Abstract

In optical diffraction particle sizing a numerical transform is sought so that a particle size distribution can be determined from angular measurements of near forward scattering. We consider the nonuniqueness and instability of this transform for discrete data. Our arguments are based on the approximation of the kernel by a function to which it is asymptotic. The results, which include an angular sampling criterion and a rescaling of the forward transform, are applied to choosing and developing algorithms for inverting experimental measurements of scattering. Measurements of scattering from distributions of polystyrene spheres are successfully inverted.

© 1991 Optical Society of America

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  1. I. N. McCave, R. J. Bryant, H. F. Cook, C. A. Coughanowr, “Evaluation of a laser-diffraction-size analyzer for use with natural sediments,” J. Sediment. Petrol. 56, 561–564 (1986).
  2. J. B. Riley, Y. C. Agrawal, “Optical particle sizing for hydrodynamics: further results,” in Ocean Optics VIII, M. A. Blizzard, ed. Proc. Soc. Photo-Opt. Instrum. Eng.637, 164– 170 (1986).
  3. J. B. Riley. “Laser diffraction particle sizing: sampling and inversion,” Ph.D. dissertation (Joint Program in Oceanographic Engineering, Massachusetts Institute of Technology and the Woods Hole Oceanographic Institution, September1987).
    [CrossRef]
  4. J. H. Chin, C. M. Sliepcevich, M. Tribus, “Particle size distributions from angular variation of forward-scattered light at very small angles,” J. Phys. Chem. 59, 841–844 (1955).
    [CrossRef]
  5. J. H. Chin, C. M. Sliepcevich, M. Tribus; “Determination of particle size distributions by means of measurements of angular variation of intensity of forward-scattered light at very small angles,” J. Phys. Chem. 59, 845–848 (1955).
    [CrossRef]
  6. A. L. Fymat, K. D. Mease, “Reconstructuring the size distribution of spherical particles from angular forward scattering data,” in Remote Sensing of the Atmosphere: Inversion Methods and Applications, A. L. Fymat, V. E. Zuev, eds. (Elsevier, Amsterdam, 1978), pp. 195–231.
  7. J. H. Koo, E. D. Hirleman, “Comparitive study of laser diffraction analysis using integral transform techniques: factors affecting the reconstruction of droplet size distributions,” in Proceedings of 1986 Joint Spring Meeting of Canadian and Western Sections (Combustion Institute, Pittsburgh, Pa., 1986), paper WSSCI-86-18.
  8. K. S. Shifrin, “The essential range of scattering angles in measuring particle-size distribution by the small-angle method,” Izv. Atmos. Oceanic Phys. 2, 559–561 (1966).
  9. K. S. Shifrin, I. B. Kolmakov, “Effect of limitation of the range of measurement of the indicatrix on the accuracy of the small-angle method,” Izv. Atmos. Oceanic Phys. 2, 514–518 (1966).
  10. A. N. Tikhononv, V. Y. Arsenin, Solutions of Ill-Posed Problems (Wiley, New York, 1977).
  11. E. D. Hirleman, “Optimal scaling of the inverse Fraunhoffer diffraction particle sizing problem: the linear system produced by quadrature,” J. Part. Charact. 4, 128–133 (1987).
    [CrossRef]
  12. C. M. Bender, S. A. Orzag. Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, New York, 1978).
  13. A. J. Jerri, “The Shannon sampling theorem—its various extensions and applications: a tutorial review,” Proc. IEEE 65, 1565–1596 (1977).
    [CrossRef]
  14. D. Slepian, H.O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—I,” Bell Syst. Tech. J. 40, 43–63 (1961).
  15. G. A. Viano, “On the extrapolation of optical image data,” J. Math. Phys. 17, 1160–1165 (1976).
    [CrossRef]
  16. M. S. Sabri, W. Steenhaart, “An approach to band-limited extrapolation: the extrapolation matrix,” IEEE Trans. Circuits Syst. CAS-25, 74–78 (1978).
    [CrossRef]
  17. J. Maeda, K. Murata, “Restoration of band-limited images by an iterative regularized pseudoinverse method,” J. Opt. Soc. Am. A 1, 28–34 (1984).
    [CrossRef]
  18. A Papoulis, “A new algorithm in spectral analysis and band-limited extrapolation,” IEEE Trans Circuits Syst. CAS-22, 735–742 (1975).
