Abstract

We combine optical remote sensing with computed tomography to determine simultaneously (a) the concentration and (b) the size distribution of particles at every pixel in a plane that slices through an aerosol. Light-extinction measurements are made along intersecting paths that pass through the plane. The spatial distribution of extinction coefficients at multiple wavelengths is obtained by an algebraic image-reconstruction technique (ART3). The size distribution of the aerosol at every pixel in the plane is obtained by inversion of the Fredholm integral equation. Computer simulations of this procedure were conducted. Extinction coefficients were found at all pixels in the plane at multiple wavelengths. Aerosol size distributions were retrieved at four pixels. Results of this analysis show that four projection angles were sufficient for reconstruction of extinction coefficient distributions in the plane. The technique can tolerate up to 10% random, normally distributed noise in the measurements. The size distributions at the four pixels were close to the true distributions. The size of the smallest feature that needs to be recovered should be larger than the ray spacing. We were able to delete three of every four rays and still get good reconstructions.

© 1994 Optical Society of America

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References

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  1. G. Ramachandran, D. Leith, “Extraction of aerosol size distributions from mulitspectral light extinction data,” Aerosol Sci. Technol. 17, 303–325 (1992).
    [Crossref]
  2. S. Kaczmarz, “Angenaherte auflosung von Systemen linearer Gleichungen,” Bull. Acad. Polit. Sci. Lett. A 6-8A, 355–357 (1937).
  3. G. T. Herman, A. Lent, S. W. Rowland, “ART: mathematics and applications. A report on the mathematical foundations and on the applicability to real data of the algebraic reconstruction techniques,”J. Theor. Biol. 42, 1–32 (1973).
    [Crossref] [PubMed]
  4. R. Gordon, “A tutorial on ART,”IEEE Trans. Nucl. Sci. NS-21, 78–93 (1974).
  5. P. Gilbert, “Iterative methods for the three-dimensional reconstruction of an object from projections,”J. Theor. Biol. 36, 105–117 (1972).
    [Crossref] [PubMed]
  6. G. T. Herman, “A relaxation method for reconstructing objects from noisy x-rays,” Math. Program. 8, 1–19 (1975).
    [Crossref]
  7. A. H. Andersen, A. C. Kak, “Simultaneous algebraic reconstruction technique (SART): a superior implementation of the ART algorithm,” Ultrason. Imaging 6, 81–94 (1984).
    [PubMed]
  8. L. Todd, G. Ramachandran, “Evaluation of algorithms for tomographic reconstruction of chemicals in indoor air,” Am. Ind. Hyg. Assoc. J. (to be published).
  9. J. Radon, “Uber die Bestimmung von Funktionen durch ihre integralwerte langs gewisser Mannigfaltigkeiten,” Ber. Verhandl. 69, 262–277 (1917).
  10. G. N. Hounsfield, “Computerized transverse axial scanning (tomography). Part I: description of system,” Brit. J. Radiol. 46, 1016–1022 (1973).
    [Crossref]
  11. A. M. Cormack, “Early two-dimensional reconstruction (CT scanning) and recent topics stemming from it. Nobel lecture Dec. 8, 1979,”J. Comput. Assist. Tomog. 4, 658–664 (1980).
    [Crossref]
  12. R. A. Brooks, G. De Chiro, “Theory of image reconstruction in computed tomography,” Radiology 117, 561–572 (1975).
    [PubMed]
  13. R. N. Bracewell, “Strip integration in radio astronomy,” Aust. J. Phys. 9, 198–217 (1956).
    [Crossref]
  14. M. D. Altschuler, “Reconstruction of the global scale three-dimensional solar corona,” in Image Reconstruction from Projections, G. T. Herman, ed., Vol. 32 of Topics in Applied Physics (Springer-Verlag, Berlin, 1979).
    [Crossref]
  15. D. J. De Rosier, A. Klug, “Reconstruction of three-dimensional structures from electron micrographs,” Nature (London) 217, 130–134 (1968).
    [Crossref]
  16. S. C. Solomon, P. B. Hays, V. J. Abreu, “Tomographic inversion of satellite photometry,” Appl. Opt. 23, 3409–3414 (1984).
    [Crossref] [PubMed]
  17. D. C. Wolfe, R. L. Byer, “Model studies of laser absorption computed tomography for remote air pollution measurement,” Appl. Opt. 21, 1165–1178 (1982).
    [Crossref] [PubMed]
  18. J. A. Weinman, “Tomographic lidar to measure the extinction coefficients of atmospheric aerosols,” Appl. Opt. 23, 3882–3888 (1984).
    [Crossref] [PubMed]
  19. G. Lorbeer, B. Siemund, I. Willms, “Opto-computer-tomographical methods as aids for characterizing local inhomogeneous aerosol distributions,” J. Aerosol Sci. 15, 287–293 (1984).
    [Crossref]
  20. H. O. Luck, B. Siemund, G. Lorbeer, “The measurement of spatial aerosol distributions in enclosures by means of computed tomography,” Part. Charact. 2, 137–142 (1985).
    [Crossref]
  21. L. Todd, D. Leith, “Remote sensing and computed tomography in industrial hygiene,” Am. Ind. Hyg. Assoc. J. 51, 224–233 (1990).
    [Crossref]
  22. A. C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (Institute of Electrical and Electronics Engineers, New York, 1988).
  23. K. Tanabe, “Projection method for solving a singular system,” Numer. Math. 17, 203–214 (1971).
    [Crossref]
  24. M. R. Trummer, “Reconstructing pictures from projections: on the convergence of the ART algorithm with relaxation,” Computing 26, 189–195 (1981).
    [Crossref]
  25. G. T. Herman, “Two direct methods for reconstructing pictures from their projections: a comparative study,” Comput. Graphics Image Process. 1, 123–144 (1972).
    [Crossref]
  26. G. T. Herman, S. Rowland, “Three methods for reconstructing objects from x-rays: a comparative study,” Comput. Graphics Image Process. 2, 151–178 (1973).
    [Crossref]
  27. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).
  28. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  29. T. Nguyen, K. Cox, “A method for the determination of aerosol particle distributions from light extinction data,” in Abstracts of the American Association for Aerosol Research Annual Meeting (American Association of Aerosol Research, Cincinnati, Ohio, 1989), p. 330.
  30. R. A. Brooks, G. De Chiro, “Principles of computer assisted tomography (CAT) in radiographic and radioisotopic imaging,” Phys. Med. Biol. 21, 689–732 (1976).
    [Crossref] [PubMed]
  31. K. Kouris, H. Tuy, A. Lent, G. T. Herman, R. M. Lewitt, “Reconstruction from sparsely sampled data by ART with interpolated rays,” IEEE Trans. Med. Imaging MI-1, 161–167 (1982).
    [Crossref]
  32. G. H. Golub, M. Heath, G. Wahba, “Generalized cross-validation as a method for choosing a good ridge parameter,” Technometrics 21, 215–223 (1979).
    [Crossref]

