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Primal-Dual Method for Hybrid Regularizers-Based Image Restoration with Impulse Noise

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Abstract

With the aim of improving the restoration accuracy, this article introduces a hybrid regularizers approach to recovering images corrupted by impulse noise. The proposed model closely incorporates the superiorities of two recently developed methods: the total generalized variation method and the wavelet frame-based method. Numerically, a highly efficient primal-dual algorithm is constructed to solve the minimization problem, which is derived from the canonical alternating minimization method and based on the Moreau decomposition. Eventually, in comparison with several well-developed numerical methods, simulation experiments are provided to demonstrate the effective performance and advantages of our proposed strategy for image reconstruction under impulse noise, in terms of both image quality assessment and visual improvement.

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References

  1. S. Alliney, Digital filters as absolute norm regularizers. IEEE Trans. Signal Process. 40(6), 1548–1562 (1992)

    Article  MATH  Google Scholar 

  2. S. Alliney, A property of the minimum vectors of a regularizing functional defined by means of the absolute norm. IEEE Trans. Signal Process. 45(4), 913–917 (1997)

    Article  Google Scholar 

  3. H.H. Bauschke, P.L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Space (Springer, New York, 2011)

    Book  MATH  Google Scholar 

  4. D. Bertsekas, J. Tsitsiklis, Parallel and Distributed Computation: Numerical Methods (Athena Scientific, Belmont, 1997)

    MATH  Google Scholar 

  5. S. Boyd, N. Parikh, E. Chu, B. Peleato, J. Eckstein, Distributed optimization and statistical learning via the alternating direction method of multipliers. Found. Trends Mach. Learn. 3(1), 1–122 (2011)

    Article  MATH  Google Scholar 

  6. S. Boyd, L. Vandenberghe, Convex Optimization (Cambridge University Press, New York, 2004)

    Book  MATH  Google Scholar 

  7. K. Bredies, Y. Dong, M. Hintermüller, Spatially dependent regularization parameter selection in total generalized variation models for image restoration. Int. J. Comput. Math. 90(1), 109–123 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  8. K. Bredies, M. Holler, A TGV regularized wavelet based zooming model. Lect. Notes Comput. Sci. 7893, 149–160 (2013)

    Article  Google Scholar 

  9. K. Bredies, K. Kunisch, T. Pock, Total generalized variation. SIAM J. Imaging Sci. 3(3), 492–526 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  10. K. Bredies, T. Valkonen, Inverse problems with second-order total generalized variation constraints, in Proceedings of SampTA 2011, 9th International Conference on Sampling Theory and Applications (2011)

  11. K. Bredies, Recovering piecewise smooth multichannel images by minimization of convex functionals with total generalized variation penalty. Lect. Notes Comput. Sci. 8293, 44–77 (2014)

    Article  Google Scholar 

  12. J.F. Cai, R. Chan, M. Nikolova, Two phase methods for deblurring image corrupted by impulse plus Gaussian noise. Inverse Prob. Imaging 2(2), 187–204 (2008)

    Article  MATH  Google Scholar 

  13. J.F. Cai, B. Dong, S. Osher, Z. Shen, Image restoration: total variation, wavelet frames, and beyond. J. Am. Math. Soc. 25(4), 1033–1089 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  14. J.F. Cai, S. Osher, Z. Shen, Split Bregman methods and frame based image restoration. Multiscale Model. Simul. 8(2), 337–369 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  15. R.H. Chan, Y. Dong, M. Hintermüller, An efficient two-phase \(L^1\)-TV method for restoring blurred images with impulse noise. IEEE Trans. Image Process. 19(7), 1731–1739 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  16. R.H. Chan, C. Hu, M. Nikolova, An iterative procedure for removing random-valued impulse noise. IEEE Signal Process. Lett. 11(12), 921–924 (2004)

    Article  Google Scholar 

  17. T.F. Chan, S. Esedoglu, Aspects of total variation regularized \(L^1\) function approximation. SIAM J. Appl. Math. 65(5), 1817–1837 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  18. D.-Q. Chen, L.-Z. Cheng, Deconvolving Poissonian images by a novel hybrid variational model. J. Vis. Commun. Image R. 22(7), 643–652 (2011)

    Article  Google Scholar 

  19. F. Chen, Y. Jiao, L. Lin, Q. Qin, Image deblurring via combined total variation and framelet. Circuits Syst. Signal Process. 33(6), 1899–1916 (2014)

    Article  Google Scholar 

  20. J. Delon, A. Desolneux, A. Viano, Mixing non-local and TV-L1 methods to remove impulse noise from images, in MAP5 2016-29 (2016)

  21. Y. Dong, M. Hintermüller, M. Neri, An efficient primal-dual method for \(L^1\)TV image restoration. SIAM J. Imaging SCI. 2(4), 1168–1189 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  22. V. Duval, J.-F. Aujol, Y. Gousseau, The TVL1 model: a geometric point of view. Multiscale Model. Simul. 8(1), 154–189 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  23. M. Fornasier, Y. Kim, A. Langer, C.-B. Schönlieb, Wavelet decomposition method for L2/TV-image deblurring. SIAM J. Imag. Sci. 5(3), 857–885 (2011)