    [CrossRef]
  19. D. C. Youla, “Generalized image restoration by the method of alternating orthogonal projections,” IEEE Trans. Circuits Syst. CAS-25, 694–702 (1978).
    [CrossRef]
  20. D. W. Oldenberg, “Calculation of Fourier transforms by the Backus-Gilbert method,” Geophys. J. R. Astron. Soc. 44, 413–431 (1976).
    [CrossRef]
  21. G. M. Jenkins, D. G. Watts, Spectral Analysis and Its Applications (Holden-Day, Oakland, Calif., 1968).
  22. F. Hagin, “On the construction of well-conditioned systems for Fredholm I problems by mesh adapting,” J. Comput. Phys. 36, 154–169 (1980).
    [CrossRef]
  23. A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).
  24. A. V. Oppenheim, A. S. Wilskey, I. T. Young, Signals and Systems (Prentice-Hall, Englewood Cliffs, N.J., 1983).
  25. J. V. Dave, “Effect of coarseness of the integration increment on the calculation of the radiation scattered by polydispersed aerosols,” Appl. Opt. 8, 1161–1167 (1969).
    [CrossRef] [PubMed]
  26. M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1970).
  27. A. L. Fymat, K. D. Mease, “Mie forward scattering: improved semiempirical approximation with application to particle size distribution inversion,” Appl. Opt. 20, 194–198 (1981).
    [CrossRef] [PubMed]
  28. L. P. Bayvel, A. R. Jones, Electromagnetic Scattering and its Applications (Applied Science Publishers, London, 1981).
    [CrossRef]
  29. D. D. Jackson, “Interpretation of inaccurate, insufficient and inconsistent data,” Geophys. J. R. Astron. Soc. 28, 97–109 (1972).
    [CrossRef]
  30. C. Lanczos, Linear Differential Operators (Van Nostrand, London, 1961).
  31. R. A. Wiggins, “The general linear inverse problem: implication of surface waves and free oscillations for earth structure,” Rev. Geophys. Space Phys. 10, 251–283 (1972).
    [CrossRef]
  32. J. N. Franklin, “Well-posed stochastic extensions of ill-posed linear problems,” J. Math. Anal. Appl. 31, 682–716 (1970).
    [CrossRef]
  33. K. Aki, P. G. Richards, “Quantitative Seismology, Theory and Methods (Freeman, San Francisco, Calif., 1980).
  34. H. Bader, “The hyperbolic distribution of particle sizes,” J. Geophys. Res. 75, 2822–2830 (1970).
    [CrossRef]
  35. L. C. Chow, C. L. Tien, “Inversion techniques for determining the droplet size distribution in clouds: numerical examination,” Appl. Opt. 15, 378–383 (1976).
    [CrossRef] [PubMed]
  36. R. W. Schafer, R. M. Mersereau, M. A. Richards, “Constrained iterative restoration algorithms,” Proc. IEEE 69, 432–450 (1981).
    [CrossRef]
  37. S. Twomey, Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurements (Elsevier, Amsterdam, 1977).
  38. T. S. Huang, D. A. Barker, S. P. Berger, “Iterative image restoration,” Appl. Opt. 14, 1165–1168 (1975).
    [CrossRef] [PubMed]
  39. M. Z. Hansen, “Atmospheric particulate analysis using angular light scattering,” Appl. Opt. 19, 3441–3448 (1980).
    [CrossRef] [PubMed]
  40. C. L. Lawson, D. J. Hanson, Solving Least Squares Problems (Prentice-Hall, Englewood Cliffs, N.J., 1974).
  41. W. Menke, Geophysical Data Analysis: Discrete Inverse Theory (Academic, New York, 1984).
  42. A. L. Fymat, “Analytical inversions in remote sensing of particle size distributions. 2: Angular and spectral scattering in diffraction approximations,” Appl. Opt. 17, 1677–1678 (1978).
    [CrossRef] [PubMed]
  43. Y. C. Agrawal, J. B. Riley, “Optical particle sizing for hydrodynamics based on near forward scattering,” Ocean Optics VII, M. A. Blizzard, ed., Proc. Soc. Photo-Opt. Instrum. Eng.489, 68–76 (1984).
    [CrossRef]

1987 (1)

E. D. Hirleman, “Optimal scaling of the inverse Fraunhoffer diffraction particle sizing problem: the linear system produced by quadrature,” J. Part. Charact. 4, 128–133 (1987).
[CrossRef]

1986 (1)

I. N. McCave, R. J. Bryant, H. F. Cook, C. A. Coughanowr, “Evaluation of a laser-diffraction-size analyzer for use with natural sediments,” J. Sediment. Petrol. 56, 561–564 (1986).

