Abstract

We report experimental results of quantitative imaging in supersonic circular jets by using a monochromatic light probe. An expanding cone of light interrogates a three-dimensional volume of a supersonic steady-state flow from a circular jet. The distortion caused to the spherical wave by the presence of the jet is determined through our measuring normal intensity transport. A cone-beam tomographic algorithm is used to invert wave-front distortion to changes in refractive index introduced by the flow. The refractive index is converted into density whose cross sections reveal shock and other characteristics of the flow.

© 2004 Optical Society of America

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  1. N. Kolmogorov, “A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reyolds number,” J. Fluid Mech. 13, 82–85 (1962).
    [CrossRef]
  2. B. Chhabra, R. V. Jensen, “Direct determination of the f(α) singularity spectrum,” Phy. Rev. Lett. 62, 1327–1330 (1989).
    [CrossRef]
  3. C. Meneveau, K. R. Srinivasan, “The multifractal nature of turbulent energy dissipation,” J. Fluid Mech. 224, 429–484 (1991).
    [CrossRef]
  4. M. Raffel, H. Richard, G. E. A. Meier, “On the applicability of background oriented optical tomography for large scale aerodynamic investigations,” Expt. Fluids 28, 477–481 (2000).
    [CrossRef]
  5. R. E. Pierson, D. F. Olson, E. Y. Chen, L. McMackin, “Comparison of reconstruction-algorithm performance for optical-phase tomography of a heated air flow,” Opt. Eng. 39, 838–846 (2000).
    [CrossRef]
  6. L. McMackin, B. Masson, N. Clark, K. Bishop, R. Pierson, E. Chen, “Hartmann wavefront sensor studies of dynamic organized structures in flow fields,” AIAA J. 33, 2158–2164 (1995).
    [CrossRef]
  7. A. C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (IEEE Press, New York, 1988).
  8. K. Creath, “Phase measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (North Holland, Amsterdam, 1988), pp. 349–392.
    [CrossRef]
  9. G. W. Faris, R. L. Byer, “Three-dimensional beam-deflection optical tomography of a supersonic jet,” Appl. Opt. 27, 5202–5212 (1988).
    [CrossRef] [PubMed]
  10. G. Keshava Datta, R. M. Vasu, “Non-interferometric methods of phase estimation for application in optical tomography,” J. Mod. Opt. 46, 1377–1388 (1999).
  11. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [CrossRef] [PubMed]
  12. M. H. Maleki, A. J. Devaney, S. Alon, “Tomographic reconstruction from optical scattered intensities,” J. Opt. Soc. Am. 9, 1356–1363 (1992).
    [CrossRef]
  13. N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 49, 6–10 (1984).
    [CrossRef]
  14. T. E. Gureyev, A. Roberts, K. A. Nugent, “Phase retrieval with the transport-of-intensity equation: matrix solution with use of Zernike polynomials,” J. Opt. Soc. Am. A 12, 1932–1941 (1995).
    [CrossRef]
  15. A. Barty, K. A. Nugent, A. Roberts, D. Peganin, “Quantitative phase tomography,” Opt. Commun. 175, 329–336 (2000).
