Abstract

The weak symmetry relationship between the relative error and solution norm holds in our developed nonnegative least-squares truncated singular value decomposition method. By using this relationship to specify the optimal regularization parameters, we applied the proposed algorithm to recover particle size distribution from dynamic light scattering (DLS) data. Simulated results and experimental validity demonstrate that the proposed method, which compliments the CONTIN algorithm, might serve as a powerful and simple approach to the inverse problem in DLS.

© 2010 Optical Society of America

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  1. R. Pecora, Dynamic Light Scattering: Application of Photon Correlation Spectroscopy (Plenum, 1985).
  2. L. Gugliotta, J. 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] [PubMed]
  3. F. Scheffold, A. Shalkevich, R. Vavrin, J. Crassous, and P. Schurtenberger, PCS Particle Sizing in Turbid Suspensions: Scope and Limitations (American Chemical Society, 2004).
  4. K. Otsuka, K. Abe, N. Sano, S. Sudo, and J. Y. Ko, “Two-channel self-mixing laser Doppler measurement with carrier-frequency-division multiplexing,” Appl. Opt. 44, 1709–1714 (2005).
    [CrossRef] [PubMed]
  5. C. Zakian, M. Dickinson, and T. King, “Dynamic light scattering by using self-mixing interferometry with a laser diode,” Appl. Opt. 45, 2240–2245 (2006).
    [CrossRef] [PubMed]
  6. H. S. Dhadwal, K. Suh, and D. A. Ross, “A direct method of particle sizing based on the statistical processing of scattered photons from particles executing Brownian motion,” Appl. Phys. B 62, 575–581 (1996).
    [CrossRef]
  7. J. Shen, G. Zheng, G. Sun, and Q. Tu, “Fractal character of dynamic light scattering of particles,” Part. Part. Syst. Charact. 21, 411–414 (2004).
    [CrossRef]
  8. D. E. Kopple, “Analysis of macromolecular polydispersity in intensity correlation spectroscopy: the method of cumulants,” J. Chem. Phys. 57, 4814–4820 (1972).
    [CrossRef]
  9. J. G. McWhirter and E. R. Pike, “On the numerical inversion of the Laplace transform and similar Fredholm integral equations of the first kind,” J. Phys. A Math. Theor. 11, 1729–1745(1978).
    [CrossRef]
  10. N. Ostrowsky, D. Sornette, P. Parker, and E. R. Pike, “Exponential sampling method for light scattering polydispersity analysis,” J. Mod. Opt. 28, 1059–1070 (1981).
    [CrossRef]
  11. S. W. Provencher, “A constrained regularization method for inverting data represented by linear algebraic or integral equations,” Commun. Comput. Phys. 27, 213–227(1982).
    [CrossRef]
  12. I. D. Morrison and E. F. Grabowski, “Improved techniques for particle size determination for quasi-elastic light scattering,” Langmuir 1, 496–501 (1985).
    [CrossRef]
  13. M. Iqbal, “On photon correlation measurements of colloidal size distributions using Bayesian strategies,” J. Comput. Appl. Math. 126, 77–89 (2000).
    [CrossRef]
  14. J. V. Ubera, J. F. Aguilar, and D. M. Gale, “Reconstruction of particle-size distributions from light-scattering patterns using three inversion methods,” Appl. Opt. 46, 124–132(2007).
    [CrossRef]
  15. B. J. Frisken, “Revisiting the method of cumulants for analysis of dynamic light-scattering data,” Appl. Opt. 40, 4087–4091(2001).
    [CrossRef]
  16. P. A. Hassan and S. K. Kulshreshtha, “Modification to the cumulant analysis of polydispersity in quasi-elastic light scattering data,” J. Colloid. Interface Sci. 300, 744–748(2006).
    [CrossRef] [PubMed]
  17. A. R. Roig and J. L. Alessandrini, “Particle size distribution from static light scattering with regularized non-negative least squares constraints,” Part. Part. Syst. Charact. 23, 431–437 (2006).
    [CrossRef]
  18. M. L. Arias and G. L. Frontini, “Particle size distribution retrieval from elastic light scattering measurement by a modified regularization method,” Part. Part. Syst. Charact. 23, 374–380 (2006).
    [CrossRef]
  19. D. A. Ligon, T. W. Chen, and J. B. Gillespie, “Determination of aerosol parameters light-scattering data using an inverse Monte Carlo technique,” Appl. Opt. 35, 4297–4303 (1996).
    [CrossRef] [PubMed]
  20. M. Ye, S. Wang, and Y. Lu, “Inversion of particle-size distribution from angular light-scattering data with genetic algorithms,” Appl. Opt. 38, 2677–2685 (1999).
    [CrossRef]
  21. L. M. Gugliotta, G. S. Stegmayer, L. A. Clementi, and V. D. G. Gonzalez, “A neural network model for estimating the particle size distribution of dilute latex from multiangle dynamic light scattering measurements,” Part. Part. Syst. Charact. 26, 41–52 (2009).
    [CrossRef]
  22. P. C. Hansen, “The truncated SVD as a method for regularization,” BIT (Nord. Tidskr. Inf.-Behandl.) 27, 534–553(1987).
    [CrossRef]
  23. P. C. Hansen and D. P. O’Leary, “The use of L-curve in the regularization of discrete ill-posed problems,” SIAM J. Sci. Comput. 14, 1487–1503 (1993).
    [CrossRef]
  24. D. Krawczyk-Stando and M. Rudnicki, “Regularization parameter selection in discrete ill-posed problems—the use of the U-curve,” Int. J. Appl. Math. Comput. Sci. 17, 157–164(2007).
    [CrossRef]
  25. A. B. Yu and N. Standish, “A study of particle size distribution,” Powder Technol. 62, 101–118 (1990).
    [CrossRef]
  26. T. F. Coleman and Y. Li, “A reflective Newton method for minimizing a quadratic function subject to bounds on some of the variables,” SIAM J. Optim. 6, 1040–1058 (1996).
    [CrossRef]
  27. R. Finsy, L. Deriemaeker, N. De Jaeger, R. Sneyers, J. Vanderdeelen, P. Van der Meeren, H. Demeyere, J. Stone-Masui, A. Haestier, J. Clauwaert, W. De Wispelaere, P. Gillioen, S. Steyfkens, and E. Gelade, “Particle sizing by photon correlation spectroscopy. Part IV: resolution of bimodals and comparison with other particle sizing methods,” Part. Part. Syst. Charact. 10, 118–128 (1993).
    [CrossRef]
  28. W. Liu, J. Shen, and X. Sun, “Design of multiple-tau photon correlation system implemented by FPGA,” in Proceedings of The International Conference on Embedded Software and Systems (IEEE, 2008), pp. 410–414.
    [CrossRef]
  29. W. Liu, J. Shen, Y. Cheng, and W. Chen, “Novel photon correlator with less hardware resource,” Proc. SPIE 7283, 72833B (2009).
    [CrossRef]

