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Multi-scale UDCT dictionary learning based highly undersampled MR image reconstruction using patch-based constraint splitting augmented Lagrangian shrinkage algorithm

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Abstract

Recently, dictionary learning (DL) based methods have been introduced to compressed sensing magnetic resonance imaging (CS-MRI), which outperforms pre-defined analytic sparse priors. However, single-scale trained dictionary directly from image patches is incapable of representing image features from multi-scale, multi-directional perspective, which influences the reconstruction performance. In this paper, incorporating the superior multi-scale properties of uniform discrete curvelet transform (UDCT) with the data matching adaptability of trained dictionaries, we propose a flexible sparsity framework to allow sparser representation and prominent hierarchical essential features capture for magnetic resonance (MR) images. Multi-scale decomposition is implemented by using UDCT due to its prominent properties of lower redundancy ratio, hierarchical data structure, and ease of implementation. Each sub-dictionary of different sub-bands is trained independently to form the multi-scale dictionaries. Corresponding to this brand-new sparsity model, we modify the constraint splitting augmented Lagrangian shrinkage algorithm (C-SALSA) as patch-based C-SALSA (PB C-SALSA) to solve the constraint optimization problem of regularized image reconstruction. Experimental results demonstrate that the trained sub-dictionaries at different scales, enforcing sparsity at multiple scales, can then be efficiently used for MRI reconstruction to obtain satisfactory results with further reduced undersampling rate. Multi-scale UDCT dictionaries potentially outperform both single-scale trained dictionaries and multi-scale analytic transforms. Our proposed sparsity model achieves sparser representation for reconstructed data, which results in fast convergence of reconstruction exploiting PB C-SALSA. Simulation results demonstrate that the proposed method outperforms conventional CS-MRI methods in maintaining intrinsic properties, eliminating aliasing, reducing unexpected artifacts, and removing noise. It can achieve comparable performance of reconstruction with the state-of-the-art methods even under substantially high undersampling factors.

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References

  • Afonso, M.V., Bioucas-Dias, J.M., Figueiredo, M.A.T., 2011. An augmented Lagrangian approach to the constrained optimization formulation of imaging inverse problems. IEEE Trans. Image Process., 20(3):681–695. [doi:10. 1109/TIP.2010.2076294]

    Article  MathSciNet  Google Scholar 

  • Aharon, M., Elad, M., Bruckstein, A., 2006. K-SVD: an algorithm for designing overcomplete dictionaries for sparse representation. IEEE Trans. Signal Process., 54(11): 4311–4322. [doi:10.1109/TSP.2006.881199]

    Article  Google Scholar 

  • Baraniuk, R., 2007. Compressive sensing. IEEE Signal Process. Mag., 24(4):118–121. [doi:10.1109/MSP.2007.4286571]

    Article  Google Scholar 

  • Candes, E.J., Donoho, D.L., 2004. New tight frames of curvelets and optimal representations of objects with piecewise C2 singularities. Commun. Pure Appl. Math., 57(2):219–266. [doi:10.1002/cpa.10116]

    Article  MathSciNet  MATH  Google Scholar 

  • Candes, E.J., Romberg, J., Tao, T., 2006a. Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information. IEEE Trans. Inform. Theory, 52(2):489–509. [doi:10.1109/TIT.2005.862083]

    Article  MathSciNet  MATH  Google Scholar 

  • Candes, E.J., Romberg, J.K., Tao, T., 2006b. Stable signal rcovery from incomplete and inaccurate measurements. Commun. Pure Appl. Math., 59(8):1207–1223. [doi:10. 1002/cpa.20124]

    Article  MathSciNet  MATH  Google Scholar 

  • Chambolle, A., 2004. An algorithm for total variation minimization and applications. J. Math. Imag. Vis., 20(1-2): 89–97. [doi:10.1023/B:JMIV.0000011325.36760.1e]

    MathSciNet  Google Scholar 

  • Chen, C., Huang, J., 2014. The benefit of tree sparsity in accelerated MRI. Med. Image Anal., 18(6):834–842. [doi:10. 1016/j.media.2013.12.004]

    Article  Google Scholar 

  • Combettes, P.L., Wajs, V.R., 2005. Signal recovery by proximal forward-backward splitting. Multiscale Model. Simul., 4(4):1168–1200. [doi:10.1137/050626090]

    Article  MathSciNet  MATH  Google Scholar 

  • Dahl, J., Hansen, P.C., Jensen, S.H., et al., 2010. Algorithms and software for total variation image reconstruction via first-order methods. Numer. Algor., 53(1):67–92. [doi:10. 1007/s11075-009-9310-3]