    Article  MATH  Google Scholar 

  24. T. Goldstein, S. Osher, The split Bregman algorithm for L1 regularized problems. SIAM J. Imaging Sci. 2(2), 323–343 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  25. X. Guo, F. Li, M.K. Ng, A fast \(\ell \)1-TV algorithm for image restoration. SIAM J. Sci. Comput. 31(1), 2322–2341 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  26. W. Guo, J. Qin, W. Yin, A new detail-preserving regularity scheme. SIAM J. Imaging Sci. 7(2), 1309–1334 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  27. B. He, L.Z. Liao, D. Han, H. Yang, A new inexact alternating directions method for monotone variational inequalities. Math. Program. 92(1), 103–118 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  28. C. He, C. Hu, X. Li, X. Yang, W. Zhang, A parallel alternating direction method with application to compound \(l_1\)-regularized imaging inverse problems. Inform. Sciences 348(20), 179–197 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  29. F. Knoll, K. Bredies, T. Pock, R. Stollberger, Second order total generalized variation (TGV) for MRI. Magn. Reson. Med. 65(2), 480–491 (2011)

    Article  Google Scholar 

  30. X. Liu, Alternating minimization method for image restoration corrupted by impulse noise. Multimed. Tools Appl. 76(10), 12505–12516 (2017)

    Article  Google Scholar 

  31. C.A. Micchelli, L. Shen, Y. Xu, X. Zeng, Proximity algorithms for image models II: L1/TV denosing. Adv. Comput. Math. 38(2), 401–426 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  32. M. Nikolova, Minimizers of cost-functions involving non-smooth data fidelity terms. Application to the processing of outliers. SIAM J. Numer. Anal. 40(3), 965–994 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  33. M. Nikolova, A variational approach to remove outliers and impulse noise. J. Math. Imaging Vis. 20(1–2), 99–120 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  34. L. Rudin, S. Osher, E. Fatemi, Nonlinear total variation based noise removal algorithms. Physica D 60(1–4), 259–268 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  35. I.W. Selesnick, A.F. Abdelnour, Symmetric wavelet tight frames with two generators. Appl. Comput. Harmon. Anal. 17(2), 211–225 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  36. S. Setzer, Operator splittings, Bregman methods and frame shrinkage in image processing. Int. J. Comput. Vis. 92(3), 265–280 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  37. T. Valkonen, K. Bredies, F. Knoll, Total generalized variation in diffusion tensor imaging. SIAM J. Imaging Sci. 6(1), 487–525 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  38. Z. Wang, A.C. Bovik, H.R. Sheikh, E.P. Simoncelli, Image quality assessment: from error measurement to structural similarity. IEEE Trans. Image Process. 13(4), 600–612 (2004)

    Article  Google Scholar 

  39. P. Weiss, L. Blanc-Féraud, G. Aubert, Efficient schemes for total variation minimization under constraints in image processing. SIAM J. Sci. Comput. 31(3), 2047–2080 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  40. C. Wu, J. Zhang, X.-C. Tai, Augmented lagrangian method for total variation restoration with non-quadratic fidelity. Inverse Probl. Imag. 5(1), 237–261 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  41. M. Yan, W. Yin, Self equivalence of the alternating direction method of multipliers, in UCLA CAM Report 14–59 (2014)

  42. J. Yang, Y. Zhang, W. Yin, An efficient TVL1 algorithm for deblurring multichannel images corrupted by impulsive noise. SIAM J. Sci. Comput. 31(4), 2842–2865 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  43. Y. Zhang, N. Kingsbury, Improved bounds for subband-adaptive iterative shrinkage/thresholding algorithms. IEEE Trans. Image Process. 22(4), 1373–1381 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  44. L. Zhang, L. Zhang, X. Mou, D. Zhang, FSIM: a feature similarity index for image quality assessment. IEEE Trans. Image Process. 20(8), 2378–2386 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  45. Z. Zuo, T. Zhang, X. Lan, L. Yan, An adaptive non-local total variation blind deconvolution employing split Bregman iteration. Circuits Syst. Signal Process. 32(5), 2407–2421 (2013)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The author would like to thank the editors and anonymous reviewers for their constructive comments and valuable suggestions.

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Authors and Affiliations

  1. School of Mathematics and Computational Science, Hunan University of Science and Technology, Xiangtan, 411201, Hunan, China

    Xinwu Liu

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  1. Xinwu Liu
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Correspondence to Xinwu Liu.

Additional information

This work was supported by National Natural Science Foundation of China (61402166, 11571102 and 61702179) and Hunan Provincial Natural Science Foundation of China (14JJ3105).

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Liu, X. Primal-Dual Method for Hybrid Regularizers-Based Image Restoration with Impulse Noise. Circuits Syst Signal Process 38, 1318–1332 (2019). https://doi.org/10.1007/s00034-018-0918-1

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  • DOI: https://doi.org/10.1007/s00034-018-0918-1

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