Abstract

An improved version of the nonlinear iterative projection algorithm is proposed and applied to the inversion of the light-scattering data for particle sizing. Compared with the original projection algorithm, the improved one is much more stable, reliable, and accurate with respect to random noise. The particle-size distributions retrieved with the improved projection algorithm are independent of the different starting solutions. The criterion for stopping iteration is fairly clearly determined and is not a user-dependent parameter of the inversion algorithm. Numerical computer simulations and experimental results for the mixture of monodispersed size distributions are presented to allow both the inversion procedure validity and the instrument performance to be evaluated.

© 2001 Optical Society of America

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References

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  1. A. H. Lefebvre, Gas Turbine Combustion (McGraw-Hill, New York, 1983).
  2. E. Trakhovsky, “Atmospheric aerosols investigated by inversion of experimental transmittance data,” Appl. Opt. 21, 3005–3010 (1982).
    [CrossRef] [PubMed]
  3. P. J. Wyatt, “Differential light scattering: a physical method for identifying living bacterial cells,” Appl. Opt. 7, 1879–1896 (1968).
    [CrossRef] [PubMed]
  4. A. A. Kokhanivsky, E. P. Zege, “Local optical parameters of spherical polydispersions: simple approximation,” Appl. Opt. 34, 5513–5519 (1995).
    [CrossRef]
  5. A. Ben-David, B. M. Herman, “Method for determining particle size distribution by nonlinear inversion of backscattered radiation,” Appl. Opt. 24, 1037–1042 (1985).
    [CrossRef]
  6. F. Ferri, A. Bassini, E. Paganini, “Commercial spectrophotometer for particle sizing,” Appl. Opt. 36, 885–891 (1997).
    [CrossRef] [PubMed]
  7. H. Stark, D. Lee, B. Dimitriadis, “Smoothing of irradiance spectra with finite-bandwidth windows, with application to particle-size analysis,” J. Opt. Soc. Am. 65, 1436–1442 (1975).
    [CrossRef]
  8. M. Kerker, D. D. Cooke, “Remote sensing of particle size and refractive index by varying the wavelength,” Appl. Opt. 15, 2105–2111 (1976).
    [CrossRef] [PubMed]
  9. D. W. Schuerman, D. E. Beeson, F. Giovane, “Coronagraphic technique to infer the nature of the skylab particulate environment,” Appl. Opt. 16, 1591–1597 (1977).
    [CrossRef] [PubMed]
  10. J. B. Riley, Y. C. Agrawal, “Sampling and inversion of data in diffraction particle sizing,” Appl. Opt. 30, 4800–4817 (1991).
    [CrossRef] [PubMed]
  11. Z. Yimo, “A study of laser diffraction measurement for particle size distribution,” Opto-Electron. Eng. 17, 1–8 (1990).
  12. M. T. Chahine, “Determination of the temperature profile in an atmosphere from its outgoing radiance,” J. Opt. Soc. Am. 58, 1634–1637 (1968).
    [CrossRef]
  13. N. Wolfson, Y. Mekler, J. H. Joseph, “Comparative study of inversion technique. Part I: accuracy and stability,” J. Appl. Meteorol. 18, 556–561 (1979).
    [CrossRef]
  14. S. Kaczmarz, “Angenachrte aufloesunf von systemen linearer gleichungen,” Bull. Ini. Acad. Po. Sci. Lett. A355–357 (1937), in German.
  15. G. Baozhen, W. Jianping, Z. Yimo, “Study on the projection non-mode algorithm of laser particle size measurement,” Chin. J. Lasers B7, 430–438 (1998).
  16. H. Gomi, “Multiple scattering correction in the measurement of particle size and number density by the diffraction method,” Appl. Opt. 25, 3552–3558 (1986).
    [CrossRef] [PubMed]
  17. S. D. Coston, N. George, “Particle sizing by inversion of the optical transform pattern,” Appl. Opt. 30, 4785–4794 (1991).
    [CrossRef] [PubMed]
  18. E. A. Bueher, “Computer simulation of light pulse propagation for communication through thick clouds,” Appl. Opt. 12, 2391–2407 (1973).
    [CrossRef]
  19. H. Schnablegger, O. Glatter, “Sizing of colloidal particles with light scattering: corrections for beginning multiple scattering,” Appl. Opt. 34, 3489–3501 (1995).
    [CrossRef] [PubMed]
  20. E. D. Hirleman, “General solution to the inverse near-forward-scattering particle-sizing problem in multiple-scattering environments: theory,” Appl. Opt. 30, 4832–4838 (1991).
    [CrossRef] [PubMed]
  21. T. S. Huang, D. A. Barker, S. P. Berger, “Iterative image restoration,” Appl. Opt. 14, 1165–1168 (1975).
    [CrossRef] [PubMed]
  22. J. G. Crump, J. H. Seinfeld, “A new algorithm for inversion of aerosol size distribution data,” Aerosol Sci. Technol. 1, 15–34 (1982).
    [CrossRef]