1992 (1)

G. Ramachandran, D. Leith, “Extraction of aerosol size distributions from mulitspectral light extinction data,” Aerosol Sci. Technol. 17, 303–325 (1992).
[Crossref]

1990 (1)

L. Todd, D. Leith, “Remote sensing and computed tomography in industrial hygiene,” Am. Ind. Hyg. Assoc. J. 51, 224–233 (1990).
[Crossref]

1985 (1)

H. O. Luck, B. Siemund, G. Lorbeer, “The measurement of spatial aerosol distributions in enclosures by means of computed tomography,” Part. Charact. 2, 137–142 (1985).
[Crossref]

1984 (4)

A. H. Andersen, A. C. Kak, “Simultaneous algebraic reconstruction technique (SART): a superior implementation of the ART algorithm,” Ultrason. Imaging 6, 81–94 (1984).
[PubMed]

S. C. Solomon, P. B. Hays, V. J. Abreu, “Tomographic inversion of satellite photometry,” Appl. Opt. 23, 3409–3414 (1984).
[Crossref] [PubMed]

J. A. Weinman, “Tomographic lidar to measure the extinction coefficients of atmospheric aerosols,” Appl. Opt. 23, 3882–3888 (1984).
[Crossref] [PubMed]

G. Lorbeer, B. Siemund, I. Willms, “Opto-computer-tomographical methods as aids for characterizing local inhomogeneous aerosol distributions,” J. Aerosol Sci. 15, 287–293 (1984).
[Crossref]

1982 (2)

D. C. Wolfe, R. L. Byer, “Model studies of laser absorption computed tomography for remote air pollution measurement,” Appl. Opt. 21, 1165–1178 (1982).
[Crossref] [PubMed]

K. Kouris, H. Tuy, A. Lent, G. T. Herman, R. M. Lewitt, “Reconstruction from sparsely sampled data by ART with interpolated rays,” IEEE Trans. Med. Imaging MI-1, 161–167 (1982).
[Crossref]

1981 (1)

