Abstract

In this paper we review and extend the state of the art in the algorithms that have been developed to tomographically reconstruct 1-D and 2-D refractive-index fields in the presence of significant refraction. A perturbation approach and two iterative procedures were tested and compared in numerical simulation of holographic interferometry experiments. Due to the nonlinearity of the problem, it is very difficult to draw general conclusions with respect to the behavior of the iterative algorithms, which is divergent in the examples presented here. In contrast, the perturbation technique, which is the easiest one to implement and the fastest to run, is shown to be very powerful in reducing refraction errors.

© 1987 Optical Society of America

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  1. C. M. Vest, Holographic Interferometry (Wiley, New York, 1979).
  2. C. M. Vest “Tomography for Properties of Materials that Bend Rays: a Tutorial,” Appl. Opt. 24, 4089 (1985).
    [CrossRef] [PubMed]
  3. R. J. Lytle, K. A. Dines, “Iterative Ray Tracing Between Boreholes for Underground Image Reconstruction,” IEEE Trans. Geosci. Remote Sensing GE-18, 234 (1980).
    [CrossRef]
  4. A. H. Andersen, A. C. Kak, “The Application of Ray Tracing Towards a Correction for Refracting Effects in Computed Tomography with Diffracting Sources,” TR-EE 84-14, School of Electrical Engineering, Purdue U. (1984).
  5. S. A. Johnson, J. F. Greenleaf, A. Chu, J. Sjostrand, B. K. Gilbert, E. H. Wood, “Reconstruction of Material Characteristics from Highly Refraction Distorted Projections by Ray Tracing,” in Technical Digest of Topical Meeting on Image Processing for 2-D and 3-D Reconstruction from Projections: Theory and Practice in Medicine and the Physical Sciences (Optical Society of America, Washington, DC, 1975), paper TuB2.
  6. M. Born, E. Wolf, Principles of Optics(Pergamon, New York, 1975).
  7. G. P. Wachtel, “Refraction Error in Interferometry of Boundary Layer in Supersonic Flow Along a Flat Plate,” Ph.D. Thesis, Princeton U. (1951).
  8. W. L. Howes, D. R. Buchele, “A Theory and Method for Applying Interferometry to the Measurement of Certain Two-Dimensional Density Fields,” NACA TN-2693 (1952).
  9. W. L. Howes, D. R. Buchele, “Generalization of Gas-Flow Interferometry Theory and Interferogram Evaluation Equations for One-Dimensional Density Fields,” NACA TN-3340 (1955).
  10. W. L. Howes, D. R. Buchele, “Practical Considerations in Specific Applications of Gas-Flow Interferometry,” NACA TN-3507 (1955).
  11. W. L. Howes, D. R. Buchele, “Optical Interferometry of Inhomogeneous Gases,” J. Opt. Soc. Am. 56, 1517 (1966).
    [CrossRef]
  12. E. E. Anderson, W. H. Stevenson, R. Viskanta, “Estimating the Refractive Error in Optical Measurements of Transport Phenomena,” Appl. Opt. 14, 185 (1975).
    [PubMed]
  13. K. W. Beach, R. H. Muller, C. W. Tobias, “Light-Deflection Effects in the Interferometry of One-Dimensional Refractive-Index Fields,” J. Opt. Soc. Am. 63, 559 (1973).
    [CrossRef]
  14. F. R. McLarnon, R. H. Muller, C. W. Tobias, “Derivation of One-Dimensional Refractive-Index Profiles from Interferograms,” J. Opt. Soc. Am. 65, 1011 (1975).
    [CrossRef]
  15. J. M. Mehta, W. Z. Black, “Errors Associated with Interferometric Measurement of Convective Heat Transfer Coefficients,” Appl. Opt. 16, 1720 (1977).
    [CrossRef] [PubMed]
  16. J. M. Mehta, W. M. Worek, “Analysis of Refraction Errors for Interferometric Measurements in Multicomponent Systems,” Appl. Opt. 23, 928 (1984).
    [CrossRef] [PubMed]
  17. F. W. Schmidt, M. E. Newell, “Evaluation of Refraction Errors in Interferometric Heat Transfer Studies,” Rev. Sci. Instrum. 39, 592 (1968).
    [CrossRef]
  18. H. Svensson, “The Second-Order Aberrations in the Interferometric Measurement of Concentration Gradients,” Opt. Acta 1, 25 (1954).
    [CrossRef]