2009 (2)

L. M. Gugliotta, G. S. Stegmayer, L. A. Clementi, and V. D. G. Gonzalez, “A neural network model for estimating the particle size distribution of dilute latex from multiangle dynamic light scattering measurements,” Part. Part. Syst. Charact. 26, 41–52 (2009).
[CrossRef]

W. Liu, J. Shen, Y. Cheng, and W. Chen, “Novel photon correlator with less hardware resource,” Proc. SPIE 7283, 72833B (2009).
[CrossRef]

2007 (2)

J. V. Ubera, J. F. Aguilar, and D. M. Gale, “Reconstruction of particle-size distributions from light-scattering patterns using three inversion methods,” Appl. Opt. 46, 124–132(2007).
[CrossRef]

D. Krawczyk-Stando and M. Rudnicki, “Regularization parameter selection in discrete ill-posed problems—the use of the U-curve,” Int. J. Appl. Math. Comput. Sci. 17, 157–164(2007).
[CrossRef]

2006 (4)

P. A. Hassan and S. K. Kulshreshtha, “Modification to the cumulant analysis of polydispersity in quasi-elastic light scattering data,” J. Colloid. Interface Sci. 300, 744–748(2006).
[CrossRef] [PubMed]

A. R. Roig and J. L. Alessandrini, “Particle size distribution from static light scattering with regularized non-negative least squares constraints,” Part. Part. Syst. Charact. 23, 431–437 (2006).
[CrossRef]

M. L. Arias and G. L. Frontini, “Particle size distribution retrieval from elastic light scattering measurement by a modified regularization method,” Part. Part. Syst. Charact. 23, 374–380 (2006).
[CrossRef]