1984 (1)

1981 (2)

1980 (2)

F. Hagin, “On the construction of well-conditioned systems for Fredholm I problems by mesh adapting,” J. Comput. Phys. 36, 154–169 (1980).
[CrossRef]

M. Z. Hansen, “Atmospheric particulate analysis using angular light scattering,” Appl. Opt. 19, 3441–3448 (1980).
[CrossRef] [PubMed]

1978 (3)

A. L. Fymat, “Analytical inversions in remote sensing of particle size distributions. 2: Angular and spectral scattering in diffraction approximations,” Appl. Opt. 17, 1677–1678 (1978).
[CrossRef] [PubMed]

D. C. Youla, “Generalized image restoration by the method of alternating orthogonal projections,” IEEE Trans. Circuits Syst. CAS-25, 694–702 (1978).
[CrossRef]

M. S. Sabri, W. Steenhaart, “An approach to band-limited extrapolation: the extrapolation matrix,” IEEE Trans. Circuits Syst. CAS-25, 74–78 (1978).
[CrossRef]

1977 (1)

A. J. Jerri, “The Shannon sampling theorem—its various extensions and applications: a tutorial review,” Proc. IEEE 65, 1565–1596 (1977).
[CrossRef]

1976 (3)

D. W. Oldenberg, “Calculation of Fourier transforms by the Backus-Gilbert method,” Geophys. J. R. Astron. Soc. 44, 413–431 (1976).
[CrossRef]

G. A. Viano, “On the extrapolation of optical image data,” J. Math. Phys. 17, 1160–1165 (1976).
[CrossRef]

L. C. Chow, C. L. Tien, “Inversion techniques for determining the droplet size distribution in clouds: numerical examination,” Appl. Opt. 15, 378–383 (1976).
[CrossRef] [PubMed]

1975 (2)

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

A Papoulis, “A new algorithm in spectral analysis and band-limited extrapolation,” IEEE Trans Circuits Syst. CAS-22, 735–742 (1975).
[CrossRef]

1972 (2)

D. D. Jackson, “Interpretation of inaccurate, insufficient and inconsistent data,” Geophys. J. R. Astron. Soc. 28, 97–109 (1972).
[CrossRef]

R. A. Wiggins, “The general linear inverse problem: implication of surface waves and free oscillations for earth structure,” Rev. Geophys. Space Phys. 10, 251–283 (1972).
[CrossRef]

1970 (2)

J. N. Franklin, “Well-posed stochastic extensions of ill-posed linear problems,” J. Math. Anal. Appl. 31, 682–716 (1970).
[CrossRef]

H. Bader, “The hyperbolic distribution of particle sizes,” J. Geophys. Res. 75, 2822–2830 (1970).
[CrossRef]

1969 (1)

1966 (2)

K. S. Shifrin, “The essential range of scattering angles in measuring particle-size distribution by the small-angle method,” Izv. Atmos. Oceanic Phys. 2, 559–561 (1966).

K. S. Shifrin, I. B. Kolmakov, “Effect of limitation of the range of measurement of the indicatrix on the accuracy of the small-angle method,” Izv. Atmos. Oceanic Phys. 2, 514–518 (1966).

1961 (1)

D. Slepian, H.O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—I,” Bell Syst. Tech. J. 40, 43–63 (1961).

1955 (2)

J. H. Chin, C. M. Sliepcevich, M. Tribus, “Particle size distributions from angular variation of forward-scattered light at very small angles,” J. Phys. Chem. 59, 841–844 (1955).
[CrossRef]

J. H. Chin, C. M. Sliepcevich, M. Tribus; “Determination of particle size distributions by means of measurements of angular variation of intensity of forward-scattered light at very small angles,” J. Phys. Chem. 59, 845–848 (1955).
[CrossRef]

Agrawal, Y. C.

Y. C. Agrawal, J. B. Riley, “Optical particle sizing for hydrodynamics based on near forward scattering,” Ocean Optics VII, M. A. Blizzard, ed., Proc. Soc. Photo-Opt. Instrum. Eng.489, 68–76 (1984).
[CrossRef]

J. B. Riley, Y. C. Agrawal, “Optical particle sizing for hydrodynamics: further results,” in Ocean Optics VIII, M. A. Blizzard, ed. Proc. Soc. Photo-Opt. Instrum. Eng.637, 164– 170 (1986).

Aki, K.

K. Aki, P. G. Richards, “Quantitative Seismology, Theory and Methods (Freeman, San Francisco, Calif., 1980).

Arsenin, V. Y.

