Abstract

The size distribution of a particle suspension can be inferred from studying the diffracted light pattern produced by a laser beam that passes through the suspension. This involves solving an ill-posed linear system. Two previous versions of projection algorithms were tested by use of computer simulations and experiments. Both algorithms showed limitations in restoring the size distribution, because of either the shape of the size distribution or the presence of noise in the scattered light signal. The generalized projection algorithm presented in this work solves these difficulties by introducing a parameter that allows the user to adjust the noise-resolution trade-off, making it suitable for the analysis of natural suspensions.

© 2006 Optical Society of America

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References

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  1. Y. C. Agrawal and H. C. Pottsmith, "Instruments for particle size and settling velocity observations in sediment transport," Mar. Geol. 168, 89-114 (2000).
    [Crossref]
  2. Y. C. Agrawal and H. C. Pottsmith, "Laser diffraction particle sizing in STRESS," Cont. Shelf Res. 14, 1101-1121 (1994).
    [Crossref]
  3. 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]
  4. E. D. Hirleman, "General solution to the inverse near-forward-scattering particles-sizing problem in multiple-scattering environments: theory," Appl. Opt. 30, 4832-4838 (1991).
    [Crossref] [PubMed]
  5. H. Mühlenweg and E. D. Hirleman, "Laser diffraction spectroscopy: Influence of particle shape and shape adaptation technique," Part. Part. Syst. Charact. 15, 163-169 (1998).
    [Crossref]
  6. T. Matsuyama, H. Yamamoto, and B. Scarlett, "Transformation of diffraction pattern due to ellipsoids into equivalent diameter distribution for spheres," Part. Part. Syst. Charact. 17, 41-46 (2000).
    [Crossref]
  7. E. D. Hirleman, "Optimal scaling of the inverse Fraunhofer diffraction particle sizing problem: the linear system produced by quadrature," Part. Part. Syst. Charact. 4, 128-133 (1987).
    [Crossref]
  8. J. B. Riley and Y. C. Agrawal, "Sampling and inversion of data in diffraction particle sizing," Appl. Opt. 30, 4800-4817 (1991).
    [Crossref] [PubMed]
  9. T. S. Huang, D. A. Barker, and S. P. Berger, "Iterative image restoration," Appl. Opt. 14, 1165-1168 (1975).
    [Crossref] [PubMed]
  10. W. Jianping, X. Shizhong, Z. Yimo, and L. Wei, "Improved projection algorithm to invert forward scattered light for particle sizing," Appl. Opt. 40, 3937-3945 (2001).
    [Crossref]
  11. J. W. Gartner, R. T. Cheng, P. Wang, and K. Richter, "Laboratory and field evaluations of the LISST-100 instrument for suspended particle size determinations," Mar. Geol. 175, 199-219 (2001).
    [Crossref]
  12. P. Traykovski, R. J. Latter, and J. D. Irish, "A laboratory evaluation of the laser in situ scattering and transmissometery instrument using natural sediments," Mar. Geol. 159, 355-367 (1999).
    [Crossref]

2001 (2)

W. Jianping, X. Shizhong, Z. Yimo, and L. Wei, "Improved projection algorithm to invert forward scattered light for particle sizing," Appl. Opt. 40, 3937-3945 (2001).
[Crossref]

J. W. Gartner, R. T. Cheng, P. Wang, and K. Richter, "Laboratory and field evaluations of the LISST-100 instrument for suspended particle size determinations," Mar. Geol. 175, 199-219 (2001).
[Crossref]

2000 (2)

Y. C. Agrawal and H. C. Pottsmith, "Instruments for particle size and settling velocity observations in sediment transport," Mar. Geol. 168, 89-114 (2000).
[Crossref]

T. Matsuyama, H. Yamamoto, and B. Scarlett, "Transformation of diffraction pattern due to ellipsoids into equivalent diameter distribution for spheres," Part. Part. Syst. Charact. 17, 41-46 (2000).
[Crossref]

1999 (1)

P. Traykovski, R. J. Latter, and J. D. Irish, "A laboratory evaluation of the laser in situ scattering and transmissometery instrument using natural sediments," Mar. Geol. 159, 355-367 (1999).
[Crossref]

