Abstract

A method to retrieve the radius and the relative refractive index of spherical homogeneous nonabsorbing particles by multiangle scattering is proposed. It is based on the formation of noise-resistant functionals of the scattered intensity, which are invariant with respect to the linear homogeneous transformations of an intensity-based signal and approximation of the retrieved parameters’ dependence on the functionals by a feed-forward neural network. The neural network was trained by minimization of the mean squared relative error in the range of particle radii from 0.6 mkm up to 13.6 mkm and relative refractive index from 1.015 up to 1.28. In comparison with training on a minimum of the mean squared error, this method enables one to increase the accuracy of the radius retrieval in the range of radii from 0.6to2μm and refractive index in the range from 1.015 to 1.1. The values of intensity of light scattered in the interval of angles 10°60° are used as input data. If the measurement error is 20%, the mean errors of the radius and relative refractive index are 0.8% and 7%, respectively. The results obtained by the proposed method and by the trial and error method with published experimental data (measured with a scanning flow cytometer) are compared. The maximal difference in the retrieval results of radius and the relative refractive index of particles obtained by both methods is under 5%.

© 2009 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. Xu, Particle Characterization: Light Scattering Methods (Kluwer, 2000).
  2. S. Twomey, Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements (Elsevier, 1977).
  3. A. N. Tichonov and V. Y. Arsenin, Solutions of Ill-Posed Problems (Wiley, 1977).
  4. E. Kissa, Dispersions: Characterization, Testing, and Measurement, Vol. 84 of Surfactant Science Series (Marcel Dekker, 1999).
  5. M. A. van Dilla, P. N. Dean, O. D. Laerum, and M. R. Melamed, Flowcytometry: Instrumentation and Data Analysis (Academic, 1985).
  6. M. Bartholdi, G. C. Salzman, R. D. Hielbert, and M. Kerker, “Differential light scattering photometer for rapid analysis of single particles in flow,” Appl. Opt. 19, 1573-1581 (1980).
    [CrossRef]
  7. V. P. Maltsev, “Scanning flow cytometry for individual particle analysis,” Rev. Sci. Instrum. 71, 243-255 (2000).
    [CrossRef]
  8. F. Girosi, M. Jones, and T. Poggio, “Regularization theory and neural networks architectutes,” Neural Comput. 7, 219-296(1995).
    [CrossRef]
  9. C. M. Bishop, “Training with noise is equivalent to Tikhonov regularization,” Neural Comput. 7, 108-116 (1995).
    [CrossRef]
  10. K. Ludlow and J. Everitt, “Inverse Mie problem,” J. Opt. Soc. Am. A 17, 2229-2235 (2000).
    [CrossRef]
  11. D. H. Tycko, M. H. Metz, E. A. Epstein, and A. Grinbaum, “Flow-cytometric light scattering measurement of red blood cell volume and hemoglobin concentration,” Appl. Opt. 24, 1355-1365 (1985).
    [CrossRef]
  12. S. Zakovic, Z. J. Ulanowski, and M. C. Bartholomew-Biggs, “Application of global optimization to particle identification using light scattering,” Inverse Prob. 14, 1053-1067(1998).
  13. S. Zakovic, Z. J. Ulanowski, and M. C. Bartholomew-Biggs, “Using global optimization for a microparticle identification problem with noise data,” J. Global Optimization 32, 325-347(2005).
  14. S. Min and A. Gomez, “High-resolutionsize measurement if single spherical particles with a fast Fourier transform of the angular scattering intensity,” Appl. Opt. 35, 4919-4926(1996).
    [CrossRef]
  15. K. A. Semyanov, P. A. Tarasov, A. E. Zharinov, A. V. Chernyshev, A. G. Hoekstra, and V. P. Maltsev, “Single-particle sizing from light scattering by spectral decomposition,” Appl. Opt. 43, 5110-5115 (2004).
    [CrossRef]
  16. www.nvidia.com
  17. S. Haykin, Neural Networks--A Comprehensive Foundation (Prentice-Hall, 1999).
  18. C. Lee Giles and T. Maxwell, “Learning, invariance, and generalization in high-order neural networks,” Appl. Opt. 26, 4972-4978 (1987).
    [CrossRef]
  19. Z. Ulanowski, Z. Wang, P. H. Kaye, and I. K. Ludlow, “Application of neural networks to the inverse light-scattering problem for spheres,” Appl. Opt. 37, 4027-4033 (1998).
    [CrossRef]
  20. Z. Wang, Z. Ulanowski, and P. H. Kaye, “On solving the inverse scattering problem with RBF neural networks: noise-free case,” Neural Comput. Applic. 8, 177-186 (1999).
  21. V. V. Berdnik, R. D. Mukhamedjarov, and V. A. Loiko, “Application of the neural network method for determining the characteristics of homogeneous spherical particles,” Opt. Spectrosc. 96, 285-291 (2004).
    [CrossRef]
  22. V. V. Berdnik, R. D. Mukhamedjarov, and V. A. Loiko, “Characterization of optically soft spheroidal particles by multiangle light-scattering data by use of the neural-networks method,” Opt. Lett. 29, 1019-1021 (2004).
    [CrossRef]
  23. V. V. Berdnik, K. Gilev, A. Shvalov, V. P. Maltsev, and V. A. Loiko, “Characterization of spherical particles using high-order neural networks and scanning flow cytometry,” J. Quant. Spectrosc. Radiative Transf. 102, 62-72(2006).
  24. V. V. Berdnik and V. A. Loiko, “Retrieval of particle characteristics with high-order neural networks: application to scanning flow cytometry,” Proc. SPIE 6734 (2007).
  25. A. Ishimaru, R. J. Marks, L. Tsang, C. M. Lam, D. C. Park, and S. Kitamura, “Particle-size distribution determination using optical sensing and neural networks,” Opt. Lett. 15, 1221-1223 (1990).
    [CrossRef]
  26. A. O. Nascimento, R. Guardani, and M. Giulietti, “Use of neural networks in the analysis of particle size distributions by laser diffraction,” Powd. Technol. 90, 89-94 (1997).
  27. P. G. Hull and Quinby-Hunt, “A neural-network to extract size parameter from light-scattering data,” Proc. SPIE 2963, 448-453 (1997).
  28. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1983).
  29. D. Dejrmendjian, Electromagnetic Scattering on Spherical Polydispersions (Elsevier, 1969).
  30. V. A. Babenko, L. A. Astafyeva, and V. N. Kuzmin, Electromagnetic Scattering in Disperse Media (Springer, , 2003).
  31. X.-P. Zhang, “Space--scale adaptive noise reduction in images based on thresholding neural network,” Mathematical Programming B 45, 503-528 (1989).
  32. D. Liu and J. Nocedal, “On the limited memory BFGS method for large scale optimization,” Mathematical Programming B 45, 503-528 (1989).
  33. V. P. Maltsev and K. A. Semyanov, Characterization of Bio-Particles from Light Scattering (VSP, 2004).