A. N. Tikhononv, V. Y. Arsenin, Solutions of Ill-Posed Problems (Wiley, New York, 1977).

Bader, H.

H. Bader, “The hyperbolic distribution of particle sizes,” J. Geophys. Res. 75, 2822–2830 (1970).
[CrossRef]

Barker, D. A.

Bayvel, L. P.

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

Bender, C. M.

C. M. Bender, S. A. Orzag. Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, New York, 1978).

Berger, S. P.

Bryant, R. J.

I. N. McCave, R. J. Bryant, H. F. Cook, C. A. Coughanowr, “Evaluation of a laser-diffraction-size analyzer for use with natural sediments,” J. Sediment. Petrol. 56, 561–564 (1986).

Chin, J. H.

J. H. Chin, C. M. Sliepcevich, M. Tribus; “Determination of particle size distributions by means of measurements of angular variation of intensity of forward-scattered light at very small angles,” J. Phys. Chem. 59, 845–848 (1955).
[CrossRef]

J. H. Chin, C. M. Sliepcevich, M. Tribus, “Particle size distributions from angular variation of forward-scattered light at very small angles,” J. Phys. Chem. 59, 841–844 (1955).
[CrossRef]

Chow, L. C.

Cook, H. F.

I. N. McCave, R. J. Bryant, H. F. Cook, C. A. Coughanowr, “Evaluation of a laser-diffraction-size analyzer for use with natural sediments,” J. Sediment. Petrol. 56, 561–564 (1986).

Coughanowr, C. A.

I. N. McCave, R. J. Bryant, H. F. Cook, C. A. Coughanowr, “Evaluation of a laser-diffraction-size analyzer for use with natural sediments,” J. Sediment. Petrol. 56, 561–564 (1986).

Dave, J. V.

Franklin, J. N.

J. N. Franklin, “Well-posed stochastic extensions of ill-posed linear problems,” J. Math. Anal. Appl. 31, 682–716 (1970).
[CrossRef]

Fymat, A. L.

A. L. Fymat, K. D. Mease, “Mie forward scattering: improved semiempirical approximation with application to particle size distribution inversion,” Appl. Opt. 20, 194–198 (1981).
[CrossRef] [PubMed]

A. L. Fymat, “Analytical inversions in remote sensing of particle size distributions. 2: Angular and spectral scattering in diffraction approximations,” Appl. Opt. 17, 1677–1678 (1978).
[CrossRef] [PubMed]

A. L. Fymat, K. D. Mease, “Reconstructuring the size distribution of spherical particles from angular forward scattering data,” in Remote Sensing of the Atmosphere: Inversion Methods and Applications, A. L. Fymat, V. E. Zuev, eds. (Elsevier, Amsterdam, 1978), pp. 195–231.

Hagin, F.

F. Hagin, “On the construction of well-conditioned systems for Fredholm I problems by mesh adapting,” J. Comput. Phys. 36, 154–169 (1980).
[CrossRef]

Hansen, M. Z.

Hanson, D. J.

C. L. Lawson, D. J. Hanson, Solving Least Squares Problems (Prentice-Hall, Englewood Cliffs, N.J., 1974).

Hirleman, E. D.

E. D. Hirleman, “Optimal scaling of the inverse Fraunhoffer diffraction particle sizing problem: the linear system produced by quadrature,” J. Part. Charact. 4, 128–133 (1987).
[CrossRef]

J. H. Koo, E. D. Hirleman, “Comparitive study of laser diffraction analysis using integral transform techniques: factors affecting the reconstruction of droplet size distributions,” in Proceedings of 1986 Joint Spring Meeting of Canadian and Western Sections (Combustion Institute, Pittsburgh, Pa., 1986), paper WSSCI-86-18.

Huang, T. S.

Jackson, D. D.

D. D. Jackson, “Interpretation of inaccurate, insufficient and inconsistent data,” Geophys. J. R. Astron. Soc. 28, 97–109 (1972).
[CrossRef]

Jenkins, G. M.

G. M. Jenkins, D. G. Watts, Spectral Analysis and Its Applications (Holden-Day, Oakland, Calif., 1968).

Jerri, A. J.

A. J. Jerri, “The Shannon sampling theorem—its various extensions and applications: a tutorial review,” Proc. IEEE 65, 1565–1596 (1977).
[CrossRef]

Jones, A. R.

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

Kolmakov, I. B.

