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Learning Non-Negativity Constrained Variation for Image Denoising and Deblurring

  • Tengda Wei (a1), Linshan Wang (a2), Ping Lin (a3), Jialing Chen (a3), Yangfan Wang (a4) and Haiyong Zheng (a5)...
Abstract

This paper presents a heuristic Learning-based Non-Negativity Constrained Variation (L-NNCV) aiming to search the coefficients of variational model automatically and make the variation adapt different images and problems by supervised-learning strategy. The model includes two terms: a problem-based term that is derived from the prior knowledge, and an image-driven regularization which is learned by some training samples. The model can be solved by classical ε-constraint method. Experimental results show that: the experimental effectiveness of each term in the regularization accords with the corresponding theoretical proof; the proposed method outperforms other PDE-based methods on image denoising and deblurring.

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Corresponding author
*Corresponding author. Email addresses: tdwei123@163.com (T. Wei), wangls@ouc.edu.cn (L. Wang), plin@maths.dundee.ac.uk (P. Lin), j.z.chen@dundee.ac.uk (J. Chen), yfwang@ouc.edu.cn (Y. Wang), zhenghaiyong@ouc.edu.cn (H. Zheng)
References
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[1] Aubert, G. and Kornprobst, P., Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations, Springer-Verlag, New York, USA, 2006.
[2] Bauschke, H. H., Burachik, R., Combettes, P. L., Elser, V., Luke, D. R. and Wolkowicz, H., Fixed-Point Algorithms for Inverse Problems in Science and Engineering, Springer Science and Business Media, 2011.
[3] Beck, A. and Teboulle, M., Fast gradient-based algorithms for constrained total variation image denoising and deblurring problems, IEEE Trans. Image Process., 18 (2009), pp. 24192434.
[4] Becker, S. and Fadili, J., A quasi-newton proximal splitting method, in NIPS, 2012, pp. 26182626.
[5] Boyd, S. and Vandenberghe, L., Convex Optimization, Cambridge University Press, New York, USA, 2004.
[6] Cai, J. F., Chan, R. H. and Shen, Z., A framelet-based image inpainting algorithm, Appl. Comput. Harmon. Anal., 24 (2008), pp. 131149.
[7] Canny, J., A computational approach to edge detection, IEEE Trans. Pattern Anal. Mach. Intell., 8 (1986), pp. 679698.
[8] Chambolle, A. and Lions, P. L., Image recovery via total variation minimization and related problems, Numer. Math., 76 (1997), pp. 167188.
[9] Chambolle, A. and Pock, T., A first-order primal-dual algorithm for convex problems with applications to imaging, J. Math. Imaging Vis., 40 (2011), pp. 120145.
[10] Chan, R. H., Wen, Y. W. and Yip, A. M., A fast optimization transfer algorithm for image inpainting in wavelet domains, IEEE Trans. Image Process., 18 (2009), pp. 14671476.
[11] Chan, T. F., Esedoglu, S. and Nikolova, M., Algorithms for finding global minimizers of image segmentation and denoising models, SIAM J. Appl. Math., 66 (2006), pp. 16321648.
[12] Chan, T. F., Golub, G. H. and Mulet, P., A nonlinear primal-dual method for total variation-based image restoration, SIAM J. Sci. Comput., 20 (1999), pp. 19641977.
[13] Chan, T. F., Ng, M. K., Yau, A. C. and Yip, A. M., Superresolution image reconstruction using fast inpainting algorithms, Appl. Comput. Harmon. Anal., 23 (2007), pp. 324.
[14] Chan, T. F. and Shen, J., Variational image inpainting, Commun. Pure Appl. Math., 58 (2005), pp. 579619.
[15] Chan, T. F., Shen, J. and Zhou, H. M., Total variation wavelet inpainting, J. Math. Imaging Vis., 25 (2006), pp. 107125.
[16] Chan, T. F. and Vese, L. A., Active contours without edges, IEEE Trans. Image Process., 10 (2001), pp. 266277.
[17] Chartrand, R. and Wohlberg, B., Total-variation regularization with bound constraints, in IEEE ICASSP, 2010, pp. 766769.
[18] Chen, X., Ng, M. K. and Zhang, C., Non-Lipschitz lp-regularization and box constrained model for image restoration, IEEE Trans. Image Process., 21 (2012), pp. 47094721.
[19] Chen, Y., Yu, W. and Pock, T., On learning optimized reaction diffusion processes for effective image restoration, in IEEE CVPR, 2015, pp. 8790.
[20] Dey, N., Blanc-Féraud, L., Zimmer, Z. K. C., Olivo-Marin, J. C. and Zerubia, J., A deconvolution method for confocal microscopy with total variation regularization, in IEEE Intern. Symp. on Biomedical Imaging: Macro to Nano, 2004, pp. 12231226.
