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Constrained numerical optimization methods for blind deconvolution

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Abstract

This paper describes a nonlinear least squares framework to solve a separable nonlinear ill-posed inverse problem that arises in blind deconvolution. It is shown that with proper constraints and well chosen regularization parameters, it is possible to obtain an objective function that is fairly well behaved and the nonlinear minimization problem can be effectively solved by a Gauss–Newton method. Although uncertainties in the data and inaccuracies of linear solvers make it unlikely to obtain a smooth and convex objective function, it is shown that implicit filtering optimization methods can be used to avoid becoming trapped in local minima. Computational considerations, such as computing the Jacobian, are discussed, and numerical experiments are used to illustrate the behavior of the algorithms. Although the focus of the paper is on blind deconvolution, the general mathematical model addressed in this paper, and the approaches discussed to solve it, arise in many other applications.

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References

  1. Andrews, H., Hunt, B.: Digital Image Restoration. Prentice-Hall, Englewood Cliffs (1977)

    Google Scholar 

  2. Bardsley, J., Vogel, C.R.: A nonnegativly constrained convex programming method for image reconstruction. SIAM J. Sci. Comput. 25(4), 1326–1343 (2004)

    Article  MathSciNet  Google Scholar 

  3. Bardsley, J.M., Goldes, J.: Regularization parameter selection methods for ill-posed Poisson maximum likelihood estimation. Inverse Probl. 25(9) (2009). doi:10.1088/0266-5611/25/9/095,005

  4. Bardsley, J.M., Merikoski, J., Vio, R.: The stabilizing properties of nonnegativity constraints in least-squares image reconstruction. IJPAM 43(1), 95–109 (2008)

    MATH  MathSciNet  Google Scholar 

  5. Bardsley, J.M., Nagy, J.G.: Covariance-preconditioned iterative methods for nonnegatively constrained astronomical imaging. SIAM J. Matrix Anal. Appl. 27, 1184–1197 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  6. Bertsekas, D.P.: Projected newton methods for optimization problems with simple constraints. SIAM J. Control Optim. 20, 221–246 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  7. Bonettini, S., Zanella, R., Zanni, L.: A scaled gradient projection method for constrained image deblurring. Inverse Probl. 25, 015002 (2009). doi:10.1088/0266-5611/25/1/015002

    Article  MathSciNet  Google Scholar 

  8. Chung, J., Haber, E., Nagy, J.: Numerical methods for coupled super-resolution. Inverse Probl. 22(4), 1261–1272 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  9. Chung, J., Nagy, J.: An efficient iterative approach for large-scale separable nonlinear inverse problems. SIAM J. Sci. Comput 31(6), 4654–4674 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  10. Chung, J., Nagy, J.G., O’Leary, D.P.: A weighted GCV method for Lanczos hybrid regularization. Electron. Trans. Numer. Anal. 28, 149–167 (2008)

    MathSciNet  Google Scholar 

  11. Chung, J., Sternberg, P., Yang, C.: High performance 3-D image reconstruction for molecular structure determination. Int. J. High Perform. Comput. Appl. 24(2), 117–135 (2010)

    Article  Google Scholar 

  12. Frank, J.: Three-dimensional Electron Microscopy of Macromolecular Assemblies. Oxford University Press, New York (2006)

    Book  Google Scholar 

  13. Golub, G.H., Pereyra, V.: The differentiation of pseudo-inverses and nonlinear least squares problems whose variables separate. SIAM J. Numer. Anal. 10(2), 413–432 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  14. Golub, G.H., Pereyra, V.: Separable nonlinear least squares: the variable projection method and its applications. Inverse Probl. 19, R1–R26 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  15. Haber, E., Ascher, U.M., Oldenburg, D.: On optimization techniques for solving nonlinear inverse problems. Inverse Probl. 16(5), 1263–1280 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  16. Hansen, P.C., Nagy, J.G., O’Leary, D.P.: Deblurring Images: Matrices, Spectra and Filtering. SIAM, Philadelphia (2006)

    Book  Google Scholar 

  17. Hohn, M., Tang, G., Goodyear, G., Baldwin, P.R., Huang, Z., Penczek, P.A., Yang, C., Glaeser, R.M., Adams, P.D., Ludtke, S.J.: SPARX, a new environment for Cryo-EM image processing. J. Struct. Biol. 157(1), 47–55 (2007)

