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Variational Multiplicative Noise Removal by DC Programming

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Abstract

This paper proposes a difference of convex algorithm (DCA) to deal with a non-convex data fidelity term, proposed by Aubert and Aujol referred to as the AA model. The AA model was adopted in many subsequent works for multiplicative noise removal, most of which focused on convex approximation so that numerical algorithms with guaranteed convergence can be designed. Noting that the AA model can be naturally split into a difference of two convex functions, we apply the DCA to solve the original AA model. Compared to the gradient projection algorithm considered by Aubert and Aujol, the DCA often converges faster and leads to a better solution. We prove that the DCA sequence converges to a stationary point, which satisfies the first order optimality condition. In the experiments, we consider two applications, image denoising and deblurring, both of which involve multiplicative Gamma noise. Numerical results demonstrate that the proposed algorithm outperforms the state-of-the-art methods for multiplicative noise removal.

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References

  1. Alvarado, A., Scutari, G., Pang, J.S.: A new decomposition method for multiuser DC-programming and its applications. IEEE Trans. Signal Process. 62(11), 2984–2998 (2014)

    Article  MathSciNet  Google Scholar 

  2. Aubert, G., Aujol, J.F.: A variational approach to removing multiplicative noise. SIAM J. Appl. Math. 68(4), 925–946 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  3. Cai, J., Ji, H., Liu, C., Shen, Z.: Framelet-based blind motion deblurring from a single image. IEEE Trans. Image Process. 21(2), 562–572 (2004)

    MathSciNet  Google Scholar 

  4. Chambolle, A.: An algorithm for total variation minimization and applications. J. Math. Imaging Vis. 20(1–2), 89–97 (2004)

    MathSciNet  Google Scholar 

  5. Chambolle, A., Pock, T.: A first-order primal-dual algorithm for convex problems with applications to imaging. J. Math. Imaging Vis. 40(1), 120–145 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chan, R., Yang, H., Zeng, T.: A two-stage image segmentation method for blurry images with poisson or multiplicative gamma noise. SIAM J. Sci. Comput. 7(1), 98–127 (2014)

    MathSciNet  MATH  Google Scholar 

  7. Chan, T.F., Shen, J.: Image processing and analysis: variational, PDE, wavelet, and stochastic methods. Society for Industrial and Applied Mathematics (2005). doi:10.1137/1.9780898717877

  8. Chen, X., Ng, M., Zhang, C.: Non-lipschitz-regularization and box constrained model for image restoration. IEEE Trans. Image Process. 21(12), 4709–4721 (2012)

    Article  MathSciNet  Google Scholar 

  9. Chen, X., Zhou, W.: Smoothing nonlinear conjugate gradient method for image restoration using nonsmooth nonconvex minimization. SIAM J. Imaging Sci. 3(4), 765–790 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  10. Dong, Y., Zeng, T.: A convex variational model for restoring blurred images with multiplicative noise. SIAM J. Imaging Sci. 6(3), 1598–1625 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  11. Dür, M., Hiriart-Urruty, J.B.: Testing copositivity with the help of difference-of-convex optimization. Math. Program. 140(1), 31–43 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  12. Ekeland, I., Témam, R.: Convex analysis and variational problems. Society for Industrial and Applied Mathematics (1999). doi:10.1137/1.9781611971088

  13. Harrington, J., Hobbs, B.F., Pang, J.S., Liu, A., Roch, G.: Collusive game solutions via optimization. Math. Program. 104(2–3), 407–435 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  14. Hiriart-Urruty, J.B., Lemaréchal, C.: Convex analysis and minimization algorithms. Springer-Verlag, Berlin, Heidelberg (1993). doi:10.1007/978-3-662-02796-7

  15. Huang, Y.M., Ng, M., Wen, Y.W.: A new total variation method for multiplicative noise removal. SIAM J. Imaging Sci. 2(1), 20–40 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  16. Le-Thi, H.A.: An efficient algorithm for globally minimizing a quadratic function under convex quadratic constraints. Math. Program. 87, 401–426 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  17. Le-Thi, H.A., Pham-Dihn, T.: The DC (difference of convex functions) programming and DCA revisited with DC models of real world nonconvex optimization problems. Ann. Oper. Res. 133(1–4), 23–46 (2005)

    MathSciNet  MATH  Google Scholar 

  18. Le-Thi, H.A., Pham-Dihn, T.: DC programming in communication systems: challenging problems and methods. Vietnam J. Comput. Sci. 1(1), 15–28 (2014)

    Article  Google Scholar 

  19. Long, X., Solna, K., Xin, J.: Two \(\ell _{1}\) based nonconvex methods for constructing sparse mean reverting portfolios. Technical Report, Department of Mathematics, University of California, Los Angeles (2014)

  20. Lou, Y., Osher, S., Xin, J.: Computational aspects of constrained \(\ell _{1}\)-\(\ell _{2}\) minimization for compressive sensing. Model. Comput. Optim. Inf. Syst. Manag. Sci. 359, 169–180 (2015)