  19. W. L. Howes, “Rainbow Schlieren vs Mach-Zehnder Interferometer: a Comparison,” Appl. Opt. 24, 816 (1985).
    [CrossRef] [PubMed]
  20. G. D. Kahl, D. C. Mylin, “Refractive Deviation Errors of Interferograms,” J. Opt. Soc. Am. 55, 364 (1965).
    [CrossRef]
  21. C. M. Vest, “Interferometry of Strongly Refracting Axisymmetric Phase Objects,” Appl. Opt. 14, 1601 (1975).
    [CrossRef] [PubMed]
  22. G. Gillman, “Interferometer Focussing Accuracy and the Effect on Interferograms for Density Measurements of Laser Produced Plasmas,” Opt. Commun. 35, 127 (1980).
    [CrossRef]
  23. G. P. Montogomery, D. L. Reuss, “Effects of Refraction on Axisymmetric Flame Temperatures Measured by Holographic Interferometry,” Appl. Opt. 21, 1373 (1982).
    [CrossRef]
  24. D. W. Sweeney, D. T. Attwood, L. W. Coleman, “Interferometric Probing of Laser Produced Plasmas,” Appl. Opt. 15, 1126 (1976).
    [CrossRef] [PubMed]
  25. Y. Maruyama, K. Iwata, R. Nagata, “A Method for Measuring Axially Symmetrical Refractive Index Distribution Using Eikonal Approximation,” Jpn. J. Appl. Phys. 15, 1921 (1976).
    [CrossRef]
  26. V. D. Zimin, P. G. Frik, “Strong Refraction by Axisymmetric Optical Inhomogeneities,” Sov. Phys. Tech. Phys. 21, 233 (1976).
  27. Y. Maruyama, K. Iwata, R. Nagata, “Determination of Axially Symmetrical Refractive Index Distribution from Directions of Emerging Rays,” Appl. Opt. 16, 2500 (1977).
    [CrossRef] [PubMed]
  28. A. M. Hunter, P. W. Schreiber, “Mach-Zehnder Interferometer Data Reduction Method for Refractively Inhomogeneous Test Objects,” Appl. Opt. 14, 634 (1975).
    [CrossRef]
  29. S. Cha, C. M. Vest, “Tomographic Reconstruction of Strongly Refracting Fields and Its Application to Interferometric Measurement of Boundary Layers,” Appl. Opt. 20, 2787 (1981).
    [CrossRef] [PubMed]
  30. I. H. Lira, C. M. Vest, “Perturbation Correction for Refraction in Interferometric Tomography,” Appl. Opt. 26, 774 (1987).
    [CrossRef] [PubMed]
  31. H. Schomberg, “An Improved Approach to Reconstructive Ultrasound Tomography,” J. Phys. D 11, L181 (1978).
    [CrossRef]
  32. R. D. Radcliff, C. A. Balanis, “Electromagnetic Geophysical Imaging Incorporating Refraction and Reflection,” IEEE Trans. Antennas Propag. AP-29, 288 (1981).
    [CrossRef]
  33. R. H. T. Bates, G. C. McKinnon, “Towards Improving Images in Ultrasonic Transmission Tomography,” Australas. Phys. Sci. Med. 2–3, 134 (1979).
  34. G. C. McKinnon, R. H. T. Bates, “A Limitation on Ultrasonic Transmission Tomography,” Ultrasonic Imaging 2, 48 (1980).
    [CrossRef] [PubMed]
  35. A. H. Anderson, A. C. Kak, “Digital Ray Tracing in Two-Dimensional Refractive Fields,” J. Acoust. Soc. Am. 72, 1593 (1982).
    [CrossRef]
  36. S. J. Norton, M. Linzer, “Correcting for Ray Refraction in Velocity and Attenuation Tomography: a Perturbation Approach,” Ultrasonic Imaging 4, 201 (1982).
    [CrossRef] [PubMed]
  37. I. H. Lira, “Correcting for Refraction Effects in Holographic Interferometry of Transparent Objects,” Ph.D. Thesis, U. Michigan (1987).
  38. Y. Censor, “Finite Series-Expansion Reconstruction Methods,” IEEE Proc. 71, 409 (1983).
    [CrossRef]
  39. G. N. Ramachandran, A. V. Lakshminarayanan, “Three Dimensional Reconstruction from Radiographs and Electron Micrographs: Application of Convolution instead of Fourier Transforms,” Proc. Natl. Acad. Sci. U.S.A. 68, 2236 (1971).
    [CrossRef] [PubMed]
  40. L. A. Shepp, B. F. Logan, “The Fourier Reconstruction of a Head Section,” IEEE Trans. Nucl. Sci. NS-21, 21 (1974).
  41. A. H. Andersen, A. C. Kak, “Simultaneous Algebraic Reconstruction Technique (SART): a Superior Implementation of the ART Algorithm,” Ultrasonic Imaging 6, 81 (1984).
    [CrossRef] [PubMed]
  42. B. Gebhart, L. Pera, “The Nature of Vertical Natural Convection Flows Resulting from the Combined Buoyancy Effects on Thermal Mass Difussion,” Int. J. Heat Mass Transfer 14, 2025 (1971).
    [CrossRef]