M. R. Trummer, “Reconstructing pictures from projections: on the convergence of the ART algorithm with relaxation,” Computing 26, 189–195 (1981).
[Crossref]

1980 (1)

A. M. Cormack, “Early two-dimensional reconstruction (CT scanning) and recent topics stemming from it. Nobel lecture Dec. 8, 1979,”J. Comput. Assist. Tomog. 4, 658–664 (1980).
[Crossref]

1979 (1)

G. H. Golub, M. Heath, G. Wahba, “Generalized cross-validation as a method for choosing a good ridge parameter,” Technometrics 21, 215–223 (1979).
[Crossref]

1976 (1)

R. A. Brooks, G. De Chiro, “Principles of computer assisted tomography (CAT) in radiographic and radioisotopic imaging,” Phys. Med. Biol. 21, 689–732 (1976).
[Crossref] [PubMed]

1975 (2)

R. A. Brooks, G. De Chiro, “Theory of image reconstruction in computed tomography,” Radiology 117, 561–572 (1975).
[PubMed]

G. T. Herman, “A relaxation method for reconstructing objects from noisy x-rays,” Math. Program. 8, 1–19 (1975).
[Crossref]

1974 (1)

R. Gordon, “A tutorial on ART,”IEEE Trans. Nucl. Sci. NS-21, 78–93 (1974).

1973 (3)

G. T. Herman, A. Lent, S. W. Rowland, “ART: mathematics and applications. A report on the mathematical foundations and on the applicability to real data of the algebraic reconstruction techniques,”J. Theor. Biol. 42, 1–32 (1973).
[Crossref] [PubMed]

G. N. Hounsfield, “Computerized transverse axial scanning (tomography). Part I: description of system,” Brit. J. Radiol. 46, 1016–1022 (1973).
[Crossref]

G. T. Herman, S. Rowland, “Three methods for reconstructing objects from x-rays: a comparative study,” Comput. Graphics Image Process. 2, 151–178 (1973).
[Crossref]

1972 (2)

G. T. Herman, “Two direct methods for reconstructing pictures from their projections: a comparative study,” Comput. Graphics Image Process. 1, 123–144 (1972).
[Crossref]

P. Gilbert, “Iterative methods for the three-dimensional reconstruction of an object from projections,”J. Theor. Biol. 36, 105–117 (1972).
[Crossref] [PubMed]

1971 (1)

K. Tanabe, “Projection method for solving a singular system,” Numer. Math. 17, 203–214 (1971).
[Crossref]

1968 (1)

D. J. De Rosier, A. Klug, “Reconstruction of three-dimensional structures from electron micrographs,” Nature (London) 217, 130–134 (1968).
[Crossref]

1956 (1)

R. N. Bracewell, “Strip integration in radio astronomy,” Aust. J. Phys. 9, 198–217 (1956).
[Crossref]

1937 (1)

S. Kaczmarz, “Angenaherte auflosung von Systemen linearer Gleichungen,” Bull. Acad. Polit. Sci. Lett. A 6-8A, 355–357 (1937).

1917 (1)

J. Radon, “Uber die Bestimmung von Funktionen durch ihre integralwerte langs gewisser Mannigfaltigkeiten,” Ber. Verhandl. 69, 262–277 (1917).

Abreu, V. J.

Altschuler, M. D.

M. D. Altschuler, “Reconstruction of the global scale three-dimensional solar corona,” in Image Reconstruction from Projections, G. T. Herman, ed., Vol. 32 of Topics in Applied Physics (Springer-Verlag, Berlin, 1979).
[Crossref]

Andersen, A. H.

A. H. Andersen, A. C. Kak, “Simultaneous algebraic reconstruction technique (SART): a superior implementation of the ART algorithm,” Ultrason. Imaging 6, 81–94 (1984).
[PubMed]

Bracewell, R. N.

R. N. Bracewell, “Strip integration in radio astronomy,” Aust. J. Phys. 9, 198–217 (1956).
[Crossref]

Brooks, R. A.

R. A. Brooks, G. De Chiro, “Principles of computer assisted tomography (CAT) in radiographic and radioisotopic imaging,” Phys. Med. Biol. 21, 689–732 (1976).
[Crossref] [PubMed]

R. A. Brooks, G. De Chiro, “Theory of image reconstruction in computed tomography,” Radiology 117, 561–572 (1975).
[PubMed]

Byer, R. L.

Cormack, A. M.

A. M. Cormack, “Early two-dimensional reconstruction (CT scanning) and recent topics stemming from it. Nobel lecture Dec. 8, 1979,”J. Comput. Assist. Tomog. 4, 658–664 (1980).
[Crossref]

Cox, K.