2007 (1)

V. V. Berdnik and V. A. Loiko, “Retrieval of particle characteristics with high-order neural networks: application to scanning flow cytometry,” Proc. SPIE 6734 (2007).

2006 (1)

V. V. Berdnik, K. Gilev, A. Shvalov, V. P. Maltsev, and V. A. Loiko, “Characterization of spherical particles using high-order neural networks and scanning flow cytometry,” J. Quant. Spectrosc. Radiative Transf. 102, 62-72(2006).

2005 (1)

S. Zakovic, Z. J. Ulanowski, and M. C. Bartholomew-Biggs, “Using global optimization for a microparticle identification problem with noise data,” J. Global Optimization 32, 325-347(2005).

2004 (3)

2000 (2)

V. P. Maltsev, “Scanning flow cytometry for individual particle analysis,” Rev. Sci. Instrum. 71, 243-255 (2000).
[CrossRef]

K. Ludlow and J. Everitt, “Inverse Mie problem,” J. Opt. Soc. Am. A 17, 2229-2235 (2000).
[CrossRef]

1999 (1)

Z. Wang, Z. Ulanowski, and P. H. Kaye, “On solving the inverse scattering problem with RBF neural networks: noise-free case,” Neural Comput. Applic. 8, 177-186 (1999).

1998 (2)

Z. Ulanowski, Z. Wang, P. H. Kaye, and I. K. Ludlow, “Application of neural networks to the inverse light-scattering problem for spheres,” Appl. Opt. 37, 4027-4033 (1998).
[CrossRef]

S. Zakovic, Z. J. Ulanowski, and M. C. Bartholomew-Biggs, “Application of global optimization to particle identification using light scattering,” Inverse Prob. 14, 1053-1067(1998).

