A combined dictionary learning and TV model for image restoration with convergence analysis. (English) Zbl 1488.94035
Summary: We consider in this paper the \(l_0\)-norm based dictionary learning approach combined with total variation regularization for the image restoration problem. It is formulated as a nonconvex nonsmooth optimization problem. Despite that this image restoration model has been proposed in many works, it remains important to ensure that the considered minimization method satisfies the global convergence property, which is the main objective of this work. Therefore, we employ the proximal alternating linearized minimization method whereby we demonstrate the global convergence of the generated sequence to a critical point. The results of several experiments demonstrate the performance of the proposed algorithm for image restoration.
MSC:
| 94A08 | Image processing (compression, reconstruction, etc.) in information and communication theory |
| 49J52 | Nonsmooth analysis |
| 65K05 | Numerical mathematical programming methods |
| 34A34 | Nonlinear ordinary differential equations and systems |
Keywords:
image deblurring; dictionary learning; sparse approximation; total variation; proximal methods; nonconvex optimizationSoftware:
FASTAReferences:
| [1] | M. Afonso, V.M. Jos D. Bioucas-Dias, M.T. Figueiredo,Fast image recovery using variable splitting and constrained optimization, IEEE Trans. Image Process.19(2010) 2345-2356. · Zbl 1371.94018 |
| [2] | A. Asaei, M. Golbabaee, H. Bourlard, V. Cevher,Structured sparsity models for reverberant speech separation, IEEE/ACM Trans. Au. Sp. Lang. Proc.22(2014) 620-633. |
| [3] | M. Aharon, M. Elad, A. Bruckstein,K-SVD: An algorithm for designing overcomplete dictionaries for sparse representation, IEEE Trans. Signal Process.54(2006) 4311-4322. · Zbl 1375.94040 |
| [4] | Z. Alexander, R. Tichatschke,Proximal point methods and nonconvex optimization, J. Global Optim.13(1998) 389-406. · Zbl 0916.90224 |
| [5] | H. Attouch, J. Bolte,On the convergence of the proximal algorithm for nonsmooth functions involving analytic features, Math. Program.116(2009) 5-16. · Zbl 1165.90018 |
| [6] | H. Attouch, J. Bolte, P. Redont, A. Soubeyran,Proximal alternating minimization and projection methods for nonconvex problems: An approach based on the Kurdyka- Lojasiewicz inequality,Math. Oper. Res.35(2) (2010) 438-457. · Zbl 1214.65036 |
| [7] | C. Bao, H. Ji, Y. Quan, Z. Shen,Dictionary learning for sparse coding: Algorithms and convergence analysis, IEEE Trans. Pattern Anal. Mach. Intell38(7) (2015) 1356-1369. |
| [8] | J.M. Bioucas-DiasA variable splitting augmented Lagrangian approach to linear spectral unmixing, First Workshop on Hyperspectral Image and Signal Processing: Evolution in Remote Sensing, 2009. |
| [9] | J.M. Bioucas-Dias, M.A.T. Figueiredo, J.P. Oliveira,Total variation-based image deconvolution: a majorization-minimization approach, IEEE Int. Conf. ICASSP, 2006. · Zbl 1178.94029 |
| [10] | J. Bolte, A. Daniilidis, A. Lewis,The kojasiewicz inequality for nonsmooth subanalytic functions with applications to subgradient dynamical systems, SIAM J. Optim.17(4) (2007) 1205-1223. · Zbl 1129.26012 |
| [11] | J. Bolte, A. Daniilidis, A. Lewis, M. Shiota,Clarke subgradients of stratifiable functions, SIAM J. Optim.18(2007) 556-572 . · Zbl 1142.49006 |
| [12] | S. Bolte, S. Sabach, M. Teboulle,Proximal alternating linearized minimization or nonconvex and nonsmooth problems, Math. Program.146(1-2) (2014) 459-494. · Zbl 1297.90125 |
| [13] | T. Chan, A. Marquina, P. Mulet,High-order total variation-based image restoration, SIAM J. Sci. Comput.22(2) (2000) 503-516. · Zbl 0968.68175 |