T. Nguyen, K. Cox, “A method for the determination of aerosol particle distributions from light extinction data,” in Abstracts of the American Association for Aerosol Research Annual Meeting (American Association of Aerosol Research, Cincinnati, Ohio, 1989), p. 330.

De Chiro, G.

R. A. Brooks, G. De Chiro, “Principles of computer assisted tomography (CAT) in radiographic and radioisotopic imaging,” Phys. Med. Biol. 21, 689–732 (1976).
[Crossref] [PubMed]

R. A. Brooks, G. De Chiro, “Theory of image reconstruction in computed tomography,” Radiology 117, 561–572 (1975).
[PubMed]

De Rosier, D. J.

D. J. De Rosier, A. Klug, “Reconstruction of three-dimensional structures from electron micrographs,” Nature (London) 217, 130–134 (1968).
[Crossref]

Gilbert, P.

P. Gilbert, “Iterative methods for the three-dimensional reconstruction of an object from projections,”J. Theor. Biol. 36, 105–117 (1972).
[Crossref] [PubMed]

Golub, G. H.

G. H. Golub, M. Heath, G. Wahba, “Generalized cross-validation as a method for choosing a good ridge parameter,” Technometrics 21, 215–223 (1979).
[Crossref]

Gordon, R.

R. Gordon, “A tutorial on ART,”IEEE Trans. Nucl. Sci. NS-21, 78–93 (1974).

Hays, P. B.

Heath, M.

G. H. Golub, M. Heath, G. Wahba, “Generalized cross-validation as a method for choosing a good ridge parameter,” Technometrics 21, 215–223 (1979).
[Crossref]

Herman, G. T.

K. Kouris, H. Tuy, A. Lent, G. T. Herman, R. M. Lewitt, “Reconstruction from sparsely sampled data by ART with interpolated rays,” IEEE Trans. Med. Imaging MI-1, 161–167 (1982).
[Crossref]

G. T. Herman, “A relaxation method for reconstructing objects from noisy x-rays,” Math. Program. 8, 1–19 (1975).
[Crossref]

G. T. Herman, A. Lent, S. W. Rowland, “ART: mathematics and applications. A report on the mathematical foundations and on the applicability to real data of the algebraic reconstruction techniques,”J. Theor. Biol. 42, 1–32 (1973).
[Crossref] [PubMed]

G. T. Herman, S. Rowland, “Three methods for reconstructing objects from x-rays: a comparative study,” Comput. Graphics Image Process. 2, 151–178 (1973).
[Crossref]

G. T. Herman, “Two direct methods for reconstructing pictures from their projections: a comparative study,” Comput. Graphics Image Process. 1, 123–144 (1972).
[Crossref]

Hounsfield, G. N.

G. N. Hounsfield, “Computerized transverse axial scanning (tomography). Part I: description of system,” Brit. J. Radiol. 46, 1016–1022 (1973).
[Crossref]

Kaczmarz, S.

S. Kaczmarz, “Angenaherte auflosung von Systemen linearer Gleichungen,” Bull. Acad. Polit. Sci. Lett. A 6-8A, 355–357 (1937).

Kak, A. C.

A. H. Andersen, A. C. Kak, “Simultaneous algebraic reconstruction technique (SART): a superior implementation of the ART algorithm,” Ultrason. Imaging 6, 81–94 (1984).
[PubMed]

A. C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (Institute of Electrical and Electronics Engineers, New York, 1988).

Kerker, M.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).

Klug, A.

D. J. De Rosier, A. Klug, “Reconstruction of three-dimensional structures from electron micrographs,” Nature (London) 217, 130–134 (1968).
[Crossref]

Kouris, K.

K. Kouris, H. Tuy, A. Lent, G. T. Herman, R. M. Lewitt, “Reconstruction from sparsely sampled data by ART with interpolated rays,” IEEE Trans. Med. Imaging MI-1, 161–167 (1982).
[Crossref]

Leith, D.

G. Ramachandran, D. Leith, “Extraction of aerosol size distributions from mulitspectral light extinction data,” Aerosol Sci. Technol. 17, 303–325 (1992).
[Crossref]

L. Todd, D. Leith, “Remote sensing and computed tomography in industrial hygiene,” Am. Ind. Hyg. Assoc. J. 51, 224–233 (1990).
[Crossref]

Lent, A.