    Article  MathSciNet  MATH  Google Scholar 

  • Donoho, D.L., 2001. Sparse components of images and optimal atomic decompositions. Constr. Approx., 17(3): 353–382. [doi:10.1007/s003650010032]

    Article  MathSciNet  MATH  Google Scholar 

  • Donoho, D.L., 2006. Compressed sensing. IEEE Trans. Inform. Theory, 52(4):1289–1306. [doi:10.1109/TIT.2006.871582]

    Article  MathSciNet  MATH  Google Scholar 

  • Eckstein, J., Bertsekas, D.P., 1992. On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators. Math. Program., 55(1): 293–318. [doi:10.1007/BF01581204]

    Article  MathSciNet  MATH  Google Scholar 

  • Elad, M., 2010. Sparse and Redundant Representations: from Theory to Applications in Signal and Image Processing. Springer, New York, USA. [doi:10.1007/978-1-4419-7011-4]

    Book  Google Scholar 

  • Elad, M., Aharon, M., 2006. Image denoising via sparse and redundant representations over learned dictionaries. IEEE Trans. Image Process., 15(12):3736–3745. [doi:10.1109/TIP.2006.881969]

    Article  MathSciNet  Google Scholar 

  • Gabay, D., Mercier, B., 1976. A dual algorithm for the solution of nonlinear variational problems via finite element approximation. Comput. Math. Appl., 2(1):17–40. [doi:10. 1016/0898-1221(76)90003-1]

    Article  MATH  Google Scholar 

  • Gho, S.M., Nam, Y., Zho, S.Y., et al., 2010. Three dimension double inversion recovery gray matter imaging using compressed sensing. Magn. Reson. Imag., 28(10): 1395–1402. [doi:10.1016/j.mri.2010.06.029]

    Article  Google Scholar 

  • Huang, J., Zhang, S., Metaxas, D., 2011. Efficient MR image reconstruction for compressed MR imaging. Med. Image Anal., 15(5):670–679. [doi:10.1016/j.media.2011.06.001]

    Article  Google Scholar 

  • Kim, Y., Altbach, M.I., Trouard, T.P., et al., 2009. Compressed sensing using dual-tree complex wavelet transform. Proc. Int. Soc. Mag. Reson. Med., 17:2814.

    Google Scholar 

  • Kim, Y., Nadar, M.S., Bilgin, A., 2012. Wavelet-based compressed sensing using a Gaussian scale mixture model. IEEE Trans. Image Process., 21(6):3102–3108. [doi:10. 1109/TIP.2012.2188807]

    Article  MathSciNet  Google Scholar 

  • Lewicki, M.S., Sejnowski, T.J., 2000. Learning overcomplete representations. Neur. Comput., 12(2):337–365. [doi:10. 1162/089976600300015826]

    Article  Google Scholar 

  • Lin, L., 1989. A concordance correlation coefficient to evaluate reproducibility. Biometrics, 45(1):255–268. [doi:10. 2307/2532051]

    Article  MATH  Google Scholar 

  • Liu, Y., Cai, J., Zhan, Z., et al., 2015. Balanced sparse model for tight frames in compressed sensing magnetic resonance imaging. PLoS ONE, 10(4):e0119584.1–e0119584.19. [doi:10.1371/journal.pone.0119584]

    Google Scholar 

  • Lustig, M., Donoho, D., Pauly, J.M., 2007. Sparse MRI: the application of compressed sensing for rapid MR imaging. Magn. Reson. Med., 58(6):1182–1195. [doi:10.1002/mrm.21391]

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Mallat, S., 2008. A Wavelet Tour of Signal Processing: the Sparse Way (3rd Ed.). Academic Press, USA.

    Google Scholar 

  • Nguyen, T.T., Chauris, H., 2010. Uniform discrete curvelet transform. IEEE Trans. Signal Process., 58(7):3618–3634. [doi:10.1109/TSP.2010.2047666]

    Article  MathSciNet  Google Scholar 

  • Ning, B., Qu, X., Guo, D., et al., 2013. Magnetic resonance image reconstruction using trained geometric directions in 2D redundant wavelets domain and non-convex optimization. Magn. Reson. Imag., 31(9):1611–1622. [doi:10.1016/j.mri.2013.07.010]

    Article  Google Scholar 

  • Ophir, B., Lustig, M., Elad, M., 2011. Multi-scale dictionary learning using wavelets. IEEE J. Sel. Topics Signal Process., 5(5):1014–1024. [doi:10.1109/JSTSP.2011.2155032]

    Article  Google Scholar 

  • Qu, G., Zhang, D., Yan, P., 2002. Information measure for performance of image fusion. Electron. Lett., 38(7): 313–315. [doi:10.1049/el:20020212]

