Abstract

We propose a minimum variation of solution method to determine the optimal regularization parameter for singular value decomposition for obtaining the initial distribution for a Chahine iterative algorithm used to determine the particle size distribution from photon correlation spectroscopy data. We impose a nonnegativity constraint to make the initial distribution more realistic. The minimum variation of solution is a single constraint method and we show that a better regularization parameter may be obtained by increasing the discrimination between adjacent values. We developed the S-R curve method as a means of determining the modest iterative solution from the Chahine algorithm. The S-R curve method requires a smoothing operator. We have used simulated data to verify our new method and applied it to real data. Both simulated and experimental data show that the method works well and that the first derivative smoothing operator in the S-R curve gives the best results.

© 2012 Optical Society of America

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  1. L. M. Gugliotta, J. R. Vega, and G. R. Meira, “Latex particle size distribution by dynamic light scattering: computer evaluation of two alternative calculation paths,” J. Colloid Interface Sci. 228, 14–17 (2000).
    [CrossRef]
  2. A. R. Jones, “Light scattering for particle characterization,” Prog. Energy Combust. Sci. 25, 1–53 (1999).
    [CrossRef]
  3. J. C. Thomas, “Photon correlation spectroscopy: technique and instrumentation,” Proc. SPIE 1430, 2–18 (1991).
    [CrossRef]
  4. J. Mroczka and D. Szczuczyński, “Inverse problems formulated in terms of first-kind Fredholm integral equations in indirect measurements,” Metrol. Meas. Syst. 16, 333–357 (2009).
  5. 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]
  6. 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 (2009).
    [CrossRef]
  7. J. Mroczka and D. Szczuczyński, “Improved regularized solution of the inverse problem in turbidimetric measurements,” Appl. Opt. 49, 4591–4603 (2010).
    [CrossRef]
  8. J. Mroczka and D. Szczuczyński, “Simulation research on improved regularized solution of the inverse problem in spectral extinction measurements,” Appl. Opt. 51, 1715–1723(2012).
    [CrossRef]
  9. T. Antony, A. Saxena, K. B. Roy, and H. B. Bohidar, “Laser light scattering immunoassay: An improved data analysis by CONTIN method,” J. Biochem. Biophys. Methods 36, 75–85 (1998).
    [CrossRef]
  10. E. R. Pike, G. Hester, B. McNally, and G. D. de Villiers, “Mathematical methods for data inversion,” in Lecture Notes in Physics, F. González and F. Moreno, eds. (Springer, 2000), pp. 41–61.
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    [CrossRef]
  12. 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).
  13. P. C. Hansen, “Numerical tools for analysis and solution for Fredholm integral equations of the first kind,” Inverse Probl. 8, 849–872 (1992).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  18. F. Ferri, A. Bassini, and E. Paganini, “Modified version of the Chahine algorithm to invert spectral extinction data for particle sizing,” Appl. Opt. 34, 5829–5839 (1995).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  21. L. Reichel and H. Sadok, “A new L-curve for ill-posed problems,” J. Comput. Appl. Math. 219, 493–508 (2008).
    [CrossRef]
  22. A. Neumaier, “Solving ill-conditioned and singular linear systems: A tutorial on regularization,” SIAM Rev. 40, 636–666 (1998).
    [CrossRef]

2012 (2)

J. Mroczka and D. Szczuczyński, “Simulation research on improved regularized solution of the inverse problem in spectral extinction measurements,” Appl. Opt. 51, 1715–1723(2012).
[CrossRef]

X. Liu, J. Shen, J. C. Thomas, L. A. Clementi, and X. Sun, “Multiangle dynamic light scattering analysis using a modified Chahine method,” J. Quant. Spectrosc. Radiat. Transfer 113, 489–497 (2012).
[CrossRef]

2010 (1)

2009 (2)

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 (2009).
[CrossRef]

J. Mroczka and D. Szczuczyński, “Inverse problems formulated in terms of first-kind Fredholm integral equations in indirect measurements,” Metrol. Meas. Syst. 16, 333–357 (2009).