1987 (1)

1985 (2)

1984 (2)

J. M. Mehta, W. M. Worek, “Analysis of Refraction Errors for Interferometric Measurements in Multicomponent Systems,” Appl. Opt. 23, 928 (1984).
[CrossRef] [PubMed]

A. H. Andersen, A. C. Kak, “Simultaneous Algebraic Reconstruction Technique (SART): a Superior Implementation of the ART Algorithm,” Ultrasonic Imaging 6, 81 (1984).
[CrossRef] [PubMed]

1983 (1)

Y. Censor, “Finite Series-Expansion Reconstruction Methods,” IEEE Proc. 71, 409 (1983).
[CrossRef]

1982 (3)

A. H. Anderson, A. C. Kak, “Digital Ray Tracing in Two-Dimensional Refractive Fields,” J. Acoust. Soc. Am. 72, 1593 (1982).
[CrossRef]

S. J. Norton, M. Linzer, “Correcting for Ray Refraction in Velocity and Attenuation Tomography: a Perturbation Approach,” Ultrasonic Imaging 4, 201 (1982).
[CrossRef] [PubMed]

G. P. Montogomery, D. L. Reuss, “Effects of Refraction on Axisymmetric Flame Temperatures Measured by Holographic Interferometry,” Appl. Opt. 21, 1373 (1982).
[CrossRef]

1981 (2)

R. D. Radcliff, C. A. Balanis, “Electromagnetic Geophysical Imaging Incorporating Refraction and Reflection,” IEEE Trans. Antennas Propag. AP-29, 288 (1981).
[CrossRef]

S. Cha, C. M. Vest, “Tomographic Reconstruction of Strongly Refracting Fields and Its Application to Interferometric Measurement of Boundary Layers,” Appl. Opt. 20, 2787 (1981).
[CrossRef] [PubMed]

1980 (3)

G. Gillman, “Interferometer Focussing Accuracy and the Effect on Interferograms for Density Measurements of Laser Produced Plasmas,” Opt. Commun. 35, 127 (1980).
[CrossRef]

G. C. McKinnon, R. H. T. Bates, “A Limitation on Ultrasonic Transmission Tomography,” Ultrasonic Imaging 2, 48 (1980).
[CrossRef] [PubMed]

R. J. Lytle, K. A. Dines, “Iterative Ray Tracing Between Boreholes for Underground Image Reconstruction,” IEEE Trans. Geosci. Remote Sensing GE-18, 234 (1980).
[CrossRef]

1979 (1)

R. H. T. Bates, G. C. McKinnon, “Towards Improving Images in Ultrasonic Transmission Tomography,” Australas. Phys. Sci. Med. 2–3, 134 (1979).

1978 (1)

H. Schomberg, “An Improved Approach to Reconstructive Ultrasound Tomography,” J. Phys. D 11, L181 (1978).
[CrossRef]

1977 (2)

1976 (3)

D. W. Sweeney, D. T. Attwood, L. W. Coleman, “Interferometric Probing of Laser Produced Plasmas,” Appl. Opt. 15, 1126 (1976).
[CrossRef] [PubMed]

Y. Maruyama, K. Iwata, R. Nagata, “A Method for Measuring Axially Symmetrical Refractive Index Distribution Using Eikonal Approximation,” Jpn. J. Appl. Phys. 15, 1921 (1976).
[CrossRef]

V. D. Zimin, P. G. Frik, “Strong Refraction by Axisymmetric Optical Inhomogeneities,” Sov. Phys. Tech. Phys. 21, 233 (1976).

