Decoupled Algorithm for MRI Reconstruction Using Nonlocal Block Matching Model: BM3D-MRI


The block matching 3D (BM3D) is an efficient image model, which has found few applications other than its niche area of denoising. We will develop a magnetic resonance imaging (MRI) reconstruction algorithm, which uses decoupled iterations alternating over a denoising step realized by the BM3D algorithm and a reconstruction step through an optimization formulation. The decoupling of the two steps allows the adoption of a strategy with a varying regularization parameter, which contributes to the reconstruction performance. This new iterative algorithm efficiently harnesses the power of the nonlocal, image-dependent BM3D model. The MRI reconstruction performance of the proposed algorithm is superior to state-of-the-art algorithms from the literature. A convergence analysis of the algorithm is also presented.

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  1. 1.

  2. 2.

  3. 3.

  4. 4.


  1. 1.

    Akçakaya, M., Basha, T.A., Goddu, B., Goepfert, L.A., Kissinger, K.V., Tarokh, V., Manning, W.J., Nezafat, R.: Low-dimensional-structure self-learning and thresholding: regularization beyond compressed sensing for MRI reconstruction. Magn. Reson. Med. 66(3), 756–767 (2011)

    Article  Google Scholar 

  2. 2.

    Byrne, C.: A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Probl. 20(1), 103–120 (2004)

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    Chambolle, A., De Vore, R., Lee, N.Y., Lucier, B.: Nonlinear wavelet image processing: variational problems, compression, and noise removal through wavelet shrinkage. IEEE Trans. Image Process. 7(3), 319–335 (1998). doi:10.1109/83.661182

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Chambolle, A., Lions, P.L.: Image recovery via total variation minimization and related problems. Numerische Mathematik 76(2), 167–188 (1997). doi:10.1007/s002110050258

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Combettes, P., Pesquet, J.: A Douglas-Rachford splitting approach to nonsmooth convex variational signal recovery. IEEE J. Sel. Topics Signal Process. 1(4), 564–574 (2007). doi:10.1109/JSTSP.2007.910264

    Article  Google Scholar 

  6. 6.

    Crandall, R., Bilgin, A.: Lossless image compression using causal block matching and 3d collaborative filtering. In: 2014 IEEE International Conference on Image Processing (ICIP), pp. 5636–5640 (2014). doi:10.1109/ICIP.2014.7026140

  7. 7.

    Dabov, K., Foi, A., Katkovnik, V., Egiazarian, K.: Image denoising by sparse 3-d transform-domain collaborative filtering. IEEE Trans. Image Process. 16(8), 2080–2095 (2007). doi:10.1109/TIP.2007.901238

    MathSciNet  Article  Google Scholar 

  8. 8.

    Dabov, K., Foi, A., Katkovnik, V., Egiazarian, K.: Image restoration by sparse 3d transform-domain collaborative filtering. In: Proc. SPIE vol. 6812, pp. 1–12 (2008)

  9. 9.

    Danielyan, A., Foi, A., Katkovnik, V., Egiazarian, K.: Image upsampling via spatially adaptive block-matching filtering. In: Signal Processing Conference, 2008 16th European, pp. 1–5 (2008)

  10. 10.

    Danielyan, A., Katkovnik, V., Egiazarian, K.: Image deblurring by augmented Lagrangian with BM3D frame prior. Workshop on Information Theoretic Methods in Science and Engineering (WITMSE), Tampere, Finland pp. 16–18 (2010)

  11. 11.

    Danielyan, A., Katkovnik, V., Egiazarian, K.: BM3D frames and variational image deblurring. IEEE Trans. Image Process. 21(4), 1715–1728 (2012). doi:10.1109/TIP.2011.2176954

    MathSciNet  Article  Google Scholar 

  12. 12.

    Egiazarian, K., Foi, A., Katkovnik, V.: Compressed sensing image reconstruction via recursive spatially adaptive filtering. In: 2007 IEEE International Conference on Image Processing (ICIP), vol. 1, pp. 549–552 (2007)

  13. 13.

    Fadili, M.J., Starck, J.L., Murtagh, F.: Inpainting and zooming using sparse representations. Comput. J. 52(1), 64–79 (2007)

    Article  Google Scholar 

  14. 14.

    Goldstein, T., Osher, S.: The split Bregman method for L1-regularized problems. SIAM J. Imaging Sci. 2(2), 323–343 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  15. 15.

