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Nonlocal elastica model for sparse reconstruction. (English) Zbl 1455.94040
Summary: In view of the exceptional ability of curvature in connecting missing edges and structures, we propose novel sparse reconstruction models via the Euler’s elastica energy. In particular, we firstly extend the Euler’s elastica regularity into the nonlocal formulation to fully take the advantages of the pattern redundancy and structural similarity in image data. Due to its non-convexity, non-smoothness and nonlinearity, we regard both local and nonlocal elastica functional as the weighted total variation for a good trade-off between the runtime complexity and performance. The splitting techniques and alternating direction method of multipliers (ADMM) are used to achieve efficient algorithms, the convergence of which is also discussed under certain assumptions. The weighting function occurred in our model can be well estimated according to the local approach. Numerical experiments demonstrate that our nonlocal elastica model achieves the state-of-the-art reconstruction results for different sampling patterns and sampling ratios, especially when the sampling rate is extremely low.
##### MSC:
 94A08 Image processing (compression, reconstruction, etc.) in information and communication theory 68U10 Computing methodologies for image processing 49M41 PDE constrained optimization (numerical aspects) 42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
##### Software:
DLMRI-Lab; EdgeCS
Full Text:
##### References:
 [1] Ambrosio, L.; Masnou, S., On a variational problem arising in image reconstruction, Isnm Int., 147, 17-26 (2006) · Zbl 1111.68732 [2] Arias, P., Caselles, V., Sapiro, G.: A variational framework for non-local image inpainting. In: International Workshop on Energy Minimization Methods in Computer Vision and Pattern Recognition, pp. 345-358 (2009) [3] Arias, P.; Facciolo, G.; Caselles, V.; Sapiro, G., A variational framework for exemplar-based image inpainting, Int. J. Comput. Vis., 93, 3, 319-347 (2011) · Zbl 1235.94015 [4] Bae, E.; Shi, J.; Tai, XC, Graph cuts for curvature based image denoising, IEEE Trans. Image Process., 20, 5, 1199-1210 (2011) · Zbl 1372.94015 [5] Bae, E.; Tai, XC; Zhu, W., Augmented Lagrangian method for an Euler’s elastica based segmentation model that promotes convex contours, Inverse Probl. Imaging, 11, 1, 1-23 (2017) · Zbl 1416.94013 [6] Bernstein, MA; Fain, SB; Riederer, SJ, Effect of windowing and zero-filled reconstruction of MRI data on spatial resolution and acquisition strategy, J. Magn. Resonance Imaging, 14, 3, 270-280 (2001) [7] Bredies, K.; Kunisch, K.; Pock, T., Total generalized variation, SIAM J. Imaging Sci., 3, 3, 492-526 (2010) · Zbl 1195.49025 [8] Buades, A.; Coll, B.; Morel, JM, A review of image denoising algorithms, with a new one, Multiscale Model. Simul., 4, 2, 490-530 (2005) · Zbl 1108.94004 [9] Chambolle, A., An algorithm for total variation minimization and applications, J. Math. Imaging Vis., 20, 1-2, 89-97 (2004) · Zbl 1366.94048 [10] Chambolle, A.; Pock, T., A first-order primal-dual algorithm for convex problems with applications to imaging, J. Math. Imaging Vis., 40, 1, 120-145 (2011) · Zbl 1255.68217 [11] Chan, TF; Kang, SH; Shen, J., Euler’s elastica and curvature-based inpainting, SIAM J. Appl. Math., 63, 2, 564-592 (2002) · Zbl 1028.68185 [12] Chen, C.; Huang, J., Exploiting the wavelet structure in compressed sensing