## Simulation research on improved regularized solution of the inverse problem in spectral extinction measurements |

Applied Optics, Vol. 51, Issue 11, pp. 1715-1723 (2012)

http://dx.doi.org/10.1364/AO.51.001715

Enhanced HTML Acrobat PDF (1072 KB)

### Abstract

We present further results of the simulation research on the constrained regularized least squares (CRLS) solution of the ill-conditioned inverse problem in spectral extinction (turbidimetric) measurements, which we originally presented in this journal [Appl. Opt. **49,** 4591 (2010)]. The inverse problem consists of determining the particle size distribution (PSD) function of a particulate system on the basis of a measured extinction coefficient as a function of wavelength. In our previous paper, it was shown that under assumed conditions the problem can be formulated in terms of the discretized Fredholm integral equation of the first kind. The CRLS method incorporates two constraints, which the PSD sought will satisfy: nonnegativity of the PSD values and normalization of the PSD to unity when integrated over the whole range of particle size, into the regularized least squares (RLS) method. This leads to the quadratic programming problem, which is solved by means of the active set algorithm within the research. The simulation research that is the subject of the present paper is a continuation and extension of the research described in our previous paper. In the present research, the performance of the CRLS method variants is compared not only to the corresponding RLS method variants but also to other regularization techniques: the truncated generalized singular value decomposition and the filtered generalized singular value decomposition, as well as nonlinear iterative algorithms: The Twomey algorithm and the Twomey—Markowski algorithm. Moreover, two methods of selecting the optimum value of the regularization parameter are considered: The

© 2012 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:**

Scattering

**History**

Original Manuscript: November 8, 2011

Manuscript Accepted: December 30, 2011

Published: April 5, 2012

**Virtual Issues**

Vol. 7, Iss. 6 *Virtual Journal for Biomedical Optics*

**Citation**

Janusz Mroczka and Damian Szczuczyński, "Simulation research on improved regularized solution of the inverse problem in spectral extinction measurements," Appl. Opt. **51**, 1715-1723 (2012)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=ao-51-11-1715

Sort: Year | Journal | Reset

### References

- J. Mroczka and D. Szczuczyński, “Improved regularized solution of the inverse problem in turbidimetric measurements,” Appl. Opt. 49, 4591–4603 (2010). [CrossRef]
- C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1983).
- A. R. Jones, “Light scattering for particle characterization,” Progr. Energy Combust. Sci. 25, 1–53 (1992).
- P. C. Hansen, “Regularization tools: A Matlab package for analysis and solution of discrete ill-posed problems,” Numerical Algorithms 6, 1–35 (1994).
- 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]
- M. Kandlikar and G. Ramachandran, “Inverse methods for analysing aerosol spectrometer measurements: A Critical Review,” J. Aerosol Sci. 30, 413–437 (1999). [CrossRef]
- F. Stout and J. H. Kalivas, “Tikhonov regularization in standardized and general form for multivariate calibration with application towards removing unwanted spectral artifacts,” J. Chemom. 20, 22–33 (2006). [CrossRef]
- 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]
- 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]
- C. L. Lawson and R. J. Hanson, Solving Least Squares Problems (Prentice Hall, 1974).
- 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]
- G. H. Golub and U. von Matt, “Quadratically constrained least squares and quadratic problems,” Numer. Math. 59, 561–580 (1991). [CrossRef]
- 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. Softw. 10, 282–298 (1984). [CrossRef]
- P. C. Hansen, “Numerical tools for analysis and solution of Fredholm integral equations of the first kind,” Inverse Probl. 8, 849–872 (1992). [CrossRef]
- P. C. Hansen, “Regularization, GSVD and truncated GSVD,” BIT (Nord. Tidskr. Inf.-behandl.) 29, 491–504 (1989). [CrossRef]
- P. C. Hansen, “Relations between SVD and GSVD of discrete regularization problems in standard and general form,” Linear Algebra Appl. 141, 165–176 (1990). [CrossRef]
- P. C. Hansen, T. Sekii, and H. Shibahashi, “The modified truncated SVD method for regularization in general form,” SIAM J. Sci. Stat. Comput. 13, 1142–1150 (1992). [CrossRef]
- 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]
- P. J. McCarthy, “Direct analytic model of the L-curve for Tikhonov regularization parameter selection,” Inverse Probl. 19, 643–663 (2003). [CrossRef]
- 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]
- S. Twomey, “Comparison of constrained linear inversion and an iterative non-linear algorithm applied to the indirect estimation of particle size distributions,” J. Comput. Phys. 18, 188–200 (1975). [CrossRef]
- W. Winklmayr, H. Wang, and W. John, “Adaptation of the Twomey Algorithm to the Inversion of Cascade Impactor Data,” Aerosol Sci. Technol. 13, 322–331 (1990). [CrossRef]
- G. R. Markowski, “Improving Twomey’s Algorithm for Inversion of Aerosol Measurement Data,” Aerosol Sci. Technol 7, 127–141 (1987). [CrossRef]
- Index of Refraction, Technical Note-007 (Duke Scientific Corporation, December 1, 1996).
- “Release on the refractive index of ordinary water substance as a function of wavelength, temperature and pressure” (International Association for the Properties of Water and Steam, 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.