K. S. Shifrin, I. B. Kolmakov, “Effect of limitation of the range of measurement of the indicatrix on the accuracy of the small-angle method,” Izv. Atmos. Oceanic Phys. 2, 514–518 (1966).

Koo, J. H.

J. H. Koo, E. D. Hirleman, “Comparitive study of laser diffraction analysis using integral transform techniques: factors affecting the reconstruction of droplet size distributions,” in Proceedings of 1986 Joint Spring Meeting of Canadian and Western Sections (Combustion Institute, Pittsburgh, Pa., 1986), paper WSSCI-86-18.

Lanczos, C.

C. Lanczos, Linear Differential Operators (Van Nostrand, London, 1961).

Lawson, C. L.

C. L. Lawson, D. J. Hanson, Solving Least Squares Problems (Prentice-Hall, Englewood Cliffs, N.J., 1974).

Maeda, J.

McCave, I. N.

I. N. McCave, R. J. Bryant, H. F. Cook, C. A. Coughanowr, “Evaluation of a laser-diffraction-size analyzer for use with natural sediments,” J. Sediment. Petrol. 56, 561–564 (1986).

Mease, K. D.

A. L. Fymat, K. D. Mease, “Mie forward scattering: improved semiempirical approximation with application to particle size distribution inversion,” Appl. Opt. 20, 194–198 (1981).
[CrossRef] [PubMed]

A. L. Fymat, K. D. Mease, “Reconstructuring the size distribution of spherical particles from angular forward scattering data,” in Remote Sensing of the Atmosphere: Inversion Methods and Applications, A. L. Fymat, V. E. Zuev, eds. (Elsevier, Amsterdam, 1978), pp. 195–231.

Menke, W.

W. Menke, Geophysical Data Analysis: Discrete Inverse Theory (Academic, New York, 1984).

Mersereau, R. M.

R. W. Schafer, R. M. Mersereau, M. A. Richards, “Constrained iterative restoration algorithms,” Proc. IEEE 69, 432–450 (1981).
[CrossRef]

Murata, K.

Oldenberg, D. W.

D. W. Oldenberg, “Calculation of Fourier transforms by the Backus-Gilbert method,” Geophys. J. R. Astron. Soc. 44, 413–431 (1976).
[CrossRef]

Oppenheim, A. V.

A. V. Oppenheim, A. S. Wilskey, I. T. Young, Signals and Systems (Prentice-Hall, Englewood Cliffs, N.J., 1983).

A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

Orzag, S. A.

C. M. Bender, S. A. Orzag. Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, New York, 1978).

Papoulis, A

A Papoulis, “A new algorithm in spectral analysis and band-limited extrapolation,” IEEE Trans Circuits Syst. CAS-22, 735–742 (1975).
[CrossRef]

Pollak, H.O.

D. Slepian, H.O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—I,” Bell Syst. Tech. J. 40, 43–63 (1961).

Richards, M. A.

R. W. Schafer, R. M. Mersereau, M. A. Richards, “Constrained iterative restoration algorithms,” Proc. IEEE 69, 432–450 (1981).
[CrossRef]

Richards, P. G.

K. Aki, P. G. Richards, “Quantitative Seismology, Theory and Methods (Freeman, San Francisco, Calif., 1980).

Riley, J. B.

J. B. Riley. “Laser diffraction particle sizing: sampling and inversion,” Ph.D. dissertation (Joint Program in Oceanographic Engineering, Massachusetts Institute of Technology and the Woods Hole Oceanographic Institution, September1987).
[CrossRef]

J. B. Riley, Y. C. Agrawal, “Optical particle sizing for hydrodynamics: further results,” in Ocean Optics VIII, M. A. Blizzard, ed. Proc. Soc. Photo-Opt. Instrum. Eng.637, 164– 170 (1986).

Y. C. Agrawal, J. B. Riley, “Optical particle sizing for hydrodynamics based on near forward scattering,” Ocean Optics VII, M. A. Blizzard, ed., Proc. Soc. Photo-Opt. Instrum. Eng.489, 68–76 (1984).
[CrossRef]

Sabri, M. S.

M. S. Sabri, W. Steenhaart, “An approach to band-limited extrapolation: the extrapolation matrix,” IEEE Trans. Circuits Syst. CAS-25, 74–78 (1978).
[CrossRef]

Schafer, R. W.

R. W. Schafer, R. M. Mersereau, M. A. Richards, “Constrained iterative restoration algorithms,” Proc. IEEE 69, 432–450 (1981).
[CrossRef]

A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

Shifrin, K. S.