    [CrossRef]
  16. N. Jayshree, G. Keshava Datta, R. M. Vasu, “Optical tomographic microscope for quantitative imaging of phase objects,” Appl. Opt. 39, 277–283 (2000).
    [CrossRef]
  17. T. E. Gureyev, K. A. Nugent, “Phase retrieval with the transport-of-intensity equation. II. Orthogonal series solution for nonuniform illumination,” J. Opt. Soc. Am. A 13, 1670–1682 (1996).
    [CrossRef]
  18. L. A. Feldkamp, L. C. Davis, J. W. Kress, “Practical cone-beam algorithm,” J. Opt. Soc. Am. A 1, 612–619 (1984).
    [CrossRef]
  19. H. S. Ko, K. D. Kihm, “An extended algebraic reconstruction technique (ART) for density-gradient projections: laser speckle photographic tomography,” Expt. Fluids 27, 542–550 (1999).
    [CrossRef]
  20. M. R. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. 73, 1434–1441 (1983).
    [CrossRef]
  21. G. Vdovin, “Reconstruction of an object shape from the near-field intensity of a reflected paraxial beam,” Appl. Opt. 36, 5508–5513 (1997).
    [CrossRef] [PubMed]
  22. M. Schweiger, S. R. Arridge, M. Hiraoka, D. T. Delpy, “The finite element method for the propagation of light in scattering media: boundary and source conditions,” Med. Phys. 22, 1779–1791 (1995).
    [CrossRef] [PubMed]
  23. A. H. Anderson, “Ray tracing for reconstructive tomography in the presence of object discontinuity boundaries: a comparative analysis of recursive schemes,” J. Opt. Soc. Am. 89, 574–582 (1991).
  24. A. V. Lakshminarayanan, “Reconstruction from divergent ray data,” in Tech. Rep. 92 (Department of Computer Science, State University of New York at Buffalo, 1975).
  25. H. Kudo, T. Saito, “Feasible cone beam scanning methods for exact reconstruction in three-dimensional tomography,” J. Opt. Soc. Am. A 7, 2169–2183 (1990).
    [CrossRef] [PubMed]
  26. B. D. Smith, “Image reconstruction from cone beam projections: necessary and sufficient conditions and reconstruction methods,” IEEE Trans. Med. Imaging MI-4, 14–25 (1985).
    [CrossRef]
  27. H. K. Tuy, “An inversion formula for cone-beam reconstruction,” SIAM J. Appl. Math. 43, 546–552 (1983).
    [CrossRef]
  28. M. Defrise, R. Clack, “A cone-beam reconstruction algorithm using shift-variant filtering and cone-beam backprojection,” IEEE Trans. Med. Imaging 13, 186–195 (1994).
    [CrossRef] [PubMed]
  29. F. Noo, M. Defrise, R. Clack, “Direct reconstruction of cone-beam data acquired with a vertex path containing a circle,” IEEE Trans. Image Process 7, 854–867 (1998).
    [CrossRef]
  30. L. A. Shepp, B. F. Logan, “The Fourier reconstruction of a head section,” IEEE Trans. Nucl. Sci. NS-21, 21–43 (1974).
    [CrossRef]
  31. R. H. Sabersky, A. J. Acosta, E. G. Hauptmann, E. M. Gates, Fluid Flow: A First Course in Fluid Dynamics (Prentice-Hall, Englewood Cliffs, N.J., 1999).
  32. A. H. Anderson, “A ray-tracing approach to restoration and resolution enhancement in experimental ultrasound tomography,” Ultrason. Imaging 12, 268–291 (1990).

