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Learning Non-Negativity Constrained Variation for Image Denoising and Deblurring

Part of: Equations of mathematical physics and other areas of application Partial differential equations

Published online by Cambridge University Press:  12 September 2017

Tengda Wei ,
Linshan Wang ,
Ping Lin ,
Jialing Chen ,
Yangfan Wang  and
Haiyong Zheng
Tengda Wei*
Affiliation:
College of Oceanic and Atmospheric Sciences, Ocean University of China, Qingdao 266100, China
Linshan Wang*
Affiliation:
College of Mathematics, Ocean University of China, Qingdao 266100, China
Ping Lin*
Affiliation:
Department of Mathematics, University of Dundee, Dundee DD1 4HN, UK
Jialing Chen*
Affiliation:
Department of Mathematics, University of Dundee, Dundee DD1 4HN, UK
Yangfan Wang*
Affiliation:
College of Marine Life Science, Ocean University of China, Qingdao 266100, China
Haiyong Zheng*
Affiliation:
College of Information Science and Engineering, Ocean University of China, Qingdao 266100, China
*
*Corresponding author. Email addresses:[email protected] (T. Wei), [email protected] (L. Wang), [email protected] (P. Lin), [email protected] (J. Chen), [email protected] (Y. Wang), [email protected] (H. Zheng)
*Corresponding author. Email addresses:[email protected] (T. Wei), [email protected] (L. Wang), [email protected] (P. Lin), [email protected] (J. Chen), [email protected] (Y. Wang), [email protected] (H. Zheng)
*Corresponding author. Email addresses:[email protected] (T. Wei), [email protected] (L. Wang), [email protected] (P. Lin), [email protected] (J. Chen), [email protected] (Y. Wang), [email protected] (H. Zheng)
*Corresponding author. Email addresses:[email protected] (T. Wei), [email protected] (L. Wang), [email protected] (P. Lin), [email protected] (J. Chen), [email protected] (Y. Wang), [email protected] (H. Zheng)
*Corresponding author. Email addresses:[email protected] (T. Wei), [email protected] (L. Wang), [email protected] (P. Lin), [email protected] (J. Chen), [email protected] (Y. Wang), [email protected] (H. Zheng)
*Corresponding author. Email addresses:[email protected] (T. Wei), [email protected] (L. Wang), [email protected] (P. Lin), [email protected] (J. Chen), [email protected] (Y. Wang), [email protected] (H. Zheng)
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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.

Keywords

Learning ideaTV-based modelconstraintε-constraint methodimage restoration

MSC classification

Secondary: 35A15: Variational methods 35Q93: PDEs in connection with control and optimization
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 10 , Issue 4 , November 2017 , pp. 852 - 871
Copyright
Copyright © Global-Science Press 2017 