K. S. Shifrin, “The essential range of scattering angles in measuring particle-size distribution by the small-angle method,” Izv. Atmos. Oceanic Phys. 2, 559–561 (1966).

K. S. Shifrin, I. B. Kolmakov, “Effect of limitation of the range of measurement of the indicatrix on the accuracy of the small-angle method,” Izv. Atmos. Oceanic Phys. 2, 514–518 (1966).

Slepian, D.

D. Slepian, H.O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—I,” Bell Syst. Tech. J. 40, 43–63 (1961).

Sliepcevich, C. M.

J. H. Chin, C. M. Sliepcevich, M. Tribus, “Particle size distributions from angular variation of forward-scattered light at very small angles,” J. Phys. Chem. 59, 841–844 (1955).
[CrossRef]

J. H. Chin, C. M. Sliepcevich, M. Tribus; “Determination of particle size distributions by means of measurements of angular variation of intensity of forward-scattered light at very small angles,” J. Phys. Chem. 59, 845–848 (1955).
[CrossRef]

Steenhaart, W.

M. S. Sabri, W. Steenhaart, “An approach to band-limited extrapolation: the extrapolation matrix,” IEEE Trans. Circuits Syst. CAS-25, 74–78 (1978).
[CrossRef]

Tien, C. L.

Tikhononv, A. N.

A. N. Tikhononv, V. Y. Arsenin, Solutions of Ill-Posed Problems (Wiley, New York, 1977).

Tribus, M.

J. H. Chin, C. M. Sliepcevich, M. Tribus, “Particle size distributions from angular variation of forward-scattered light at very small angles,” J. Phys. Chem. 59, 841–844 (1955).
[CrossRef]

J. H. Chin, C. M. Sliepcevich, M. Tribus; “Determination of particle size distributions by means of measurements of angular variation of intensity of forward-scattered light at very small angles,” J. Phys. Chem. 59, 845–848 (1955).
[CrossRef]

Twomey, S.

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

Viano, G. A.

G. A. Viano, “On the extrapolation of optical image data,” J. Math. Phys. 17, 1160–1165 (1976).
[CrossRef]

Watts, D. G.

G. M. Jenkins, D. G. Watts, Spectral Analysis and Its Applications (Holden-Day, Oakland, Calif., 1968).

Wiggins, R. A.

R. A. Wiggins, “The general linear inverse problem: implication of surface waves and free oscillations for earth structure,” Rev. Geophys. Space Phys. 10, 251–283 (1972).
[CrossRef]

Wilskey, A. S.

A. V. Oppenheim, A. S. Wilskey, I. T. Young, Signals and Systems (Prentice-Hall, Englewood Cliffs, N.J., 1983).

Youla, D. C.

D. C. Youla, “Generalized image restoration by the method of alternating orthogonal projections,” IEEE Trans. Circuits Syst. CAS-25, 694–702 (1978).
[CrossRef]

Young, I. T.

A. V. Oppenheim, A. S. Wilskey, I. T. Young, Signals and Systems (Prentice-Hall, Englewood Cliffs, N.J., 1983).

Appl. Opt. (6)

Bell Syst. Tech. J. (1)

D. Slepian, H.O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—I,” Bell Syst. Tech. J. 40, 43–63 (1961).

Geophys. J. R. Astron. Soc. (2)

D. W. Oldenberg, “Calculation of Fourier transforms by the Backus-Gilbert method,” Geophys. J. R. Astron. Soc. 44, 413–431 (1976).
[CrossRef]

D. D. Jackson, “Interpretation of inaccurate, insufficient and inconsistent data,” Geophys. J. R. Astron. Soc. 28, 97–109 (1972).
[CrossRef]

IEEE Trans Circuits Syst. (1)

A Papoulis, “A new algorithm in spectral analysis and band-limited extrapolation,” IEEE Trans Circuits Syst. CAS-22, 735–742 (1975).
[CrossRef]

IEEE Trans. Circuits Syst. (2)

D. C. Youla, “Generalized image restoration by the method of alternating orthogonal projections,” IEEE Trans. Circuits Syst. CAS-25, 694–702 (1978).
[CrossRef]

M. S. Sabri, W. Steenhaart, “An approach to band-limited extrapolation: the extrapolation matrix,” IEEE Trans. Circuits Syst. CAS-25, 74–78 (1978).
[CrossRef]

Izv. Atmos. Oceanic Phys. (2)

K. S. Shifrin, “The essential range of scattering angles in measuring particle-size distribution by the small-angle method,” Izv. Atmos. Oceanic Phys. 2, 559–561 (1966).