2000 (4)

M. Raffel, H. Richard, G. E. A. Meier, “On the applicability of background oriented optical tomography for large scale aerodynamic investigations,” Expt. Fluids 28, 477–481 (2000).
[CrossRef]

R. E. Pierson, D. F. Olson, E. Y. Chen, L. McMackin, “Comparison of reconstruction-algorithm performance for optical-phase tomography of a heated air flow,” Opt. Eng. 39, 838–846 (2000).
[CrossRef]

A. Barty, K. A. Nugent, A. Roberts, D. Peganin, “Quantitative phase tomography,” Opt. Commun. 175, 329–336 (2000).
[CrossRef]

N. Jayshree, G. Keshava Datta, R. M. Vasu, “Optical tomographic microscope for quantitative imaging of phase objects,” Appl. Opt. 39, 277–283 (2000).
[CrossRef]

1999 (2)

G. Keshava Datta, R. M. Vasu, “Non-interferometric methods of phase estimation for application in optical tomography,” J. Mod. Opt. 46, 1377–1388 (1999).

H. S. Ko, K. D. Kihm, “An extended algebraic reconstruction technique (ART) for density-gradient projections: laser speckle photographic tomography,” Expt. Fluids 27, 542–550 (1999).
[CrossRef]

1998 (1)

F. Noo, M. Defrise, R. Clack, “Direct reconstruction of cone-beam data acquired with a vertex path containing a circle,” IEEE Trans. Image Process 7, 854–867 (1998).
[CrossRef]

1997 (1)

1996 (1)

1995 (3)

T. E. Gureyev, A. Roberts, K. A. Nugent, “Phase retrieval with the transport-of-intensity equation: matrix solution with use of Zernike polynomials,” J. Opt. Soc. Am. A 12, 1932–1941 (1995).
[CrossRef]

L. McMackin, B. Masson, N. Clark, K. Bishop, R. Pierson, E. Chen, “Hartmann wavefront sensor studies of dynamic organized structures in flow fields,” AIAA J. 33, 2158–2164 (1995).
[CrossRef]

M. Schweiger, S. R. Arridge, M. Hiraoka, D. T. Delpy, “The finite element method for the propagation of light in scattering media: boundary and source conditions,” Med. Phys. 22, 1779–1791 (1995).
[CrossRef] [PubMed]

1994 (1)

M. Defrise, R. Clack, “A cone-beam reconstruction algorithm using shift-variant filtering and cone-beam backprojection,” IEEE Trans. Med. Imaging 13, 186–195 (1994).
[CrossRef] [PubMed]

1992 (1)

M. H. Maleki, A. J. Devaney, S. Alon, “Tomographic reconstruction from optical scattered intensities,” J. Opt. Soc. Am. 9, 1356–1363 (1992).
[CrossRef]

1991 (2)

C. Meneveau, K. R. Srinivasan, “The multifractal nature of turbulent energy dissipation,” J. Fluid Mech. 224, 429–484 (1991).
[CrossRef]

A. H. Anderson, “Ray tracing for reconstructive tomography in the presence of object discontinuity boundaries: a comparative analysis of recursive schemes,” J. Opt. Soc. Am. 89, 574–582 (1991).

1990 (2)

H. Kudo, T. Saito, “Feasible cone beam scanning methods for exact reconstruction in three-dimensional tomography,” J. Opt. Soc. Am. A 7, 2169–2183 (1990).
[CrossRef] [PubMed]

A. H. Anderson, “A ray-tracing approach to restoration and resolution enhancement in experimental ultrasound tomography,” Ultrason. Imaging 12, 268–291 (1990).

1989 (1)

B. Chhabra, R. V. Jensen, “Direct determination of the f(α) singularity spectrum,” Phy. Rev. Lett. 62, 1327–1330 (1989).
[CrossRef]

1988 (1)

1985 (1)

B. D. Smith, “Image reconstruction from cone beam projections: necessary and sufficient conditions and reconstruction methods,” IEEE Trans. Med. Imaging MI-4, 14–25 (1985).
[CrossRef]

1984 (2)

N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 49, 6–10 (1984).
[CrossRef]

L. A. Feldkamp, L. C. Davis, J. W. Kress, “Practical cone-beam algorithm,” J. Opt. Soc. Am. A 1, 612–619 (1984).
[CrossRef]

1983 (2)

H. K. Tuy, “An inversion formula for cone-beam reconstruction,” SIAM J. Appl. Math. 43, 546–552 (1983).
[CrossRef]

M. R. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. 73, 1434–1441 (1983).
[CrossRef]

1982 (1)

1974 (1)

L. A. Shepp, B. F. Logan, “The Fourier reconstruction of a head section,” IEEE Trans. Nucl. Sci. NS-21, 21–43 (1974).
[CrossRef]

1962 (1)

N. Kolmogorov, “A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reyolds number,” J. Fluid Mech. 13, 82–85 (1962).
[CrossRef]

Acosta, A. J.

R. H. Sabersky, A. J. Acosta, E. G. Hauptmann, E. M. Gates, Fluid Flow: A First Course in Fluid Dynamics (Prentice-Hall, Englewood Cliffs, N.J., 1999).

Alon, S.

M. H. Maleki, A. J. Devaney, S. Alon, “Tomographic reconstruction from optical scattered intensities,” J. Opt. Soc. Am. 9, 1356–1363 (1992).
[CrossRef]

Anderson, A. H.