    Griswold, M.A., Jakob, P.M., Heidemann, R.M., Nittka, M., Jellus, V., Wang, J., Kiefer, B., Haase, A.: Generalized autocalibrating partially parallel acquisitions (grappa). Magn. Reson. Med. 47(6), 1202–1210 (2002). doi:10.1002/mrm.10171

    Article  Google Scholar 

  16. 16.

    Gunturk, B.K., Li, X.: Image Restoration: Fundamentals and Advances. CRC Press, New York (2012)

    MATH  Google Scholar 

  17. 17.

    Huang, J., Zhang, S., Metaxas, D.: Efficient MR image reconstruction for compressed MR imaging. Med. Image Anal. 15(5), 670–679 (2011). doi:10.1016/

    Article  Google Scholar 

  18. 18.

    Li, X.: Fine-granularity and spatially-adaptive regularization for projection-based image deblurring. IEEE Trans. Image Process. 20(4), 971–983 (2011). doi:10.1109/TIP.2010.2081681

    MathSciNet  Article  Google Scholar 

  19. 19.

    Li, X.: Image recovery via hybrid sparse representations: a deterministic annealing approach. IEEE J. Sel. Top. Signal Process. 5(5), 953–962 (2011). doi:10.1109/JSTSP.2011.2138676

    Article  Google Scholar 

  20. 20.

    Lustig, M., Donoho, D., Pauly, J.: Sparse MRI: the application of compressed sensing for rapid MR imaging. Magn. Reson. Med. 58(6), 1182–1195 (2007)

    Article  Google Scholar 

  21. 21.

    Lustig, M., Donoho, D., Santos, J., Pauly, J.: Compressed sensing MRI. IEEE Signal Process. Mag. 25(2), 72–82 (2008). doi:10.1109/MSP.2007.914728

    Article  Google Scholar 

  22. 22.

    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, 2008. CVPR 2008. pp. 1–8 (2008). doi:10.1109/CVPR.2008.4587391

  23. 23.

    Pruessmann, K.P., Weiger, M., Scheidegger, M.B., Boesiger, P.: Sense: sensitivity encoding for fast mri. Magn. Reson. Med. 42(5), 952–962 (1999)

    Article  Google Scholar 

  24. 24.

    Qu, X., Cao, X., Guo, D., Hu, C., Chen, Z.: Combined sparsifying transforms for compressed sensing MRI. Electron. Lett. 46(2), 121–123 (2010). doi:10.1049/el.2010.1845

    Article  Google Scholar 

  25. 25.

    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. 18(6), 843–856 (2014)

    Article  Google Scholar 

  26. 26.

    Ravishankar, S., Bresler, Y.: MR image reconstruction from highly undersampled k-space data by dictionary learning. IEEE Trans. Med. Imag. 30, 1028–1041 (2011)

    Article  Google Scholar 

  27. 27.

    Setzer, S.: Operator splittings, Bregman methods and frame shrinkage in image processing. Int. J. Comput. Vis. 92(3), 265–280 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  28. 28.

    Sreehari, S., Venkatakrishnan, S.V., Drummy, L.F., Simmons, J.P., Bouman, C.A.: Advanced prior modeling for 3d bright field electron tomography. Proc. SPIE 9401, 940,108–1–940,108–12 (2015)

    Google Scholar 

  29. 29.

    Tan, J., Ma, Y., Baron, D.: Compressive imaging via approximate message passing with image denoising. IEEE Trans. Signal Process. 63(8), 2085–2092 (2015)

    MathSciNet  Article  Google Scholar 

  30. 30.

    Wen, Y.W., Ng, M.K., Ching, W.K.: Iterative algorithms based on decoupling of deblurring and denoising for image restoration. SIAM J. Sci. Comput. 30(5), 2655–2674 (2008)

  31. 31.

    Yang, J., Zhang, Y., Yin, W.: A fast alternating direction method for TVL1-L2 signal reconstruction from partial Fourier data. IEEE J. Sel. Topics Signal Process. 4(2), 288–297 (2010). doi:10.1109/JSTSP.2010.2042333

    Article  Google Scholar 

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Correspondence to Ender M. Eksioglu.

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Eksioglu, E.M. Decoupled Algorithm for MRI Reconstruction Using Nonlocal Block Matching Model: BM3D-MRI. J Math Imaging Vis 56, 430–440 (2016).

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  • Image reconstruction
  • Magnetic resonance
  • Block matching
  • BM3D
  • Compressed sensing
  • Sparsity