C. Zakian, M. Dickinson, and T. King, “Dynamic light scattering by using self-mixing interferometry with a laser diode,” Appl. Opt. 45, 2240–2245 (2006).
[CrossRef] [PubMed]

2005 (1)

2004 (1)

J. Shen, G. Zheng, G. Sun, and Q. Tu, “Fractal character of dynamic light scattering of particles,” Part. Part. Syst. Charact. 21, 411–414 (2004).
[CrossRef]

2001 (1)

2000 (2)

M. Iqbal, “On photon correlation measurements of colloidal size distributions using Bayesian strategies,” J. Comput. Appl. Math. 126, 77–89 (2000).
[CrossRef]

L. Gugliotta, J. 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] [PubMed]

1999 (1)

1996 (3)

D. A. Ligon, T. W. Chen, and J. B. Gillespie, “Determination of aerosol parameters light-scattering data using an inverse Monte Carlo technique,” Appl. Opt. 35, 4297–4303 (1996).
[CrossRef] [PubMed]

T. F. Coleman and Y. Li, “A reflective Newton method for minimizing a quadratic function subject to bounds on some of the variables,” SIAM J. Optim. 6, 1040–1058 (1996).
[CrossRef]

H. S. Dhadwal, K. Suh, and D. A. Ross, “A direct method of particle sizing based on the statistical processing of scattered photons from particles executing Brownian motion,” Appl. Phys. B 62, 575–581 (1996).
[CrossRef]

1993 (2)

R. Finsy, L. Deriemaeker, N. De Jaeger, R. Sneyers, J. Vanderdeelen, P. Van der Meeren, H. Demeyere, J. Stone-Masui, A. Haestier, J. Clauwaert, W. De Wispelaere, P. Gillioen, S. Steyfkens, and E. Gelade, “Particle sizing by photon correlation spectroscopy. Part IV: resolution of bimodals and comparison with other particle sizing methods,” Part. Part. Syst. Charact. 10, 118–128 (1993).
[CrossRef]

P. C. Hansen and D. P. O’Leary, “The use of L-curve in the regularization of discrete ill-posed problems,” SIAM J. Sci. Comput. 14, 1487–1503 (1993).
[CrossRef]

1990 (1)

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

1987 (1)

P. C. Hansen, “The truncated SVD as a method for regularization,” BIT (Nord. Tidskr. Inf.-Behandl.) 27, 534–553(1987).
[CrossRef]

1985 (1)

I. D. Morrison and E. F. Grabowski, “Improved techniques for particle size determination for quasi-elastic light scattering,” Langmuir 1, 496–501 (1985).
[CrossRef]

1982 (1)

S. W. Provencher, “A constrained regularization method for inverting data represented by linear algebraic or integral equations,” Commun. Comput. Phys. 27, 213–227(1982).
[CrossRef]

1981 (1)

N. Ostrowsky, D. Sornette, P. Parker, and E. R. Pike, “Exponential sampling method for light scattering polydispersity analysis,” J. Mod. Opt. 28, 1059–1070 (1981).
[CrossRef]

1978 (1)

J. G. McWhirter and E. R. Pike, “On the numerical inversion of the Laplace transform and similar Fredholm integral equations of the first kind,” J. Phys. A Math. Theor. 11, 1729–1745(1978).
[CrossRef]

1972 (1)

D. E. Kopple, “Analysis of macromolecular polydispersity in intensity correlation spectroscopy: the method of cumulants,” J. Chem. Phys. 57, 4814–4820 (1972).
[CrossRef]

Abe, K.

Aguilar, J. F.

Alessandrini, J. L.

A. R. Roig and J. L. Alessandrini, “Particle size distribution from static light scattering with regularized non-negative least squares constraints,” Part. Part. Syst. Charact. 23, 431–437 (2006).
[CrossRef]

Arias, M. L.

M. L. Arias and G. L. Frontini, “Particle size distribution retrieval from elastic light scattering measurement by a modified regularization method,” Part. Part. Syst. Charact. 23, 374–380 (2006).
[CrossRef]

Chen, T. W.

Chen, W.

W. Liu, J. Shen, Y. Cheng, and W. Chen, “Novel photon correlator with less hardware resource,” Proc. SPIE 7283, 72833B (2009).
[CrossRef]

Cheng, Y.