1998 (1)

H. Mühlenweg and E. D. Hirleman, "Laser diffraction spectroscopy: Influence of particle shape and shape adaptation technique," Part. Part. Syst. Charact. 15, 163-169 (1998).
[Crossref]

1994 (1)

Y. C. Agrawal and H. C. Pottsmith, "Laser diffraction particle sizing in STRESS," Cont. Shelf Res. 14, 1101-1121 (1994).
[Crossref]

1991 (2)

1987 (1)

E. D. Hirleman, "Optimal scaling of the inverse Fraunhofer diffraction particle sizing problem: the linear system produced by quadrature," Part. Part. Syst. Charact. 4, 128-133 (1987).
[Crossref]

1986 (1)

1975 (1)

Agrawal, Y. C.

Y. C. Agrawal and H. C. Pottsmith, "Instruments for particle size and settling velocity observations in sediment transport," Mar. Geol. 168, 89-114 (2000).
[Crossref]

Y. C. Agrawal and H. C. Pottsmith, "Laser diffraction particle sizing in STRESS," Cont. Shelf Res. 14, 1101-1121 (1994).
[Crossref]

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

Barker, D. A.

Berger, S. P.

Cheng, R. T.

J. W. Gartner, R. T. Cheng, P. Wang, and K. Richter, "Laboratory and field evaluations of the LISST-100 instrument for suspended particle size determinations," Mar. Geol. 175, 199-219 (2001).
[Crossref]

Gartner, J. W.

J. W. Gartner, R. T. Cheng, P. Wang, and K. Richter, "Laboratory and field evaluations of the LISST-100 instrument for suspended particle size determinations," Mar. Geol. 175, 199-219 (2001).
[Crossref]

Gomi, H.

Hirleman, E. D.

H. Mühlenweg and E. D. Hirleman, "Laser diffraction spectroscopy: Influence of particle shape and shape adaptation technique," Part. Part. Syst. Charact. 15, 163-169 (1998).
[Crossref]

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

E. D. Hirleman, "Optimal scaling of the inverse Fraunhofer diffraction particle sizing problem: the linear system produced by quadrature," Part. Part. Syst. Charact. 4, 128-133 (1987).
[Crossref]

Huang, T. S.

Irish, J. D.

P. Traykovski, R. J. Latter, and J. D. Irish, "A laboratory evaluation of the laser in situ scattering and transmissometery instrument using natural sediments," Mar. Geol. 159, 355-367 (1999).
[Crossref]

Jianping, W.

Latter, R. J.

P. Traykovski, R. J. Latter, and J. D. Irish, "A laboratory evaluation of the laser in situ scattering and transmissometery instrument using natural sediments," Mar. Geol. 159, 355-367 (1999).
[Crossref]

Matsuyama, T.

T. Matsuyama, H. Yamamoto, and B. Scarlett, "Transformation of diffraction pattern due to ellipsoids into equivalent diameter distribution for spheres," Part. Part. Syst. Charact. 17, 41-46 (2000).
[Crossref]

Mühlenweg, H.

H. Mühlenweg and E. D. Hirleman, "Laser diffraction spectroscopy: Influence of particle shape and shape adaptation technique," Part. Part. Syst. Charact. 15, 163-169 (1998).
[Crossref]

Pottsmith, H. C.

Y. C. Agrawal and H. C. Pottsmith, "Instruments for particle size and settling velocity observations in sediment transport," Mar. Geol. 168, 89-114 (2000).
[Crossref]

Y. C. Agrawal and H. C. Pottsmith, "Laser diffraction particle sizing in STRESS," Cont. Shelf Res. 14, 1101-1121 (1994).
[Crossref]

Richter, K.

J. W. Gartner, R. T. Cheng, P. Wang, and K. Richter, "Laboratory and field evaluations of the LISST-100 instrument for suspended particle size determinations," Mar. Geol. 175, 199-219 (2001).
[Crossref]

Riley, J. B.

Scarlett, B.

T. Matsuyama, H. Yamamoto, and B. Scarlett, "Transformation of diffraction pattern due to ellipsoids into equivalent diameter distribution for spheres," Part. Part. Syst. Charact. 17, 41-46 (2000).
[Crossref]

Shizhong, X.