A. H. Anderson, “Ray tracing for reconstructive tomography in the presence of object discontinuity boundaries: a comparative analysis of recursive schemes,” J. Opt. Soc. Am. 89, 574–582 (1991).

A. H. Anderson, “A ray-tracing approach to restoration and resolution enhancement in experimental ultrasound tomography,” Ultrason. Imaging 12, 268–291 (1990).

Arridge, S. R.

M. Schweiger, S. R. Arridge, M. Hiraoka, D. T. Delpy, “The finite element method for the propagation of light in scattering media: boundary and source conditions,” Med. Phys. 22, 1779–1791 (1995).
[CrossRef] [PubMed]

Barty, A.

A. Barty, K. A. Nugent, A. Roberts, D. Peganin, “Quantitative phase tomography,” Opt. Commun. 175, 329–336 (2000).
[CrossRef]

Bishop, K.

L. McMackin, B. Masson, N. Clark, K. Bishop, R. Pierson, E. Chen, “Hartmann wavefront sensor studies of dynamic organized structures in flow fields,” AIAA J. 33, 2158–2164 (1995).
[CrossRef]

Byer, R. L.

Chen, E.

L. McMackin, B. Masson, N. Clark, K. Bishop, R. Pierson, E. Chen, “Hartmann wavefront sensor studies of dynamic organized structures in flow fields,” AIAA J. 33, 2158–2164 (1995).
[CrossRef]

Chen, E. Y.

R. E. Pierson, D. F. Olson, E. Y. Chen, L. McMackin, “Comparison of reconstruction-algorithm performance for optical-phase tomography of a heated air flow,” Opt. Eng. 39, 838–846 (2000).
[CrossRef]

Chhabra, B.

B. Chhabra, R. V. Jensen, “Direct determination of the f(α) singularity spectrum,” Phy. Rev. Lett. 62, 1327–1330 (1989).
[CrossRef]

Clack, R.

F. Noo, M. Defrise, R. Clack, “Direct reconstruction of cone-beam data acquired with a vertex path containing a circle,” IEEE Trans. Image Process 7, 854–867 (1998).
[CrossRef]

M. Defrise, R. Clack, “A cone-beam reconstruction algorithm using shift-variant filtering and cone-beam backprojection,” IEEE Trans. Med. Imaging 13, 186–195 (1994).
[CrossRef] [PubMed]

Clark, N.

L. McMackin, B. Masson, N. Clark, K. Bishop, R. Pierson, E. Chen, “Hartmann wavefront sensor studies of dynamic organized structures in flow fields,” AIAA J. 33, 2158–2164 (1995).
[CrossRef]

Creath, K.

K. Creath, “Phase measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (North Holland, Amsterdam, 1988), pp. 349–392.
[CrossRef]

Davis, L. C.

Defrise, M.

F. Noo, M. Defrise, R. Clack, “Direct reconstruction of cone-beam data acquired with a vertex path containing a circle,” IEEE Trans. Image Process 7, 854–867 (1998).
[CrossRef]

M. Defrise, R. Clack, “A cone-beam reconstruction algorithm using shift-variant filtering and cone-beam backprojection,” IEEE Trans. Med. Imaging 13, 186–195 (1994).
[CrossRef] [PubMed]

Delpy, D. T.

M. Schweiger, S. R. Arridge, M. Hiraoka, D. T. Delpy, “The finite element method for the propagation of light in scattering media: boundary and source conditions,” Med. Phys. 22, 1779–1791 (1995).
[CrossRef] [PubMed]

Devaney, A. J.

M. H. Maleki, A. J. Devaney, S. Alon, “Tomographic reconstruction from optical scattered intensities,” J. Opt. Soc. Am. 9, 1356–1363 (1992).
[CrossRef]

Faris, G. W.

Feldkamp, L. A.

Fienup, J. R.

Gates, E. M.

R. H. Sabersky, A. J. Acosta, E. G. Hauptmann, E. M. Gates, Fluid Flow: A First Course in Fluid Dynamics (Prentice-Hall, Englewood Cliffs, N.J., 1999).

