Numerical Algorithms

, Volume 60, Issue 1, pp 169–188

An improved Newton projection method for nonnegative deblurring of Poisson-corrupted images with Tikhonov regularization

Original Paper


In this paper a quasi-Newton projection method for image deblurring is presented. The image restoration problem is mathematically formulated as a nonnegatively constrained minimization problem where the objective function is the sum of the Kullback–Leibler divergence, used to express fidelity to the data in the presence of Poisson noise, and of a Tikhonov regularization term. The Hessian of the objective function is approximated so that the Newton system can be efficiently solved by using Fast Fourier Transforms. The numerical results show the potential of the proposed method both in terms of relative error reduction and computational efficiency.


Nonnegatively constrained minimization Regularization Image deblurring Newton projection method Poisson noise 


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  1. 1.
    Anconelli, B., Bertero, M., Boccacci, P., Carbillet, M., Lanteri, H.: Iterative methods for the reconstruction of astronomical images with high dynamic range. J. Comput. Appl. Math. 198, 321–331 (2007)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Bardsley, J., N’djekornom, L.: Tikhonov regularized poisson likelihood estimation: theoretical justification and a computational method. Inverse Probl. Sci. Eng. 16(2), 199–215 (2008)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bardsley, J., N’djekornom, L.: An analysis of regularization by diffusion for ill-posed poisson likelihood estimation. Inverse Probl. Sci. Eng. 17(4), 537–550 (2009)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Bardsley, J.M., Vogel, C.R.: A nonnegatively constrained convex programming method for image reconstruction. SIAM J. Sci. Comput. 25, 1326–1343 (2003)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Bertsekas, D.: Constrained Optimization and Lagrange Multiplier Methods. Academic, New York (1982)MATHGoogle Scholar
  6. 6.
    Bertsekas, D.: Projected Newton methods for optimization problem with simple constraints. SIAM J. Control Optim. 20(2), 221–245 (1982)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Bertsekas, D.: Nonlinear Programming, 2nd ed. Athena Scientific, Belmont, Massachusetts (1999)Google Scholar
  8. 8.
    Conchello, J.A., McNally, J.G.: Fast regularization technique for expectation maximization algorithm for optical sectioning microscopy. Proc. SPIE 2655, 199–208 (1996)CrossRefGoogle Scholar
  9. 9.
    Gafni, E.M., Bertsekas, D.P.: Two-metric projection methods for constrained optimization. SIAM J. Control. Optim. 22, 936–964 (1984)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Green, P.J.: Bayesian reconstructions from emission tomography data using a modified EM algorithm. IEEE Trans. Med. Imag. 9, 84–93 (1990)CrossRefGoogle Scholar
  11. 11.
    Hanke, M., Nagy, J.G., Vogel, C.: Quasi Newton approach to nonnegative image restorations. Linear Algebra Appl. 316, 222–333 (2000)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Hansen, P.C., Nagy, J.G., O’Leary, D.P.: Deblurring images: matrices, spectra and filtering. SIAM, Philadelphia (2006)MATHGoogle Scholar
  13. 13.
    Kelley, C.T.: Iterative Methods for Optimization. SIAM, Philadelphia (1999)MATHCrossRefGoogle Scholar
  14. 14.
    Landi, G., Loli Piccolomini, E.: A projected Newton-CG method for nonnegative astronomical image deblurring. Numer. Algorithms 48, 279–300 (2008)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Lant’eri, H., Roche, M., Aime, C.: Penalized maximum likelihood image restoration with positivity constraints: multiplicative algorithms. Inverse Probl. 18, 1397–1419 (2002)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Bertero, M, Boccacci, P., Desiderà, G., Vicidomini, G.: Image deblurring with Poisson data: from cells to galaxies. Inverse Probl. 25, 123006 (2009)CrossRefGoogle Scholar
  17. 17.
    Lanteri, H., Bertero, M., Zanni, L.: Iterative image reconstruction: a point of view. In: Mathematical Methods in Biomedical Imaging and Intensity-Modulated Radiation Therapy (IMRT) (CRM Series, vol. 7), pp. 37–63 (2008)Google Scholar
  18. 18.
    Markham, J., Conchello, J.A.: Fast maximum-likelihood image-restoration algorithms for threedimensional fluorescence microscopy. J. Opt. Soc. Am. A 18, 1062–1071 (2001).CrossRefGoogle Scholar
  19. 19.
    Nocedal, J., Wright, S.J.: Numerical Optimization. Springer, New York (1999)MATHCrossRefGoogle Scholar
  20. 20.
    Shepp, L.A., Vardi, Y.: Maximum likelihood reconstruction for emission tomography. IEEE Trans. Med. Imag. 1, 113–122 (1982)CrossRefGoogle Scholar
  21. 21.
    Vogel, C.R.: Computational Methods for Inverse Problems. SIAM, Philadelphia (2002)MATHCrossRefGoogle Scholar
  22. 22.
    Vogel, C.R., Oman, M.E.: Fast, robust total variation–based reconstruction of noisy, blurred images. IEEE Trans. Image Process. 7, 813–824 (1998)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Wang, Z., Bovik, A.C.: Mean squared error: love it or leave it? IEEE Signal Process. Mag. 98, 98–117 (2009)CrossRefGoogle Scholar
  24. 24.
    Wang, Z., Bovik, A.C., Sheikh, H.R., Simoncelli, E.P.: Image quality assessment: from error visibility to structural similarity. IEEE Trans. Image Process. 13(4), 600–612 (2004)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC. 2011

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of BolognaBolognaItaly

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