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

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.

Keywords

Compressed sensing (CS) / Magnetic resonance imaging (MRI) / Uniform discrete curvelet transform (UDCT) / Multi-scale dictionary learning (MSDL) / Patch-based constraint splitting augmented Lagrangian shrinkage algorithm(PB C-SALSA)

References

[1] Afonso, M.V., Bioucas-Dias, J.M., Figueiredo, M.A.T., 2011.An augmented Lagrangian approach to the constrainedoptimization formulation of imaging inverse problems.IEEE Trans.Image Process., 20(3):681–695.

[2] 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.

[3] Baraniuk, R., 2007. Compressive sensing. IEEE Signal Process.Mag., 24(4):118–121.

[4] 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.

[5] 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.

[6] 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.

[7] Chambolle, A., 2004. An algorithm for total variation minimization and applications. J. Math. Imag. Vis., 20(1-2):89–97.

[8] Chen, C., Huang, J., 2014. The benefit of tree sparsity in accelerated MRI. Med. Image Anal., 18(6):834–842.

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

[10] Dahl, J., Hansen, P.C., Jensen, S.H., et al., 2010. Algorithms and software for total variation image reconstruction viafirst-order methods. Numer. Algor., 53(1):67–92.

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

[12] Donoho, D.L., 2006. Compressed sensing. IEEE Trans. Inform.Theory, 52(4):1289–1306.

[13] 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.

[14] Elad, M., 2010. Sparse and Redundant Representations: from Theory to Applications in Signal and Image Processing.Springer, New York, USA.

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

[16] 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.

[17] 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.

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

[19] 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.

[20] 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.

[21] Lewicki, M.S., Sejnowski, T.J., 2000. Learning overcomplete representations. Neur. Comput., 12(2):337–365.

[22] Lin, L., 1989. A concordance correlation coefficient to evaluate reproducibility. Biometrics, 45(1):255–268.

[23] 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.

[24] 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.

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

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

[27] Nguyen, T.T., Chauris, H., 2010. Uniform discrete curvelet transform. IEEE Trans. Signal Process., 58(7):3618–3634.

[28] 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.

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

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

[31] 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.

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

[33] 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.

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

[35] 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.

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

[37] Rudin, L.I., Osher, S., Fatemi, E., 1992. Nonlinear total variation based noise removal algorithms. Phys. D, 60(1-4):259–268.

[38] Trzasko, J., Manduca, A., 2009. Highly undersampled magnetic resonance image reconstruction via homotopic l0-minimization. IEEE Trans. Med. Imag., 28(1):106–121.

[39] 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.

[40] Xydeas, C.S., Petrović, V., 2000. Objective image fusion performance measure. Electron. Lett., 36(4):308–309.

[41] 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.