Gureyev, T. E.

Hauptmann, E. G.

R. H. Sabersky, A. J. Acosta, E. G. Hauptmann, E. M. Gates, Fluid Flow: A First Course in Fluid Dynamics (Prentice-Hall, Englewood Cliffs, N.J., 1999).

Hiraoka, M.

M. Schweiger, S. R. Arridge, M. Hiraoka, D. T. Delpy, “The finite element method for the propagation of light in scattering media: boundary and source conditions,” Med. Phys. 22, 1779–1791 (1995).
[CrossRef] [PubMed]

Jayshree, N.

Jensen, R. V.

B. Chhabra, R. V. Jensen, “Direct determination of the f(α) singularity spectrum,” Phy. Rev. Lett. 62, 1327–1330 (1989).
[CrossRef]

Kak, A. C.

A. C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (IEEE Press, New York, 1988).

Keshava Datta, G.

N. Jayshree, G. Keshava Datta, R. M. Vasu, “Optical tomographic microscope for quantitative imaging of phase objects,” Appl. Opt. 39, 277–283 (2000).
[CrossRef]

G. Keshava Datta, R. M. Vasu, “Non-interferometric methods of phase estimation for application in optical tomography,” J. Mod. Opt. 46, 1377–1388 (1999).

Kihm, K. D.

H. S. Ko, K. D. Kihm, “An extended algebraic reconstruction technique (ART) for density-gradient projections: laser speckle photographic tomography,” Expt. Fluids 27, 542–550 (1999).
[CrossRef]

Ko, H. S.

H. S. Ko, K. D. Kihm, “An extended algebraic reconstruction technique (ART) for density-gradient projections: laser speckle photographic tomography,” Expt. Fluids 27, 542–550 (1999).
[CrossRef]

Kolmogorov, N.

N. Kolmogorov, “A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reyolds number,” J. Fluid Mech. 13, 82–85 (1962).
[CrossRef]

Kress, J. W.

Kudo, H.

Lakshminarayanan, A. V.

A. V. Lakshminarayanan, “Reconstruction from divergent ray data,” in Tech. Rep. 92 (Department of Computer Science, State University of New York at Buffalo, 1975).

Logan, B. F.

L. A. Shepp, B. F. Logan, “The Fourier reconstruction of a head section,” IEEE Trans. Nucl. Sci. NS-21, 21–43 (1974).
[CrossRef]

Maleki, M. H.

M. H. Maleki, A. J. Devaney, S. Alon, “Tomographic reconstruction from optical scattered intensities,” J. Opt. Soc. Am. 9, 1356–1363 (1992).
[CrossRef]

Masson, B.

L. McMackin, B. Masson, N. Clark, K. Bishop, R. Pierson, E. Chen, “Hartmann wavefront sensor studies of dynamic organized structures in flow fields,” AIAA J. 33, 2158–2164 (1995).
[CrossRef]

McMackin, L.

R. E. Pierson, D. F. Olson, E. Y. Chen, L. McMackin, “Comparison of reconstruction-algorithm performance for optical-phase tomography of a heated air flow,” Opt. Eng. 39, 838–846 (2000).
[CrossRef]

L. McMackin, B. Masson, N. Clark, K. Bishop, R. Pierson, E. Chen, “Hartmann wavefront sensor studies of dynamic organized structures in flow fields,” AIAA J. 33, 2158–2164 (1995).
[CrossRef]

Meier, G. E. A.

M. Raffel, H. Richard, G. E. A. Meier, “On the applicability of background oriented optical tomography for large scale aerodynamic investigations,” Expt. Fluids 28, 477–481 (2000).
[CrossRef]

Meneveau, C.

C. Meneveau, K. R. Srinivasan, “The multifractal nature of turbulent energy dissipation,” J. Fluid Mech. 224, 429–484 (1991).
[CrossRef]

Noo, F.

F. Noo, M. Defrise, R. Clack, “Direct reconstruction of cone-beam data acquired with a vertex path containing a circle,” IEEE Trans. Image Process 7, 854–867 (1998).
[CrossRef]

Nugent, K. A.