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References

[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.CrossRefGoogle Scholar
[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.Google Scholar
[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. 2419–2434.Google Scholar
[4] Becker, S. and Fadili, J., A quasi-newton proximal splitting method, in NIPS, 2012, pp. 2618–2626.Google Scholar
[5] Boyd, S. and Vandenberghe, L., Convex Optimization, Cambridge University Press, New York, USA, 2004.Google Scholar
[6] Cai, J. F., Chan, R. H. and Shen, Z., A framelet-based image inpainting algorithm, Appl. Comput. Harmon. Anal., 24 (2008), pp. 131–149.Google Scholar
[7] Canny, J., A computational approach to edge detection, IEEE Trans. Pattern Anal. Mach. Intell., 8 (1986), pp. 679–698.Google Scholar
[8] Chambolle, A. and Lions, P. L., Image recovery via total variation minimization and related problems, Numer. Math., 76 (1997), pp. 167–188.Google Scholar
[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. 120–145.CrossRefGoogle Scholar
[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. 1467–1476.CrossRefGoogle ScholarPubMed
[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. 1632–1648.Google Scholar
[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. 1964–1977.CrossRefGoogle Scholar
[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. 3–24.CrossRefGoogle Scholar
[14] Chan, T. F. and Shen, J., Variational image inpainting, Commun. Pure Appl. Math., 58 (2005), pp. 579–619.Google Scholar
[15] Chan, T. F., Shen, J. and Zhou, H. M., Total variation wavelet inpainting, J. Math. Imaging Vis., 25 (2006), pp. 107–125.CrossRefGoogle Scholar
[16] Chan, T. F. and Vese, L. A., Active contours without edges, IEEE Trans. Image Process., 10 (2001), pp. 266–277.Google Scholar
[17] Chartrand, R. and Wohlberg, B., Total-variation regularization with bound constraints, in IEEE ICASSP, 2010, pp. 766–769.Google Scholar
[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. 4709–4721.Google Scholar
[19] Chen, Y., Yu, W. and Pock, T., On learning optimized reaction diffusion processes for effective image restoration, in IEEE CVPR, 2015, pp. 87–90.Google Scholar
[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. 1223–1226.Google Scholar
[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. 1015–1046.Google Scholar
[22] Goldstein, T. and Osher, S., The split bregman method for l1-regularized problems, SIAM J. Imaging Sci., 2 (2009), pp. 323–343.Google Scholar
[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. 296–297.Google Scholar
[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. 865–888.Google Scholar
[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. 2766–2777.Google Scholar
[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. 237–266.CrossRefGoogle Scholar
[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.Google Scholar
[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. 2289–2300.Google Scholar
[29] Liu, R., Lin, Z., Zhang, W. and Su, Z., Learning PDEs for Image Restoration via Optimal Control, in ECCV, 2010, pp. 115–128.Google Scholar
[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.Google Scholar
[31] Mersereau, R. M. and Schafer, R. W., Comparative study of iterative deconvolution algorithms, in IEEE ICASSP, 1978, pp. 192–195.Google Scholar
[32] Morel, J. M. and Solimini, S., Variational Methods in Image Segmentation, Birkhauser Boston Inc., Cambridge, USA, 1995.Google Scholar
[33] Paragios, N., Chen, Y. and Faugeras, O. D., Handbook of Mathematical Models in Computer Vision, Springer Science and Business Media, 2006.CrossRefGoogle Scholar
[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. 853–866.Google Scholar
[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. 57–77.Google Scholar
[36] Rudin, L. I., Osher, S. and Fatemi, E., Nonlinear total variation based noise removal algorithms, Phys. D, 60 (1992), pp. 259–268.Google Scholar
[37] Schafer, R. W., Mersereau, R. M. and Richards, M. A., Constrained iterative restoration algorithms, Proc. IEEE, 69 (1981), pp. 432–450.Google Scholar
[38] Scherzer, O., Handbook of Mathematical Methods in Imaging, Springer-Verlag, New York, USA, 2011.Google Scholar
[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. 3440–3451.Google Scholar
[40] Tai, C., Zhang, X. and Shen, Z., Wavelet frame based multiphase image segmentation, SIAM J. Imaging Sci., 6 (2013), pp. 2521–2546.Google Scholar
[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.Google Scholar
[42] Tai, X. C., Mørken, K., Lysaker, M. and Lie, K. A., Scale Space and Variational Methods in Computer Vision, Springer Berlin Heidelberg, 2009.CrossRefGoogle Scholar
[43] Tikhonov, A. N. and Arsenin, V. Y., Solutions of Ill-Posed Problems, Winston and Sons, Washington, D. C., 1977.Google Scholar
[44] Vogel, C. R., Computational Methods for Inverse Problems, SIAM, Philadelphia, USA, 2002.Google Scholar
[45] Weickert, J., Ishikawa, S. and Imiya, A., Linear scale-space has first been proposed in Japan, J. Math. Imaging Vis., 10 (1999), pp. 237–252.Google Scholar
[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. 300–339.Google Scholar
[47] You, Y. L. and Kaveh, M., Blind image restoration by anisotropic regularization, IEEE Trans. Image Process., 8 (1999), pp. 396–407.Google Scholar