MRI, Magn. Resonance Imaging, 32, 10, 1377-1389 (2014) [13] Chen, Y.; Hager, WW; Yashtini, M.; Ye, X.; Zhang, H., Bregman operator splitting with variable stepsize for total variation image reconstruction, Comput. Optim. Appl., 54, 2, 317-342 (2013) · Zbl 1290.90071 [14] Compton, R.; Osher, S.; Bouchard, LS, Hybrid regularization for MRI reconstruction with static field inhomogeneity correction, Inverse Probl. Imaging, 7, 4, 1215-1233 (2013) · Zbl 1302.65153 [15] Deng, LJ; Glowinski, R.; Tai, XC, A new operator splitting method for the Euler elastica model for image smoothing, SIAM J. Imaging Sci., 12, 2, 1190-1230 (2019) [16] Duan, Y., Huang, W., Zhou, J., Chang, H., Zeng, T.: A two-stage image segmentation method using Euler’s elastica regularized Mumford-Shah model. In: 2014 22nd International Conference on Pattern Recognition, pp. 118-123 (2014) [17] Eksioglu, EM, Decoupled algorithm for MRI reconstruction using nonlocal block matching model: BM3D-MRI, J. Math. Imaging Vis., 56, 3, 430-440 (2016) · Zbl 1386.68204 [18] Esedoglu, S.; March, R., Segmentation with depth but without detecting junctions, J. Math. Imaging Vis., 18, 1, 7-15 (2003) · Zbl 1033.68132 [19] Gilboa, G.; Osher, S., Nonlocal linear image regularization and supervised segmentation, Multiscale Model. Simul., 6, 2, 595-630 (2007) · Zbl 1140.68517 [20] Gilboa, G.; Osher, S., Nonlocal operators with applications to image processing, SIAM J. Multiscale Model. Simul., 7, 3, 1005-1028 (2007) · Zbl 1181.35006 [21] Goldstein, T.; Osher, S., The split Bregman method for $$L_1$$ regularized problems, SIAM J. Imaging Sci., 2, 2, 323-343 (2009) · Zbl 1177.65088 [22] Guo, W.; Qin, J.; Yin, W., A new detail-preserving regularization scheme, SIAM J. Imaging Sci., 7, 2, 1309-1334 (2014) · Zbl 1299.65130 [23] Guo, W.; Yin, W., Edge guided reconstruction for compressive imaging, SIAM J. Imaging Sci., 5, 3, 809-834 (2012) · Zbl 1259.65102 [24] Haldar, JP; Diego, H.; Zhi-Pei, L., Compressed-sensing MRI with random encoding, IEEE Trans. Med. Imaging, 30, 4, 893-903 (2011) [25] Hammernik, K.; Klatzer, T.; Kobler, E.; Recht, MP; Sodickson, DK; Pock, T.; Knoll, F., Learning a variational network for reconstruction of accelerated MRI data, Magn. Resonance Med., 79, 6, 3055-3071 (2018) [26] He, X.; Zhu, W.; Tai, XC, Segmentation by elastica energy with $${L}_1$$ and $${L}_2$$ curvatures: a performance comparison, Numer. Math. Theory Methods Appl., 12, 1, 285-311 (2019) · Zbl 1449.94011 [27] Huang, J., Yang, F.: Compressed magnetic resonance imaging based on wavelet sparsity and nonlocal total variation. In: 2012 9th IEEE International Symposium on Biomedical Imaging (ISBI), pp. 968-971 (2012) [28] Kang, SH; Zhu, W.; Jianhong, J., Illusory shapes via corner fusion, SIAM J. Imaging Sci., 7, 4, 1907-1936 (2014) · Zbl 1320.68217 [29] Knoll, F.; Bredies, K.; Pock, T.; Stollberger, R., Second order total generalized variation (TGV) for MRI, Magn. Resonance Med., 65, 2, 480-491 (2011) [30] Liu, J.; Zheng, X., A block nonlocal TV method for image restoration, SIAM J. Imaging Sci., 10, 2, 920-941 (2017) · Zbl 1437.94021 [31] Lou, Y.; Zhang, X.; Osher, S.; Bertozzi, A., Image recovery via nonlocal operators, J. Sci. Comput., 42, 2, 185-197 (2010) · Zbl 1203.65088 [32] Lustig, M.; Donoho, D.; Pauly, JM, Sparse MRI: the application of compressed sensing for rapid MR imaging, Magn. Resonance Med., 58, 6, 1182-1195 (2007) [33] Lustig, M.; Donoho, DL; Santos, JM; Pauly, JM, Compressed sensing MRI, IEEE