W. Liu, J. Shen, Y. Cheng, and W. Chen, “Novel photon correlator with less hardware resource,” Proc. SPIE 7283, 72833B (2009).
[CrossRef]

Clauwaert, J.

R. Finsy, L. Deriemaeker, N. De Jaeger, R. Sneyers, J. Vanderdeelen, P. Van der Meeren, H. Demeyere, J. Stone-Masui, A. Haestier, J. Clauwaert, W. De Wispelaere, P. Gillioen, S. Steyfkens, and E. Gelade, “Particle sizing by photon correlation spectroscopy. Part IV: resolution of bimodals and comparison with other particle sizing methods,” Part. Part. Syst. Charact. 10, 118–128 (1993).
[CrossRef]

Clementi, L. A.

L. M. Gugliotta, G. S. Stegmayer, L. A. Clementi, and V. D. G. Gonzalez, “A neural network model for estimating the particle size distribution of dilute latex from multiangle dynamic light scattering measurements,” Part. Part. Syst. Charact. 26, 41–52 (2009).
[CrossRef]

Coleman, T. F.

T. F. Coleman and Y. Li, “A reflective Newton method for minimizing a quadratic function subject to bounds on some of the variables,” SIAM J. Optim. 6, 1040–1058 (1996).
[CrossRef]

Crassous, J.

F. Scheffold, A. Shalkevich, R. Vavrin, J. Crassous, and P. Schurtenberger, PCS Particle Sizing in Turbid Suspensions: Scope and Limitations (American Chemical Society, 2004).

De Jaeger, N.

R. Finsy, L. Deriemaeker, N. De Jaeger, R. Sneyers, J. Vanderdeelen, P. Van der Meeren, H. Demeyere, J. Stone-Masui, A. Haestier, J. Clauwaert, W. De Wispelaere, P. Gillioen, S. Steyfkens, and E. Gelade, “Particle sizing by photon correlation spectroscopy. Part IV: resolution of bimodals and comparison with other particle sizing methods,” Part. Part. Syst. Charact. 10, 118–128 (1993).
[CrossRef]

De Wispelaere, W.

R. Finsy, L. Deriemaeker, N. De Jaeger, R. Sneyers, J. Vanderdeelen, P. Van der Meeren, H. Demeyere, J. Stone-Masui, A. Haestier, J. Clauwaert, W. De Wispelaere, P. Gillioen, S. Steyfkens, and E. Gelade, “Particle sizing by photon correlation spectroscopy. Part IV: resolution of bimodals and comparison with other particle sizing methods,” Part. Part. Syst. Charact. 10, 118–128 (1993).
[CrossRef]

Demeyere, H.

R. Finsy, L. Deriemaeker, N. De Jaeger, R. Sneyers, J. Vanderdeelen, P. Van der Meeren, H. Demeyere, J. Stone-Masui, A. Haestier, J. Clauwaert, W. De Wispelaere, P. Gillioen, S. Steyfkens, and E. Gelade, “Particle sizing by photon correlation spectroscopy. Part IV: resolution of bimodals and comparison with other particle sizing methods,” Part. Part. Syst. Charact. 10, 118–128 (1993).
[CrossRef]

Deriemaeker, L.

R. Finsy, L. Deriemaeker, N. De Jaeger, R. Sneyers, J. Vanderdeelen, P. Van der Meeren, H. Demeyere, J. Stone-Masui, A. Haestier, J. Clauwaert, W. De Wispelaere, P. Gillioen, S. Steyfkens, and E. Gelade, “Particle sizing by photon correlation spectroscopy. Part IV: resolution of bimodals and comparison with other particle sizing methods,” Part. Part. Syst. Charact. 10, 118–128 (1993).
[CrossRef]

Dhadwal, H. S.

H. S. Dhadwal, K. Suh, and D. A. Ross, “A direct method of particle sizing based on the statistical processing of scattered photons from particles executing Brownian motion,” Appl. Phys. B 62, 575–581 (1996).
[CrossRef]

Dickinson, M.

Finsy, R.

