A fast relaxed normal two split method and an effective weighted TV approach for Euler’s elastica image inpainting.

*(English)*Zbl 1366.94075##### MSC:

94A08 | Image processing (compression, reconstruction, etc.) in information and communication theory |

90C26 | Nonconvex programming, global optimization |

##### Keywords:

Euler’s elastica model; numerical optimization methods; convergence analysis; KKT conditions; image inpainting##### Software:

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\textit{M. Yashtini} and \textit{S. H. Kang}, SIAM J. Imaging Sci. 9, No. 4, 1552--1581 (2016; Zbl 1366.94075)

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[1] | P. Arias, V. Caselles, and G. Sapiro, A variational framework for non-local image inpainting, in Energy Minimization Methods in Computer Vision and Pattern Recognition, Lecture Notes in Comput. Sci. 5681, Springer, New York, 2009, pp. 345–358. |

[2] | J.-F. Aujol, S. Ladjal, and S. Masnou, Exemplar-based inpainting from a variational point of view, SIAM J. Math. Anal., 42 (2010), pp. 1246–1285. · Zbl 1210.49002 |

[3] | D. Auroux and M. Masmoudi, A one-shot inpainting algorithm based on the topological asymptotic analysis, Comput. Appl. Math., 25 (2006), pp. 251–267. · Zbl 1182.94006 |

[4] | E. Bae, J. Shi, and X.-C. Tai, Graph cuts for curvature based image denoising, IEEE Trans. Image Process., 20 (2011), pp. 1199–1210. · Zbl 1372.94015 |

[5] | C. Ballester, M. Bertalmio, V. Caselles, G. Sapiro, and J. J. Verdera, Filling-in by joint interpolation of vector fields and gray levels, IEEE Trans. Image Process., 10 (2001), pp. 1200–1211. · Zbl 1037.68771 |

[6] | M. Bertalmio, G. Sapiro, V. Caselles, and C. Ballester, Image inpainting, in Proceedings of the 27th Annual Conference on Computer Graphics and Interactive Techniques, 2000, pp. 417–424. |

[7] | A. Bertozzi, S. Esedoglu, and A. Gillette, Analysis of a two-scale Cahn-Hilliard model for binary image inpainting, Multiscale Model. Simul., 6 (2007), pp. 913–936. · Zbl 1149.35309 |

[8] | A. Bertozzi, S. Esedoglu, and A. Gillette, Inpainting of binary images using the Cahn-Hilliard equation, IEEE Trans. Image Process., 16 (2007), pp. 285–291. · Zbl 1279.94008 |

[9] | F. Bornemann and T. März, Fast image inpainting based on coherence transport, J. Math. Imaging Vision, 28 (2007), pp. 259–278. |

[10] | S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, Distributed optimization and statistical learning via the alternating direction method of multipliers, Found. Trends Machine Learning, 3 (2010), pp. 1–122. · Zbl 1229.90122 |

[11] | K. Bredies, T. Pock, and B. Wirth, A convex, lower semi-continuous approximation of Euler’s elastica energy, SIAM J. Math. Anal., 47 (2015), pp. 566–613. · Zbl 1319.49018 |

[12] | C. Brito-Loeza and K. Chen, Fast numerical algorithms for Euler’s elastica inpainting model, Int. J. Modern Math., 2 (2010), pp. 157–182. · Zbl 1205.68475 |

[13] | M. Burger, L. He, and C. B. Schönlieb, Cahn-Hilliard inpainting and a generalization for grayvalue images, SIAM J. Imaging Sci., 2 (2009), pp. 1129–1167. · Zbl 1180.49007 |

[14] | F. Cao, Y. Gousseau, S. Masnou, and P. Péres, Geometrically guided exemplar-based inpainting, SIAM J. Imaging Sci., 4 (2011), pp. 1143–1179. · Zbl 1235.94017 |

[15] | T. Chan, S. H. Kang, and J. Shen, Euler’s elastica and curvature based inpaintings, SIAM J. Appl. Math., 63 (2002), pp. 564–592. · Zbl 1028.68185 |

[16] | T. Chan and J. Shen, Non-texture inpainting by curvature-driven diffusion, J. Visual Commun. Image Representation, 12 (2001), pp. 436–449. |

[17] | T. Chan and J. Shen, Mathematical models for local non-texture inpaintings, SIAM J. Appl. Math., 62 (2002), pp. 1019–1043. · Zbl 1050.68157 |

[18] | T. Chan and J. Shen, Variational image inpainting, Comm. Pure Appl. Math., 58 (2005), pp. 579–619. · Zbl 1067.68168 |

