Abstract

The iterative projection method, originally proposed by Kaczmarz and Cimmino, was recently applied to particle size analysis with forward light scattering. Modification was made to improve the convergent procedure. However, there are still limitations. It is found that the solutions are oscillatory when the method is applied to a set of underdetermined linear equations without any additional constraints. A new version of the projection method, combined with the constraints of nonnegativity and smoothness, is proposed and is studied in the presence of experimental noise. Compared with the early versions of the projection method, this one shows fast convergence, is more stable against random noise, and is insensitive to particle size distributions.

© 2008 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2006 (1)

2005 (2)

D. Gordon and R. Gordon, “Component-averaged row projections: a robust block-parallel scheme for sparse linear systems,” SIAM J. Sci. Comput. 27, 1092-1117 (2005).
[CrossRef]

G. Appleby and D. C. Smolarski, “A linear acceleration row action method for projecting onto subspaces,” Electron. Trans. Numer. Anal. 20, 253-275 (2005).

2003 (1)

J. Shen and U. Riebel, “Particle size analysis by transmission fluctuation spectrometry: experimental results obtained with a Gaussian beam and analog signal processing,” Part. Part. Syst. Charact. 20, 250-258 (2003).
[CrossRef]

2001 (1)

1999 (1)

M. Kandlikar and G. Ramachandran, “Inverse methods for analyzing aerosol spectrometer measurements: a critical review,” J. Aerosp. Sci. 30, 413-437 (1999).
[CrossRef]

1997 (1)

M. A. Popovici, N. Mincu, and C. Plosceanu, “Use of linear programming in near forward light scattering data processing,” Rom. J. Optoelectron. 5, 43-48 (1997).

1995 (1)

1991 (2)

1990 (1)

M. Alderliesten, “Mean particle diameters. Part I: evaluation of definition systems,” Part. Part. Syst. Charact. 7, 233-241(1990).
[CrossRef]

1983 (1)

Y. Censor, P. P. B. Eggermont, and D. Gordon, “Strong underrelaxation in Kaczmarz's method for inconsistent systems,” Numer. Math. 41, 83-92 (1983).
[CrossRef]

1979 (1)

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

1975 (1)

1971 (1)

K. Tanabe, “Projection method for solving a singular system of linear equations and its applications,” Numer. Math. 17, 203-214 (1971).
[CrossRef]

1968 (1)

1957 (1)

F. G. Shuman, “Numerical methods in weather prediction: II. smoothing and filtering,” Mon. Weather Rev. 85, 357-361(1957).
[CrossRef]

1938 (1)

G. Cimmino, “Calcolo approssimato per le soluzioni dei sistemi di equazioni lineari,” Ric. Sci. Progr. Tec. 41, 326-333(1938).

1937 (1)

S. Kaczmarz, “Angenaeherte Aufloesung von Systemen linearen Gleichungen,” Bull. Int. Acad. Pol. Sci. Lett. Cl. Sci. Math. Nat. Ser. A 35, 355-357 (1937).

Agrawal, Y. C.

Alderliesten, M.

M. Alderliesten, “Mean particle diameters. Part I: evaluation of definition systems,” Part. Part. Syst. Charact. 7, 233-241(1990).
[CrossRef]

Allen, T.

T. Allen, “Particle Size Measurement,” in Powder Sampling and Particle Size Measurement, 5th ed. (Chapman & Hall, 1997), Vol. 1.

Appleby, G.

G. Appleby and D. C. Smolarski, “A linear acceleration row action method for projecting onto subspaces,” Electron. Trans. Numer. Anal. 20, 253-275 (2005).

Barker, D. A.

Bassini, A.

Bayvel, L. P.

L. P. Bayvel and A. R. Jones, Electromagnetic Scattering and its Applications (Applied Science, 1981).

Beer, J. M.

J. Swithenbank, J. M. Beer, D. S. Tayor, and G. C. McCreath, “A Laser Diagnostic Technique for the Measurement of Droplet and Particle Size Distribution,” presented at 14th Aerospace Sciences Meeting, Washington, D.C., 26-28 January 1976 (American Institute of Aeronautics and Astronautics), paper 76-79.

Berger, S. P.

Bohren, C.

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

Censor, Y.

Y. Censor, P. P. B. Eggermont, and D. Gordon, “Strong underrelaxation in Kaczmarz's method for inconsistent systems,” Numer. Math. 41, 83-92 (1983).
[CrossRef]

Chahine, M. T.

Cimmino, G.

G. Cimmino, “Calcolo approssimato per le soluzioni dei sistemi di equazioni lineari,” Ric. Sci. Progr. Tec. 41, 326-333(1938).

Eggermont, P. P. B.

