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Journal of Scientific Computing

, Volume 79, Issue 2, pp 1214–1240 | Cite as

A Fast Algorithm for Solving Linear Inverse Problems with Uniform Noise Removal

  • Xiongjun ZhangEmail author
  • Michael K. Ng
Article
  • 456 Downloads

Abstract

In this paper, we develop a fast algorithm for solving an unconstrained optimization model for uniform noise removal which is an important task in inverse problems. The optimization model consists of an \(\ell _\infty \) data fitting term and a total variation regularization term. By utilizing the alternating direction method of multipliers (ADMM) for such optimization model, we demonstrate that one of the ADMM subproblems can be formulated by involving a projection onto \(\ell _1\) ball which can be solved efficiently by iterations. The convergence of the ADMM method can be established under some mild conditions. In practice, the balance between the \(\ell _\infty \) data fitting term and the total variation regularization term is controlled by a regularization parameter. We present numerical experiments by using the L-curve method of the logarithms of data fitting term and total variation regularization term to select regularization parameters for uniform noise removal. Numerical results for image denoising and deblurring, inverse source, inverse heat conduction problems and second derivative problems have shown the effectiveness of the proposed model.

Keywords

Uniform noise Linear inverse problems Total variation \(\ell _\infty \)-Norm Alternating direction method of multipliers 

Mathematics Subject Classification

49N45 65F22 90C25 

Notes

Acknowledgements

The authors would like to thank the anonymous referees for their constructive comments and suggestions that have helped improve the presentation of the paper greatly.