Olson, D. F.

R. E. Pierson, D. F. Olson, E. Y. Chen, L. McMackin, “Comparison of reconstruction-algorithm performance for optical-phase tomography of a heated air flow,” Opt. Eng. 39, 838–846 (2000).
[CrossRef]

Peganin, D.

A. Barty, K. A. Nugent, A. Roberts, D. Peganin, “Quantitative phase tomography,” Opt. Commun. 175, 329–336 (2000).
[CrossRef]

Pierson, R.

L. McMackin, B. Masson, N. Clark, K. Bishop, R. Pierson, E. Chen, “Hartmann wavefront sensor studies of dynamic organized structures in flow fields,” AIAA J. 33, 2158–2164 (1995).
[CrossRef]

Pierson, R. E.

R. E. Pierson, D. F. Olson, E. Y. Chen, L. McMackin, “Comparison of reconstruction-algorithm performance for optical-phase tomography of a heated air flow,” Opt. Eng. 39, 838–846 (2000).
[CrossRef]

Raffel, M.

M. Raffel, H. Richard, G. E. A. Meier, “On the applicability of background oriented optical tomography for large scale aerodynamic investigations,” Expt. Fluids 28, 477–481 (2000).
[CrossRef]

Richard, H.

M. Raffel, H. Richard, G. E. A. Meier, “On the applicability of background oriented optical tomography for large scale aerodynamic investigations,” Expt. Fluids 28, 477–481 (2000).
[CrossRef]

Roberts, A.

Sabersky, R. H.

R. H. Sabersky, A. J. Acosta, E. G. Hauptmann, E. M. Gates, Fluid Flow: A First Course in Fluid Dynamics (Prentice-Hall, Englewood Cliffs, N.J., 1999).

Saito, T.

Schweiger, M.

M. Schweiger, S. R. Arridge, M. Hiraoka, D. T. Delpy, “The finite element method for the propagation of light in scattering media: boundary and source conditions,” Med. Phys. 22, 1779–1791 (1995).
[CrossRef] [PubMed]

Shepp, L. A.

L. A. Shepp, B. F. Logan, “The Fourier reconstruction of a head section,” IEEE Trans. Nucl. Sci. NS-21, 21–43 (1974).
[CrossRef]

Slaney, M.

A. C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (IEEE Press, New York, 1988).

Smith, B. D.

B. D. Smith, “Image reconstruction from cone beam projections: necessary and sufficient conditions and reconstruction methods,” IEEE Trans. Med. Imaging MI-4, 14–25 (1985).
[CrossRef]

Srinivasan, K. R.

C. Meneveau, K. R. Srinivasan, “The multifractal nature of turbulent energy dissipation,” J. Fluid Mech. 224, 429–484 (1991).
[CrossRef]

Streibl, N.

N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 49, 6–10 (1984).
[CrossRef]

Teague, M. R.

Tuy, H. K.

H. K. Tuy, “An inversion formula for cone-beam reconstruction,” SIAM J. Appl. Math. 43, 546–552 (1983).
[CrossRef]

Vasu, R. M.

N. Jayshree, G. Keshava Datta, R. M. Vasu, “Optical tomographic microscope for quantitative imaging of phase objects,” Appl. Opt. 39, 277–283 (2000).
[CrossRef]

G. Keshava Datta, R. M. Vasu, “Non-interferometric methods of phase estimation for application in optical tomography,” J. Mod. Opt. 46, 1377–1388 (1999).

Vdovin, G.