1997 (2)

A. O. Nascimento, R. Guardani, and M. Giulietti, “Use of neural networks in the analysis of particle size distributions by laser diffraction,” Powd. Technol. 90, 89-94 (1997).

P. G. Hull and Quinby-Hunt, “A neural-network to extract size parameter from light-scattering data,” Proc. SPIE 2963, 448-453 (1997).

1996 (1)

1995 (2)

F. Girosi, M. Jones, and T. Poggio, “Regularization theory and neural networks architectutes,” Neural Comput. 7, 219-296(1995).
[CrossRef]

C. M. Bishop, “Training with noise is equivalent to Tikhonov regularization,” Neural Comput. 7, 108-116 (1995).
[CrossRef]

1990 (1)

1989 (2)

X.-P. Zhang, “Space--scale adaptive noise reduction in images based on thresholding neural network,” Mathematical Programming B 45, 503-528 (1989).

D. Liu and J. Nocedal, “On the limited memory BFGS method for large scale optimization,” Mathematical Programming B 45, 503-528 (1989).

1987 (1)

1985 (1)

1980 (1)

Arsenin, V. Y.

A. N. Tichonov and V. Y. Arsenin, Solutions of Ill-Posed Problems (Wiley, 1977).

Astafyeva, L. A.

V. A. Babenko, L. A. Astafyeva, and V. N. Kuzmin, Electromagnetic Scattering in Disperse Media (Springer, , 2003).

Babenko, V. A.

V. A. Babenko, L. A. Astafyeva, and V. N. Kuzmin, Electromagnetic Scattering in Disperse Media (Springer, , 2003).

Bartholdi, M.

Bartholomew-Biggs, M. C.

S. Zakovic, Z. J. Ulanowski, and M. C. Bartholomew-Biggs, “Using global optimization for a microparticle identification problem with noise data,” J. Global Optimization 32, 325-347(2005).

S. Zakovic, Z. J. Ulanowski, and M. C. Bartholomew-Biggs, “Application of global optimization to particle identification using light scattering,” Inverse Prob. 14, 1053-1067(1998).

Berdnik, V. V.

V. V. Berdnik and V. A. Loiko, “Retrieval of particle characteristics with high-order neural networks: application to scanning flow cytometry,” Proc. SPIE 6734 (2007).

V. V. Berdnik, K. Gilev, A. Shvalov, V. P. Maltsev, and V. A. Loiko, “Characterization of spherical particles using high-order neural networks and scanning flow cytometry,” J. Quant. Spectrosc. Radiative Transf. 102, 62-72(2006).

V. V. Berdnik, R. D. Mukhamedjarov, and V. A. Loiko, “Application of the neural network method for determining the characteristics of homogeneous spherical particles,” Opt. Spectrosc. 96, 285-291 (2004).
[CrossRef]

V. V. Berdnik, R. D. Mukhamedjarov, and V. A. Loiko, “Characterization of optically soft spheroidal particles by multiangle light-scattering data by use of the neural-networks method,” Opt. Lett. 29, 1019-1021 (2004).
[CrossRef]

Bishop, C. M.

C. M. Bishop, “Training with noise is equivalent to Tikhonov regularization,” Neural Comput. 7, 108-116 (1995).
[CrossRef]

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1983).

Chernyshev, A. V.

Dean, P. N.

M. A. van Dilla, P. N. Dean, O. D. Laerum, and M. R. Melamed, Flowcytometry: Instrumentation and Data Analysis (Academic, 1985).

Dejrmendjian, D.

D. Dejrmendjian, Electromagnetic Scattering on Spherical Polydispersions (Elsevier, 1969).

Epstein, E. A.

Everitt, J.

Giles, C. Lee

Gilev, K.

V. V. Berdnik, K. Gilev, A. Shvalov, V. P. Maltsev, and V. A. Loiko, “Characterization of spherical particles using high-order neural networks and scanning flow cytometry,” J. Quant. Spectrosc. Radiative Transf. 102, 62-72(2006).

Girosi, F.

F. Girosi, M. Jones, and T. Poggio, “Regularization theory and neural networks architectutes,” Neural Comput. 7, 219-296(1995).
[CrossRef]

Giulietti, M.

A. O. Nascimento, R. Guardani, and M. Giulietti, “Use of neural networks in the analysis of particle size distributions by laser diffraction,” Powd. Technol. 90, 89-94 (1997).

Gomez, A.

Grinbaum, A.

Guardani, R.

