sparse MRI: the application of compressed sensing for rapid MR imaging. Magn. Reson. Med. 58, 1182–1195 (2007)CrossRefGoogle Scholar
  43. 43.
    Ma, S., Yin, W., Zhang, Y., Chakraborty, A.: An efficient algorithm for compressed MR imaging using total variation and wavelets. In: IEEE Conference on Computer Vision and Pattern Recognition (CVPR 2008), pp. 1–8. Anchorage, AK (2008)Google Scholar
  44. 44.
    Majumdar, A.: Compressed Sensing for Magnetic Resonance Image Reconstruction. Cambridge University Press, India (2015)CrossRefGoogle Scholar
  45. 45.
    Mallat, S., Zhang, Z.: Matching pursuits with time-frequency dictionaries. IEEE Trans. Signal Process. 41, 3397–3415 (1993)zbMATHCrossRefGoogle Scholar
  46. 46.
    Moreau, J.J.: Fonctions convexes duales et points proximaux dans un espace hilbertien. Comptes Rendus de lAcad emie des Sciences (Paris), S erie A 255, 2897–2899 (1962)MathSciNetzbMATHGoogle Scholar
  47. 47.
    Nesterov, Y.: A method of solving a convex programming problem with convergence rate O(1/sqr(k)). Sov. Math. Dokl. 27, 372–376 (1983)Google Scholar
  48. 48.
    Osher, S., Burger, M., Goldfarb, D., Xu, J., Yin, W.: An iterative regularization method for total variation-based image restoration. Multiscale Model. Simul. 4(2), 460–489 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  49. 49.
    Parikh, N., Boyd, S.: Proximal algorithms. Found. Trends Optim. 1(3), 127–239 (2014)CrossRefGoogle Scholar
  50. 50.
    Pati, Y.C., Rezaiifar, R., Pati, Y.C., Rezaiifar, R., Krishnaprasad, P.S.: Orthogonal Matching Pursuit: Recursive Function Approximation with Applications to Wavelet Decomposition. pp. 40–44(1993)Google Scholar
  51. 51.
    Powell, M.J.D.: A method for nonlinear constraints in minimization problems. In: Fletcher, R. (ed.) Optimization, pp. 283–298. Academic Press, New York (1969)Google Scholar
  52. 52.
    Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Phys. D 60, 259–268 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  53. 53.
    Wang, Y., Yang, J., Yin, W., Zhang, Y.: A new alternating minimization algorithm for total variation image reconstruction. SIAM J. Imaging Sci. 1(3), 248–272 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  54. 54.
    Wright, S.J., Nowak, R.D., Figueiredo, M.A.T.: Sparse reconstruction by separable approximation. IEEE Trans. Signal Process. 57(7), 2479–2493 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  55. 55.
    Xiao, Y., Yang, J., Yuan, X.: Alternating algorithms for total variation image reconstruction from random projections. Inverse Probl. Imaging (IPI) 6(3), 547–563 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  56. 56.
    Yang, A.Y., Ganesh, A., Zhou, Z., Sastry, S., Ma, Y.: A review of fast \(L_1\)-minimization algorithms for robust face recognition. CoRR (2010). arXiv:1007.3753
  57. 57.
    Yang, J., Zhang, Y.: Alternating direction algorithms for \(L_1\)-problems in compressive sensing. SIAM J. Sci. Comput. 33(1), 250–278 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  58. 58.
    Yang, J., Zhang, Y., Yin, W.: A fast alternating direction method for TV\(L_1\)-\(L_2\) signal reconstruction from partial Fourier data. IEEE J. Sel. Top. Signal Process. 4(2), 288–297 (2010)CrossRefGoogle Scholar
  59. 59.
    Yin, W., Osher, S., Goldfarb, D., Darbon, J.: Bregman iterative algorithms for \(L_1\)-minimization with applications to compressed sensing. SIAM J. Imaging Sci., 143–168 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  60. 60.
    Youla, D., Webb, H.: Image restoration by the method of convex projections: Part 1-theory. IEEE Trans. Med. Imaging 1(2), 81–94 (1982)CrossRefGoogle Scholar

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© Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  1. 1.Department of Electronics and Communication EngineeringTezpur UniversityTezpurIndia

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