Y. Censor, P. P. B. Eggermont, and D. Gordon, “Strong underrelaxation in Kaczmarz's method for inconsistent systems,” Numer. Math. 41, 83-92 (1983).
[CrossRef]

Ferri, F.

Garcia, M. H.

Glatter, O.

Gordon, D.

D. Gordon and R. Gordon, “Component-averaged row projections: a robust block-parallel scheme for sparse linear systems,” SIAM J. Sci. Comput. 27, 1092-1117 (2005).
[CrossRef]

Y. Censor, P. P. B. Eggermont, and D. Gordon, “Strong underrelaxation in Kaczmarz's method for inconsistent systems,” Numer. Math. 41, 83-92 (1983).
[CrossRef]

Gordon, R.

D. Gordon and R. Gordon, “Component-averaged row projections: a robust block-parallel scheme for sparse linear systems,” SIAM J. Sci. Comput. 27, 1092-1117 (2005).
[CrossRef]

Huang, T. S.

Huffman, D.

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

Jones, A. R.

L. P. Bayvel and A. R. Jones, Electromagnetic Scattering and its Applications (Applied Science, 1981).

Joseph, J. H.

N. Wolfson, Y. Mekler, and 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, “Angenaeherte Aufloesung von Systemen linearen Gleichungen,” Bull. Int. Acad. Pol. Sci. Lett. Cl. Sci. Math. Nat. Ser. A 35, 355-357 (1937).

Kandlikar, M.

M. Kandlikar and G. Ramachandran, “Inverse methods for analyzing aerosol spectrometer measurements: a critical review,” J. Aerosp. Sci. 30, 413-437 (1999).
[CrossRef]

Li, W.

McCreath, G. S.

J. Swithenbank, J. M. Beer, D. S. Tayor, and G. C. McCreath, “A Laser Diagnostic Technique for the Measurement of Droplet and Particle Size Distribution,” presented at 14th Aerospace Sciences Meeting, Washington, D.C., 26-28 January 1976 (American Institute of Aeronautics and Astronautics), paper 76-79.

Mekler, Y.

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

Mincu, N.

M. A. Popovici, N. Mincu, and C. Plosceanu, “Use of linear programming in near forward light scattering data processing,” Rom. J. Optoelectron. 5, 43-48 (1997).

Paganini, E.

Pedocchi, F.

Plosceanu, C.

M. A. Popovici, N. Mincu, and C. Plosceanu, “Use of linear programming in near forward light scattering data processing,” Rom. J. Optoelectron. 5, 43-48 (1997).

Popovici, M. A.

M. A. Popovici, N. Mincu, and C. Plosceanu, “Use of linear programming in near forward light scattering data processing,” Rom. J. Optoelectron. 5, 43-48 (1997).

Ramachandran, G.

M. Kandlikar and G. Ramachandran, “Inverse methods for analyzing aerosol spectrometer measurements: a critical review,” J. Aerosp. Sci. 30, 413-437 (1999).
[CrossRef]

Riebel, U.

J. Shen and U. Riebel, “Particle size analysis by transmission fluctuation spectrometry: experimental results obtained with a Gaussian beam and analog signal processing,” Part. Part. Syst. Charact. 20, 250-258 (2003).
[CrossRef]

Riley, J. B.

Schnablegger, H.

Shen, J.

J. Shen and U. Riebel, “Particle size analysis by transmission fluctuation spectrometry: experimental results obtained with a Gaussian beam and analog signal processing,” Part. Part. Syst. Charact. 20, 250-258 (2003).
[CrossRef]

Shuman, F. G.

F. G. Shuman, “Numerical methods in weather prediction: II. smoothing and filtering,” Mon. Weather Rev. 85, 357-361(1957).
[CrossRef]

Smolarski, D. C.

G. Appleby and D. C. Smolarski, “A linear acceleration row action method for projecting onto subspaces,” Electron. Trans. Numer. Anal. 20, 253-275 (2005).

Swithenbank, J.

J. Swithenbank, J. M. Beer, D. S. Tayor, and G. C. McCreath, “A Laser Diagnostic Technique for the Measurement of Droplet and Particle Size Distribution,” presented at 14th Aerospace Sciences Meeting, Washington, D.C., 26-28 January 1976 (American Institute of Aeronautics and Astronautics), paper 76-79.

Tanabe, K.

K. Tanabe, “Projection method for solving a singular system of linear equations and its applications,” Numer. Math. 17, 203-214 (1971).
[CrossRef]

Tayor, D. S.

J. Swithenbank, J. M. Beer, D. S. Tayor, and G. C. McCreath, “A Laser Diagnostic Technique for the Measurement of Droplet and Particle Size Distribution,” presented at 14th Aerospace Sciences Meeting, Washington, D.C., 26-28 January 1976 (American Institute of Aeronautics and Astronautics), paper 76-79.