K. Kouris, H. Tuy, A. Lent, G. T. Herman, R. M. Lewitt, “Reconstruction from sparsely sampled data by ART with interpolated rays,” IEEE Trans. Med. Imaging MI-1, 161–167 (1982).
[Crossref]

G. T. Herman, A. Lent, S. W. Rowland, “ART: mathematics and applications. A report on the mathematical foundations and on the applicability to real data of the algebraic reconstruction techniques,”J. Theor. Biol. 42, 1–32 (1973).
[Crossref] [PubMed]

Lewitt, R. M.

K. Kouris, H. Tuy, A. Lent, G. T. Herman, R. M. Lewitt, “Reconstruction from sparsely sampled data by ART with interpolated rays,” IEEE Trans. Med. Imaging MI-1, 161–167 (1982).
[Crossref]

Lorbeer, G.

H. O. Luck, B. Siemund, G. Lorbeer, “The measurement of spatial aerosol distributions in enclosures by means of computed tomography,” Part. Charact. 2, 137–142 (1985).
[Crossref]

G. Lorbeer, B. Siemund, I. Willms, “Opto-computer-tomographical methods as aids for characterizing local inhomogeneous aerosol distributions,” J. Aerosol Sci. 15, 287–293 (1984).
[Crossref]

Luck, H. O.

H. O. Luck, B. Siemund, G. Lorbeer, “The measurement of spatial aerosol distributions in enclosures by means of computed tomography,” Part. Charact. 2, 137–142 (1985).
[Crossref]

Nguyen, T.

T. Nguyen, K. Cox, “A method for the determination of aerosol particle distributions from light extinction data,” in Abstracts of the American Association for Aerosol Research Annual Meeting (American Association of Aerosol Research, Cincinnati, Ohio, 1989), p. 330.

Radon, J.

J. Radon, “Uber die Bestimmung von Funktionen durch ihre integralwerte langs gewisser Mannigfaltigkeiten,” Ber. Verhandl. 69, 262–277 (1917).

Ramachandran, G.

G. Ramachandran, D. Leith, “Extraction of aerosol size distributions from mulitspectral light extinction data,” Aerosol Sci. Technol. 17, 303–325 (1992).
[Crossref]

L. Todd, G. Ramachandran, “Evaluation of algorithms for tomographic reconstruction of chemicals in indoor air,” Am. Ind. Hyg. Assoc. J. (to be published).

Rowland, S.

G. T. Herman, S. Rowland, “Three methods for reconstructing objects from x-rays: a comparative study,” Comput. Graphics Image Process. 2, 151–178 (1973).
[Crossref]

Rowland, S. W.

G. T. Herman, A. Lent, S. W. Rowland, “ART: mathematics and applications. A report on the mathematical foundations and on the applicability to real data of the algebraic reconstruction techniques,”J. Theor. Biol. 42, 1–32 (1973).
[Crossref] [PubMed]

Siemund, B.

H. O. Luck, B. Siemund, G. Lorbeer, “The measurement of spatial aerosol distributions in enclosures by means of computed tomography,” Part. Charact. 2, 137–142 (1985).
[Crossref]

G. Lorbeer, B. Siemund, I. Willms, “Opto-computer-tomographical methods as aids for characterizing local inhomogeneous aerosol distributions,” J. Aerosol Sci. 15, 287–293 (1984).
[Crossref]

Slaney, M.

A. C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (Institute of Electrical and Electronics Engineers, New York, 1988).

Solomon, S. C.

Tanabe, K.

K. Tanabe, “Projection method for solving a singular system,” Numer. Math. 17, 203–214 (1971).
[Crossref]

Todd, L.

L. Todd, D. Leith, “Remote sensing and computed tomography in industrial hygiene,” Am. Ind. Hyg. Assoc. J. 51, 224–233 (1990).
[Crossref]

L. Todd, G. Ramachandran, “Evaluation of algorithms for tomographic reconstruction of chemicals in indoor air,” Am. Ind. Hyg. Assoc. J. (to be published).

Trummer, M. R.

M. R. Trummer, “Reconstructing pictures from projections: on the convergence of the ART algorithm with relaxation,” Computing 26, 189–195 (1981).
[Crossref]

Tuy, H.

K. Kouris, H. Tuy, A. Lent, G. T. Herman, R. M. Lewitt, “Reconstruction from sparsely sampled data by ART with interpolated rays,” IEEE Trans. Med. Imaging MI-1, 161–167 (1982).
[Crossref]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

Wahba, G.

G. H. Golub, M. Heath, G. Wahba, “Generalized cross-validation as a method for choosing a good ridge parameter,” Technometrics 21, 215–223 (1979).
[Crossref]

Weinman, J. A.