1975 (4)

1974 (1)

L. A. Shepp, B. F. Logan, “The Fourier Reconstruction of a Head Section,” IEEE Trans. Nucl. Sci. NS-21, 21 (1974).

1973 (1)

1971 (2)

G. N. Ramachandran, A. V. Lakshminarayanan, “Three Dimensional Reconstruction from Radiographs and Electron Micrographs: Application of Convolution instead of Fourier Transforms,” Proc. Natl. Acad. Sci. U.S.A. 68, 2236 (1971).
[CrossRef] [PubMed]

B. Gebhart, L. Pera, “The Nature of Vertical Natural Convection Flows Resulting from the Combined Buoyancy Effects on Thermal Mass Difussion,” Int. J. Heat Mass Transfer 14, 2025 (1971).
[CrossRef]

1968 (1)

F. W. Schmidt, M. E. Newell, “Evaluation of Refraction Errors in Interferometric Heat Transfer Studies,” Rev. Sci. Instrum. 39, 592 (1968).
[CrossRef]

1966 (1)

1965 (1)

1955 (2)

W. L. Howes, D. R. Buchele, “Generalization of Gas-Flow Interferometry Theory and Interferogram Evaluation Equations for One-Dimensional Density Fields,” NACA TN-3340 (1955).

W. L. Howes, D. R. Buchele, “Practical Considerations in Specific Applications of Gas-Flow Interferometry,” NACA TN-3507 (1955).

1954 (1)

H. Svensson, “The Second-Order Aberrations in the Interferometric Measurement of Concentration Gradients,” Opt. Acta 1, 25 (1954).
[CrossRef]

1952 (1)

W. L. Howes, D. R. Buchele, “A Theory and Method for Applying Interferometry to the Measurement of Certain Two-Dimensional Density Fields,” NACA TN-2693 (1952).

Andersen, A. H.

A. H. Andersen, A. C. Kak, “Simultaneous Algebraic Reconstruction Technique (SART): a Superior Implementation of the ART Algorithm,” Ultrasonic Imaging 6, 81 (1984).
[CrossRef] [PubMed]

A. H. Andersen, A. C. Kak, “The Application of Ray Tracing Towards a Correction for Refracting Effects in Computed Tomography with Diffracting Sources,” TR-EE 84-14, School of Electrical Engineering, Purdue U. (1984).

Anderson, A. H.

A. H. Anderson, A. C. Kak, “Digital Ray Tracing in Two-Dimensional Refractive Fields,” J. Acoust. Soc. Am. 72, 1593 (1982).
[CrossRef]

Anderson, E. E.

Attwood, D. T.

Balanis, C. A.

R. D. Radcliff, C. A. Balanis, “Electromagnetic Geophysical Imaging Incorporating Refraction and Reflection,” IEEE Trans. Antennas Propag. AP-29, 288 (1981).
[CrossRef]

Bates, R. H. T.

G. C. McKinnon, R. H. T. Bates, “A Limitation on Ultrasonic Transmission Tomography,” Ultrasonic Imaging 2, 48 (1980).
[CrossRef] [PubMed]

R. H. T. Bates, G. C. McKinnon, “Towards Improving Images in Ultrasonic Transmission Tomography,” Australas. Phys. Sci. Med. 2–3, 134 (1979).

Beach, K. W.

Black, W. Z.

Born, M.

M. Born, E. Wolf, Principles of Optics(Pergamon, New York, 1975).

Buchele, D. R.

W. L. Howes, D. R. Buchele, “Optical Interferometry of Inhomogeneous Gases,” J. Opt. Soc. Am. 56, 1517 (1966).
[CrossRef]

W. L. Howes, D. R. Buchele, “Practical Considerations in Specific Applications of Gas-Flow Interferometry,” NACA TN-3507 (1955).

W. L. Howes, D. R. Buchele, “Generalization of Gas-Flow Interferometry Theory and Interferogram Evaluation Equations for One-Dimensional Density Fields,” NACA TN-3340 (1955).

W. L. Howes, D. R. Buchele, “A Theory and Method for Applying Interferometry to the Measurement of Certain Two-Dimensional Density Fields,” NACA TN-2693 (1952).

Censor, Y.

Y. Censor, “Finite Series-Expansion Reconstruction Methods,” IEEE Proc. 71, 409 (1983).
[CrossRef]

Cha, S.