    Google Scholar 

  21. Lou, Y., Yin, P., He, Q., Xin, J.: Computing sparse representation in a highly coherent dictionary based on difference of L1 and L2. J. Sci. Comput. 64(1), 178–196 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  22. Lou, Y., Zeng, T., Osher, S., Xin, J.: A weighted difference of anisotropic and isotropic total variation model for image processing. SIAM J. Imaging Sci. 8(3), 1798–1823 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  23. Lou, Y., Zhang, X., Osher, S., Bertozzi, A.: Image recovery via nonlocal operators. J. Sci. Comput. 42(2), 185–197 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  24. Maranas, C.D., Floudas, C.A.: A global optimization approach for Lennard–Jones microclusters. J. Chem. Phys. 97(10), 7667–7678 (1992)

    Article  Google Scholar 

  25. Neumann, J., Schnörr, C., Steidl, G.: Combined SVM-based feature selection and classification. Mach. Learn. 61(1–3), 129–150 (2005)

    Article  MATH  Google Scholar 

  26. Nikolova, M., Ng, M., Tam, C.P.: Fast nonconvex nonsmooth minimization methods for image restoration and reconstruction. IEEE Trans. Image Process. 19(12), 3073–3088 (2010)

    Article  MathSciNet  Google Scholar 

  27. Ochs, P., Dosovitskiy, A., Brox, T., Pock, T.: An iterated L1 algorithm for non-smooth non-convex optimization in computer vision. In: IEEE conference on computer vision and pattern recognition, pp. 1759–1766 (2013)

  28. Parikh, N., Boyd, S.: Proximal algorithms. Found. Trends Optim. 1(3), 123–231 (2013)

    Google Scholar 

  29. Pham-Dihn, T., Le-Thi, H.A.: Convex analysis approach to DC programming: theory, algorithms and applications. Acta Math. Vietnam 22(1), 289–355 (1997)

    MathSciNet  MATH  Google Scholar 

  30. Pham-Dihn, T., Le-Thi, H.A.: A DC optimization algorithm for solving the trust-region subproblem. SIAM J. Optim. 8(2), 476–505 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  31. Punithakumar, K., Yuan, J., Ayed, I.B., Li, S., Boykov, Y.: A convex max-flow approach to distribution-based figure-ground separation. SIAM J. Imaging Sci. 5(4), 1333–1354 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  32. Repetti, A., Chouzenoux, E., Pesquet, J.C.: A nonconvex regularized approach for phase retrieval. In: IEEE international conference on image processing, pp. 1753–1757 (2014)

  33. Rudin, L., Lions, P.L., Osher, S.: Multiplicative denoising and deblurring: theory and algorithms. In: Geometric Level Set Methods in Imaging, Vision, and Graphics, pp. 103–119. Springer, New York (2003)

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

    Article  MathSciNet  MATH  Google Scholar 

  35. Schnörr, C., Schüle, T., Weber, S.: Variational reconstruction with DC-programming. In: Advances in Discrete Tomography and Its Applications, pp. 227–243. Springer, Berlin (2007)

  36. Shi, J., Osher, S.: A nonlinear inverse scale space method for a convex multiplicative noise model. SIAM J. Imaging Sci. 1(3), 294–321 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  37. Tang, M., Ayed, I.B., Boykov,Y.: Pseudo-bound optimization for binary energies. In: European Conference on Computer Vision, pp. 691–707. Springer (2014)

  38. Tseng, P.: A modified forward–backward splitting method for maximal monotone mappings. SIAM J. Control Optim. 38(2), 431–446 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  39. Xiao, J., Ng, M., Yang, Y.F.: On the convergence of nonconvex minimization methods for image recovery. IEEE Trans. Image Process. 24(5), 1587–1598 (2015)

    Article  MathSciNet  Google Scholar 

  40. Yin, P., Lou, Y., He, Q., Xin, J.: Minimization of L1–L2 for compressed sensing. SIAM J. Sci. Comput. 37(1), A536–A563 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

We would like to thank the anonymous referees for valuable comments and suggestions, which significantly improved the content of the paper.

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Authors and Affiliations

  1. Department of Mathematics, Hong Kong Baptist University, Hong Kong, China

    Zhi Li & Tieyong Zeng

  2. Department of Mathematical Sciences, University of Texas at Dallas, Richardson, TX, USA

    Yifei Lou

Authors
  1. Zhi Li
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  2. Yifei Lou
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  3. Tieyong Zeng
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Corresponding author

Correspondence to Tieyong Zeng.

Additional information

Yifei Lou is partially supported by NSF Grant DMS-1522786. Tieyong Zeng is partially supported by NSFC 11271049, RGC 12302714 and RFGs of HKBU.

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Li, Z., Lou, Y. & Zeng, T. Variational Multiplicative Noise Removal by DC Programming. J Sci Comput 68, 1200–1216 (2016). https://doi.org/10.1007/s10915-016-0175-z

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