Traykovski, P.

P. Traykovski, R. J. Latter, and J. D. Irish, "A laboratory evaluation of the laser in situ scattering and transmissometery instrument using natural sediments," Mar. Geol. 159, 355-367 (1999).
[Crossref]

Wang, P.

J. W. Gartner, R. T. Cheng, P. Wang, and K. Richter, "Laboratory and field evaluations of the LISST-100 instrument for suspended particle size determinations," Mar. Geol. 175, 199-219 (2001).
[Crossref]

Wei, L.

Yamamoto, H.

T. Matsuyama, H. Yamamoto, and B. Scarlett, "Transformation of diffraction pattern due to ellipsoids into equivalent diameter distribution for spheres," Part. Part. Syst. Charact. 17, 41-46 (2000).
[Crossref]

Yimo, Z.

Appl. Opt. (5)

Cont. Shelf Res. (1)

Y. C. Agrawal and H. C. Pottsmith, "Laser diffraction particle sizing in STRESS," Cont. Shelf Res. 14, 1101-1121 (1994).
[Crossref]

Mar. Geol. (3)

Y. C. Agrawal and H. C. Pottsmith, "Instruments for particle size and settling velocity observations in sediment transport," Mar. Geol. 168, 89-114 (2000).
[Crossref]

J. W. Gartner, R. T. Cheng, P. Wang, and K. Richter, "Laboratory and field evaluations of the LISST-100 instrument for suspended particle size determinations," Mar. Geol. 175, 199-219 (2001).
[Crossref]

P. Traykovski, R. J. Latter, and J. D. Irish, "A laboratory evaluation of the laser in situ scattering and transmissometery instrument using natural sediments," Mar. Geol. 159, 355-367 (1999).
[Crossref]

Part. Part. Syst. Charact. (3)

H. Mühlenweg and E. D. Hirleman, "Laser diffraction spectroscopy: Influence of particle shape and shape adaptation technique," Part. Part. Syst. Charact. 15, 163-169 (1998).
[Crossref]

T. Matsuyama, H. Yamamoto, and B. Scarlett, "Transformation of diffraction pattern due to ellipsoids into equivalent diameter distribution for spheres," Part. Part. Syst. Charact. 17, 41-46 (2000).
[Crossref]

E. D. Hirleman, "Optimal scaling of the inverse Fraunhofer diffraction particle sizing problem: the linear system produced by quadrature," Part. Part. Syst. Charact. 4, 128-133 (1987).
[Crossref]

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Figures (15)

Fig. 1
Fig. 1

Size distributions normalized with the Rosin–Rammler model with means D ¯ = 10 μ m (squares) and D ¯ = 60 μ m (circles) and widths δ = 1.5 (open symbols) and δ = 4 (filled symbols).

Fig. 2
Fig. 2

Scattering power in arbitrary units over the 32 power detectors from the size distributions in Fig. 1. Symbols are the same as in Fig. 1.

Fig. 3
Fig. 3

Original (filled symbols) and restored (open symbols) size distributions with 3% noise added to the scattering power signal, with α = 0. The criterion to stop the iteration was ε = 10−6 [condition (12)]. The distributions had means D ¯ = 10 μ m (squares) and D ¯ = 60 μ m (circles) and width δ = 1.5. The residual, the total number of iterations, and the relative volume were R = 4.42 × 10−4, M = 21, and V = 1.12 for D ¯ = 10 μ m , respectively, and R = 5.20 × 10−4, M = 85, and V = 1.04 for D ¯ = 60 μ m , respectively.

Fig. 4
Fig. 4

Original (filled symbols) and restored (open symbols) size distributions with 3% noise added to the scattering power signal, with α = 0. The criterion to stop the iteration was ε = 10−6 [condition (12)]. The distributions had means D ¯ = 10 μ m (squares) and D ¯ = 60 μ m (circles) and width δ = 4. The residual, the total number of iterations, and the relative volume were R = 0.0016, M = 155, and V = 1.06 for D ¯ = 10 μ m , respectively, and R = 8.08 × 10−4, M = 74, and V = 1.04 for D ¯ = 60 μ m , respectively.