R. Finsy, L. Deriemaeker, N. De Jaeger, R. Sneyers, J. Vanderdeelen, P. Van der Meeren, H. Demeyere, J. Stone-Masui, A. Haestier, J. Clauwaert, W. De Wispelaere, P. Gillioen, S. Steyfkens, and E. Gelade, “Particle sizing by photon correlation spectroscopy. Part IV: resolution of bimodals and comparison with other particle sizing methods,” Part. Part. Syst. Charact. 10, 118–128 (1993).
[CrossRef]

Frisken, B. J.

Frontini, G. L.

M. L. Arias and G. L. Frontini, “Particle size distribution retrieval from elastic light scattering measurement by a modified regularization method,” Part. Part. Syst. Charact. 23, 374–380 (2006).
[CrossRef]

Gale, D. M.

Gelade, E.

R. Finsy, L. Deriemaeker, N. De Jaeger, R. Sneyers, J. Vanderdeelen, P. Van der Meeren, H. Demeyere, J. Stone-Masui, A. Haestier, J. Clauwaert, W. De Wispelaere, P. Gillioen, S. Steyfkens, and E. Gelade, “Particle sizing by photon correlation spectroscopy. Part IV: resolution of bimodals and comparison with other particle sizing methods,” Part. Part. Syst. Charact. 10, 118–128 (1993).
[CrossRef]

Gillespie, J. B.

Gillioen, P.

R. Finsy, L. Deriemaeker, N. De Jaeger, R. Sneyers, J. Vanderdeelen, P. Van der Meeren, H. Demeyere, J. Stone-Masui, A. Haestier, J. Clauwaert, W. De Wispelaere, P. Gillioen, S. Steyfkens, and E. Gelade, “Particle sizing by photon correlation spectroscopy. Part IV: resolution of bimodals and comparison with other particle sizing methods,” Part. Part. Syst. Charact. 10, 118–128 (1993).
[CrossRef]

Gonzalez, V. D. G.

L. M. Gugliotta, G. S. Stegmayer, L. A. Clementi, and V. D. G. Gonzalez, “A neural network model for estimating the particle size distribution of dilute latex from multiangle dynamic light scattering measurements,” Part. Part. Syst. Charact. 26, 41–52 (2009).
[CrossRef]

Grabowski, E. F.

I. D. Morrison and E. F. Grabowski, “Improved techniques for particle size determination for quasi-elastic light scattering,” Langmuir 1, 496–501 (1985).
[CrossRef]

Gugliotta, L.

L. Gugliotta, J. 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] [PubMed]

Gugliotta, L. M.

L. M. Gugliotta, G. S. Stegmayer, L. A. Clementi, and V. D. G. Gonzalez, “A neural network model for estimating the particle size distribution of dilute latex from multiangle dynamic light scattering measurements,” Part. Part. Syst. Charact. 26, 41–52 (2009).
[CrossRef]

Haestier, A.

R. Finsy, L. Deriemaeker, N. De Jaeger, R. Sneyers, J. Vanderdeelen, P. Van der Meeren, H. Demeyere, J. Stone-Masui, A. Haestier, J. Clauwaert, W. De Wispelaere, P. Gillioen, S. Steyfkens, and E. Gelade, “Particle sizing by photon correlation spectroscopy. Part IV: resolution of bimodals and comparison with other particle sizing methods,” Part. Part. Syst. Charact. 10, 118–128 (1993).
[CrossRef]

Hansen, P. C.

P. C. Hansen and D. P. O’Leary, “The use of L-curve in the regularization of discrete ill-posed problems,” SIAM J. Sci. Comput. 14, 1487–1503 (1993).
[CrossRef]

P. C. Hansen, “The truncated SVD as a method for regularization,” BIT (Nord. Tidskr. Inf.-Behandl.) 27, 534–553(1987).
[CrossRef]

Hassan, P. A.

P. A. Hassan and S. K. Kulshreshtha, “Modification to the cumulant analysis of polydispersity in quasi-elastic light scattering data,” J. Colloid. Interface Sci. 300, 744–748(2006).
[CrossRef] [PubMed]

Iqbal, M.

M. Iqbal, “On photon correlation measurements of colloidal size distributions using Bayesian strategies,” J. Comput. Appl. Math. 126, 77–89 (2000).
[CrossRef]

King, T.

Ko, J. Y.

Kopple, D. E.

D. E. Kopple, “Analysis of macromolecular polydispersity in intensity correlation spectroscopy: the method of cumulants,” J. Chem. Phys. 57, 4814–4820 (1972).
[CrossRef]

Krawczyk-Stando, D.