1998

G. Baozhen, W. Jianping, Z. Yimo, “Study on the projection non-mode algorithm of laser particle size measurement,” Chin. J. Lasers B7, 430–438 (1998).

1997

1995

A. A. Kokhanivsky, E. P. Zege, “Local optical parameters of spherical polydispersions: simple approximation,” Appl. Opt. 34, 5513–5519 (1995).
[CrossRef]

H. Schnablegger, O. Glatter, “Sizing of colloidal particles with light scattering: corrections for beginning multiple scattering,” Appl. Opt. 34, 3489–3501 (1995).
[CrossRef] [PubMed]

1991

1990

Z. Yimo, “A study of laser diffraction measurement for particle size distribution,” Opto-Electron. Eng. 17, 1–8 (1990).

1986

1985

1982

E. Trakhovsky, “Atmospheric aerosols investigated by inversion of experimental transmittance data,” Appl. Opt. 21, 3005–3010 (1982).
[CrossRef] [PubMed]

J. G. Crump, J. H. Seinfeld, “A new algorithm for inversion of aerosol size distribution data,” Aerosol Sci. Technol. 1, 15–34 (1982).
[CrossRef]

1979

N. Wolfson, Y. Mekler, J. H. Joseph, “Comparative study of inversion technique. Part I: accuracy and stability,” J. Appl. Meteorol. 18, 556–561 (1979).
[CrossRef]

1977

D. W. Schuerman, D. E. Beeson, F. Giovane, “Coronagraphic technique to infer the nature of the skylab particulate environment,” Appl. Opt. 16, 1591–1597 (1977).
[CrossRef] [PubMed]

1976

M. Kerker, D. D. Cooke, “Remote sensing of particle size and refractive index by varying the wavelength,” Appl. Opt. 15, 2105–2111 (1976).
[CrossRef] [PubMed]

1975

1973

1968

M. T. Chahine, “Determination of the temperature profile in an atmosphere from its outgoing radiance,” J. Opt. Soc. Am. 58, 1634–1637 (1968).
[CrossRef]

P. J. Wyatt, “Differential light scattering: a physical method for identifying living bacterial cells,” Appl. Opt. 7, 1879–1896 (1968).
[CrossRef] [PubMed]

1937

S. Kaczmarz, “Angenachrte aufloesunf von systemen linearer gleichungen,” Bull. Ini. Acad. Po. Sci. Lett. A355–357 (1937), in German.

Agrawal, Y. C.

Baozhen, G.

G. Baozhen, W. Jianping, Z. Yimo, “Study on the projection non-mode algorithm of laser particle size measurement,” Chin. J. Lasers B7, 430–438 (1998).

Barker, D. A.

Bassini, A.

Beeson, D. E.

D. W. Schuerman, D. E. Beeson, F. Giovane, “Coronagraphic technique to infer the nature of the skylab particulate environment,” Appl. Opt. 16, 1591–1597 (1977).
[CrossRef] [PubMed]

Ben-David, A.

Berger, S. P.

Bueher, E. A.

Chahine, M. T.

M. T. Chahine, “Determination of the temperature profile in an atmosphere from its outgoing radiance,” J. Opt. Soc. Am. 58, 1634–1637 (1968).
[CrossRef]

Cooke, D. D.