Chu, A.

S. A. Johnson, J. F. Greenleaf, A. Chu, J. Sjostrand, B. K. Gilbert, E. H. Wood, “Reconstruction of Material Characteristics from Highly Refraction Distorted Projections by Ray Tracing,” in Technical Digest of Topical Meeting on Image Processing for 2-D and 3-D Reconstruction from Projections: Theory and Practice in Medicine and the Physical Sciences (Optical Society of America, Washington, DC, 1975), paper TuB2.

Coleman, L. W.

Dines, K. A.

R. J. Lytle, K. A. Dines, “Iterative Ray Tracing Between Boreholes for Underground Image Reconstruction,” IEEE Trans. Geosci. Remote Sensing GE-18, 234 (1980).
[CrossRef]

Frik, P. G.

V. D. Zimin, P. G. Frik, “Strong Refraction by Axisymmetric Optical Inhomogeneities,” Sov. Phys. Tech. Phys. 21, 233 (1976).

Gebhart, B.

B. Gebhart, L. Pera, “The Nature of Vertical Natural Convection Flows Resulting from the Combined Buoyancy Effects on Thermal Mass Difussion,” Int. J. Heat Mass Transfer 14, 2025 (1971).
[CrossRef]

Gilbert, B. K.

S. A. Johnson, J. F. Greenleaf, A. Chu, J. Sjostrand, B. K. Gilbert, E. H. Wood, “Reconstruction of Material Characteristics from Highly Refraction Distorted Projections by Ray Tracing,” in Technical Digest of Topical Meeting on Image Processing for 2-D and 3-D Reconstruction from Projections: Theory and Practice in Medicine and the Physical Sciences (Optical Society of America, Washington, DC, 1975), paper TuB2.

Gillman, G.

G. Gillman, “Interferometer Focussing Accuracy and the Effect on Interferograms for Density Measurements of Laser Produced Plasmas,” Opt. Commun. 35, 127 (1980).
[CrossRef]

Greenleaf, J. F.

S. A. Johnson, J. F. Greenleaf, A. Chu, J. Sjostrand, B. K. Gilbert, E. H. Wood, “Reconstruction of Material Characteristics from Highly Refraction Distorted Projections by Ray Tracing,” in Technical Digest of Topical Meeting on Image Processing for 2-D and 3-D Reconstruction from Projections: Theory and Practice in Medicine and the Physical Sciences (Optical Society of America, Washington, DC, 1975), paper TuB2.

Howes, W. L.

W. L. Howes, “Rainbow Schlieren vs Mach-Zehnder Interferometer: a Comparison,” Appl. Opt. 24, 816 (1985).
[CrossRef] [PubMed]

W. L. Howes, D. R. Buchele, “Optical Interferometry of Inhomogeneous Gases,” J. Opt. Soc. Am. 56, 1517 (1966).
[CrossRef]

W. L. Howes, D. R. Buchele, “Practical Considerations in Specific Applications of Gas-Flow Interferometry,” NACA TN-3507 (1955).

W. L. Howes, D. R. Buchele, “Generalization of Gas-Flow Interferometry Theory and Interferogram Evaluation Equations for One-Dimensional Density Fields,” NACA TN-3340 (1955).

W. L. Howes, D. R. Buchele, “A Theory and Method for Applying Interferometry to the Measurement of Certain Two-Dimensional Density Fields,” NACA TN-2693 (1952).

Hunter, A. M.

Iwata, K.

Y. Maruyama, K. Iwata, R. Nagata, “Determination of Axially Symmetrical Refractive Index Distribution from Directions of Emerging Rays,” Appl. Opt. 16, 2500 (1977).
[CrossRef] [PubMed]

Y. Maruyama, K. Iwata, R. Nagata, “A Method for Measuring Axially Symmetrical Refractive Index Distribution Using Eikonal Approximation,” Jpn. J. Appl. Phys. 15, 1921 (1976).
[CrossRef]

Johnson, S. A.

S. A. Johnson, J. F. Greenleaf, A. Chu, J. Sjostrand, B. K. Gilbert, E. H. Wood, “Reconstruction of Material Characteristics from Highly Refraction Distorted Projections by Ray Tracing,” in Technical Digest of Topical Meeting on Image Processing for 2-D and 3-D Reconstruction from Projections: Theory and Practice in Medicine and the Physical Sciences (Optical Society of America, Washington, DC, 1975), paper TuB2.