Wang, J.

Wolfson, N.

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

Xie, S.

Zhang, Y.

Appl. Opt. (6)

Bull. Int. Acad. Pol. Sci. Lett. Cl. Sci. Math. Nat. Ser. A (1)

S. Kaczmarz, “Angenaeherte Aufloesung von Systemen linearen Gleichungen,” Bull. Int. Acad. Pol. Sci. Lett. Cl. Sci. Math. Nat. Ser. A 35, 355-357 (1937).

Electron. Trans. Numer. Anal. (1)

G. Appleby and D. C. Smolarski, “A linear acceleration row action method for projecting onto subspaces,” Electron. Trans. Numer. Anal. 20, 253-275 (2005).

J. Aerosp. Sci. (1)

M. Kandlikar and G. Ramachandran, “Inverse methods for analyzing aerosol spectrometer measurements: a critical review,” J. Aerosp. Sci. 30, 413-437 (1999).
[CrossRef]

J. Appl. Meteorol. (1)

N. Wolfson, Y. Mekler, and 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. (1)

Mon. Weather Rev. (1)

F. G. Shuman, “Numerical methods in weather prediction: II. smoothing and filtering,” Mon. Weather Rev. 85, 357-361(1957).
[CrossRef]

Numer. Math. (2)

K. Tanabe, “Projection method for solving a singular system of linear equations and its applications,” Numer. Math. 17, 203-214 (1971).
[CrossRef]

Y. Censor, P. P. B. Eggermont, and D. Gordon, “Strong underrelaxation in Kaczmarz's method for inconsistent systems,” Numer. Math. 41, 83-92 (1983).
[CrossRef]

Part. Part. Syst. Charact. (2)

J. Shen and U. Riebel, “Particle size analysis by transmission fluctuation spectrometry: experimental results obtained with a Gaussian beam and analog signal processing,” Part. Part. Syst. Charact. 20, 250-258 (2003).
[CrossRef]

M. Alderliesten, “Mean particle diameters. Part I: evaluation of definition systems,” Part. Part. Syst. Charact. 7, 233-241(1990).
[CrossRef]

Ric. Sci. Progr. Tec. (1)

G. Cimmino, “Calcolo approssimato per le soluzioni dei sistemi di equazioni lineari,” Ric. Sci. Progr. Tec. 41, 326-333(1938).

Rom. J. Optoelectron. (1)

M. A. Popovici, N. Mincu, and C. Plosceanu, “Use of linear programming in near forward light scattering data processing,” Rom. J. Optoelectron. 5, 43-48 (1997).

SIAM J. Sci. Comput. (1)

D. Gordon and R. Gordon, “Component-averaged row projections: a robust block-parallel scheme for sparse linear systems,” SIAM J. Sci. Comput. 27, 1092-1117 (2005).
[CrossRef]

Other (5)

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

T. Allen, “Particle Size Measurement,” in Powder Sampling and Particle Size Measurement, 5th ed. (Chapman & Hall, 1997), Vol. 1.

Inverse Scattering Problems in Optics, H. P. Baltes, ed. (Springer-Verlag, 1980).
[CrossRef]

L. P. Bayvel and A. R. Jones, Electromagnetic Scattering and its Applications (Applied Science, 1981).

J. Swithenbank, J. M. Beer, D. S. Tayor, and G. C. McCreath, “A Laser Diagnostic Technique for the Measurement of Droplet and Particle Size Distribution,” presented at 14th Aerospace Sciences Meeting, Washington, D.C., 26-28 January 1976 (American Institute of Aeronautics and Astronautics), paper 76-79.

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Figures (13)

Fig. 1
Fig. 1

Schematic of the laser particle analyzer.

Fig. 2
Fig. 2

PSDs of a monodispersion obtained with different versions of the projection method, in which the measurement errors are within the scale of ± 5 % and only the nonnegativity constraint is used. (a)  M = 32 , N = 32 ; (b)  M = 32 , N = 64 .

Fig. 3
Fig. 3

Inverse procedure of the projection method in the case M = 32 , N = 32 , corresponding to Fig. 2a. Points A and B denote the best fit between the retrieved PSD and the input one. (a) Residual R ( p ) . (b) Fit of the inversion result to the input PSD Φ ( p ) .

Fig. 4
Fig. 4

Inverse procedure of the projection method in the case M = 32 , N = 64 , corresponding to Fig. 2b. Points A and B denote the best fit between the retrieved PSD and the input one. (a) Residual R ( p ) . (b) Fit of the inversion result to the input PSD Φ ( p ) .