References

  1. 1.
    Alliney, S.: 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)Google Scholar
  2. 2.
    Aubert, G., Aujol, J.-F.: A variational approach to removing multiplicative noise. SIAM J. Appl. Math. 68(4), 925–946 (2008)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Bai, M., Zhang, X., Shao, Q.: Adaptive correction procedure for TVL1 image deblurring under impulse noise. Inverse Probl. 32(8), 085004 (2016)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Bertero, M., Boccacci, P.: Introduction to Inverse Problems in Imaging. CRC Press, Boca Raton (1998)zbMATHGoogle Scholar
  5. 5.
    Bovik, A.: Handbook of Image and Video Processing. Academic Press, New York (2000)zbMATHGoogle Scholar
  6. 6.
    Boyd, S., Parikh, N., Chu, E., Peleato, B., Eckstein, J.: Distributed optimization and statistical learning via the alternating direction method of multipliers. Found. Trends Mach. Learn. 3(1), 1–122 (2011)zbMATHGoogle Scholar
  7. 7.
    Castellanos, J.L., Gómez, S., Guerra, V.: The triangle method for finding the corner of the L-curve. Appl. Numer. Math. 43(4), 359–373 (2002)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Chan, T.F., Esedoglu, S.: Aspects of total variation regularized \({L}^1\) function approximation. SIAM J. Appl. Math. 65(5), 1817–1837 (2005)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Clason, C.: \({L}^{\infty }\) fitting for inverse problems with uniform noise. Inverse Probl. 28(10), 104007 (2012)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Clason, C., Jin, B., Kunisch, K.: A semismooth Newton method for \(\text{ L }^1\) data fitting with automatic choice of regularization parameters and noise calibration. SIAM J. Imaging Sci. 3(2), 199–231 (2010)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Colton, D., Coyle, J., Monk, P.: Recent developments in inverse acoustic scattering theory. SIAM Rev. 42(3), 369–414 (2000)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Condat, L.: Fast projection onto the simplex and the \(l_1\) ball. Math. Program. 158(1–2), 575–585 (2016)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Dong, Y., Zeng, T.: A convex variational model for restoring blurred images with multiplicative noise. SIAM J. Imaging Sci. 6(3), 1598–1625 (2013)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Duchi, J., Shalev-Shwartz, S., Singer, Y., Chandra, T.: Efficient projections onto the \(\ell _1\)-ball for learning in high dimensions. In: Proceedings of the International Conference on Machine Learning, pp. 272–279. ACM (2008)Google Scholar
  15. 15.
    Durand, S., Nikolova, M.: Denoising of frame coefficients using \(\ell ^1\) data-fidelity term and edge-preserving regularization. Multiscale Model. Simul. 6(2), 547–576 (2007)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Fazel, M., Pong, T.K., Sun, D., Tseng, P.: Hankel matrix rank minimization with applications to system identification and realization. SIAM J. Matrix Anal. Appl. 34(3), 946–977 (2013)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Gabay, D., Mercier, B.: A dual algorithm for the solution of nonlinear variational problems via finite element approximation. Comput. Math. Appl. 2(1), 17–40 (1976)zbMATHGoogle Scholar
  18. 18.
    Glowinski, R., Marrocco, A.: Sur l’approximation par éléments finis d’ordre un, et la résolution, par pénalisation-dualité, d’une classe de problèmes de Dirichlet non linéaires. Revue Française d’Autom. Informat. Rech. Opér. Anal. Numér 9(R2), 41–76 (1975)Google Scholar
  19. 19.
    Gonzalez, R.C., Woods, R.E.: Digital Image Processing, 3rd edn. Pearson, London (2008)Google Scholar
  20. 20.
    Hansen, P.C.: Analysis of discrete ill-posed problems by means of the L-curve. SIAM Rev. 34(4), 561–580 (1992)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Hansen, P.C.: Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion. SIAM, Philadephia (1998)Google Scholar
  22. 22.
    Hansen, P.C.: Regularization tools version 4.0 for Matlab 7.3. Numer. Algorithms 46(2), 189–194 (2007)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Hansen, P.C., O’Leary, D.P.: The use of the L-curve in the regularization of discrete ill-posed problems. SIAM J. Sci. Comput. 14(6), 1487–1503 (1993)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Held, M., Wolfe, P., Crowder, H.P.: Validation of subgradient optimization. Math. Program. 6(1), 62–88 (1974)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Huang, Y.-M., Lu, D.-Y., Zeng, T.: Two-step approach for the restoration of images corrupted by multiplicative noise. SIAM J. Sci. Comput. 35(6), A2856–A2873 (2013)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Huang, Y.-M., Moisan, L., Ng, M.K., Zeng, T.: Multiplicative noise removal via a learned dictionary. IEEE Trans. Image Process. 21(11), 4534–4543 (2012)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Huang, Y.-M., Ng, M.K., Wen, Y.-W.: A fast total variation minimization method for image restoration. Multiscale Model. Simul. 7(2), 774–795 (2008)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Kay, S.M.: Fundamentals of Statistical Signal Processing: Estimation Theory. Prentice-Hall PTR, Englewood Cliffs (1993)zbMATHGoogle Scholar