2008 (1)

L. Reichel and H. Sadok, “A new L-curve for ill-posed problems,” J. Comput. Appl. Math. 219, 493–508 (2008).
[CrossRef]

2006 (1)

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]

2000 (1)

L. M. Gugliotta, J. R. Vega, and G. R. Meira, “Latex particle size distribution by dynamic light scattering: computer evaluation of two alternative calculation paths,” J. Colloid Interface Sci. 228, 14–17 (2000).
[CrossRef]

1999 (1)

A. R. Jones, “Light scattering for particle characterization,” Prog. Energy Combust. Sci. 25, 1–53 (1999).
[CrossRef]

1998 (2)

T. Antony, A. Saxena, K. B. Roy, and H. B. Bohidar, “Laser light scattering immunoassay: An improved data analysis by CONTIN method,” J. Biochem. Biophys. Methods 36, 75–85 (1998).
[CrossRef]

A. Neumaier, “Solving ill-conditioned and singular linear systems: A tutorial on regularization,” SIAM Rev. 40, 636–666 (1998).
[CrossRef]

1995 (1)

1992 (2)

P. C. Hansen, “Numerical tools for analysis and solution for Fredholm integral equations of the first kind,” Inverse Probl. 8, 849–872 (1992).
[CrossRef]

P. C. Hansen, “Analysis of discrete ill-posed problems by means of the L-curve,” SIAM Rev. 34, 561–580 (1992).
[CrossRef]

1991 (1)

J. C. Thomas, “Photon correlation spectroscopy: technique and instrumentation,” Proc. SPIE 1430, 2–18 (1991).
[CrossRef]

1990 (2)

R. K. Bryan, “Maximum entropy analysis of oversampled data problems,” Eur. Biophys. J. 18, 165–174 (1990).
[CrossRef]

A. B. Yu and N. Standish, “A study of particle size distribution,” Powder Technol. 62, 101–118 (1990).
[CrossRef]

1983 (1)

1971 (1)

1968 (1)

Alessandrini, J. L.

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]

Antony, T.

T. Antony, A. Saxena, K. B. Roy, and H. B. Bohidar, “Laser light scattering immunoassay: An improved data analysis by CONTIN method,” J. Biochem. Biophys. Methods 36, 75–85 (1998).
[CrossRef]

Bassini, A.

Bohidar, H. B.

T. Antony, A. Saxena, K. B. Roy, and H. B. Bohidar, “Laser light scattering immunoassay: An improved data analysis by CONTIN method,” J. Biochem. Biophys. Methods 36, 75–85 (1998).
[CrossRef]

Bryan, R. K.

R. K. Bryan, “Maximum entropy analysis of oversampled data problems,” Eur. Biophys. J. 18, 165–174 (1990).
[CrossRef]

Cai, X.

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 (2009).
[CrossRef]

Chahine, M. T.

Christophersen, N.

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).

Clementi, L. A.

X. Liu, J. Shen, J. C. Thomas, L. A. Clementi, and X. Sun, “Multiangle dynamic light scattering analysis using a modified Chahine method,” J. Quant. Spectrosc. Radiat. Transfer 113, 489–497 (2012).
[CrossRef]

de Villiers, G. D.

E. R. Pike, G. Hester, B. McNally, and G. D. de Villiers, “Mathematical methods for data inversion,” in Lecture Notes in Physics, F. González and F. Moreno, eds. (Springer, 2000), pp. 41–61.

Dorey, J. M.

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 (2009).
[CrossRef]

Ferri, F.

Grassl, H.

Gugliotta, L. M.

L. M. Gugliotta, J. R. Vega, and G. R. Meira, “Latex particle size distribution by dynamic light scattering: computer evaluation of two alternative calculation paths,” J. Colloid Interface Sci. 228, 14–17 (2000).
[CrossRef]

Hansen, P. C.

P. C. Hansen, “Numerical tools for analysis and solution for Fredholm integral equations of the first kind,” Inverse Probl. 8, 849–872 (1992).
[CrossRef]

P. C. Hansen, “Analysis of discrete ill-posed problems by means of the L-curve,” SIAM Rev. 34, 561–580 (1992).
[CrossRef]

Herman, M.

Hester, G.

E. R. Pike, G. Hester, B. McNally, and G. D. de Villiers, “Mathematical methods for data inversion,” in Lecture Notes in Physics, F. González and F. Moreno, eds. (Springer, 2000), pp. 41–61.

Jones, A. R.

A. R. Jones, “Light scattering for particle characterization,” Prog. Energy Combust. Sci. 25, 1–53 (1999).
[CrossRef]

Lingjearde, O. C.

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).

Liu, X.

X. Liu, J. Shen, J. C. Thomas, L. A. Clementi, and X. Sun, “Multiangle dynamic light scattering analysis using a modified Chahine method,” J. Quant. Spectrosc. Radiat. Transfer 113, 489–497 (2012).
[CrossRef]

McNally, B.

E. R. Pike, G. Hester, B. McNally, and G. D. de Villiers, “Mathematical methods for data inversion,” in Lecture Notes in Physics, F. González and F. Moreno, eds. (Springer, 2000), pp. 41–61.

Meira, G. R.