Willms, I.

G. Lorbeer, B. Siemund, I. Willms, “Opto-computer-tomographical methods as aids for characterizing local inhomogeneous aerosol distributions,” J. Aerosol Sci. 15, 287–293 (1984).
[Crossref]

Wolfe, D. C.

Aerosol Sci. Technol. (1)

G. Ramachandran, D. Leith, “Extraction of aerosol size distributions from mulitspectral light extinction data,” Aerosol Sci. Technol. 17, 303–325 (1992).
[Crossref]

Am. Ind. Hyg. Assoc. J. (1)

L. Todd, D. Leith, “Remote sensing and computed tomography in industrial hygiene,” Am. Ind. Hyg. Assoc. J. 51, 224–233 (1990).
[Crossref]

Appl. Opt. (3)

Aust. J. Phys. (1)

R. N. Bracewell, “Strip integration in radio astronomy,” Aust. J. Phys. 9, 198–217 (1956).
[Crossref]

Ber. Verhandl. (1)

J. Radon, “Uber die Bestimmung von Funktionen durch ihre integralwerte langs gewisser Mannigfaltigkeiten,” Ber. Verhandl. 69, 262–277 (1917).

Brit. J. Radiol. (1)

G. N. Hounsfield, “Computerized transverse axial scanning (tomography). Part I: description of system,” Brit. J. Radiol. 46, 1016–1022 (1973).
[Crossref]

Bull. Acad. Polit. Sci. Lett. A (1)

S. Kaczmarz, “Angenaherte auflosung von Systemen linearer Gleichungen,” Bull. Acad. Polit. Sci. Lett. A 6-8A, 355–357 (1937).

Comput. Graphics Image Process. (2)

G. T. Herman, “Two direct methods for reconstructing pictures from their projections: a comparative study,” Comput. Graphics Image Process. 1, 123–144 (1972).
[Crossref]

G. T. Herman, S. Rowland, “Three methods for reconstructing objects from x-rays: a comparative study,” Comput. Graphics Image Process. 2, 151–178 (1973).
[Crossref]

Computing (1)

M. R. Trummer, “Reconstructing pictures from projections: on the convergence of the ART algorithm with relaxation,” Computing 26, 189–195 (1981).
[Crossref]

IEEE Trans. Med. Imaging (1)

K. Kouris, H. Tuy, A. Lent, G. T. Herman, R. M. Lewitt, “Reconstruction from sparsely sampled data by ART with interpolated rays,” IEEE Trans. Med. Imaging MI-1, 161–167 (1982).
[Crossref]

IEEE Trans. Nucl. Sci. (1)

R. Gordon, “A tutorial on ART,”IEEE Trans. Nucl. Sci. NS-21, 78–93 (1974).

J. Aerosol Sci. (1)

G. Lorbeer, B. Siemund, I. Willms, “Opto-computer-tomographical methods as aids for characterizing local inhomogeneous aerosol distributions,” J. Aerosol Sci. 15, 287–293 (1984).
[Crossref]

J. Comput. Assist. Tomog. (1)

A. M. Cormack, “Early two-dimensional reconstruction (CT scanning) and recent topics stemming from it. Nobel lecture Dec. 8, 1979,”J. Comput. Assist. Tomog. 4, 658–664 (1980).
[Crossref]

J. Theor. Biol. (2)

P. Gilbert, “Iterative methods for the three-dimensional reconstruction of an object from projections,”J. Theor. Biol. 36, 105–117 (1972).
[Crossref] [PubMed]

G. T. Herman, A. Lent, S. W. Rowland, “ART: mathematics and applications. A report on the mathematical foundations and on the applicability to real data of the algebraic reconstruction techniques,”J. Theor. Biol. 42, 1–32 (1973).
[Crossref] [PubMed]

Math. Program. (1)

G. T. Herman, “A relaxation method for reconstructing objects from noisy x-rays,” Math. Program. 8, 1–19 (1975).
[Crossref]

Nature (London) (1)

D. J. De Rosier, A. Klug, “Reconstruction of three-dimensional structures from electron micrographs,” Nature (London) 217, 130–134 (1968).
[Crossref]

Numer. Math. (1)

K. Tanabe, “Projection method for solving a singular system,” Numer. Math. 17, 203–214 (1971).
[Crossref]

Part. Charact. (1)

H. O. Luck, B. Siemund, G. Lorbeer, “The measurement of spatial aerosol distributions in enclosures by means of computed tomography,” Part. Charact. 2, 137–142 (1985).
[Crossref]

