Weakly Constrained Lucy–Richardson with Applications to Inversion of Light Scattering Data
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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.
KeywordsLucy–Richardson algorithm Light scattering Particle sizing
Mathematics Subject Classification65R32 78A46 78A10
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).
- 7.Brown, W.: Dynamic Light Scattering. Clarendon, Oxford (1993)Google Scholar
- 14.Glasse, B., Riefler, N., Fritsching, U.: Intercomparison of numerical inversion algorithms for particle size determination of polystyrene suspensions using spectral turbidimetry. J. Spectrosc. 2015, 645–879 (2015)Google Scholar
- 15.Glatter, O.: Fourier transformation and deconvolution. In: Linder, P., Zemb, T. (eds.) Neutrons, X-rays and Light: Scattering Methods Applied to Soft Condensed Matter. Elsevier, North Holland (2002)Google Scholar
- 16.Glatter, O.: The inverse scattering problem in small angle scattering. In: Linder, P., Zemb, T. (eds.) Neutrons, X-rays and Light: Scattering Methods Applied to Soft Condensed Matter. Elsevier, North Holland (2002)Google Scholar
- 17.Glatter, O.: Static light scattering of large systems. In: Linder, P., Zemb, T. (eds.) Neutrons, X-rays and Light: Scattering Methods Applied to Soft Condensed Matter. Elsevier, North Holland (2002)Google Scholar
- 22.Hulst, H.C.: Light Scattering by Small Particles. Dover Publications, New York (1981)Google Scholar
- 23.Kerker, M.: The scattering of light and other electromagnetic radiation. Academic Press, New York (1969)Google Scholar
- 27.Nagy, J.G., Strakoš, Z.: Enforcing nonnegativity in image reconstruction algorithms. In: International Symposium on Optical Science and Technology, pp. 182–190. International Society for Optics and Photonics (2000)Google Scholar