OSA's Digital Library

Applied Optics

Applied Optics


  • Editor: Joseph N. Mait
  • Vol. 49, Iss. 24 — Aug. 20, 2010
  • pp: 4591–4603

Improved regularized solution of the inverse problem in turbidimetric measurements

Janusz Mroczka and Damian Szczuczyński  »View Author Affiliations

Applied Optics, Vol. 49, Issue 24, pp. 4591-4603 (2010)

View Full Text Article

Enhanced HTML    Acrobat PDF (1445 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



We present results of simulation research on the constrained regularized least-squares (RLS) solution of the ill-conditioned inverse problem in turbidimetric measurements. The problem is formulated in terms of the discretized Fredholm integral equation of the first kind. The inverse problem in turbidimetric measurements consists in determining particle size distribution (PSD) function of particulate system on the basis of turbidimetric measurements. The desired PSD should satisfy two constraints: non negativity of PSD values and normalization of PSD to unity when integrated over the whole range of particle size. Incorporating the constraints into the RLS method leads to the constrained regularized least-squares (CRLS) method, which is realized by means of an active set algorithm of quadratic programming. Results of simulation research prove that the CRLS method performs considerably better with reconstruction of PSD than the RLS method in terms of better fidelity and smaller uncertainty.

© 2010 Optical Society of America

OCIS Codes
(290.3200) Scattering : Inverse scattering
(290.4020) Scattering : Mie theory
(290.5820) Scattering : Scattering measurements
(290.5850) Scattering : Scattering, particles
(290.7050) Scattering : Turbid media
(290.2558) Scattering : Forward scattering

ToC Category:

Original Manuscript: February 23, 2010
Revised Manuscript: June 4, 2010
Manuscript Accepted: July 7, 2010
Published: August 16, 2010

Virtual Issues
Vol. 5, Iss. 13 Virtual Journal for Biomedical Optics

Janusz Mroczka and Damian Szczuczyński, "Improved regularized solution of the inverse problem in turbidimetric measurements," Appl. Opt. 49, 4591-4603 (2010)

Sort:  Author  |  Year  |  Journal  |  Reset  


  1. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1983).
  2. A. R. Jones, “Light scattering for particle characterization,” Prog. Energy Combust. Sci. 25, 1–53 (1999). [CrossRef]
  3. R. Xu, Particle Characterization: Light Scattering Methods (Kluwer Academic, 2000).
  4. A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems (Winston, 1977).
  5. O. C. Lingjearde and N. Christophersen, Regularization Principles: Solving Ill-Posed Inverse Problems, Lecture Notes to IN INVPAR, Department of Infomatics, University of Oslo (1998).
  6. P. C. Hansen, “Regularization tools: a MATLAB package for analysis and solution of discrete ill-posed problems,” Numer. Algorithms 6, 1–35 (1994). [CrossRef]
  7. J. G. Crump and J. H. Seinfeld, “A new algorithm for inversion of aerosol size distribution data,” Aerosol Sci. Technol. 1, 15–34 (1981). [CrossRef]
  8. M. Kandlikar and G. Ramachandran, “Inverse methods for analysing aerosol spectrometer measurements: a critical review,” J. Aerosol Sci. 30, 413–437 (1999). [CrossRef]
  9. F. Stout and J. H. Kalivas, “Tikhonov regularization in standardized and general form for multivariate calibration with application towards removing unwanted spectral artifacts,” J. Chemometrics 20, 22–33 (2006). [CrossRef]
  10. S. Twomey, “On the numerical solution of Fredholm integral equations of the first kind by the inversion of the linear system produced by quadrature,” J. Assoc. Comput. Mach. 10, 97–101(1963). [CrossRef]
  11. D. L. Phillips, “A technique for the numerical solution of certain integral equations of the first kind,” J. Assoc. Comput. Mach. 9, 84–97 (1962). [CrossRef]
  12. P. C. Hansen and D. P. O’Leary, “The use of the L-curve in the regularization of discrete ill-posed problems,” SIAM J. Sci. Comput. 14, 1487–1503 (1993). [CrossRef]
  13. P. J. Mc Carthy, “Direct analytic model of the L-curve for Tikhonov regularization parameter selection,” Inverse Probl. 19, 643–663 (2003). [CrossRef]
  14. L. Eldén, “A weighted pseudoinverse, generalized singular values, and constrained least squares problems,” BIT Numer. Math 22, 487–502 (1982). [CrossRef]
  15. G. H. Golub and U. von Matt, “Quadratically constrained least squares and quadratic problems,” Numer. Math. 59, 561–580 (1991). [CrossRef]
  16. B. Martos, Nonlinear Programming: Theory and Methods (Akadémiai Kiadó, 1975).
  17. W. I. Zangwill, Nonlinear Programming: a Unified Approach (Prentice-Hall, 1969).
  18. P. E. Gill, W. Murray, M. A. Saunders, and M. H. Wright, “Procedures for optimization problems with a mixture of bounds and general linear constraints,” ACM Trans. Math. Software 10, 282–298 (1984). [CrossRef]
  19. P. E. Gill, W. Murray, and M. H. Wright, Numerical Linear Algebra and Optimization (Addison-Wesley, 1991), Vol. 1.
  20. C. L. Lawson and R. J. Hanson, Solving Least Squares Problems (Prentice-Hall, 1974).
  21. A. R. Roig and J. L. Alessandrini, “Particle size distributions from static light scattering with regularized non-negative least squares constraints,” Part. Part. Syst. Charact. 23, 431–437 (2006). [CrossRef]
  22. K. F. Ren, F. Xu, X. Cai, and J. M. Dorey, “Development of a precise and in situ turbidity measurement system,” Chem. Eng. Commun. 197, 250–259 (2010). [CrossRef]
  23. G. H. Golub, M. Heath, and H. Wahba, “Generalized cross validation as a method for choosing a good ridge parameter,” Technometrics 21, 215–224 (1979). [CrossRef]
  24. “Index of refraction,” Technical note 007 (Duke Scientific Corporation, 1 December 1996).
  25. “Release on the refractive index of ordinary water substance as a function of wavelength, temperature and pressure” (The International Association for the Properties of Water and Steam, September 1997).

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited