Abstract

A new approach suitable for solving inverse problems in multiangle light scattering is presented. The method takes advantage of multidimensional function approximation capability of radial basis function neural networks. An algorithm for training the networks is described in detail. It is shown that the radius and refractive index of homogeneous spheres can be recovered accurately and quickly, with maximum relative errors of the order of 10-3 and mean errors as low as 10-5. The influence of the angular range of available scattering data on the loss of information and inversion accuracy is investigated, and it is shown that more than two thirds of input data can be removed before substantial degradation of accuracy occurs.

© 1998 Optical Society of America

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References

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1997 (2)

1996 (1)

I. K. Ludlow, J. Everitt, “Systematic behavior of the Mie scattering coefficients of spheres as a function of order,” Phys. Rev. E 53, 2909–2924 (1996).
[CrossRef]

1995 (2)

R. Naimimohasses, D. M. Barnett, D. A. Green, P. R. Smith, “Sensor optimization using neural-network sensitivity measures,” Meas. Sci. Technol. 6, 1291–1300 (1995).
[CrossRef]

I. K. Ludlow, J. Everitt, “Application of Gegenbauer analysis to light scattering from spheres: theory,” Phys. Rev. E 51, 2516–2526 (1995).
[CrossRef]

1994 (1)

1993 (1)

L. A. de Pieri, I. K. Ludlow, W. M. Waites, “The application of laser diffractometry to study the water content of spores of Bacillus sphaericus with different heat resistances,” J. Appl. Bacteriol. 74, 578–582 (1993).
[PubMed]

1991 (1)

J. Everitt, I. K. Ludlow, “Particle sizing using methods of discrete Legendre analysis,” Biochem. Soc. Trans. 19, 504–505 (1991).
[PubMed]

1990 (1)

1989 (1)

Z. Ulanowski, I. K. Ludlow, “Water distribution, size and wall thickness in Lycoperdon pyriforme spores,” Mycol. Res. 93, 28–32 (1989).
[CrossRef]

1987 (1)

Z. Ulanowski, I. K. Ludlow, W. M. Waites, “Water content and size of bacterial spore components determined from laser diffractometry,” FEMS Microbiol. Lett. 40, 229–232 (1987).
[CrossRef]

1985 (1)

1980 (1)

Barber, P. W.

P. W. Barber, S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, Singapore, 1990).

Barnett, D. M.

R. Naimimohasses, D. M. Barnett, D. A. Green, P. R. Smith, “Sensor optimization using neural-network sensitivity measures,” Meas. Sci. Technol. 6, 1291–1300 (1995).
[CrossRef]

Bartholomew-Biggs, M. C.

M. C. Bartholomew-Biggs, Z. Ulanowski, S. Zakovic, “Further experience with least squares solutions of an inverse light-scattering problem,” Technical Rep. 294 (Numerical Optimization Centre, University of Hertfordshire, UK, 1994).

S. Zakovic, Z. Ulanowski, M. C. Bartholomew-Biggs, “Application of global optimisation to particle identification using light scattering,” Technical Rep. 325 (Numerical Optimization Centre, University of Hertfordshire, UK, 1997).

M. C. Bartholomew-Biggs, Z. Ulanowski, S. Zakovic, “A parameter estimation problem with multiple solutions arising in laser diffractometry,” Technical Rep. 281 (Numerical Optimization Centre, University of Hertfordshire, UK, 1994).

Bayvel, L. P.

L. P. Bayvel, A. R. Jones, Electromagnetic Scattering and Its Applications (Applied Science, London, 1981).
[CrossRef]

Bohren, C. F.

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

Brewster, M. Q.

Curry, B. P.

de Pieri, L. A.

L. A. de Pieri, I. K. Ludlow, W. M. Waites, “The application of laser diffractometry to study the water content of spores of Bacillus sphaericus with different heat resistances,” J. Appl. Bacteriol. 74, 578–582 (1993).
[PubMed]

Everitt, J.

I. K. Ludlow, J. Everitt, “Systematic behavior of the Mie scattering coefficients of spheres as a function of order,” Phys. Rev. E 53, 2909–2924 (1996).
[CrossRef]

I. K. Ludlow, J. Everitt, “Application of Gegenbauer analysis to light scattering from spheres: theory,” Phys. Rev. E 51, 2516–2526 (1995).
[CrossRef]

J. Everitt, I. K. Ludlow, “Particle sizing using methods of discrete Legendre analysis,” Biochem. Soc. Trans. 19, 504–505 (1991).
[PubMed]

Green, D. A.

R. Naimimohasses, D. M. Barnett, D. A. Green, P. R. Smith, “Sensor optimization using neural-network sensitivity measures,” Meas. Sci. Technol. 6, 1291–1300 (1995).
[CrossRef]

Hill, S. C.

P. W. Barber, S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, Singapore, 1990).

Hirst, E.

Huffman, D. R.

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

Ishimaru, A.

Jones, A. R.

L. P. Bayvel, A. R. Jones, Electromagnetic Scattering and Its Applications (Applied Science, London, 1981).
[CrossRef]

Jones, M. R.

Kaye, P. H.

Kitamura, S.

Lam, C. M.

Leong, K. H.

Lopatin, V. N.

Ludlow, I. K.

I. K. Ludlow, J. Everitt, “Systematic behavior of the Mie scattering coefficients of spheres as a function of order,” Phys. Rev. E 53, 2909–2924 (1996).
[CrossRef]

I. K. Ludlow, J. Everitt, “Application of Gegenbauer analysis to light scattering from spheres: theory,” Phys. Rev. E 51, 2516–2526 (1995).
[CrossRef]

L. A. de Pieri, I. K. Ludlow, W. M. Waites, “The application of laser diffractometry to study the water content of spores of Bacillus sphaericus with different heat resistances,” J. Appl. Bacteriol. 74, 578–582 (1993).
[PubMed]

J. Everitt, I. K. Ludlow, “Particle sizing using methods of discrete Legendre analysis,” Biochem. Soc. Trans. 19, 504–505 (1991).
[PubMed]

Z. Ulanowski, I. K. Ludlow, “Water distribution, size and wall thickness in Lycoperdon pyriforme spores,” Mycol. Res. 93, 28–32 (1989).
[CrossRef]

Z. Ulanowski, I. K. Ludlow, W. M. Waites, “Water content and size of bacterial spore components determined from laser diffractometry,” FEMS Microbiol. Lett. 40, 229–232 (1987).
[CrossRef]

Maltsev, V. P.

Marks, R. J.

Naimimohasses, R.

R. Naimimohasses, D. M. Barnett, D. A. Green, P. R. Smith, “Sensor optimization using neural-network sensitivity measures,” Meas. Sci. Technol. 6, 1291–1300 (1995).
[CrossRef]

Park, D. C.

Quist, G. M.

Smith, P. R.

R. Naimimohasses, D. M. Barnett, D. A. Green, P. R. Smith, “Sensor optimization using neural-network sensitivity measures,” Meas. Sci. Technol. 6, 1291–1300 (1995).
[CrossRef]

Tsang, L.

Ulanowski, Z.

Z. Ulanowski, I. K. Ludlow, “Water distribution, size and wall thickness in Lycoperdon pyriforme spores,” Mycol. Res. 93, 28–32 (1989).
[CrossRef]

Z. Ulanowski, I. K. Ludlow, W. M. Waites, “Water content and size of bacterial spore components determined from laser diffractometry,” FEMS Microbiol. Lett. 40, 229–232 (1987).
[CrossRef]

M. C. Bartholomew-Biggs, Z. Ulanowski, S. Zakovic, “Further experience with least squares solutions of an inverse light-scattering problem,” Technical Rep. 294 (Numerical Optimization Centre, University of Hertfordshire, UK, 1994).

Z. Ulanowski, “Investigations of microbial physiology and cell structure using laser diffractometry,” Ph.D. dissertation (Hatfield Polytechnic, Hatfield, UK, 1988).

M. C. Bartholomew-Biggs, Z. Ulanowski, S. Zakovic, “A parameter estimation problem with multiple solutions arising in laser diffractometry,” Technical Rep. 281 (Numerical Optimization Centre, University of Hertfordshire, UK, 1994).

S. Zakovic, Z. Ulanowski, M. C. Bartholomew-Biggs, “Application of global optimisation to particle identification using light scattering,” Technical Rep. 325 (Numerical Optimization Centre, University of Hertfordshire, UK, 1997).

Waites, W. M.

L. A. de Pieri, I. K. Ludlow, W. M. Waites, “The application of laser diffractometry to study the water content of spores of Bacillus sphaericus with different heat resistances,” J. Appl. Bacteriol. 74, 578–582 (1993).
[PubMed]

Z. Ulanowski, I. K. Ludlow, W. M. Waites, “Water content and size of bacterial spore components determined from laser diffractometry,” FEMS Microbiol. Lett. 40, 229–232 (1987).
[CrossRef]

Wang, Z.

Wyatt, P. J.

Zakovic, S.

S. Zakovic, Z. Ulanowski, M. C. Bartholomew-Biggs, “Application of global optimisation to particle identification using light scattering,” Technical Rep. 325 (Numerical Optimization Centre, University of Hertfordshire, UK, 1997).

M. C. Bartholomew-Biggs, Z. Ulanowski, S. Zakovic, “A parameter estimation problem with multiple solutions arising in laser diffractometry,” Technical Rep. 281 (Numerical Optimization Centre, University of Hertfordshire, UK, 1994).

M. C. Bartholomew-Biggs, Z. Ulanowski, S. Zakovic, “Further experience with least squares solutions of an inverse light-scattering problem,” Technical Rep. 294 (Numerical Optimization Centre, University of Hertfordshire, UK, 1994).

Appl. Opt. (4)

Biochem. Soc. Trans. (1)

J. Everitt, I. K. Ludlow, “Particle sizing using methods of discrete Legendre analysis,” Biochem. Soc. Trans. 19, 504–505 (1991).
[PubMed]

FEMS Microbiol. Lett. (1)

Z. Ulanowski, I. K. Ludlow, W. M. Waites, “Water content and size of bacterial spore components determined from laser diffractometry,” FEMS Microbiol. Lett. 40, 229–232 (1987).
[CrossRef]

J. Appl. Bacteriol. (1)

L. A. de Pieri, I. K. Ludlow, W. M. Waites, “The application of laser diffractometry to study the water content of spores of Bacillus sphaericus with different heat resistances,” J. Appl. Bacteriol. 74, 578–582 (1993).
[PubMed]

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

Meas. Sci. Technol. (1)

R. Naimimohasses, D. M. Barnett, D. A. Green, P. R. Smith, “Sensor optimization using neural-network sensitivity measures,” Meas. Sci. Technol. 6, 1291–1300 (1995).
[CrossRef]

Mycol. Res. (1)

Z. Ulanowski, I. K. Ludlow, “Water distribution, size and wall thickness in Lycoperdon pyriforme spores,” Mycol. Res. 93, 28–32 (1989).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. E (2)

I. K. Ludlow, J. Everitt, “Application of Gegenbauer analysis to light scattering from spheres: theory,” Phys. Rev. E 51, 2516–2526 (1995).
[CrossRef]

I. K. Ludlow, J. Everitt, “Systematic behavior of the Mie scattering coefficients of spheres as a function of order,” Phys. Rev. E 53, 2909–2924 (1996).
[CrossRef]

Other (8)

M. C. Bartholomew-Biggs, Z. Ulanowski, S. Zakovic, “A parameter estimation problem with multiple solutions arising in laser diffractometry,” Technical Rep. 281 (Numerical Optimization Centre, University of Hertfordshire, UK, 1994).