M. Kerker, D. D. Cooke, “Remote sensing of particle size and refractive index by varying the wavelength,” Appl. Opt. 15, 2105–2111 (1976).
[CrossRef] [PubMed]

Coston, S. D.

Crump, J. G.

J. G. Crump, J. H. Seinfeld, “A new algorithm for inversion of aerosol size distribution data,” Aerosol Sci. Technol. 1, 15–34 (1982).
[CrossRef]

Dimitriadis, B.

Ferri, F.

George, N.

Giovane, F.

D. W. Schuerman, D. E. Beeson, F. Giovane, “Coronagraphic technique to infer the nature of the skylab particulate environment,” Appl. Opt. 16, 1591–1597 (1977).
[CrossRef] [PubMed]

Glatter, O.

Gomi, H.

Herman, B. M.

Hirleman, E. D.

Huang, T. S.

Jianping, W.

G. Baozhen, W. Jianping, Z. Yimo, “Study on the projection non-mode algorithm of laser particle size measurement,” Chin. J. Lasers B7, 430–438 (1998).

Joseph, J. H.

N. Wolfson, Y. Mekler, J. H. Joseph, “Comparative study of inversion technique. Part I: accuracy and stability,” J. Appl. Meteorol. 18, 556–561 (1979).
[CrossRef]

Kaczmarz, S.

S. Kaczmarz, “Angenachrte aufloesunf von systemen linearer gleichungen,” Bull. Ini. Acad. Po. Sci. Lett. A355–357 (1937), in German.

Kerker, M.

M. Kerker, D. D. Cooke, “Remote sensing of particle size and refractive index by varying the wavelength,” Appl. Opt. 15, 2105–2111 (1976).
[CrossRef] [PubMed]

Kokhanivsky, A. A.

A. A. Kokhanivsky, E. P. Zege, “Local optical parameters of spherical polydispersions: simple approximation,” Appl. Opt. 34, 5513–5519 (1995).
[CrossRef]

Lee, D.

Lefebvre, A. H.

A. H. Lefebvre, Gas Turbine Combustion (McGraw-Hill, New York, 1983).

Mekler, Y.

N. Wolfson, Y. Mekler, J. H. Joseph, “Comparative study of inversion technique. Part I: accuracy and stability,” J. Appl. Meteorol. 18, 556–561 (1979).
[CrossRef]

Paganini, E.

Riley, J. B.

Schnablegger, H.

Schuerman, D. W.

D. W. Schuerman, D. E. Beeson, F. Giovane, “Coronagraphic technique to infer the nature of the skylab particulate environment,” Appl. Opt. 16, 1591–1597 (1977).
[CrossRef] [PubMed]

Seinfeld, J. H.

J. G. Crump, J. H. Seinfeld, “A new algorithm for inversion of aerosol size distribution data,” Aerosol Sci. Technol. 1, 15–34 (1982).
[CrossRef]

Stark, H.

Trakhovsky, E.

Wolfson, N.

N. Wolfson, Y. Mekler, J. H. Joseph, “Comparative study of inversion technique. Part I: accuracy and stability,” J. Appl. Meteorol. 18, 556–561 (1979).
[CrossRef]

Wyatt, P. J.

Yimo, Z.

G. Baozhen, W. Jianping, Z. Yimo, “Study on the projection non-mode algorithm of laser particle size measurement,” Chin. J. Lasers B7, 430–438 (1998).

Z. Yimo, “A study of laser diffraction measurement for particle size distribution,” Opto-Electron. Eng. 17, 1–8 (1990).

Zege, E. P.

A. A. Kokhanivsky, E. P. Zege, “Local optical parameters of spherical polydispersions: simple approximation,” Appl. Opt. 34, 5513–5519 (1995).
[CrossRef]

Aerosol Sci. Technol.

J. G. Crump, J. H. Seinfeld, “A new algorithm for inversion of aerosol size distribution data,” Aerosol Sci. Technol. 1, 15–34 (1982).
[CrossRef]

Appl. Opt.