A. O. Nascimento, R. Guardani, and M. Giulietti, “Use of neural networks in the analysis of particle size distributions by laser diffraction,” Powd. Technol. 90, 89-94 (1997).

Haykin, S.

S. Haykin, Neural Networks--A Comprehensive Foundation (Prentice-Hall, 1999).

Hielbert, R. D.

Hoekstra, A. G.

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1983).

Hull, P. G.

P. G. Hull and Quinby-Hunt, “A neural-network to extract size parameter from light-scattering data,” Proc. SPIE 2963, 448-453 (1997).

Ishimaru, A.

Jones, M.

F. Girosi, M. Jones, and T. Poggio, “Regularization theory and neural networks architectutes,” Neural Comput. 7, 219-296(1995).
[CrossRef]

Kaye, P. H.

Z. Wang, Z. Ulanowski, and P. H. Kaye, “On solving the inverse scattering problem with RBF neural networks: noise-free case,” Neural Comput. Applic. 8, 177-186 (1999).

Z. Ulanowski, Z. Wang, P. H. Kaye, and I. K. Ludlow, “Application of neural networks to the inverse light-scattering problem for spheres,” Appl. Opt. 37, 4027-4033 (1998).
[CrossRef]

Kerker, M.

Kissa, E.

E. Kissa, Dispersions: Characterization, Testing, and Measurement, Vol. 84 of Surfactant Science Series (Marcel Dekker, 1999).

Kitamura, S.

Kuzmin, V. N.

V. A. Babenko, L. A. Astafyeva, and V. N. Kuzmin, Electromagnetic Scattering in Disperse Media (Springer, , 2003).

Laerum, O. D.

M. A. van Dilla, P. N. Dean, O. D. Laerum, and M. R. Melamed, Flowcytometry: Instrumentation and Data Analysis (Academic, 1985).

Lam, C. M.

Liu, D.

D. Liu and J. Nocedal, “On the limited memory BFGS method for large scale optimization,” Mathematical Programming B 45, 503-528 (1989).

Loiko, V. A.

V. V. Berdnik and V. A. Loiko, “Retrieval of particle characteristics with high-order neural networks: application to scanning flow cytometry,” Proc. SPIE 6734 (2007).

V. V. Berdnik, K. Gilev, A. Shvalov, V. P. Maltsev, and V. A. Loiko, “Characterization of spherical particles using high-order neural networks and scanning flow cytometry,” J. Quant. Spectrosc. Radiative Transf. 102, 62-72(2006).

V. V. Berdnik, R. D. Mukhamedjarov, and V. A. Loiko, “Application of the neural network method for determining the characteristics of homogeneous spherical particles,” Opt. Spectrosc. 96, 285-291 (2004).
[CrossRef]

V. V. Berdnik, R. D. Mukhamedjarov, and V. A. Loiko, “Characterization of optically soft spheroidal particles by multiangle light-scattering data by use of the neural-networks method,” Opt. Lett. 29, 1019-1021 (2004).
[CrossRef]

Ludlow, I. K.

Ludlow, K.

Maltsev, V. P.

V. V. Berdnik, K. Gilev, A. Shvalov, V. P. Maltsev, and V. A. Loiko, “Characterization of spherical particles using high-order neural networks and scanning flow cytometry,” J. Quant. Spectrosc. Radiative Transf. 102, 62-72(2006).

K. A. Semyanov, P. A. Tarasov, A. E. Zharinov, A. V. Chernyshev, A. G. Hoekstra, and V. P. Maltsev, “Single-particle sizing from light scattering by spectral decomposition,” Appl. Opt. 43, 5110-5115 (2004).
[CrossRef]

V. P. Maltsev, “Scanning flow cytometry for individual particle analysis,” Rev. Sci. Instrum. 71, 243-255 (2000).
[CrossRef]

V. P. Maltsev and K. A. Semyanov, Characterization of Bio-Particles from Light Scattering (VSP, 2004).

Marks, R. J.

Maxwell, T.

Melamed, M. R.

M. A. van Dilla, P. N. Dean, O. D. Laerum, and M. R. Melamed, Flowcytometry: Instrumentation and Data Analysis (Academic, 1985).

Metz, M. H.

Min, S.

Mukhamedjarov, R. D.