    Article  Google Scholar 

  18. Kang, M.G., Chaudhuri, S.: Super-resolution image reconstruction. IEEE Signal Proc. Mag. 20(3), 19–20 (2003)

    Article  Google Scholar 

  19. Kaufman, L.: A variable projection method for solving separable nonlinear least squares problems. BIT 15(1), 49–57 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  20. Kaufman, L., Pereyra, V.: A method for separable nonlinear least squares problems with separable equality constraints. SIAM J. Numer. Anal. 15, 12–20 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  21. Kelley, C.T.: Iterative Methods for Optimization. SIAM, Philadelphia (1999)

    Book  MATH  Google Scholar 

  22. Kelley, C.T.: Implicit Filtering. SIAM, Philadelphia (2011). Software and manual can be found at: http://www4.ncsu.edu/ctk/imfil.html

    Book  MATH  Google Scholar 

  23. Lagendijk, R.L., Biemond, J.: Iterative Identification and Restoration of Images. Kluwer Academic Publishers, Boston (1991)

    Book  MATH  Google Scholar 

  24. Landi, G., Piccolomini, E.L.: A projected newton-cg method for nonnegative astronomical image deblurring. Numer. Alg. 48, 279–300 (2008)

    Article  MATH  Google Scholar 

  25. Landi, G., Piccolomini, E.L.: Quasi-newton projection methods and the discrepancy principle in image restoration. Appl. Math. Comput. 118, 2091–2107 (2011)

    Article  Google Scholar 

  26. Marabini, R., Herman, G.T., Carazo, J.M.: 3D reconstruction in electron microscopy using ART with smooth spherically symmetric volume elements (blobs). Ultramicroscopy 72(1–2), 53–65 (1998)

    Article  Google Scholar 

  27. Nocedal, J., Wright, S.: Numerical Optimization. Springer, New York (1999)

    Book  MATH  Google Scholar 

  28. Osborne, M.R.: Separable least squares, variable projection, and the Gauss–Newton algorithm. Electron. Trans. Numer. Anal. 28, 1–15 (2007)

    MATH  MathSciNet  Google Scholar 

  29. Penczek, P.A., Radermacher, M., Frank, J.: Three-dimensional reconstruction of single particles embedded in ice. Ultramicroscopy 40(1), 33–53 (1992)

    Article  Google Scholar 

  30. Pruessner, A., O’Leary, D.P.: Blind deconvolution using a regularized structured total least norm algorithm. SIAM J. Matrix Anal. Appl. 24, 1018–1037 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  31. Ruhe, A., Wedin, P.: Algorithms for separable nonlinear least squares problems. SIAM Rev. 22(3), 318–337 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  32. Saban, S.D., Silvestry, M., Nemerow, G.R., Stewart, P.L.: Visualization of α-helices in a 6-Ångstrom resolution cryoelectron microscopy structure of adenovirus allows refinement of capsid protein assignments. J. Virol. 80(24), 49–59 (2006)

    Article  Google Scholar 

  33. Sima, D.M., Huffel, S.V.: Separable nonlinear least squares fitting with linear bound constraints and its application in magnetic resonance spectroscopy data quantification. J. Comput. Appl. Math. 203, 264–278 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  34. Snyder, D.L., Hammoud, A.M., White, R.L.: Image recovery from data acquired with a charge-coupled-device camera. J. Opt. Soc. Am. A 10, 1014–1023 (1993)

    Article  Google Scholar 

  35. Snyder, D.L., Helstrom, C.W., Lanterman, A.D., Faisal, M., White, R.L.: Compensation for readout noise in ccd images. J. Opt. Soc. Am. A 12, 272–283 (1995)

    Article  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Pure and Applied Mathematics, University of Modena and Reggio Emilia, Modena, Italy

    Anastasia Cornelio

  2. Department of Mathematics, University of Bologna, Bologna, Italy

    Elena Loli Piccolomini

  3. Department of Mathematics and Computer Science, Emory University, Atlanta, GA, USA

    James G. Nagy

Authors
  1. Anastasia Cornelio
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  2. Elena Loli Piccolomini
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  3. James G. Nagy
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Corresponding author

Correspondence to James G. Nagy.

Additional information

The work of J. Nagy was supported by the NSF under grant DMS-0811031 and the AFOSR under grant FA9550-12-1-0084

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Cornelio, A., Loli Piccolomini, E. & Nagy, J.G. Constrained numerical optimization methods for blind deconvolution. Numer Algor 65, 23–42 (2014). https://doi.org/10.1007/s11075-013-9693-z

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  • DOI: https://doi.org/10.1007/s11075-013-9693-z

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