K. S. Shifrin, I. B. Kolmakov, “Effect of limitation of the range of measurement of the indicatrix on the accuracy of the small-angle method,” Izv. Atmos. Oceanic Phys. 2, 514–518 (1966).

J. Comput. Phys. (1)

F. Hagin, “On the construction of well-conditioned systems for Fredholm I problems by mesh adapting,” J. Comput. Phys. 36, 154–169 (1980).
[CrossRef]

J. Geophys. Res. (1)

H. Bader, “The hyperbolic distribution of particle sizes,” J. Geophys. Res. 75, 2822–2830 (1970).
[CrossRef]

J. Math. Anal. Appl. (1)

J. N. Franklin, “Well-posed stochastic extensions of ill-posed linear problems,” J. Math. Anal. Appl. 31, 682–716 (1970).
[CrossRef]

J. Math. Phys. (1)

G. A. Viano, “On the extrapolation of optical image data,” J. Math. Phys. 17, 1160–1165 (1976).
[CrossRef]

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

J. Part. Charact. (1)

E. D. Hirleman, “Optimal scaling of the inverse Fraunhoffer diffraction particle sizing problem: the linear system produced by quadrature,” J. Part. Charact. 4, 128–133 (1987).
[CrossRef]

J. Phys. Chem. (2)

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Figures (12)

Fig. 1
Fig. 1

Simplified schematic of a diffraction particle-sizing instrument.

Fig. 2
Fig. 2

J12 (solid line) compared with its asymptote (dashed curve).

Fig. 3
Fig. 3

First several rows of a matrix G constructed through the diffraction approximation plotted with [1(πk3)] (1 − sin 2xθ) and gi (x) computed from the diffraction approximation.

Fig. 4
Fig. 4

Plot of some of the eigenvectors of G.

Fig. 5
Fig. 5

Schematic of the optical particle-sizing instrument.

Fig. 6
Fig. 6

Scattering measured from the distribution 12dp400.

Fig. 7
Fig. 7

Scattering measured from the distribution 12dp400 presented as the function θ3I(θ).

Fig. 8
Fig. 8

Comparison of inversion algorithms on 12dp400. Curves are offset, appearing from top to bottom in the same order as in Table IV.

Fig. 9
Fig. 9

Inversions of 12dp400 with xmin = 150 and xmax = 250. Also shown are Coulter count measurements.

Fig. 10
Fig. 10

Observed scattering from the trimodal distribution of polystyrene spheres named 204050.

Fig. 11
Fig. 11

Term d(θ) for the trimodal distribution of polystyrene spheres named 204050.

Fig. 12
Fig. 12

Results of inversions and estimates of size distribution n(x) for the distribution 204050.

Tables (9)

Tables Icon

Table I Relevant Data Regarding the Distribution 12dp400

Tables Icon

Table II Relevant Data Regarding the Measurement of the Distribution 12dp400

Tables Icon

Table III Information Common to all Inversions of 12dp400

Tables Icon

Table IV Summary: Comparison of Inversion Algorithms on 12dp400 data, Which is Missing 21 Points of Small-Angle Data a

Tables Icon

Table VI Relevant Data Regarding the Distribution Named 204050

Tables Icon

Table VII Relevant Data Regarding the Measurement of the Distribution 204050

Tables Icon

Table VIII Information Common to all Inversions of 204050

Equations (78)