Phys. Med. Biol. (1)

R. A. Brooks, G. De Chiro, “Principles of computer assisted tomography (CAT) in radiographic and radioisotopic imaging,” Phys. Med. Biol. 21, 689–732 (1976).
[Crossref] [PubMed]

Radiology (1)

R. A. Brooks, G. De Chiro, “Theory of image reconstruction in computed tomography,” Radiology 117, 561–572 (1975).
[PubMed]

Technometrics (1)

G. H. Golub, M. Heath, G. Wahba, “Generalized cross-validation as a method for choosing a good ridge parameter,” Technometrics 21, 215–223 (1979).
[Crossref]

Ultrason. Imaging (1)

A. H. Andersen, A. C. Kak, “Simultaneous algebraic reconstruction technique (SART): a superior implementation of the ART algorithm,” Ultrason. Imaging 6, 81–94 (1984).
[PubMed]

Other (6)

L. Todd, G. Ramachandran, “Evaluation of algorithms for tomographic reconstruction of chemicals in indoor air,” Am. Ind. Hyg. Assoc. J. (to be published).

M. D. Altschuler, “Reconstruction of the global scale three-dimensional solar corona,” in Image Reconstruction from Projections, G. T. Herman, ed., Vol. 32 of Topics in Applied Physics (Springer-Verlag, Berlin, 1979).
[Crossref]

A. C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (Institute of Electrical and Electronics Engineers, New York, 1988).

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

T. Nguyen, K. Cox, “A method for the determination of aerosol particle distributions from light extinction data,” in Abstracts of the American Association for Aerosol Research Annual Meeting (American Association of Aerosol Research, Cincinnati, Ohio, 1989), p. 330.

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Figures (15)

Fig. 1
Fig. 1

Geometry of reconstruction from projections. The square grid is the reconstruction space divided into pixels, which are assigned values σ1, σ2, … σN. pi are the ray sums at a projection angle θ. aij is the area of the triangle ABC.

Fig. 2
Fig. 2

Light-detector setup for equal-angle, parallel-projection configuration. Representation of a 10 × 10-pixel room with four projection angles, 0° 90° and ±45°.003.

Fig. 3
Fig. 3

Concentration profile in a room divided as a 40 × 40 pixel grid. The three peaks are located at (10, 25), (30, 20), and (20, 10), with heights of 30,000, 20,000, and 20,000 particles/cm3 and standard deviations of 3, 4, and 5, respectively.

Fig. 4
Fig. 4

Distribution of extinction coefficients in the room with the concentration profile as shown in Fig. 3 and at a wavelength of 540 nm.

Fig. 5
Fig. 5

Reconstruction of the distribution of extinction coefficients at 540 nm in the ideal case with no noise and a full complement of rays for four projection angles.

Fig. 6
Fig. 6

Aerosol size distributions at (a) location 1, (b) location 2, (c) location 3, and (d) location 4. The true distribution, the distribution calculated from true extinction coefficients, and the distribution calculated from extinction coefficients in the reconstructed picture of Fig. 5 are given.

Fig. 7
Fig. 7

Extinction coefficients at location 2 for 10 wavelengths for 2%, 5%, and 10% measurement error.

Fig. 8
Fig. 8

Nearness versus iteration number for reconstruction of the picture at 540 nm for different levels of noise.

Fig. 9
Fig. 9

Reconstruction of the distribution of extinction coefficients for (a) 2% noise, (b) 5% noise, and (c) 10% noise.

Fig. 10
Fig. 10

Reconstruction of aerosol size distribution at location 2 for 2%, 5%, and 10% noise.

Fig. 11
Fig. 11

Extinction coefficients at location 1 for 10 wavelengths when 50% of the rays are used and when 25% of the rays are used.

Fig. 12
Fig. 12

Nearness versus iteration number for reconstruction of the picture when 50% of the rays are used and when 25% of the rays are used.

Fig. 13
Fig. 13

Reconstruction of the distribution of extinction coefficients when (a) 50% of the rays are used and (b) 25% of the rays are used.

Fig. 14
Fig. 14

Reconstruction of the aerosol size distribution at location 1 when 50% of the rays are used and when 25% of the rays are used.

Fig. 15
Fig. 15

Reconstruction of the aerosol size distribution at location 4 when 50% of the rays are used and when 25% of the rays are used.