A. A. Kokhanivsky, E. P. Zege, “Local optical parameters of spherical polydispersions: simple approximation,” Appl. Opt. 34, 5513–5519 (1995).
[CrossRef]

M. Kerker, D. D. Cooke, “Remote sensing of particle size and refractive index by varying the wavelength,” Appl. Opt. 15, 2105–2111 (1976).
[CrossRef] [PubMed]

D. W. Schuerman, D. E. Beeson, F. Giovane, “Coronagraphic technique to infer the nature of the skylab particulate environment,” Appl. Opt. 16, 1591–1597 (1977).
[CrossRef] [PubMed]

Appl. Opt.

J. B. Riley, Y. C. Agrawal, “Sampling and inversion of data in diffraction particle sizing,” Appl. Opt. 30, 4800–4817 (1991).
[CrossRef] [PubMed]

A. Ben-David, B. M. Herman, “Method for determining particle size distribution by nonlinear inversion of backscattered radiation,” Appl. Opt. 24, 1037–1042 (1985).
[CrossRef]

F. Ferri, A. Bassini, E. Paganini, “Commercial spectrophotometer for particle sizing,” Appl. Opt. 36, 885–891 (1997).
[CrossRef] [PubMed]

E. Trakhovsky, “Atmospheric aerosols investigated by inversion of experimental transmittance data,” Appl. Opt. 21, 3005–3010 (1982).
[CrossRef] [PubMed]

P. J. Wyatt, “Differential light scattering: a physical method for identifying living bacterial cells,” Appl. Opt. 7, 1879–1896 (1968).
[CrossRef] [PubMed]

H. Gomi, “Multiple scattering correction in the measurement of particle size and number density by the diffraction method,” Appl. Opt. 25, 3552–3558 (1986).
[CrossRef] [PubMed]

S. D. Coston, N. George, “Particle sizing by inversion of the optical transform pattern,” Appl. Opt. 30, 4785–4794 (1991).
[CrossRef] [PubMed]

E. A. Bueher, “Computer simulation of light pulse propagation for communication through thick clouds,” Appl. Opt. 12, 2391–2407 (1973).
[CrossRef]

H. Schnablegger, O. Glatter, “Sizing of colloidal particles with light scattering: corrections for beginning multiple scattering,” Appl. Opt. 34, 3489–3501 (1995).
[CrossRef] [PubMed]

E. D. Hirleman, “General solution to the inverse near-forward-scattering particle-sizing problem in multiple-scattering environments: theory,” Appl. Opt. 30, 4832–4838 (1991).
[CrossRef] [PubMed]

T. S. Huang, D. A. Barker, S. P. Berger, “Iterative image restoration,” Appl. Opt. 14, 1165–1168 (1975).
[CrossRef] [PubMed]

Bull. Ini. Acad. Po. Sci. Lett. A

S. Kaczmarz, “Angenachrte aufloesunf von systemen linearer gleichungen,” Bull. Ini. Acad. Po. Sci. Lett. A355–357 (1937), in German.

Chin. J. Lasers

G. Baozhen, W. Jianping, Z. Yimo, “Study on the projection non-mode algorithm of laser particle size measurement,” Chin. J. Lasers B7, 430–438 (1998).

J. Appl. Meteorol.

N. Wolfson, Y. Mekler, J. H. Joseph, “Comparative study of inversion technique. Part I: accuracy and stability,” J. Appl. Meteorol. 18, 556–561 (1979).
[CrossRef]

J. Opt. Soc. Am.

M. T. Chahine, “Determination of the temperature profile in an atmosphere from its outgoing radiance,” J. Opt. Soc. Am. 58, 1634–1637 (1968).
[CrossRef]

J. Opt. Soc. Am.

Opto-Electron. Eng.

Z. Yimo, “A study of laser diffraction measurement for particle size distribution,” Opto-Electron. Eng. 17, 1–8 (1990).

Other

A. H. Lefebvre, Gas Turbine Combustion (McGraw-Hill, New York, 1983).

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Figures (19)

Fig. 1
Fig. 1

Schematic of the laser particle sizer.

Fig. 2
Fig. 2

Fraunhofer diffraction from a single spherical particle.