V. V. Berdnik, R. D. Mukhamedjarov, and V. A. Loiko, “Application of the neural network method for determining the characteristics of homogeneous spherical particles,” Opt. Spectrosc. 96, 285-291 (2004).
[CrossRef]

V. V. Berdnik, R. D. Mukhamedjarov, and V. A. Loiko, “Characterization of optically soft spheroidal particles by multiangle light-scattering data by use of the neural-networks method,” Opt. Lett. 29, 1019-1021 (2004).
[CrossRef]

Nascimento, A. O.

A. O. Nascimento, R. Guardani, and M. Giulietti, “Use of neural networks in the analysis of particle size distributions by laser diffraction,” Powd. Technol. 90, 89-94 (1997).

Nocedal, J.

D. Liu and J. Nocedal, “On the limited memory BFGS method for large scale optimization,” Mathematical Programming B 45, 503-528 (1989).

Park, D. C.

Poggio, T.

F. Girosi, M. Jones, and T. Poggio, “Regularization theory and neural networks architectutes,” Neural Comput. 7, 219-296(1995).
[CrossRef]

Quinby-Hunt,

P. G. Hull and Quinby-Hunt, “A neural-network to extract size parameter from light-scattering data,” Proc. SPIE 2963, 448-453 (1997).

Salzman, G. C.

Semyanov, K. A.

Shvalov, A.

V. V. Berdnik, K. Gilev, A. Shvalov, V. P. Maltsev, and V. A. Loiko, “Characterization of spherical particles using high-order neural networks and scanning flow cytometry,” J. Quant. Spectrosc. Radiative Transf. 102, 62-72(2006).

Tarasov, P. A.

Tichonov, A. N.

A. N. Tichonov and V. Y. Arsenin, Solutions of Ill-Posed Problems (Wiley, 1977).

Tsang, L.

Twomey, S.

S. Twomey, Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements (Elsevier, 1977).

Tycko, D. H.

Ulanowski, Z.

Z. Wang, Z. Ulanowski, and P. H. Kaye, “On solving the inverse scattering problem with RBF neural networks: noise-free case,” Neural Comput. Applic. 8, 177-186 (1999).

Z. Ulanowski, Z. Wang, P. H. Kaye, and I. K. Ludlow, “Application of neural networks to the inverse light-scattering problem for spheres,” Appl. Opt. 37, 4027-4033 (1998).
[CrossRef]

Ulanowski, Z. J.

S. Zakovic, Z. J. Ulanowski, and M. C. Bartholomew-Biggs, “Using global optimization for a microparticle identification problem with noise data,” J. Global Optimization 32, 325-347(2005).

S. Zakovic, Z. J. Ulanowski, and M. C. Bartholomew-Biggs, “Application of global optimization to particle identification using light scattering,” Inverse Prob. 14, 1053-1067(1998).

van Dilla, M. A.

M. A. van Dilla, P. N. Dean, O. D. Laerum, and M. R. Melamed, Flowcytometry: Instrumentation and Data Analysis (Academic, 1985).

Wang, Z.

Z. Wang, Z. Ulanowski, and P. H. Kaye, “On solving the inverse scattering problem with RBF neural networks: noise-free case,” Neural Comput. Applic. 8, 177-186 (1999).

Z. Ulanowski, Z. Wang, P. H. Kaye, and I. K. Ludlow, “Application of neural networks to the inverse light-scattering problem for spheres,” Appl. Opt. 37, 4027-4033 (1998).
[CrossRef]

Xu, R.

R. Xu, Particle Characterization: Light Scattering Methods (Kluwer, 2000).

Zakovic, S.

S. Zakovic, Z. J. Ulanowski, and M. C. Bartholomew-Biggs, “Using global optimization for a microparticle identification problem with noise data,” J. Global Optimization 32, 325-347(2005).

S. Zakovic, Z. J. Ulanowski, and M. C. Bartholomew-Biggs, “Application of global optimization to particle identification using light scattering,” Inverse Prob. 14, 1053-1067(1998).

Zhang, X.-P.

X.-P. Zhang, “Space--scale adaptive noise reduction in images based on thresholding neural network,” Mathematical Programming B 45, 503-528 (1989).

Zharinov, A. E.

Appl. Opt. (6)

Inverse Prob. (1)

S. Zakovic, Z. J. Ulanowski, and M. C. Bartholomew-Biggs, “Application of global optimization to particle identification using light scattering,” Inverse Prob. 14, 1053-1067(1998).