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I ( θ ) = 0 I ( θ , x ) n ( x ) d x .
I ( θ ) = 1 k 2 θ 2 0 J 1 2 ( x θ ) x 2 n ( x ) d x ,
n ( x ) = 2 π k 2 x 2 0 J 1 ( x θ ) Y 1 ( x θ ) x θ d d θ [ θ 3 I ( θ ) ] d θ .
I ( θ , x ) = α ( θ ) x 6 ,
I ( θ ) = α ( θ ) 0 x 6 n ( x ) d x .
J 1 2 ( x u θ ) ~ 1 π x u θ ( 1 sin 2 x u θ ) .
I ( θ ) = 1 π θ 3 k 2 [ 0 x n ( x ) d x 0 sin ( 2 x θ ) x n ( x ) d x ] .
d ( θ ) = θ 3 I ( θ )
m ( x ) = x n ( x )
G ( θ , x ) = θ 3 I ( θ , x ) / x .
G ( θ , x ) 1 π k 2 ( 1 sin ( 2 x θ ) ,
d ( θ ) 1 π k 2 0 x u m ( x ) d x 1 π k 2 0 x u m ( x ) sin ( 2 x θ ) d x .
D ( ω ) = F { d ( θ ) } = d ( θ ) exp ( i ω θ ) d θ ,
Δ θ π 2 x u .
Δ θ ̅ = θ u M 1
Δ θ ̅ π 2 x u .
Δ θ π ( M 1 ) 2 x u ,
m ( x ) K δ ( x ) k 2 i [ D ( 2 x ) D * ( 2 x ) ] .
Δ f = 1 / T ,
Δ x = 1 2 ( θ max θ min ) ,
d = Gm ,
G i j Δ x π k 3 ( 1 sin 2 x j θ i ) .
G i j = ( 1 sin 2 x j θ i ) .
G = U V ,
V i j = sin 2 x j θ i .
F k = n = 0 N 1 f n exp ( i 2 π k n / N ) ,
f n = 1 N k = 0 N 1 F k exp ( i 2 π k n / N ) ,
F k = 2 n = 1 M f n sin 2 π k n N ,
f n = 2 N n = 1 M F k sin 2 π k n N ,
N = { 2 M + 2 for M odd 2 M + 1 for M even
F = Sf ,
f = S 1 F ,
S i j = sin 2 π i j N ,
S i j 1 = 4 N sin 2 π i j N ,
x j θ i = π i j N for i , j = 1 , , M .
θ i = i θ u M for i = 1 , , M
x j = j Δ x o ,
Δ x o = π M N θ u .
d Δ x π k 3 ( Um Vm )
[ Um ] i 0 x Nyquist m ( x ) d x .
i m ( i Δ x ) Δ x 0 x Nyquist m ( x ) d x
[ Vm ] i 0 x Nyquist M ( x ) sin 2 π x θ i d x .
f ( θ ) = 1 ( 2 π ) 1 / 2 F ( x ) exp ( i θ x ) d x ,
F ( x ) = 1 ( 2 π ) 1 / 2 f ( θ ) exp ( i θ x ) d x .
F r = F ( r Δ x ) ,
Δ x 2 π θ u ,
f ( θ ) = Δ x r = F ( r Δ x ) exp ( i r Δ x θ ) for π Δ x θ π Δ x .
f ( θ ) = 0 for | θ | M Δ θ ,
f ( θ ) = Δ x r = M M F ( r Δ x ) exp ( i r Δ x θ ) .
f ( θ ) = Δ x r = 1 M F ( r Δ x ) sin r Δ x θ .
x j = j Δ x o , j = 1 , , m .
I ( θ , x ) = I F ( θ , x ) [ Q ext ( x , η ) / 2 ] 2
Q ext = 2 4 sin ( 2 x | η 1 | ) 2 x | η 1 | + 4 [ 1 cos ( 2 x | η 1 | ) ] 2 x | η 1 | 2 .
Δ x o = π M N θ u .
G = A Λ A T ,
G = A Λ B T ,
G a i = λ i b i ,
G T b j = λ j a j ,
G = A Λ B T ,
m ̂ j = k = 1 q i = 1 M B i k λ k 1 A j k d i ,
H = { G T ( G G T + I ) 1 for M p N ( G T G + I ) 1 G T for M p N .
Λ i j 1 = δ i j λ i λ i 2 + ,
m ( x ) = C x τ for x > x 0 ,
m ̂ = ( G T G + ν K ) 1 G T d ,
K = [ 1 2 2 5 4 1 1 4 6 4 1 1 4 6 4 1 1 4 6 4 1 2 1 ]
m = Cm
d = GCm .
m = Cm + λ ( d GCm )
m ̂ k + 1 = C m ̂ k + α G T ( d GC m ̂ k ) ,
m ̂ k + 1 = C [ m ̂ k ( m ̂ k g i d i ) g i g i g i ]
m ̂ j k + 1 = C [ 1 + ξ 1 G 1 j ] [ 1 + ξ 2 G 2 j ] [ 1 + ξ M G M j ] m ̂ j k ,
ξ i = d i ( G m ̂ k ) i .
G i j Δ x π k 3 ( 1 sin 2 x θ ) .
i m ( i Δ x ) Δ x 0 x u m ( x ) d x
Δ θ ω ( M 1 ) 2 Δ x ,
R = i = 1 M ( I i meas I i calc I i meas ) 2 for I i meas 0 .
θ u π ( M 1 ) 2 ( x max x min ) .
Δ x = 1 2 ( θ max θ min ) = 13.3 .

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