D. Krawczyk-Stando and M. Rudnicki, “Regularization parameter selection in discrete ill-posed problems—the use of the U-curve,” Int. J. Appl. Math. Comput. Sci. 17, 157–164(2007).
[CrossRef]

Kulshreshtha, S. K.

P. A. Hassan and S. K. Kulshreshtha, “Modification to the cumulant analysis of polydispersity in quasi-elastic light scattering data,” J. Colloid. Interface Sci. 300, 744–748(2006).
[CrossRef] [PubMed]

Li, Y.

T. F. Coleman and Y. Li, “A reflective Newton method for minimizing a quadratic function subject to bounds on some of the variables,” SIAM J. Optim. 6, 1040–1058 (1996).
[CrossRef]

Ligon, D. A.

Liu, W.

W. Liu, J. Shen, Y. Cheng, and W. Chen, “Novel photon correlator with less hardware resource,” Proc. SPIE 7283, 72833B (2009).
[CrossRef]

W. Liu, J. Shen, and X. Sun, “Design of multiple-tau photon correlation system implemented by FPGA,” in Proceedings of The International Conference on Embedded Software and Systems (IEEE, 2008), pp. 410–414.
[CrossRef]

Lu, Y.

McWhirter, J. G.

J. G. McWhirter and E. R. Pike, “On the numerical inversion of the Laplace transform and similar Fredholm integral equations of the first kind,” J. Phys. A Math. Theor. 11, 1729–1745(1978).
[CrossRef]

Meira, G. R.

L. Gugliotta, J. 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] [PubMed]

Morrison, I. D.

I. D. Morrison and E. F. Grabowski, “Improved techniques for particle size determination for quasi-elastic light scattering,” Langmuir 1, 496–501 (1985).
[CrossRef]

O’Leary, D. P.

P. C. Hansen and D. P. O’Leary, “The use of L-curve in the regularization of discrete ill-posed problems,” SIAM J. Sci. Comput. 14, 1487–1503 (1993).
[CrossRef]

Ostrowsky, N.

N. Ostrowsky, D. Sornette, P. Parker, and E. R. Pike, “Exponential sampling method for light scattering polydispersity analysis,” J. Mod. Opt. 28, 1059–1070 (1981).
[CrossRef]

Otsuka, K.

Parker, P.

N. Ostrowsky, D. Sornette, P. Parker, and E. R. Pike, “Exponential sampling method for light scattering polydispersity analysis,” J. Mod. Opt. 28, 1059–1070 (1981).
[CrossRef]

Pecora, R.

R. Pecora, Dynamic Light Scattering: Application of Photon Correlation Spectroscopy (Plenum, 1985).

Pike, E. R.

N. Ostrowsky, D. Sornette, P. Parker, and E. R. Pike, “Exponential sampling method for light scattering polydispersity analysis,” J. Mod. Opt. 28, 1059–1070 (1981).
[CrossRef]

J. G. McWhirter and E. R. Pike, “On the numerical inversion of the Laplace transform and similar Fredholm integral equations of the first kind,” J. Phys. A Math. Theor. 11, 1729–1745(1978).
[CrossRef]

Provencher, S. W.

S. W. Provencher, “A constrained regularization method for inverting data represented by linear algebraic or integral equations,” Commun. Comput. Phys. 27, 213–227(1982).
[CrossRef]

Roig, A. R.

A. R. Roig and J. L. Alessandrini, “Particle size distribution from static light scattering with regularized non-negative least squares constraints,” Part. Part. Syst. Charact. 23, 431–437 (2006).
[CrossRef]

Ross, D. A.

H. S. Dhadwal, K. Suh, and D. A. Ross, “A direct method of particle sizing based on the statistical processing of scattered photons from particles executing Brownian motion,” Appl. Phys. B 62, 575–581 (1996).
[CrossRef]

Rudnicki, M.

D. Krawczyk-Stando and M. Rudnicki, “Regularization parameter selection in discrete ill-posed problems—the use of the U-curve,” Int. J. Appl. Math. Comput. Sci. 17, 157–164(2007).
[CrossRef]

Sano, N.

Scheffold, F.