Fig. 5
Fig. 5

Original (filled symbols) and restored (open symbols) size distributions with 3% noise added to the scattering power signal, with α = 1. The criterion to stop the iteration was ε = 10−6 [condition (12)]. The distributions had means D ¯ = 10 μ m (squares) and D ¯ = 60 μ m (circles) and width δ = 1.5. The residual, the total number of iterations, and the relative volume were R = 6.94 × 10−4, M = 31, and V = 0.95 for D ¯ = 10 μ m , respectively, and R = 6.97 × 10−4, M = 33, and V = 1.02 for D ¯ = 60 μ m , respectively.

Fig. 6
Fig. 6

Original (filled symbols) and restored (open symbols) size distributions with 3% noise added to the scattering power signal, with α = 1. The criterion to stop the iteration was ε = 10−6 [condition (12)]. The distributions had means D ¯ = 10 μ m (squares) and D ¯ = 60 μ m (circles) and width δ = 4. The residual, the total number of iterations, and the relative volume were R = 0.0031, M = 334, and V = 1.16 for D ¯ = 10 μ m , respectively, and R = 0.0023, M = 430, and V = 1.02 for D ¯ = 60 μ m , respectively.

Fig. 7
Fig. 7

Original (filled symbols) and restored (open symbols) bimodal size distributions with no noise added to the scattering power signal, with α = 0. The criterion to stop the iteration was ε = 10−6 [condition (12)]. Two empty size classes in between the original peaks are necessary to capture the two peaks in the restored distribution. The residual, the total number of iterations, and the relative volume were R = 8.83 × 10−4, M = 264, and V = 1.00, respectively.

Fig. 8
Fig. 8

Original (filled circles) and restored (open symbols) bimodal size distributions with no noise added to the scattering power signal, with α = 1. The criterion to stop the iteration was ε = 10−6 [condition (12)]. Five empty size classes in between the original peaks are necessary to capture the two peaks in the restored distribution. The residual, the total number of iterations, and the relative volume were R = 0.0118, M = 425, and V = 1.01, respectively.

Fig. 9
Fig. 9

Final residual R and number of iterations M as functions of the difference between two consecutive residuals ε. The size distributions have means D ¯ = 10 μ m (circles) and D ¯ = 60 μ m (triangles) and variance δ = 1.5. The noise levels added to the power signal were 0% (solid lines), 1% (dashed lines), and 3% (dash–dotted lines). The values of α used in the restoration process were α = 0 (filled symbols) and α = 1 (open symbols).

Fig. 10
Fig. 10

Final residual R and number of iterations M as functions of the difference between two consecutive residuals ε. The size distributions have means D ¯ = 10 μ m (circles) and D ¯ = 60 μ m (triangles) and variance δ = 4. The noise levels added to the power signal were 0% (solid lines), 1% (dashed lines), and 3% (dash–dotted lines). The values of α used in the restoration process were α = 0 (filled symbols) and α = 1 (open symbols).

Fig. 11
Fig. 11

Measured size distribution of ground silica retained between sieves with 53- and 74-μm apertures. The measured power signal was inverted with α = 0 (open squares over dashed line), α = 1 (open triangles over dash–dotted line), and α = 0.3 (filled circles over solid line). The criterion to stop the iteration was ε = 10−6 [condition (12)]. The residual, the total number of iterations, and the concentration were R = 0.0319, M = 27, and V = 445 for α = 0, respectively; R = 0.0203, M = 448, and V = 397 for α = 1, respectively; and R = 0.0181, M = 57, and V = 414 for α = 0.3, respectively.

Fig. 12
Fig. 12

Measured size distributions of raw ground silica. The measured power signal was inverted with α = 0 (open squares over dashed line), α = 1 (open triangles over dash-dotted line), and α = 0.3 (filled circles over solid line). The criterion to stop the iteration was ε = 10−6 [condition (12)]. The residual, the total number of iterations, and the concentration were R = 0.0066, M = 299, and V = 332 for α = 0, respectively; R = 0.0066, M = 424, and V = 291 for α = 1, respectively; and R = 0.0045, M = 197, and V = 299 for α = 0.3, respectively.