  29. 29.
    Le, T., Chartrand, R., Asaki, T.J.: A variational approach to reconstructing images corrupted by Poisson noise. J. Math. Imaging Vis. 27(3), 257–263 (2007)MathSciNetGoogle Scholar
  30. 30.
    Ng, M.K.: Iterative Methods for Toeplitz Systems. Oxford University Press, London (2004)zbMATHGoogle Scholar
  31. 31.
    Ng, M.K., Chan, R.H., Tang, W.-C.: A fast algorithm for deblurring models with Neumann boundary conditions. SIAM J. Sci. Comput. 21(3), 851–866 (1999)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Nikolova, M.: 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)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Nikolova, M.: A variational approach to remove outliers and impulse noise. J. Math. Imaging Vis. 20(1–2), 99–120 (2004)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Nikolova, M.: Weakly constrained minimization: application to the estimation of images and signals involving constant regions. J. Math. Imaging Vis. 21(2), 155–175 (2004)MathSciNetGoogle Scholar
  35. 35.
    Parikh, N., Boyd, S.: Proximal algorithms. Found. Trends Optim. 1(3), 127–239 (2014)Google Scholar
  36. 36.
    Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)zbMATHGoogle Scholar
  37. 37.
    Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis, 3rd edn. Springer, Berlin (2009)zbMATHGoogle Scholar
  38. 38.
    Rudin, L.I., Osher, S.: Total variation based image restoration with free local constraints. In: Proceedings of the IEEE International Conference on Image Processing, volume 1, pp. 31–35. Austin, TX (1994)Google Scholar
  39. 39.
    Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Phys. D 60(1–4), 259–268 (1992)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Sciacchitano, F., Dong, Y., Zeng, T.: Variational approach for restoring blurred images with Cauchy noise. SIAM J. Imaging Sci. 8(3), 1894–1922 (2015)MathSciNetzbMATHGoogle Scholar
  41. 41.
    Setzer, S., Steild, G., Teuber, T.: Deblurring Poissonian images by split Bregman techniques. J. Vis. Commun. Image Rep. 21(3), 193–199 (2010)Google Scholar
  42. 42.
    Steidl, G., Teuber, T.: Removing multiplicative noise by Douglas–Rachford splitting methods. J. Math. Imaging Vis. 36(2), 168–184 (2010)MathSciNetzbMATHGoogle Scholar
  43. 43.
    Tikhonov, A.N., Arsenin, V.Y.: Solutions of Ill-Posed Problems. Winston, Washington, DC (1977)zbMATHGoogle Scholar
  44. 44.
    van den Berg, E., Friedlander, M.P.: Probing the pareto frontier for basis pursuit solutions. SIAM J. Sci. Comput. 31(2), 890–912 (2008)MathSciNetzbMATHGoogle Scholar
  45. 45.
    Wan, T., Canagarajah, N., Achim, A.: Segmentation of noisy colour images using Cauchy distribution in the complex wavelet domain. IET Image Process. 5(2), 159–170 (2011)Google Scholar
  46. 46.
    Wang, F., Zhao, X.-L., Ng, M.K.: Multiplicative noise and blur removal by framelet decomposition and \(\ell _1\)-based L-curve method. IEEE Trans. Image Process. 25(9), 4222–4232 (2016)MathSciNetzbMATHGoogle Scholar
  47. 47.
    Weiss, P., Aubert, G., Blanc-Féraud, L.: Some Applications of \(\ell ^{\infty }\)-Constraints in Image Processing. INRIA Resarch Report 6115 (2006)Google Scholar
  48. 48.
    Wen, Y.-W., Chan, R.H., Zeng, T.: Primal-dual algorithms for total variation based image restoration under Poisson noise. Sci. China Math. 59(1), 141–160 (2016)MathSciNetzbMATHGoogle Scholar
  49. 49.
    Wen, Y.-W., Ching, W.-K., Ng, M.K.: A semi-smooth Newton method for inverse problem with uniform noise. J. Sci. Comput. 75(2), 713–732 (2018)MathSciNetzbMATHGoogle Scholar
  50. 50.
    Yang, J., Zhang, Y., Yin, W.: An efficient TVL1 algorithm for deblurring multichannel images corrupted by impulsive noise. SIAM J. Sci. Comput. 31(4), 2842–2865 (2009)MathSciNetzbMATHGoogle Scholar
  51. 51.
    Zhang, X., Bai, M., Ng, M.K.: Nonconvex-TV based image restoration with impulse noise removal. SIAM J. Imaging Sci. 10(3), 1627–1667 (2017)MathSciNetzbMATHGoogle Scholar
  52. 52.
    Zhang, X., Ng, M.K., Bai, M.: A fast algorithm for deconvolution and Poisson noise removal. J. Sci. Comput. 75(3), 1535–1554 (2018)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematics and Statistics, Hubei Key Laboratory of Mathematical SciencesCentral China Normal UniversityWuhanChina
  2. 2.Department of Mathematics, Centre for Mathematical Imaging and VisionHong Kong Baptist UniversityKowloon TongHong Kong

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