J. Global Optimization (1)

S. Zakovic, Z. J. Ulanowski, and M. C. Bartholomew-Biggs, “Using global optimization for a microparticle identification problem with noise data,” J. Global Optimization 32, 325-347(2005).

J. Opt. Soc. Am. A (1)

J. Quant. Spectrosc. Radiative Transf. (1)

V. V. Berdnik, K. Gilev, A. Shvalov, V. P. Maltsev, and V. A. Loiko, “Characterization of spherical particles using high-order neural networks and scanning flow cytometry,” J. Quant. Spectrosc. Radiative Transf. 102, 62-72(2006).

Mathematical Programming B (2)

X.-P. Zhang, “Space--scale adaptive noise reduction in images based on thresholding neural network,” Mathematical Programming B 45, 503-528 (1989).

D. Liu and J. Nocedal, “On the limited memory BFGS method for large scale optimization,” Mathematical Programming B 45, 503-528 (1989).

Neural Comput. (2)

F. Girosi, M. Jones, and T. Poggio, “Regularization theory and neural networks architectutes,” Neural Comput. 7, 219-296(1995).
[CrossRef]

C. M. Bishop, “Training with noise is equivalent to Tikhonov regularization,” Neural Comput. 7, 108-116 (1995).
[CrossRef]

Neural Comput. Applic. (1)

Z. Wang, Z. Ulanowski, and P. H. Kaye, “On solving the inverse scattering problem with RBF neural networks: noise-free case,” Neural Comput. Applic. 8, 177-186 (1999).

Opt. Lett. (2)

Opt. Spectrosc. (1)

V. V. Berdnik, R. D. Mukhamedjarov, and V. A. Loiko, “Application of the neural network method for determining the characteristics of homogeneous spherical particles,” Opt. Spectrosc. 96, 285-291 (2004).
[CrossRef]

Powd. Technol. (1)

A. O. Nascimento, R. Guardani, and M. Giulietti, “Use of neural networks in the analysis of particle size distributions by laser diffraction,” Powd. Technol. 90, 89-94 (1997).

Proc. SPIE (2)

P. G. Hull and Quinby-Hunt, “A neural-network to extract size parameter from light-scattering data,” Proc. SPIE 2963, 448-453 (1997).

V. V. Berdnik and V. A. Loiko, “Retrieval of particle characteristics with high-order neural networks: application to scanning flow cytometry,” Proc. SPIE 6734 (2007).

Rev. Sci. Instrum. (1)

V. P. Maltsev, “Scanning flow cytometry for individual particle analysis,” Rev. Sci. Instrum. 71, 243-255 (2000).
[CrossRef]

Other (11)

R. Xu, Particle Characterization: Light Scattering Methods (Kluwer, 2000).

S. Twomey, Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements (Elsevier, 1977).

A. N. Tichonov and V. Y. Arsenin, Solutions of Ill-Posed Problems (Wiley, 1977).

E. Kissa, Dispersions: Characterization, Testing, and Measurement, Vol. 84 of Surfactant Science Series (Marcel Dekker, 1999).

M. A. van Dilla, P. N. Dean, O. D. Laerum, and M. R. Melamed, Flowcytometry: Instrumentation and Data Analysis (Academic, 1985).

www.nvidia.com

S. Haykin, Neural Networks--A Comprehensive Foundation (Prentice-Hall, 1999).

V. P. Maltsev and K. A. Semyanov, Characterization of Bio-Particles from Light Scattering (VSP, 2004).

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1983).

D. Dejrmendjian, Electromagnetic Scattering on Spherical Polydispersions (Elsevier, 1969).

V. A. Babenko, L. A. Astafyeva, and V. N. Kuzmin, Electromagnetic Scattering in Disperse Media (Springer, , 2003).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (12)

Fig. 1
Fig. 1

Angular dependence of signal U ¯ ( θ ) averaged over particle parameters.

Fig. 2
Fig. 2

Angular dependence of the first four eigenvectors of the correlation matrix C l m Y .

Fig. 3
Fig. 3

Correlation dependence between characteristic frequency k and particle radius R.

Fig. 4
Fig. 4

Dependence of functional y 0 n on particle radius and relative refractive index.

Fig. 5
Fig. 5

Scheme of the feed-forward neural network with two internal layers.

Fig. 6
Fig. 6

Neuron activation function at x min = 0.5 , x max = 0.5 , β = 0.01 (curve 1); x min = 0.5 , x max = 0.5 , β = 0.05 (curve 2); x min = 1 , x max = 1 , β = 0.1 (curve 3).