Signal Process. Mag., 25, 2, 72-82 (2008) [34] Masnou, S., Morel, J.M.: Level lines based disocclusion. In: Proceedings 1998 International Conference on Image Processing, pp. 259-263 (1998) [35] Masnou, S.; Morel, JM, On a variational theory of image amodal completion, Rendiconti Del Seminario Matematico Della Universita Di Padova, 116, 4, 211-252 (2005) · Zbl 1150.49023 [36] Mumford, D.: Elastica and computer vision. In: Algebraic Geometry and its Applications, pp. 491-506 (1994) · Zbl 0798.53003 [37] Nesterov, YE, A method for solving the convex programming problem with convergence rate $${O}(1/k^2)$$, Dokl. Akad. Nauk SSSR, 269, 3, 543-547 (1983) [38] Panić, M.; Aelterman, J.; Crnojević, V.; Pižurica, A., Sparse recovery in magnetic resonance imaging with a Markov random field prior, IEEE Trans. Med. Imaging, 36, 10, 2104-2115 (2017) [39] Qu, X.; Guo, D.; Ning, B.; Hou, Y.; Lin, Y.; Cai, S.; Chen, Z., Undersampled MRI reconstruction with patch-based directional wavelets, Magn. Resonance Imaging, 30, 7, 964-977 (2012) [40] 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) [41] Ravishankar, S.; Bresler, Y., MR image reconstruction from highly undersampled k-space data by dictionary learning, IEEE Trans. Med. Imaging, 30, 5, 1028-1041 (2011) [42] Romberg, J., Imaging via compressive sampling, IEEE Signal Process. Mag., 25, 2, 14-20 (2008) [43] Tai, XC; Hahn, J.; Chung, GJ, A fast algorithm for Euler’s elastica model using augmented Lagrangian method, SIAM J. Imaging Sci., 4, 1, 313-344 (2011) · Zbl 1215.68262 [44] Tomasi, C., Manduchi, R.: Bilateral filtering for gray and color images. In: International Conference on Computer Vision, pp. 839-846 (1998) [45] Wang, W.; Li, F.; Ng, MK, Structural similarity based nonlocal variational models for image restoration, IEEE Trans. Image Process., 28, 9, 4260-4272 (2019) · Zbl 07122976 [46] Wu, C.; Tai, XC, Augmented Lagrangian method, dual methods, and split Bregman iteration for ROF, vectorial TV, and high order models, SIAM J. Imaging Sci., 3, 3, 300-339 (2010) · Zbl 1206.90245 [47] Yang, G.; Yu, S.; Dong, H.; Slabaugh, G.; Dragotti, PL; Ye, X.; Liu, F.; Arridge, S.; Keegan, J.; Guo, Y., Dagan: deep de-aliasing generative adversarial networks for fast compressed sensing MRI reconstruction, IEEE Trans. Med. Imaging, 37, 6, 1310-1321 (2018) [48] Yang, Y., Sun, J., Li, H., Xu, Z.: Deep ADMM-Net for compressive sensing MRI. Advances in Neural Information Processing Systems pp. 10-18 (2016) [49] Yaroslavsky, LP, Digital Picture Processing: An Introduction (1985), Berlin: Springer, Berlin [50] Yashtini, M.; Kang, SH, A fast relaxed normal two split method and an effective weighted TV approach for Euler’s elastica image inpainting, SIAM J. Imaging Sci., 9, 4, 1552-1581 (2016) · Zbl 1366.94075 [51] Zhang, X.; Burger, M.; Bresson, X.; Osher, S., Bregmanized nonlocal regularization for deconvolution and sparse reconstruction, SIAM J. Imaging Sci., 3, 3, 253-276 (2010) · Zbl 1191.94030 [52] Zhili, Y.; Mathews, J., Nonlocal regularization of inverse problems: a unified variational framework, IEEE Trans. Image Process., 22, 8, 3192-3203 (2013) [53] Zhu, W.; Chan, T.; Esedoglu, S., Segmentation with depth: a level set approach, SIAM J. Sci. Comput., 28, 5, 1957-1973 (2006) · Zbl 1344.68266 [54] Zhu, W.; Tai, XC; Chan, T., Image segmentation using Euler’s elastica as the regularization, J. Sci. Comput., 57, 2, 414-438 (2013) · Zbl 1282.65037
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