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Weakly Constrained Lucy–Richardson with Applications to Inversion of Light Scattering Data

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Abstract

Lucy–Richardson (LR) is a classical iterative regularization method largely used for the restoration of nonnegative solutions. LR finds applications in many physical problems, such as for the inversion of light scattering data. In these problems, there is often additional information on the true solution that is usually ignored by many restoration methods because the related measurable quantities are likely to be affected by non-negligible noise. In this article we propose a novel Weakly Constrained Lucy–Richardson (WCLR) method which adds a weak constraint to the classical LR by introducing a penalization term, whose strength can be varied over a very large range. The WCLR method is simple and robust as the standard LR, but offers the great advantage of widely stretching the domain range over which the solution can be reliably recovered. Some selected numerical examples prove the performances of the proposed algorithm.

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Notes

  1. After the publication of [12], it was realized that the method called “modification of the Chahine algorithm” proposed in that work, is identical to the LR algorithm.

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Acknowledgements

The authors would like to thank the reviewers for their insightful comments that greatly improved the readability and the overall quality of this work. The work of the first two authors is supported in part by MIUR - PRIN 2012 N. 2012MTE38N and by a grant of the group GNCS of INdAM. The work of the last author was partially supported by Fondazione Cariplo, Grant n. 2016-0648, Project: Romancing the stone: size controlled HYdroxyaPATItes for sustainable Agriculture (HYPATIA).

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Authors and Affiliations

  1. Department of Mathematical Sciences, Kent State University, 233 MSB, 1300 Lefton Esplanade, Kent, OH, 44242, USA

    Alessandro Buccini

  2. Dipartimento di Scienza e Alta Tecnologia, Università degli Studi dell’Insubria, Via Valleggio 11, 22100, Como, Italy

    Marco Donatelli

  3. Dipartimento di Scienza e Alta Tecnologia, To.Sca.Lab, Università degli Studi dell’Insubria, Via Valleggio 11, 22100, Como, Italy

    Fabio Ferri

Authors
  1. Alessandro Buccini
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  2. Marco Donatelli
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  3. Fabio Ferri
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Correspondence to Marco Donatelli.

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Buccini, A., Donatelli, M. & Ferri, F. Weakly Constrained Lucy–Richardson with Applications to Inversion of Light Scattering Data. J Sci Comput 74, 786–804 (2018). https://doi.org/10.1007/s10915-017-0461-4

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  • DOI: https://doi.org/10.1007/s10915-017-0461-4

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