Fig. 13
Fig. 13

Measured size distributions of a mixture of ground-silica particles retained between 53- and 74-μm aperture sieves and between 125- and 149-μm aperture sieves. The measured power signal was inverted with α = 0 (open squares over dashed line), α = 1 (open triangles over dash-dotted line), and α = 0.3 (filled circles over solid line) and ε = 10−6. The residual, the total number of iterations, and the relative volume were R = 0.0387, M = 40, and V = 656 for α = 0, respectively; R = 0.0170, M = 355, and V = 626 for α = 1, respectively; and R = 0.0177, M = 549, and V = 629 for α = 0.3, respectively.

Fig. 14
Fig. 14

Original (filled symbols) and restored (open symbols) size distributions with 3% noise added to the scattering power signal, with α = 0.3. The criterion to stop the iteration was ε = 10−6 [condition (12)]. The distributions have means D ¯ = 10 μ m (squares) and D ¯ = 60 μ m (circles) and width δ = 1.5. The residual, the total number of iterations, and the relative volume were R = 4.91 × 10−4, M = 34, and V = 1.02 for D ¯ = 10 μ m , respectively, and R = 2.95 × 10−4, M = 17, and V = 1.00 for D ¯ = 60 μ m , respectively.

Fig. 15
Fig. 15

Original (filled symbols) and restored (open symbols) size distributions with 3% noise added to the scattering power signal, with α = 0.3. The criterion to stop the iteration was ε = 10−6 [condition (12)]. The distributions have means D ¯ = 10 μ m (squares) and D ¯ = 60 μ m (circles) and width δ = 4. The residual, the total number of iterations, and the relative volume were R = 5.91 × 10−4, M = 131, and V = 1.04 for D ¯ = 10 μ m , respectively, and R = 6.07 × 10−4, M = 124, and V = 1.01 for D ¯ = 60 μ m , respectively.

Equations (21)

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E ( θ ) i ( θ ) θ 2 J 1     2 ( σ θ ) σ θ D 3 .
E ( θ ) 2 J 1     2 ( σ θ ) σ θ D 3 n ( D ) d D ,
E i = K i j V j ,
V j D j , min D j , max D 3 n ( D ) d D ,
K i j 1 ( σ j , max σ j , min ) σ j , min σ j , max θ i , min θ i , max 2 J 1     2 ( σ θ ) σ θ d θ d σ .
E 1 = K 11 V 1 + K 12 V 2 + + K 1 N V N ,
E 2 = K 21 V 1 + K 22 V 2 + + K 2 N V N ,
E N = K N 1 V 1 + K N 2 V 2 + + K N V N .
V j     1 , 1 = V j     0 , 1 ( K 1 j V j     0 , 1 E 1 K 1 j K 1 j ) K 1 j .
V j     I , n = V j     ( I 1 ) , n [ K I j V j ( I 1 ) , n E I K I j K I j ] K I j .
V j    I , n = V j     ( I 1 ) , n 1 n [ K I j V j    ( I 1 ) , n E I K I j K I j ] K I j .
V j     I , n = V j ( I 1 ) , n 1 n α [ K I j V j     ( I 1 ) , n E E K I j K I j ] K I j ,
R I , N = [ ( E i K i j V j     I , N ) ( E i K i j V j    I , N ) ] ( E i E i )   for   E i 0.
| R I , N R I 1 , N | < ε .
R = [ ( E i K i j V j     M , N ) ( E i K i j V j     M , N ) ] ( E i E i ) .
V ( x < D ) = 0 D V ( x ) d x = 1 exp [ ( D D ¯ ) δ ] ,
V j = exp [ ( D j , min D ¯ ) δ ] exp [ ( D j , max D ¯ ) δ ] .
cond ( K ij ) = K ij K ij = 1 [ max ( K ij V j V j ) ] [ min ( K ij V j V j ) ] 1 with V j 0 ,
V j V j     M , N V j cond ( K i j ) E i K i j V j     M , N E i .
[ ( V j     M , N V j ) ( V j     M , N V j ) ] ( V j V j ) [ cond ( K i j ) ] 2 R .

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