M. C. Bartholomew-Biggs, Z. Ulanowski, S. Zakovic, “Further experience with least squares solutions of an inverse light-scattering problem,” Technical Rep. 294 (Numerical Optimization Centre, University of Hertfordshire, UK, 1994).

S. Zakovic, Z. Ulanowski, M. C. Bartholomew-Biggs, “Application of global optimisation to particle identification using light scattering,” Technical Rep. 325 (Numerical Optimization Centre, University of Hertfordshire, UK, 1997).

Z. Ulanowski, “Investigations of microbial physiology and cell structure using laser diffractometry,” Ph.D. dissertation (Hatfield Polytechnic, Hatfield, UK, 1988).

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

L. P. Bayvel, A. R. Jones, Electromagnetic Scattering and Its Applications (Applied Science, London, 1981).
[CrossRef]

P. W. Barber, S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, Singapore, 1990).

MathWorks, Inc., 24 Prime Park Way, Natick, Mass. 01760.

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Figures (4)

Fig. 1
Fig. 1

Architecture of RBF networks.

Fig. 2
Fig. 2

Approximation errors for the global network shown as a function of the output parameters (radius r and refractive index n). Absolute errors are represented as lines joining the test point and the corresponding approximation (the line length represents the magnitude of the error). The training data points are shown as small rectangles and the boundaries of the ten overlapping local networks as vertical lines. The input vectors were scaled linearly and the weighting function used was g(θ) = (sin θ)4. The value of the width constant d was 1.4.

Fig. 3
Fig. 3

Approximation errors for the local networks shown as a function of the output parameters r (radius) and n (refractive index). Results from ten local networks were combined and relative errors calculated separately for (a) r and (b) n. The input vectors were scaled linearly, the weighting function used was g(θ) = (sin θ)4, and the width constant d was 1.

Fig. 4
Fig. 4

Approximation errors for the local networks trained and tested with incomplete (truncated) data. The errors are shown as a function of the truncation angle. The data are between (a) the forward scattering truncation angle and 180° (b) 0° and the backscattering truncation angle, (c) the forward scattering truncation angle and 120°. Mean (open symbols) and maximum relative errors (filled symbols) for the radius (r) and the refractive index (n) are given. The input vectors were scaled linearly, the weighting function used was g(θ) = (sin θ)4, and the width constant d was 1.

Tables (2)

Tables Icon

Table 1 Relative Approximation Errors for Local Networks by Use of Five Data Scaling Schemesa

Tables Icon

Table 2 Computation Time for Global and Local RBF Networks

Equations (13)

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

1 N i = 1 N I θ i - cM θ i 2 ,
I θ = F θ ,   r ,   n ,
y = f ¯ x = i = 1 N w i ω i x ,   c i + b ,
ω i = exp - x - c i 2 d i 2 ,
y j = i = 1 N w i exp - x j - c i 2 d i 2 + b ,   j = 1 ,   2 , ,   S ,
y 1 y 2     y S = w 1 w 2     w N Φ + ba ,
Φ = ϕ 11 ϕ 12 ϕ 1 S ϕ 21 ϕ 22 ϕ 2 S ϕ N 1 ϕ N 2 ϕ NS ,   ϕ ij = exp - x j - c i 2 d i 2 .
Y = W b Φ a ,
Y = y 1 y 2     y S ,   W b = w 1 w 2     w N b ,   Φ a = Φ a .
W b   =   Y Φ a T Φ a   Φ a T - 1 .
d i = d max j i c i - c j - min j i c i - c j ,   j = 1 ,   2 , ,   S .
x θ = 1 - log | S 1 0 | 2 + log | S 1 θ | 2 ,
x θ = g θ | S 1 θ | 2 ,

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