[19] | T. Chan, J. Shen, and H. M. Zhou, Total variation wavelet inpainting, J. Math. Imaging Vision, 25 (2006), pp. 107–125. |

[20] | Y. Chen, W. W. Hager, M. Yashtini, X. Ye, and H. Zhang, Bregman operator splitting with variable stepsize for total variation image reconstruction, Comput. Optim. Appl., 54 (2013), pp. 317–342. · Zbl 1290.90071 |

[21] | P. Combettes and V. Wajs, Signal recovery by proximal forward-backward splitting, Multiscale Model. Simul., 4 (2005), pp. 1168–1200. · Zbl 1179.94031 |

[22] | J. Dobrosotskaya and A. Bertozzi, A wavelet-Laplace variational technique for image deconvolution and inpainting, IEEE Trans. Image Process., 17 (2008), pp. 657–663. |

[23] | Y. Duan, Y. Wang, and J. Hahn, A fast augmented lagrangian method for Euler’s elastica models, Numer. Math. Theory Methods Appl., 6 (2013), pp. 47–71. · Zbl 1289.94006 |

[24] | J. Eckstein and D. Bertsekas, On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators, Math. Program., 55 (1992), pp. 293–318. · Zbl 0765.90073 |

[25] | S. Esedoglu and R. March, Segmentation with depth but without detecting junctions, J. Math. Imaging Vision, 18 (2003), pp. 7–15. · Zbl 1033.68132 |

[26] | S. Esedoglu and J. Shen, Digital inpainting based on the Mumford-Shah-Euler image model, European J. Appl. Math., 13 (2002), pp. 353–370. · Zbl 1017.94505 |

[27] | L. Euler, Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, Lausanne, 1744. · Zbl 0788.01072 |

[28] | M. J. Fadili, J. L. Starck, and F. Murtagh, Inpainting and zooming using sparse representations, Computer J., 52 (2009), pp. 64–79. |

[29] | E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Monogr. Math. 80, Birkhäuser Boston, Boston, MA, 1984. · Zbl 0545.49018 |

[30] | R. Glowinski, T.-W. Pan, and X.-C. Tai, Some Facts About Operator-Splitting and Alternating Direction Methods, UCLA CAM Report 16–10, 2016. · Zbl 1372.65205 |

[31] | B. Goldluecke and D. Cremers, Introducing total curvature for image processing, in Proceedings of the IEEE International Conference Computer Vision (ICCV), 2011, pp. 1267–1274, |

[32] | T. Goldstein and S. Osher, The split Bregman method for L1 regularized problems, SIAM J. Imaging Sci., 2 (2009), pp. 323–343. · Zbl 1177.65088 |

[33] | H. Grossauer and O. Scherzer, Using the complex Ginzburg-Landau equation for digital inpainting in 2D and 3D, in Scale Space Methods in Computer Vision, Lecture Notes in Comput. Sci. 2695, Springer, New York, 2003, pp. 225–236. · Zbl 1067.68739 |

[34] | W. W. Hager, C. Ngo, M. Yashtini, and H. Zhang, Alternating direction approximate Newton (ADAN) algorithm for ill-conditioned inverse problems with application to Parallel MRI, J. Oper. Res. Soc. China, 3 (2015), pp. 139–162. · Zbl 1317.90235 |

[35] | B. Horn, The curve of least energy, ACM Trans. Math. Software, 9 (1983), pp. 441–460. · Zbl 0533.41007 |

[36] | M. Kallay, Plane curves of minimal energy, ACM Trans. Math. Software, 12 (1986), pp. 219–222. · Zbl 0624.73102 |

[37] | M. Kallay, Method to approximate the space curve of least energy and prescribed length, Comput. Aided Design., 19 (1987), pp. 73–76. · Zbl 0655.65028 |

[38] | S. H. Kang, W. Zhu, and J. Shen, Illusory shapes via corner fusion, SIAM J. Imaging Sci., 7 (2014), pp. 1907–1936. · Zbl 1320.68217 |

[39] | F. Knoll, K. Bredies, and T. Pock, Second order total generalized variation (TGV) for MRI, Magnetic Resonance Medicine, 65 (2011), pp. 480–491. |

[40] | A. Marquina and S. Osher, A new time dependent model based on level set motion for nonlinear deblurring and noise removal, in Scale-Space Theories in Computer Vision, Lecture Notes in Comput. Sci. Springer, Berlin, 1999, pp. 429–434. |