    Article  Google Scholar 

  • Qu, X., Zhang, W., Guo, D., et al., 2010. Iterative thresholding compressed sensing MRI based on contourlet transform. Inv. Probl. Sci. Eng., 18(6):737–758. [doi:10.1080/17415977.2010.492509]

    Article  MathSciNet  MATH  Google Scholar 

  • Qu, X., Guo, D., Ning, B., et al., 2012. Undersampled MRI reconstruction with patch-based directional wavelets. Magn. Reson. Imag., 30(7):964–977. [doi:10.1016/j.mri. 2012.02.019]

    Article  Google Scholar 

  • Qu, X., Hou, Y., Lam, F., et al., 2014. Magnetic resonance image reconstruction from undersampled measurements using a patch-based nonlocal operator. Med. Image Anal., 18(6):843–856. [doi:10.1016/j.media.2013.09.007]

    Article  Google Scholar 

  • Rauhut, H., Schnass, K., Vandergheynst, P., 2008. Compressed sensing and redundant dictionaries. IEEE Trans. Inform. Theory, 54(5):2210–2219. [doi:10.1109/TIT.2008. 920190]

    Article  MathSciNet  MATH  Google Scholar 

  • Ravishankar, S., Bresler, Y., 2011. MR image reconstruction from highly undersampled k-space data by dictionary learning. IEEE Trans. Med. Imag., 30(5):1028–1041. [doi:10.1109/TMI.2010.2090538]

    Article  Google Scholar 

  • Rubinstein, R., Zibulevsky, M., Elad, M., 2010. Double sparsity: learning sparse dictionaries for sparse signal approximation. IEEE Trans. Signal Process., 58(3): 1553–1564. [doi:10.1109/TSP.2009.2036477]

    Article  MathSciNet  Google Scholar 

  • Rudin, L.I., Osher, S., Fatemi, E., 1992. Nonlinear total variation based noise removal algorithms. Phys. D, 60(1-4): 259–268. [doi:10.1016/0167-2789(92)90242-F]

    Article  MATH  Google Scholar 

  • Trzasko, J., Manduca, A., 2009. Highly undersampled magnetic resonance image reconstruction via homotopic l 0-minimization. IEEE Trans. Med. Imag., 28(1):106–121. [doi:10.1109/TMI.2008.927346]

    Article  Google Scholar 

  • Wang, Z., Bovik, A.C., Sheikh, H.R., et al., 2004. Image quality assessment: from error visibility to structural similarity. IEEE Trans. Image Process., 13(4):600–612. [doi:10.1109/TIP.2003.819861]

    Article  Google Scholar 

  • Xydeas, C.S., Petrovic, V., 2000. Objective image fusion performance measure. Electron. Lett., 36(4):308–309. [doi:10.1049/el:20000267]

    Article  Google Scholar 

  • Zhu, Z., Wahid, K., Babyn, P., et al., 2013. Compressed sensing-based MRI reconstruction using complex doubledensity dual-tree DWT. Int. J. Biomed. Imag., 2013. 907501.1–907501.12. [doi:10.1155/2013/907501]

    Article  Google Scholar 

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Authors and Affiliations

  1. School of Information Science & Engineering, Lanzhou University, Lanzhou, 730000, China

    Min Yuan, Bing-xin Yang, Yi-de Ma, Jiu-wen Zhang, Fu-xiang Lu & Tong-feng Zhang

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  1. Min Yuan
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  2. Bing-xin Yang
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  3. Yi-de Ma
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  4. Jiu-wen Zhang
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  5. Fu-xiang Lu
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  6. Tong-feng Zhang
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Corresponding author

Correspondence to Yi-de Ma.

Additional information

Project supported by the National Natural Science Foundation of China (Nos. 61175012 and 61201422), the Natural Science Foundation of Gansu Province of China (No. 1208RJ-ZA265), the Specialized Research Fund for the Doctoral Program of Higher Education of China (No. 2011021111-0026), and the Fundamental Research Funds for the Central Universities of China (Nos. lzujbky-2015-108 and lzujbky-2015-197)

ORCID: Min YUAN, http://orcid.org/0000-0001-7855-8678

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Yuan, M., Yang, Bx., Ma, Yd. et al. Multi-scale UDCT dictionary learning based highly undersampled MR image reconstruction using patch-based constraint splitting augmented Lagrangian shrinkage algorithm. Frontiers Inf Technol Electronic Eng 16, 1069–1087 (2015). https://doi.org/10.1631/FITEE.1400423

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  • DOI: https://doi.org/10.1631/FITEE.1400423

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