Fig. 3
Fig. 3

Geometrical explanation of the projection algorithm.

Fig. 4
Fig. 4

(a) Given standard particle-size distribution. (b) The retrieved size distribution with the projection algorithm.

Fig. 5
Fig. 5

Iterative procedure of the original projection algorithm.

Fig. 6
Fig. 6

Iterative procedure of the improved projection algorithm.

Fig. 7
Fig. 7

Retrieved size distribution with the improved projection algorithm.

Fig. 8
Fig. 8

Iterative procedure of the improved projection algorithm with different starting distributions: circle, the starting distribution is constant with W i = 1/32 (i = 1, 2, … , 32); triangle, the starting distribution equals to the energy distribution; inverted triangle, the starting distribution is random data.

Fig. 9
Fig. 9

Logarithmic error of the retrieved distribution corresponding to different threshold ε.

Fig. 10
Fig. 10

Contrast of broader particle-size distribution. (a) The input distribution and (b) the retrieved distribution with the improved projection algorithm.

Fig. 11
Fig. 11

Contrast of narrower particle-size distribution. (a) The input distribution and (b) the retrieved distribution with the improved projection algorithm.

Fig. 12
Fig. 12

Input particle-size distribution.

Fig. 13
Fig. 13

Retrieved size distribution with the original projection algorithm. (a) No noise, (b) ±5% random noise, and (c) ±10% random noise.

Fig. 14
Fig. 14

Retrieved size distribution with the improved projection algorithm. (a) No noise, (b) ±5% random noise, and (c) ±10% random noise.

Fig. 15
Fig. 15

Retrieved size distribution with the original projection algorithm for the mixture of GBW(E)120005 and GBW(E)120008.

Fig. 16
Fig. 16

Retrieved size distribution with the improved projection algorithm for the mixture of GBW(E)120005 and GBW(E)120008.

Fig. 17
Fig. 17

Retrieved size distribution with the improved projection algorithm for the mixture of GBW(E)120005, GBW(E)120007, and GBW(E)120008.

Fig. 18
Fig. 18

Explanation of the characteristic diameter X of the particles.

Fig. 19
Fig. 19

Explanation of the narrowness index N of the distribution.

Tables (2)

Tables Icon

Table 1 Dimension of the 32 Annuli of the Optoelectronic Detector

Tables Icon

Table 2 Logarithmic Error of the Retrieved Distribution Corresponding to Different Threshold ε

Equations (22)

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

E θ I θ θ     2 J 1 2 α θ α θ ,
E θ       2 J 1 2 α θ α θ   D 3 N D d D .
N = 0 D max   N D d D ,
π Dr 0 / λ f = 1.357 .
W j     D j , min D j , max   D 3 N D d D ,
E i = i = 1 32   T ij W j ,
i = 1 ,   2 , ,   32 ,
j = 1 ,   2 , ,   32 ,
T ij     α j , min α j , max θ i , min θ i , max 2 J 1 2 α θ α θ d θ d α / α j , max - α j , min .
  E - TW * 2 min .
T 1 , 1 W 1 + T 1 , 2 W 2 + + T 1 , 32 W 32 = E 1 , T 2 , 1 W 1 + T 2 , 2 W 2 + + T 2 , 32 W 32 = E 2 , T 32 , 1 W 1 + T 32 , 2 W 2 + + T 32 , 32 W 32 = E 32 .
W 1 = W 0 - W 0 T 1 - E 1 T 1 T 1   T 1 ,
W k + 1 = W k - W k T i - E i T i T i   T i .
LE k = log 10 i = 1 32 E i k cal - E i in 2 ,
LE k = 3.765 + 1.155 k - 1 - 2.719 k - 2 + 4.489 k - 3 - 2.435 k - 4 .
W k + 1 = W k - ω k W k T i - E i T i T i   T i .
ω k = 1 / k ,
LE k = 3.461 + 1.180 k - 1 - 1.682 k - 2 + 4.268 k - 3 - 2.989 k - 4 .
| LE k - LE k - 1 | < ε .
E i meas = E i + Δ E i .
Δ E i = E i e i % .
V = exp - D X N ,

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