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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
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.
References
Bai, Z.Z., Buccini, A., Hayami, K., Reichel, L., Ying, J.F., Zheng, N.: Modulus-based iterative methods for constrained Tikhonov regularization. J. Comput. Appl. Math. 319, 1–13 (2017)
Baltes, H.P.: Inverse Scattering Problems in Optics. Springer, Berlin (1980)
Bertero, M., Boccacci, P.: Introduction to Inverse Problems in Imaging. CRC Press, Boca Raton (1998)
Biggs, D.S.C., Andrews, M.: Acceleration of iterative image restoration algorithms. Appl. Opt. 36(8), 1766–1775 (1997)
Björck, Å.: Numerical Methods for Least Squares Problems. Siam, Philadelphia (1996)
Bohren, C.F., Hirleman, E.D.: Feature on optical particle sizing. Appl. Opt. 30, 4685–4987 (1991)
Brown, W.: Dynamic Light Scattering. Clarendon, Oxford (1993)
Calvetti, D., Landi, G., Reichel, L., Sgallari, F.: Non-negativity and iterative methods for ill-posed problems. Inverse Probl. 20(6), 1747 (2004)
Engl, H.W., Hanke, M., Neubauer, A.: Regularization of Inverse Problems, vol. 375. Springer, Berlin (1996)
Ferri, F., Bassini, A., Paganini, E.: Modified version of the Chahine algorithm to invert spectral extinction data for particle sizing. Appl. Optic. 34(25), 5829–5839 (1995)
Ferri, F., Bassini, A., Paganini, E.: Commercial spectrophotometer for particle sizing. Appl. Optic. 36(4), 885–891 (1997)
Ferri, F., Righini, G., Paganini, E.: Inversion of low-angle elastic light-scattering data with a new method devised by modification of the Chahine algorithm. Appl. Optic. 36(30), 7539–7550 (1997)
Gazzola, S., Nagy, J.G.: Generalized Arnoldi–Tikhonov method for sparse reconstruction. SIAM J. Sci. Comput. 36(2), B225–B247 (2014)
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)
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)
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)
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)
Gouesbet, G., Gréhan, G.: Optical Particle Sizing: Theory and Practice. Academic press, New York (1969)
Hanke, M., Hansen, P.C.: Regularization methods for large-scale problems. Surv. Math. Ind. 3(4), 253–315 (1993)
Hilliard, J.E., Lawson, L.: Stereology and Stochastic Geometry, vol. 28. Kluwer Academic Publisher, The Netherlands (2003)
Hu, B., Shen, J., Duan, T.: Vector similarity measure for particle size analysis based on forward light scattering. Appl. Opt. 54, 3855–3862 (2015)
Hulst, H.C.: Light Scattering by Small Particles. Dover Publications, New York (1981)
Kerker, M.: The scattering of light and other electromagnetic radiation. Academic Press, New York (1969)
Khan, M., Morigi, S., Reichel, L., Sgallari, F.: Iterative methods of Richardson–Lucy-type for image deblurring. Numer. Math. Theory Methods Appl 6(01), 262–275 (2013)
Liu, X., Shen, J., Thomas, J.C., Clementi, L.A., Sun, X.: Multiangle dynamic light scattering analysis using a modified chahine method. J. Quant. Spectrosc. Radiat. Transf. 113, 489–497 (2012)
Lucy, L.B.: An iterative technique for the rectification of observed distributions. Astron. J. 79, 745 (1974)
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)
Naiim, M., Boualem, A., Ferre, C., Jabloun, M., Jalocha, A., Ravier, P.: Multiangle multiangle dynamic light scattering for the improvement of multimodal particle size distribution measurements. Soft Matter 11, 28 (2015)
Richardson, W.H.: Bayesian-based iterative method of image restoration. JOSA 62(1), 55–59 (1972)
Roig, A.R., Alessandrini, J.L.: Particle size distributions from static light scattering with regularizednon-negative least squares constraints. Part. Part. Syst. Charact. 23, 431–437 (2006)
Wang, L., Sun, X.G., Xing, J.: Determination of particle size distribution by light extinction method using improved pattern search algorithm with tikhonov smoothing functional. J. Mod. Opt. 59, 1829–1840 (2012)
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).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
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
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10915-017-0461-4