AIAA J. (1)

L. McMackin, B. Masson, N. Clark, K. Bishop, R. Pierson, E. Chen, “Hartmann wavefront sensor studies of dynamic organized structures in flow fields,” AIAA J. 33, 2158–2164 (1995).
[CrossRef]

Appl. Opt. (4)

Expt. Fluids (2)

H. S. Ko, K. D. Kihm, “An extended algebraic reconstruction technique (ART) for density-gradient projections: laser speckle photographic tomography,” Expt. Fluids 27, 542–550 (1999).
[CrossRef]

M. Raffel, H. Richard, G. E. A. Meier, “On the applicability of background oriented optical tomography for large scale aerodynamic investigations,” Expt. Fluids 28, 477–481 (2000).
[CrossRef]

IEEE Trans. Image Process (1)

F. Noo, M. Defrise, R. Clack, “Direct reconstruction of cone-beam data acquired with a vertex path containing a circle,” IEEE Trans. Image Process 7, 854–867 (1998).
[CrossRef]

IEEE Trans. Med. Imaging (2)

M. Defrise, R. Clack, “A cone-beam reconstruction algorithm using shift-variant filtering and cone-beam backprojection,” IEEE Trans. Med. Imaging 13, 186–195 (1994).
[CrossRef] [PubMed]

B. D. Smith, “Image reconstruction from cone beam projections: necessary and sufficient conditions and reconstruction methods,” IEEE Trans. Med. Imaging MI-4, 14–25 (1985).
[CrossRef]

IEEE Trans. Nucl. Sci. (1)

L. A. Shepp, B. F. Logan, “The Fourier reconstruction of a head section,” IEEE Trans. Nucl. Sci. NS-21, 21–43 (1974).
[CrossRef]

J. Fluid Mech. (2)

C. Meneveau, K. R. Srinivasan, “The multifractal nature of turbulent energy dissipation,” J. Fluid Mech. 224, 429–484 (1991).
[CrossRef]

N. Kolmogorov, “A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reyolds number,” J. Fluid Mech. 13, 82–85 (1962).
[CrossRef]

J. Mod. Opt. (1)

G. Keshava Datta, R. M. Vasu, “Non-interferometric methods of phase estimation for application in optical tomography,” J. Mod. Opt. 46, 1377–1388 (1999).

J. Opt. Soc. Am. (3)

M. R. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. 73, 1434–1441 (1983).
[CrossRef]

M. H. Maleki, A. J. Devaney, S. Alon, “Tomographic reconstruction from optical scattered intensities,” J. Opt. Soc. Am. 9, 1356–1363 (1992).
[CrossRef]

A. H. Anderson, “Ray tracing for reconstructive tomography in the presence of object discontinuity boundaries: a comparative analysis of recursive schemes,” J. Opt. Soc. Am. 89, 574–582 (1991).

J. Opt. Soc. Am. A (4)

Med. Phys. (1)

M. Schweiger, S. R. Arridge, M. Hiraoka, D. T. Delpy, “The finite element method for the propagation of light in scattering media: boundary and source conditions,” Med. Phys. 22, 1779–1791 (1995).
[CrossRef] [PubMed]

Opt. Commun. (2)

A. Barty, K. A. Nugent, A. Roberts, D. Peganin, “Quantitative phase tomography,” Opt. Commun. 175, 329–336 (2000).
[CrossRef]

N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 49, 6–10 (1984).
[CrossRef]

Opt. Eng. (1)

R. E. Pierson, D. F. Olson, E. Y. Chen, L. McMackin, “Comparison of reconstruction-algorithm performance for optical-phase tomography of a heated air flow,” Opt. Eng. 39, 838–846 (2000).
[CrossRef]

Phy. Rev. Lett. (1)

B. Chhabra, R. V. Jensen, “Direct determination of the f(α) singularity spectrum,” Phy. Rev. Lett. 62, 1327–1330 (1989).
[CrossRef]

SIAM J. Appl. Math. (1)

H. K. Tuy, “An inversion formula for cone-beam reconstruction,” SIAM J. Appl. Math. 43, 546–552 (1983).
[CrossRef]

Ultrason. Imaging (1)

A. H. Anderson, “A ray-tracing approach to restoration and resolution enhancement in experimental ultrasound tomography,” Ultrason. Imaging 12, 268–291 (1990).

Other (4)

R. H. Sabersky, A. J. Acosta, E. G. Hauptmann, E. M. Gates, Fluid Flow: A First Course in Fluid Dynamics (Prentice-Hall, Englewood Cliffs, N.J., 1999).