L. M. Gugliotta, J. R. Vega, and G. R. Meira, “Latex particle size distribution by dynamic light scattering: computer evaluation of two alternative calculation paths,” J. Colloid Interface Sci. 228, 14–17 (2000).
[CrossRef]

Mroczka, J.

Neumaier, A.

A. Neumaier, “Solving ill-conditioned and singular linear systems: A tutorial on regularization,” SIAM Rev. 40, 636–666 (1998).
[CrossRef]

Paganini, E.

Pike, E. R.

E. R. Pike, G. Hester, B. McNally, and G. D. de Villiers, “Mathematical methods for data inversion,” in Lecture Notes in Physics, F. González and F. Moreno, eds. (Springer, 2000), pp. 41–61.

Reichel, L.

L. Reichel and H. Sadok, “A new L-curve for ill-posed problems,” J. Comput. Appl. Math. 219, 493–508 (2008).
[CrossRef]

Ren, K. F.

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 (2009).
[CrossRef]

Roig, R.

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]

Roy, K. B.

T. Antony, A. Saxena, K. B. Roy, and H. B. Bohidar, “Laser light scattering immunoassay: An improved data analysis by CONTIN method,” J. Biochem. Biophys. Methods 36, 75–85 (1998).
[CrossRef]

Sadok, H.

L. Reichel and H. Sadok, “A new L-curve for ill-posed problems,” J. Comput. Appl. Math. 219, 493–508 (2008).
[CrossRef]

Santer, R.

Saxena, A.

T. Antony, A. Saxena, K. B. Roy, and H. B. Bohidar, “Laser light scattering immunoassay: An improved data analysis by CONTIN method,” J. Biochem. Biophys. Methods 36, 75–85 (1998).
[CrossRef]

Shen, J.

X. Liu, J. Shen, J. C. Thomas, L. A. Clementi, and X. Sun, “Multiangle dynamic light scattering analysis using a modified Chahine method,” J. Quant. Spectrosc. Radiat. Transfer 113, 489–497 (2012).
[CrossRef]

Standish, N.

A. B. Yu and N. Standish, “A study of particle size distribution,” Powder Technol. 62, 101–118 (1990).
[CrossRef]

Sun, X.

X. Liu, J. Shen, J. C. Thomas, L. A. Clementi, and X. Sun, “Multiangle dynamic light scattering analysis using a modified Chahine method,” J. Quant. Spectrosc. Radiat. Transfer 113, 489–497 (2012).
[CrossRef]

Szczuczynski, D.

Thomas, J. C.

X. Liu, J. Shen, J. C. Thomas, L. A. Clementi, and X. Sun, “Multiangle dynamic light scattering analysis using a modified Chahine method,” J. Quant. Spectrosc. Radiat. Transfer 113, 489–497 (2012).
[CrossRef]

J. C. Thomas, “Photon correlation spectroscopy: technique and instrumentation,” Proc. SPIE 1430, 2–18 (1991).
[CrossRef]

Vega, J. R.

L. M. Gugliotta, J. R. Vega, and G. R. Meira, “Latex particle size distribution by dynamic light scattering: computer evaluation of two alternative calculation paths,” J. Colloid Interface Sci. 228, 14–17 (2000).
[CrossRef]

Xu, F.

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 (2009).
[CrossRef]

Yu, A. B.

A. B. Yu and N. Standish, “A study of particle size distribution,” Powder Technol. 62, 101–118 (1990).
[CrossRef]

Appl. Opt. (5)

Chem. Eng. Commun. (1)

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 (2009).
[CrossRef]

Eur. Biophys. J. (1)

R. K. Bryan, “Maximum entropy analysis of oversampled data problems,” Eur. Biophys. J. 18, 165–174 (1990).
[CrossRef]

Inverse Probl. (1)

P. C. Hansen, “Numerical tools for analysis and solution for Fredholm integral equations of the first kind,” Inverse Probl. 8, 849–872 (1992).
[CrossRef]

J. Biochem. Biophys. Methods (1)

T. Antony, A. Saxena, K. B. Roy, and H. B. Bohidar, “Laser light scattering immunoassay: An improved data analysis by CONTIN method,” J. Biochem. Biophys. Methods 36, 75–85 (1998).
[CrossRef]

J. Colloid Interface Sci. (1)

L. M. Gugliotta, J. R. Vega, and G. R. Meira, “Latex particle size distribution by dynamic light scattering: computer evaluation of two alternative calculation paths,” J. Colloid Interface Sci. 228, 14–17 (2000).
[CrossRef]

J. Comput. Appl. Math. (1)

L. Reichel and H. Sadok, “A new L-curve for ill-posed problems,” J. Comput. Appl. Math. 219, 493–508 (2008).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Quant. Spectrosc. Radiat. Transfer (1)

X. Liu, J. Shen, J. C. Thomas, L. A. Clementi, and X. Sun, “Multiangle dynamic light scattering analysis using a modified Chahine method,” J. Quant. Spectrosc. Radiat. Transfer 113, 489–497 (2012).
[CrossRef]

Metrol. Meas. Syst. (1)

J. Mroczka and D. Szczuczyński, “Inverse problems formulated in terms of first-kind Fredholm integral equations in indirect measurements,” Metrol. Meas. Syst. 16, 333–357 (2009).