[41] | S. Masnou and J.-M. Morel, On a variational theory of image amodal completion, Rend. Sem. Mat. Univ. Padova, 116 (2006), pp. 211–252. · Zbl 1150.49023 |

[42] | S. Masnou and M. Morel, Level lines based disocclusion, in Proceedings of the International Conference on Image Processing, 1998, pp. 259–263. |

[43] | D. Mumford, Elastica and computer vision, Algebraic Geometry and its Applications, Springer, Berlin, 1994, pp. 491–506. · Zbl 0798.53003 |

[44] | Y. E. Nesterov, A method for solving the convex programming problem with convergence rate \(\mathcal{O}(1/k^2)\) (in Russian), Dokl. Akad. Nauk SSSR, 269 (1983), pp. 543–547. |

[45] | M. Nitzberg, D. Mumford, and T. Shiota, Filtering, Segmentation, and Depth, Lecture Notes in Comput. Sci. 662, Springer-Verlag, Berlin, 1993. |

[46] | M. Oliveira, B. Bowen, R. Mckenna, and Y.-S. Chang, Fast digital image inpainting, in Proceedings of the International Conference on Visualization, Imaging and Image Processing (VIIP 2001), Marbella, Spain, 2001, pp. 106–107. |

[47] | K. Papafitsoros, C. B. Schöenlieb, and B. Sengul, Combined first and second order total variation inpainting using split Bregman, Image Processing On Line (2013), pp. 112–136. |

[48] | R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, NJ, 1970. · Zbl 0193.18401 |

[49] | R. T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J. Control Optim., 14 (1976), pp. 877–898. · Zbl 0358.90053 |

[50] | T. Schoenemann, F. Kahl, S. Masnou, and D. Cremers, A linear framework for region based image segmentation and inpainting involving curvature penalization, Int. J. Comput. Vis., 99 (2012), pp. 53–68. · Zbl 1254.68282 |

[51] | T. Schoenemann, S. Masnou, and D. Cremers, The elastica ratio: Introducing curvature into ratio-based image segmentation, IEEE Trans. Image Process., 20 (2011), pp. 2565–2581. · Zbl 1372.94227 |

[52] | C.-B. Schönlieb and A. Bertozzi, Unconditionally stable schemes for higher order inpainting, Commun. Math. Sci., 9 (2011), pp. 413–457. · Zbl 1216.94016 |

[53] | X.-C. Tai, J. Hahn, and G. J. Chung, A fast algorithm for Euler’s elastica model using augmented Lagrangian method, SIAM J. Imaging Sci., 4 (2011), pp. 313–344. · Zbl 1215.68262 |

[54] | Y. Wang, J. Yang, W. Yin, and Y. Zhang, A new alternating minimization algorithm for total variation image reconstruction, SIAM J. Imaging Sci., 1 (2008), pp. 248–272. · Zbl 1187.68665 |

[55] | C. Wu and X.-C. Tai, Augmented Lagrangian method, dual methods, and split Bregman iteration for ROF, vectorial TV, and high order models, SIAM J. Imaging Sci., 3 (2010), pp. 300–339. · Zbl 1206.90245 |

[56] | C. Wu, J. Zhang, and X.-C. Tai, Augmented Lagrangian method for total variation restoration with non-quadratic fidelity, Inverse Probl. Imaging, 5 (2011), pp. 237–261. · Zbl 1225.80013 |

[57] | M. Yashtini and S. H. Kang, Alternating direction method of multiplier for Euler’s elastica-based denoising, Scale Space and Variational Methods in Computer Vision, Lecture Notes in Comput. Sci. 9087, Springer, New York, 2015, pp. 690–701. |

[58] | X. Zhang and T. F. Chan, Wavelet inpainting by nonlocal total variation, Inverse Probl. Imaging, 4 (2010), pp. 191–210. · Zbl 1185.42040 |

[59] | W. Zhu, T. Chan, and S. Esedoglu, Segmentation with depth: A level set approach, SIAM J. Sci. Comput., 28 (2006), pp. 1957–1973. · Zbl 1344.68266 |

[60] | W. Zhu, X.-C. Tai, and T. Chan, Augmented Lagrangian method for a mean curvature based image denoising model, Inverse Probl. Imaging, 7 (2013), pp. 1409–1432. · Zbl 1311.94015 |

[61] | W. Zhu, X.-C. Tai, and T. Chan, Image segmentation using Euler’s elastica as the regularization, J. Sci. Comput., 57 (2013), pp. 414–438. · Zbl 1282.65037 |

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