A. V. Lakshminarayanan, “Reconstruction from divergent ray data,” in Tech. Rep. 92 (Department of Computer Science, State University of New York at Buffalo, 1975).

A. C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (IEEE Press, New York, 1988).

K. Creath, “Phase measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (North Holland, Amsterdam, 1988), pp. 349–392.
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

Geometry used in the Feldkamp-Davis-Kress cone-beam reconstruction algorithm. S is the source that illuminates the object whose center is at O, the origin of the coordinate system. The source trajectory is shown as a dotted circle around the object. Data collection on the detector array at a view angle φ is schematically shown. The axis of the cone at this view is shown as X φ. The coordinates of any point on the detector system are (Y d , Z d ).

Fig. 2
Fig. 2

Experimental setup used for recording transmitted intensity. Light from a blue LED (λ = 474 nm) is expanded by use of a beam expander (BE), and, by the suitable positioning of the lens L1, a cone beam of the required radius of curvature is obtained, which illuminates the object O. The intensity distribution at two planes marked as I and II are recorded on the CCD with the help of lens L2.

Fig. 3
Fig. 3

Surface plot of the phase data obtained for a spherical wave front in the absence of the flow. These data are also used for calculating the radius of curvature of the spherical wave front.

Fig. 4
Fig. 4

Surface plot of the transmitted wave front reconstructed after the flow is introduced.

Fig. 5
Fig. 5

Some typical cross sections of the phase profile with and without flow. Change in the wave front due to the introduction of the flow is clearly seen.

Fig. 6
Fig. 6

Intensity image of the differential phase change.

Fig. 7
Fig. 7

Reconstructed density distributions in the X-Z planes at distances (y) from the nozzle tip: (a) y = 0.436 cm, (b) y = 1.823 cm, and (c) at y = 3.211 cm. Data obtained from the phase reconstructed, (i) assuming I = constant and (ii) without the assumption of constant intensity.

Fig. 8
Fig. 8

Reconstructed density distributions in the X-Y planes at distances (a) z = -0.5 cm, (b) z = 0 cm, and (c) at z = 0.5 cm. Data obtained from the phase reconstructed, (i) assuming I = constant and (ii) without the assumption of constant intensity.

Fig. 9
Fig. 9

Result of the experiment done for testing the accuracy of the reconstructions. A conical obstruction of a half-angle of 30° is introduced into the jet, and the density in the vicinity is reconstructed. The reconstructed density in the central plane is shown in the figure. Theoretically the oblique shock angle of 46° obtained with a cone of a half-angle of 30° corresponds to a density ratio of 2.42 across the shock. Tomographic reconstruction gave a density ratio to be approximately 2.3.

Equations (20)

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ũx, y, z=ux, y, zexpjkz,
Izx, y1/2 expjϕx, y
k Iz=- · Iϕ,
i,j ΦijAmnij=abDmn,
D=kI/z,
Wmn=expj2πmxaexpj2πnyb, Amnij=2π2imbainabIˆm-i,n-j,
Φij=ab m,nAmnij-1Dmn.
2ϕ=-kI0Iz,
Φij=ab22π2i2b2+j2a2I0 Di,j.
ϕpr=inodesej ΦiΨir,
j ΦjΩ ·IΨjrΨirdΩ+Ω k Iz ΨirdΩ=0; i, j=1, 2,N,
KIϕ=S,
Ki,j=Ω IΨir·Ψjrdr, Si=Ω k Iz ΨirdΩ, ϕ=Φ1, Φ2,, ΦNT
Pt, ξ=nxδt-x·ξdx,
ξ=sin θ cos ϕ, sin θ sin ϕ, cos θ.
nr=14π2  SO2SO+r·xˆφ2 P˜φYdr, Zdrdφ.
Ix, y, Δz/2-Ix, y, -Δz/2Δz,
Ix, y, Δz/2+Ix, y, -Δz/22.
n-1ρ=Gλ,
Gλ=2.2244×10-41+6.7132×10-2λ2,

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