Part. Part. Syst. Charact. (1)

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]

Powder Technol. (1)

A. B. Yu and N. Standish, “A study of particle size distribution,” Powder Technol. 62, 101–118 (1990).
[CrossRef]

Proc. SPIE (1)

J. C. Thomas, “Photon correlation spectroscopy: technique and instrumentation,” Proc. SPIE 1430, 2–18 (1991).
[CrossRef]

Prog. Energy Combust. Sci. (1)

A. R. Jones, “Light scattering for particle characterization,” Prog. Energy Combust. Sci. 25, 1–53 (1999).
[CrossRef]

SIAM Rev. (2)

P. C. Hansen, “Analysis of discrete ill-posed problems by means of the L-curve,” SIAM Rev. 34, 561–580 (1992).
[CrossRef]

A. Neumaier, “Solving ill-conditioned and singular linear systems: A tutorial on regularization,” SIAM Rev. 40, 636–666 (1998).
[CrossRef]

Other (2)

E. R. Pike, G. Hester, B. McNally, and G. D. de Villiers, “Mathematical methods for data inversion,” in Lecture Notes in Physics, F. González and F. Moreno, eds. (Springer, 2000), pp. 41–61.

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).

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Figures (9)

Fig. 1.
Fig. 1.

Variation norm xNxNC versus regularization parameter index, q, for noiseless data and data with a noise level of 0.02 for (a) a unimodal and (b) a bimodal particle size distribution.

Fig. 2.
Fig. 2.

RME versus Chahine iteration count for (a) unimodal and (b) bimodal PSDs at a noise level of 0.02.

Fig. 3.
Fig. 3.

(a) Unimodal PSD zero derivative norm versus RME; and (b) unimodal PSD first derivative norm versus RME for a noise level of 0.02.

Fig. 4.
Fig. 4.

(a) Bimodal PSD zero derivative norm versus RME; and (b) bimodal PSD first derivative norm versus RME for a noise level of 0.02.

Fig. 5.
Fig. 5.

Unimodal PSD estimated from the zero derivative operator and the first derivative operator using the S-R curve and from the minimum RME at a noise level of 0.02.

Fig. 6.
Fig. 6.

Bimodal PSD estimated from the zero derivative operator and the first derivative operator using the S-R curve and from the minimum RME at a noise level of 0.02.

Fig. 7.
Fig. 7.

Experimental setup.

Fig. 8.
Fig. 8.

PSDs recovered for the (a) unimodal particle sample and the (b) bimodal sample using the minimum RME operator (○) and the first derivative operator (•).

Fig. 9.
Fig. 9.

Optimized PSDs recovered for (a) the unimodal particle sample and (b) the bimodal sample using the minimum RME operator (○) and the first derivative operator (•).

Tables (2)

Tables Icon

Table 1. Particle Size Distribution Results for Unimodal and Bimodal PSDs with a Noise Level of 0.02a

Tables Icon

Table 2. Particle Size Distribution Results for Unimodal and Bimodal Suspensions of Polystyrene Latex Spheres

Equations (18)

Equations on this page are rendered with MathJax. Learn more.

g(1)(τ)=exp(Γτ),
g(1)(τ)=0G(Γ)exp(Γτ)dΓ=i=1nGiexp(Γiτ).
g=Ax,
A=USVT=i=1nuisiviT,
x=i=1nfiuiTgsivi.
fi=si2si2+2{1sisi2/2si,
s1s2sn0.
x=i=1quiTgsivi.
g(p1)=Ax(p1),
xj(p)=xj(p1)i=1mWi·j·gi*gi(p1),
Wi.j=Ai·ji=1mAi·j.
RME={1mi=1m[gi*gi(p)]2[gi(p)]2}1/2
FSB(D)=σ2π(DmaxDmin)[t(1t)]1×exp{0.5[μ+σln(t1t)]2}.
t=DDminDmaxDmin,
gi(N)gi(N)+δg(N)ε/N,
L(1)=[10000001],
L(n1)×n(2)=[110011].
F(x(p))=Ax(p)gLx(p),L=L(1)orL(n1)×n(2).

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