| [14] | A. Chambolle,An algorithm for total variation minimization and applications, J. Math. Imaging Vision20(2004) 89-97. · Zbl 1366.94048 |
| [15] | A. Chambolle, P. Lions,Image recovery via total variation minimization and related problems,Numer. Math.76(2) (1997) 167-188. · Zbl 0874.68299 |
| [16] | Y. Chen, W. W. Hager, M. Yashtini, X. Ye, H. Zhang,Bregman operator splitting with variable stepsize for total variation image reconstruction,Comput. Optim. Appl.54(2) (2013) 317-342. · Zbl 1290.90071 |
| [17] | K. Dabov, S. Foi, V. Katkovnik, K. Egiazarian,Image restoration by sparse 3D transformdomain collaborative filtering,IPAS VI.6812(2008) 681207 . |
| [18] | D.L. Donoho, M. Elad, V.N. Temlyakov,Stable recovery of sparse overcomplete representations in the presence of noise,IEEE Trans. Inform. Theory52(2006) 6-18. · Zbl 1288.94017 |
| [19] | M. Elad, M. Aharon,Image denoising via sparse and redundant representations over learned dictionaries, IEEE Trans. Image Process.15(2006) 3736-3745. |
| [20] | S. Esedoglu, S.J. Osher.Decomposition of images by the anisotropic Rudin-Osher-Fatemi model,Comm. Pure Appl. Math.57(2004) 1609-1626. · Zbl 1083.49029 |
| [21] | T. Goldstein, S. Osher,The split Bregman method for L1-regularized problems,SIAM J. Imaging Sci.2(2009) 323-343. · Zbl 1177.65088 |
| [22] | T. Goldstein, C. Studer, R. Baraniuk,A field guide to forward-backward splitting with a FASTA implementation,Available: http://arxiv.org/abs/1411.3406, 2014. |
| [23] | Z. Huan, Z. Lin,Accelerated proximal gradient methods for nonconvex programming,Adv. Neural. Inf. Process. Syst. (2015) 379-387. |
| [24] | A. Kaplan, W. Tichatschke,Proximal point methods and nonconvex optimization,J. Global Optim.13(4) (1998) 389-406. · Zbl 0916.90224 |
| [25] | H. Lu, J. Wei, Q. Liu, Y. Wang, X. Deng,A dictionary learning method with total generalized variation for MRI reconstruction,Int. J. Biomed. Imaging., Volume 2016, Article ID 7512471, 13 pages. |
| [26] | L. Rebollo-Neira, D. Lowe.Optimized orthogonal matching pursuit approach, IEEE Signal Process. Lett.9(4) (2002) 137-140. |
| [27] | Y.C. Pati, R. Rezaiifar, P.S. Krishnaprasad,Orthogonal matching pursuit: recursive function approximation with applications to wavelet decomposition, Proc. 27th Asilomar Conf. Signals, Syst.. and Comput., Pacific Grove, CA, USA,1(1993) 40-44. |
| [28] | I. Ramirez, P. Sprechmann, G. Sapiro,Classification and clustering via dictionary learning with structured incoherence and shared features,Proc. IEEE Int. Conf. Comput. Vis. Pattern Recognit., (2011) 35013508 |
| [29] | L.I. Rudin, S. Osher, E. Fatemi,Nonlinear total variation based noise removal algorithms, Physica D.60(1992) 259-268. · Zbl 0780.49028 |
| [30] | R. Gribonval, M. Nikolova,A characterization of proximity operators, J. Math. Imaging Vision (2020) 1-17. · Zbl 07255773 |
| [31] | D. Strong, T. Chan,Edge-preserving and scale-dependent properties of total variation regularization,Inverse Probl.19(2006) S165. · Zbl 1043.94512 |
| [32] | T. Blumensath, M.E. Davies,Iterative hard thresholding for compressed sensing,Appl. Comput. Harmon. Anal.27(3) (2009) 265-274. · Zbl 1174.94008 |
| [33] | J.-L. Starck, Y. Moudden, J. Bobin, M. Elad, D.L. Donoho,Morphological component analysis, Proc. SPIE 5914, Wavelets XI, 59140Q, 17 September 2005. |
| [34] | P. Jain, P. Kar,Non-convex optimization for machine learning, Found. Trends Mach. Learn. 10(3-4) (2017) 142-336. · Zbl 1388.68251 |
| [35] | T. Poggio, V. Torre,Ill-posed problems and regularization analysis in early vision,MIT AI Lab, Tech. Rep. AI Memo. (1984) 773. |
| [36] | M. |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