Fig. 7
Fig. 7

Correlation dependence between radius and retrieved value of radius (a) and relative refractive index and retrieved value of relative refractive index (b).

Fig. 8
Fig. 8

Dependence of the retrieval error of radius on radius and relative refractive index.

Fig. 9
Fig. 9

Dependence of the retrieval error of relative refractive index on radius and relative refractive index.

Fig. 10
Fig. 10

Dependence of mean relative error of radius (a) and relative refractive index (b) retrieval on measurement error at the root-mean-square deviation in training noise δ l = 0 (curve 1), 0.2 (curve 2), 0.3 (curve 3).

Fig. 11
Fig. 11

Angular dependence of input signal (it is proportional to the scattered light intensity) for a particle of type1: experimental data obtained by scanning flow cytometry [33] (curve 1); calculated under the Mie theory with the particle parameters obtained by the neural network (curve 2); calculated under the least-squares method (curve 3).

Fig. 12
Fig. 12

Angular dependence of input signal (it is proportional to the scattered light intensity) for a particle of type 2: experimental data obtained by scanning flow cytometry [33] (curve 1); calculated under the Mie theory with the particle parameters obtained by the neural network (curve 2); calculated under the least-squares method (curve 3).

Tables (1)

Tables Icon

Table 1 Results of Parameter Retrieval by Experimental Data on Multiangle Light Scattering [33]

Equations (26)

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

U l ( p ) = a I ( θ l , p ) , l = 1 , 2 , L ,
U l ( p α ) = i 1 ( θ l , p α ) + i 2 ( θ l , p α )
Y l ( p ) = lg U l ( p ) 1 L l = 1 L U l ( p ) U ¯ i .
k = argmax | l = 0 L 1 Z l ( p ) e i 2 π k l L | ,
Z l ( p ) = Y l ( p ) m = 0 K 1 y m ( p ) H l m Y .
y m ( p ) = l = 0 K 1 Y l ( p ) H l m Y , m = 0 , 1 , ... , K 1 ,
z m ( p ) = l = 1 L | Z l ( p ) | H l m | Z | , m = 0 , 1 , ... K 1.
y i i n = y n 0 ( p ) , ... , y n 3 ( p ) , z n 0 ( p ) , ... , z n 3 ( p ) , k n ( p ) , i = 0 , 1 , 8.
y l out = F ( k w l k N y k N = F ( j w t j 2 y j 2 = F ( i w j i 1 y i 1 = y i in ) ) ) .
F ( x ) = 1 x max x min ( β + ( x x min ) 2 β + ( x x max ) 2 ) ,
F ( x ) = 1 x max x min ( x x min β + ( x x min ) 2 x x max β + ( x x max ) 2 ) .
Φ = 1 2 α = 1 N B ( y out ( w j i 1 , ... , w k N , y i α in ) R α 1 ) 2 ,
Φ = 1 2 α = 1 N B ( y out ( w j i 1 , ... , w k N , y i α in ) 1 n α 1 1 ) 2 ,
Φ = 1 2 α = 1 N B 1 s α 2 ( y out ( w j 1 , ... , w l k N , y i α i n ) p n α ) 2 .
y i α 1 = y i α in ,
y j α 2 = F ( w j 1 + i w j i 1 y i α 1 ) ,
y k α N = F ( w k N 1 + s w k s N 1 y s α N 1 ) ,
y l α out = y l α N + 1 = F ( w l N + k w l k N y k α N ) .
d l α out = d l α N + 1 = Φ y l α out = 1 s α 2 y l α out p l α ,
d k α N = Φ y k α N = l d l α N + 1 F ( k w l k N y k α N ) w l k N ,
Φ w l k N = α d l α N + 1 F ( k w l k N y k α N ) d k α N ,
d i α 1 = Φ y i α 1 = j d j α 2 F ( i w j i 1 y i α 1 ) w j i 1 ,
Φ w j i 1 = α d j α 2 F ( w j 1 + i w j i 1 y i α 1 ) d i α 1 .
δ R = 1 A α = 1 A | 1 R e α R α | ,
δ n = 1 A α = 1 A | 1 n e α 1 n α 1 | .
F = l = 1 L ( lg U e l 1 L l = 1 L U e l U ¯ e i 1 g U l ( p ) 1 L l = 1 L U l ( p ) U ¯ i ) 2 .

Metrics