Equations (38)

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p i = j = 1 n 2 a i j σ j ,             i = 1 , 2 , M ,
σ ( q + 1 ) = σ ( q ) + [ p i - σ ( q ) · a i ] a i a i · a i .
p i - ɛ i j = 1 n 2 a i j σ j p i + ɛ i ,             i = 1 , 2 , M .
Nearness = [ j = 1 n 2 ( σ j * - σ j ) 2 / j = 1 n 2 ( σ j * - σ avg * ) 2 ] 1 / 2 ,
D ( k ) = [ i = 1 m ( p i - j = 1 n 2 a i j σ j ) 2 ] 1 / 2 / M .
| v ( k + 1 ) - v ( k ) v ( k ) | < 0.0001 ,
v ( k ) = 1 n 2 j = 1 n 2 ( σ j k - σ avg ) ,
σ i + ɛ i = a b K i ( d , λ , m ) f ( d ) d d ,             i = 1 , 2 , N ,
K i = π 4 d 2 Q ext ( d , λ i , m ) .
0 f ( d ) dd = particle concentration .
G ( λ ) = 0 1 K ( λ , x ) f ( x ) d x ,
g ( λ ) = 0 1 Ψ ( x ) f ( x ) d x ,
g ( λ ) = G ( λ ) = f ( 0 ) [ M ( 1 ) ( 1 ) - M ( 0 ) ( 1 ) ] - f ( 1 ) M ( 1 ) ( 1 ) ,
Ψ ( x ) = x [ M ( 0 ) ( x ) - M ( 0 ) ( 1 ) + M ( 1 ) ( 1 ) ] - M ( 1 ) ( x ) ,
M ( n ) ( x ) = 0 x K ( x ) ( x ) n d x .
f ( x ) = j = 1 n C j Φ j ( x ) .
A i j = 0 1 Ψ i ( x ) Ψ j ( x ) d x .
[ γ ] N , N = [ γ 1 0 γ 2 0 γ N ] [ A ] [ B ] = [ B ] [ γ ]
f ( x ) = C T Φ ( x ) ,
g = 0 1 Ψ ( x ) f ( x ) d x = 0 1 Ψ ( x ) C T Φ ( x ) d x .
Φ ( x ) = [ B ] T Ψ ( x ) , g = 0 1 Ψ ( x ) C T Φ ( x ) d x = 0 1 Ψ ( x ) Φ ( x ) C T d x = 0 1 Ψ ( x ) Ψ ( x ) T [ B ] C d x = 0 1 Ψ ( x ) Ψ ( x ) T d x [ B ] C = [ A ] [ B ] C .
g = [ B ] [ γ ] C .
[ B ] T g = [ B ] T [ B ] [ γ ] C = [ γ ] C .
ϕ = [ B ] T g .
ϕ = [ γ ] C
C i = ϕ i γ i .
f ( x ) = f ( 0 ) + x [ f ( 1 ) - f ( 0 ) ] + 1 x d x 0 x f ( x ) d x .
f ( x ) = f ( 0 ) + x [ f ( 1 ) - f ( 0 ) ] + j = 1 n C j Ω j ( x ) ,
Ω j ( x ) = i = 1 n { 1 6 x 3 [ M i ( 0 ) ( x ) - M i ( 0 ) ( 1 ) - M i ( 1 ) ( 1 ) ] - 1 2 x 2 M i ( 1 ) ( x ) + 1 6 x [ 3 M i ( 2 ) ( x ) - 3 M i ( 2 ) ( 1 ) + 2 M i ( 1 ) ( 1 ) + M i ( 3 ) ( 1 ) ] - 1 6 M i ( 3 ) ( x ) } B i j .
θ = [ R ] ϕ ,
[ R ] = θ i ϕ i if i m = 0 if i > m ,
V ( m ) = 1 n [ j = 1 n g j 2 + j = 1 m ( θ j 2 - 2 θ j ϕ j ) ] / ( 1 - 1 n j = 1 m θ j ϕ j ) 2 .
p i - ɛ i j = 1 n 2 a i j σ j p i + ɛ i ,             i = 1 , 2 , M ,
σ ϕ ( q + 1 ) = σ ϕ ( q ) + γ i ( q ) a i a i · a i ,
γ i ( q ) = p i - σ ( q ) · a i             if p i - σ ( q ) · a i > 2 ɛ i
= 2 ( p i - σ ( q ) · a i - ɛ i )             if ɛ i < p i - σ ( q ) · a i 2 ɛ i
= 2 ( p i - σ ( q ) · a i + ɛ i )             if ɛ i > p i - σ ( q ) · a i 2 ɛ i
= 0             if p i - σ ( q ) · a i < ɛ i .

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