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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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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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