[21] Esser, E., Zhang, X. and Chan, T. F., A general framework for a class of first order primal-dual algorithms for convex optimization in imaging science, SIAM J. Imaging Sci., 3 (2010), pp. 10151046.
[22] Goldstein, T. and Osher, S., The split bregman method for l1-regularized problems, SIAM J. Imaging Sci., 2 (2009), pp. 323343.
[23] Haimes, Y. Y., Lasdon, L. S. and Wismer, D. A., On a bicriterion formulation of the problems of integrated system identification and system optimization, IEEE Trans. Syst. Man Cyber., 1 (1971), pp. 296297.
[24] Hintermüller, M., Ito, K. and Kunisch, K., The primal-dual active set strategy as a semismooth Newtons method, SIAM J. Optim., 13 (2003), pp. 865888.
[25] Krishnan, D., Lin, P. and Yip, A. M., A primal-dual active-set method for non-negativity constrained total variation deblurring problems, IEEE Trans. Image Process., 16 (2007), pp. 27662777.
[26] Krishnan, D., Pham, Q. V. and Yip, A. M., A primal dual active set algorithm for bilaterally constrained total variation deblurring and piecewise constant Mumford-Shah segmentation problems, Adv. Comput. Math., 31 (2009), pp. 237266.
[27] Larson, E. C. and Chandler, D. M., Most apparent distortion: full-reference image quality assessment and the role of strategy, J. Electron. Imaging, 19 (2010), 011006.
[28] Law, Y. N., Lee, H. K. and Yip, A. M., A multi-resolution stochastic level set method for Mumford-Shah image segmentation, IEEE Trans. Image Process., 17 (2008), pp. 22892300.
[29] Liu, R., Lin, Z., Zhang, W. and Su, Z., Learning PDEs for Image Restoration via Optimal Control, in ECCV, 2010, pp. 115128.
[30] Martin, D., Fowlkes, C., Tal, D. and Malik, J., A database of human segmented natural images and its application to evaluating segmentation algorithms and measuring ecological statistics, in IEEE ICCV, 2001.
[31] Mersereau, R. M. and Schafer, R. W., Comparative study of iterative deconvolution algorithms, in IEEE ICASSP, 1978, pp. 192195.
[32] Morel, J. M. and Solimini, S., Variational Methods in Image Segmentation, Birkhauser Boston Inc., Cambridge, USA, 1995.
[33] Paragios, N., Chen, Y. and Faugeras, O. D., Handbook of Mathematical Models in Computer Vision, Springer Science and Business Media, 2006.
[34] Persson, M., Bone, D. and Elmqvist, H., Total variation norm for three-dimension iterative reconstruction in limited view angle tomography, Phys. Med. Biol., 46 (2001), pp. 853866.
[35] Ponomarenko, N., Jin, L., Ieremeiev, O., Lukin, V., Egiazarian, K., Astola, J., Vozel, B., Chehdi, K., Carli, M., Battisti, F. et al., Image database TID2013: peculiarities, results and perspectives, Signal Process. Image Commun., 30 (2015), pp. 5777.
[36] Rudin, L. I., Osher, S. and Fatemi, E., Nonlinear total variation based noise removal algorithms, Phys. D, 60 (1992), pp. 259268.
[37] Schafer, R. W., Mersereau, R. M. and Richards, M. A., Constrained iterative restoration algorithms, Proc. IEEE, 69 (1981), pp. 432450.
[38] Scherzer, O., Handbook of Mathematical Methods in Imaging, Springer-Verlag, New York, USA, 2011.
[39] Sheikh, H. R., Sabir, M. F. and Bovik, A. C., A statistical evaluation of recent full reference image quality assessment algorithms, IEEE Trans. Image Process., 15 (2006), pp. 34403451.
[40] Tai, C., Zhang, X. and Shen, Z., Wavelet frame based multiphase image segmentation, SIAM J. Imaging Sci., 6 (2013), pp. 25212546.
[41] Tai, X. C., Lie, K. A., Chan, T. F. and Osher, S., Image Processing Based on Partial Differential Equations, Springer-Verlag, New York, USA, 2007.
[42] Tai, X. C., Mørken, K., Lysaker, M. and Lie, K. A., Scale Space and Variational Methods in Computer Vision, Springer Berlin Heidelberg, 2009.
[43] Tikhonov, A. N. and Arsenin, V. Y., Solutions of Ill-Posed Problems, Winston and Sons, Washington, D. C., 1977.
[44] Vogel, C. R., Computational Methods for Inverse Problems, SIAM, Philadelphia, USA, 2002.
[45] Weickert, J., Ishikawa, S. and Imiya, A., Linear scale-space has first been proposed in Japan, J. Math. Imaging Vis., 10 (1999), pp. 237252.
[46] Wu, C. and Tai, X. C., Augmented Lagrangian method, dual methods, and split Bregman iteration for ROF, vectorial TV, and high order models, SIAM J. Imaging Sci., 3 (2010), pp. 300339.
[47] You, Y. L. and Kaveh, M., Blind image restoration by anisotropic regularization, IEEE Trans. Image Process., 8 (1999), pp. 396407.
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Numerical Mathematics: Theory, Methods and Applications
  • ISSN: 1004-8979
  • EISSN: 2079-7338
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