F. Scheffold, A. Shalkevich, R. Vavrin, J. Crassous, and P. Schurtenberger, PCS Particle Sizing in Turbid Suspensions: Scope and Limitations (American Chemical Society, 2004).

Schurtenberger, P.

F. Scheffold, A. Shalkevich, R. Vavrin, J. Crassous, and P. Schurtenberger, PCS Particle Sizing in Turbid Suspensions: Scope and Limitations (American Chemical Society, 2004).

Shalkevich, A.

F. Scheffold, A. Shalkevich, R. Vavrin, J. Crassous, and P. Schurtenberger, PCS Particle Sizing in Turbid Suspensions: Scope and Limitations (American Chemical Society, 2004).

Shen, J.

W. Liu, J. Shen, Y. Cheng, and W. Chen, “Novel photon correlator with less hardware resource,” Proc. SPIE 7283, 72833B (2009).
[CrossRef]

J. Shen, G. Zheng, G. Sun, and Q. Tu, “Fractal character of dynamic light scattering of particles,” Part. Part. Syst. Charact. 21, 411–414 (2004).
[CrossRef]

W. Liu, J. Shen, and X. Sun, “Design of multiple-tau photon correlation system implemented by FPGA,” in Proceedings of The International Conference on Embedded Software and Systems (IEEE, 2008), pp. 410–414.
[CrossRef]

Sneyers, R.

R. Finsy, L. Deriemaeker, N. De Jaeger, R. Sneyers, J. Vanderdeelen, P. Van der Meeren, H. Demeyere, J. Stone-Masui, A. Haestier, J. Clauwaert, W. De Wispelaere, P. Gillioen, S. Steyfkens, and E. Gelade, “Particle sizing by photon correlation spectroscopy. Part IV: resolution of bimodals and comparison with other particle sizing methods,” Part. Part. Syst. Charact. 10, 118–128 (1993).
[CrossRef]

Sornette, D.

N. Ostrowsky, D. Sornette, P. Parker, and E. R. Pike, “Exponential sampling method for light scattering polydispersity analysis,” J. Mod. Opt. 28, 1059–1070 (1981).
[CrossRef]

Standish, N.

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

Stegmayer, G. S.

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R. Finsy, L. Deriemaeker, N. De Jaeger, R. Sneyers, J. Vanderdeelen, P. Van der Meeren, H. Demeyere, J. Stone-Masui, A. Haestier, J. Clauwaert, W. De Wispelaere, P. Gillioen, S. Steyfkens, and E. Gelade, “Particle sizing by photon correlation spectroscopy. Part IV: resolution of bimodals and comparison with other particle sizing methods,” Part. Part. Syst. Charact. 10, 118–128 (1993).
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[CrossRef]

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J. Shen, G. Zheng, G. Sun, and Q. Tu, “Fractal character of dynamic light scattering of particles,” Part. Part. Syst. Charact. 21, 411–414 (2004).
[CrossRef]

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R. Finsy, L. Deriemaeker, N. De Jaeger, R. Sneyers, J. Vanderdeelen, P. Van der Meeren, H. Demeyere, J. Stone-Masui, A. Haestier, J. Clauwaert, W. De Wispelaere, P. Gillioen, S. Steyfkens, and E. Gelade, “Particle sizing by photon correlation spectroscopy. Part IV: resolution of bimodals and comparison with other particle sizing methods,” Part. Part. Syst. Charact. 10, 118–128 (1993).
[CrossRef]

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R. Finsy, L. Deriemaeker, N. De Jaeger, R. Sneyers, J. Vanderdeelen, P. Van der Meeren, H. Demeyere, J. Stone-Masui, A. Haestier, J. Clauwaert, W. De Wispelaere, P. Gillioen, S. Steyfkens, and E. Gelade, “Particle sizing by photon correlation spectroscopy. Part IV: resolution of bimodals and comparison with other particle sizing methods,” Part. Part. Syst. Charact. 10, 118–128 (1993).
[CrossRef]

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J. Shen, G. Zheng, G. Sun, and Q. Tu, “Fractal character of dynamic light scattering of particles,” Part. Part. Syst. Charact. 21, 411–414 (2004).
[CrossRef]

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[CrossRef]

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J. Shen, G. Zheng, G. Sun, and Q. Tu, “Fractal character of dynamic light scattering of particles,” Part. Part. Syst. Charact. 21, 411–414 (2004).
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[CrossRef]