Fig. 5
Fig. 5

PSDs of the mono-dispersion obtained with different versions of the projection method, in which the nonnegativity constraint is used and the stopping criterion p is chosen according to Figs. 3, 4. The measurement errors are within the scale of ± 5 % . Compared with Fig. 2, these results are much smoother. (a)  M = 32 , N = 32 ; (b)  M = 32 , N = 64 .

Fig. 6
Fig. 6

PSDs of a monodispersion obtained with the projection method, combined with the constraints of nonnegativity and smoothness. The measurement errors are within the scale of ± 5 % . (a) Number of iterations is 1000; (b) the selected iteration number ( p 105 ). (In Figs. 6, 7, 8, 10, 11, the number of measurement points M is 32 and the number of particle size fractions N is 64.)

Fig. 7
Fig. 7

Inversion procedure of the projection method, corresponding to Fig. 6. (a) Residual R ( p ) . (b) Fit of the inversion result to the input PSD Φ ( p ) .

Fig. 8
Fig. 8

Inversion procedure of the projection method for the same monodispersion plotted in Fig. 6. The constraints of nonnegativity and smoothness are used. The measurement errors are within the scale of ± 10 % and ± 20 % . (a), (b) Residual R ( p ) . (c), (d) Fit of the inversion result to the input PSD Φ ( p ) .

Fig. 9
Fig. 9

Geometrical explanation of the projection between two hyperplanes.

Fig. 10
Fig. 10

Bimodal PSDs obtained with different versions of the projection method, combined with the constraints of nonnegativity and smoothness. The measurement errors are within the scales ± 5 % and ± 20 % . The number of iteration steps p is 120 or 2000.

Fig. 11
Fig. 11

Trimodal PSDs obtained with different versions of the projection method, combined with the constraints of nonnegativity and smoothness. The measurement errors are within the scales ± 5 % and ± 20 % . The number of iteration steps p is 120 or 2000.

Fig. 12
Fig. 12

PSDs obtained with different versions of the projection method from experiments, in which the constraints of nonnegativity and smoothness are used and the number of iteration steps p is 120. The PSDs obtained with the modified Chahine iterations are also plotted for comparison.

Fig. 13
Fig. 13

Comparison of the PSDs for the measurement on latex standard 5 μm . A different smoothness constraint is imposed in the Kaczmarz–random method, and the number of iteration steps p is 120.

Equations (14)

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

0 K i ( x ) q ( x ) d x = E i ( i = 1 , 2 , , M ) ,
j = 1 N K i , j X j = E i ( i = 1 , 2 , , M ) ,
E i = π θ i , in θ i , out I sca ( m , α , θ ) f 2 sin θ d θ ( i = 1 , 2 , , M ) ,
E i = x min x max { π θ i , in θ i , out I sca ( m , α , θ ) f 2 sin θ d θ } q 0 ( x ) d x ,
E i = C x min x max { π θ i , in θ i , out I sca ( m , α , θ ) f 2 sin θ d θ } q k ( x ) x k d x .
K i , j = x ¯ j 2 π θ i , in θ i , out I sca ( m , α ¯ j , θ ) f 2 sin θ d θ , X j = q 2 ( x ¯ j ) Δ x j .
X j ( i , p ) P ^ { X j ( i 1 , p ) l = 1 N K i , l X l ( i 1 , p ) E i l = 1 N K i , l K i , l K i , j } ( i = 1 , 2 , , M j = 1 , 2 , , N ) ,
P ^ { u } = { u if     u > 0 0 else .
X j ( p + 1 ) P ^ { X j ( p ) 1 M i = 1 M ( l = 1 N K i , l X l ( p ) E i l = 1 N K i , l K i , l K i , j ) } ( j = 1 , 2 , , N ) .
R ( p ) = { 1 M i = 1 M ( j = 1 N K i , j X j ( p ) E i ) 2 } 1 / 2 .
X j ( i , p ) P ^ { X j ( i 1 , p ) 1 p β l = 1 N K i , l X l E i l = 1 N K i , l K i , l K i , j } ( j = 1 , 2 , , N ) ,
Φ ( p ) = { 1 N j = 1 N ( X j ( p ) l = 1 N X l ( p ) X j ( input ) l = 1 N X l ( input ) ) 2 } 1 / 2 .
X 1 ( p ) ω X 1 ( p ) + 2 X 2 ( p ) 2 + ω , X j ( p ) X j 1 ( p ) + ω X j ( p ) + X j + 1 ( p ) 2 + ω , X N ( p ) 2 X N 1 ( p ) + ω X N ( p ) 2 + ω ( j = 2 , 3 , , N 1 ) .
cos ϕ l , m = j = 1 N K l , j K m , j , l , m ( 1 , 2 , , M ) .

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