L. M. Gugliotta, G. S. Stegmayer, L. A. Clementi, and V. D. G. Gonzalez, “A neural network model for estimating the particle size distribution of dilute latex from multiangle dynamic light scattering measurements,” Part. Part. Syst. Charact. 26, 41–52 (2009).
[CrossRef]

R. Finsy, L. Deriemaeker, N. De Jaeger, R. Sneyers, J. Vanderdeelen, P. Van der Meeren, H. Demeyere, J. Stone-Masui, A. Haestier, J. Clauwaert, W. De Wispelaere, P. Gillioen, S. Steyfkens, and E. Gelade, “Particle sizing by photon correlation spectroscopy. Part IV: resolution of bimodals and comparison with other particle sizing methods,” Part. Part. Syst. Charact. 10, 118–128 (1993).
[CrossRef]

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

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W. Liu, J. Shen, Y. Cheng, and W. Chen, “Novel photon correlator with less hardware resource,” Proc. SPIE 7283, 72833B (2009).
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Other (3)

F. Scheffold, A. Shalkevich, R. Vavrin, J. Crassous, and P. Schurtenberger, PCS Particle Sizing in Turbid Suspensions: Scope and Limitations (American Chemical Society, 2004).

R. Pecora, Dynamic Light Scattering: Application of Photon Correlation Spectroscopy (Plenum, 1985).

W. Liu, J. Shen, and X. Sun, “Design of multiple-tau photon correlation system implemented by FPGA,” in Proceedings of The International Conference on Embedded Software and Systems (IEEE, 2008), pp. 410–414.
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

Relative error distribution (solid line with filled circles) and solution norm distribution (solid line with open circles) versus regularization parameter p at a noise level of (a) 0, (b) 0.01, and (c) 0.02 from unimodal particles inversion with Johnson’s S B pa rameters of u 1 = 3.8 , σ 1 = 3.0 , α min = 250 nm , and α max = 600 nm .

Fig. 2
Fig. 2

Recovered PSDs (PSD1, PSD2, PSD3, PSD4, PSD5, and PSD6) with solution norms in increased order of unimodal particles with Johnson’s S B parameters of u = 3.8 , σ = 3.0 , α min = 250 , and α max = 600 at noise levels of 0.

Fig. 3
Fig. 3

Recovered PSDs of unimodal particles with Johnson’s S B parameters of u = 3.8 , σ = 3.0 , α min = 250 , and α max = 600 at noise levels of 0 and 0.02.

Fig. 4
Fig. 4

Recovered PSDs of bimodal particles with Johnson’s S B parameters of u 1 = 3.8 , σ 1 = 3.0 , u 2 = 4.2 , σ 2 = 2.8 , α min = 250 , and α max = 600 at noise levels of 0 and 0.02.

Fig. 5
Fig. 5

Recovered PSDs of bimodal particles with Johnson’s S B parameters of u 1 = 3.8 , σ 1 = 3.0 , u 2 = 4.2 , σ 2 = 2.8 , α min = 2 , and α max = 100 at noise levels of 0 and 0.02.

Fig. 6
Fig. 6

Recovered PSDs of 100 and 300 nm bimodal particles at noise levels of 0 and 0.02.

Fig. 7
Fig. 7

Experimental setup.

Fig. 8
Fig. 8

(a) Correlation function data of 60 nm particles and (b) inversion PSDs, using nonnegative least-squares TSVD (solid line with open circles) and CONTIN (solid line with filled circles).

Fig. 9
Fig. 9

(a) Correlation function data of 100 and 500 nm bimodal particles and (b) inversion PSDs, using nonnegative least-squares TSVD (solid line with open circles) and CONTIN (solid line with filled circles).

Tables (1)

Tables Icon

Table 1 Minimum Relative Error Versus Maximum Solution Norm at Noise Levels of 0, 0.01, and 0.02

Equations (6)

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

g ( 1 ) ( τ ) = 0 G ( Γ ) exp ( Γ τ ) d Γ ,
A x = z ,
A = U Σ V T = i = 1 n σ i μ i ν i T ,
A p = i = 1 p σ i μ i ν i T ,
x p = i = 1 p μ i T z σ i ν i .
f ( x ) = min A p x z 2 , s . t .     x 0.

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