Kahl, G. D.

Kak, A. C.

A. H. Andersen, A. C. Kak, “Simultaneous Algebraic Reconstruction Technique (SART): a Superior Implementation of the ART Algorithm,” Ultrasonic Imaging 6, 81 (1984).
[CrossRef] [PubMed]

A. H. Anderson, A. C. Kak, “Digital Ray Tracing in Two-Dimensional Refractive Fields,” J. Acoust. Soc. Am. 72, 1593 (1982).
[CrossRef]

A. H. Andersen, A. C. Kak, “The Application of Ray Tracing Towards a Correction for Refracting Effects in Computed Tomography with Diffracting Sources,” TR-EE 84-14, School of Electrical Engineering, Purdue U. (1984).

Lakshminarayanan, A. V.

G. N. Ramachandran, A. V. Lakshminarayanan, “Three Dimensional Reconstruction from Radiographs and Electron Micrographs: Application of Convolution instead of Fourier Transforms,” Proc. Natl. Acad. Sci. U.S.A. 68, 2236 (1971).
[CrossRef] [PubMed]

Linzer, M.

S. J. Norton, M. Linzer, “Correcting for Ray Refraction in Velocity and Attenuation Tomography: a Perturbation Approach,” Ultrasonic Imaging 4, 201 (1982).
[CrossRef] [PubMed]

Lira, I. H.

I. H. Lira, C. M. Vest, “Perturbation Correction for Refraction in Interferometric Tomography,” Appl. Opt. 26, 774 (1987).
[CrossRef] [PubMed]

I. H. Lira, “Correcting for Refraction Effects in Holographic Interferometry of Transparent Objects,” Ph.D. Thesis, U. Michigan (1987).

Logan, B. F.

L. A. Shepp, B. F. Logan, “The Fourier Reconstruction of a Head Section,” IEEE Trans. Nucl. Sci. NS-21, 21 (1974).

Lytle, R. J.

R. J. Lytle, K. A. Dines, “Iterative Ray Tracing Between Boreholes for Underground Image Reconstruction,” IEEE Trans. Geosci. Remote Sensing GE-18, 234 (1980).
[CrossRef]

Maruyama, Y.

Y. Maruyama, K. Iwata, R. Nagata, “Determination of Axially Symmetrical Refractive Index Distribution from Directions of Emerging Rays,” Appl. Opt. 16, 2500 (1977).
[CrossRef] [PubMed]

Y. Maruyama, K. Iwata, R. Nagata, “A Method for Measuring Axially Symmetrical Refractive Index Distribution Using Eikonal Approximation,” Jpn. J. Appl. Phys. 15, 1921 (1976).
[CrossRef]

McKinnon, G. C.

G. C. McKinnon, R. H. T. Bates, “A Limitation on Ultrasonic Transmission Tomography,” Ultrasonic Imaging 2, 48 (1980).
[CrossRef] [PubMed]

R. H. T. Bates, G. C. McKinnon, “Towards Improving Images in Ultrasonic Transmission Tomography,” Australas. Phys. Sci. Med. 2–3, 134 (1979).

McLarnon, F. R.

Mehta, J. M.

Montogomery, G. P.

Muller, R. H.

Mylin, D. C.

Nagata, R.

Y. Maruyama, K. Iwata, R. Nagata, “Determination of Axially Symmetrical Refractive Index Distribution from Directions of Emerging Rays,” Appl. Opt. 16, 2500 (1977).
[CrossRef] [PubMed]

Y. Maruyama, K. Iwata, R. Nagata, “A Method for Measuring Axially Symmetrical Refractive Index Distribution Using Eikonal Approximation,” Jpn. J. Appl. Phys. 15, 1921 (1976).
[CrossRef]

Newell, M. E.

F. W. Schmidt, M. E. Newell, “Evaluation of Refraction Errors in Interferometric Heat Transfer Studies,” Rev. Sci. Instrum. 39, 592 (1968).
[CrossRef]

Norton, S. J.

S. J. Norton, M. Linzer, “Correcting for Ray Refraction in Velocity and Attenuation Tomography: a Perturbation Approach,” Ultrasonic Imaging 4, 201 (1982).
[CrossRef] [PubMed]

Pera, L.

B. Gebhart, L. Pera, “The Nature of Vertical Natural Convection Flows Resulting from the Combined Buoyancy Effects on Thermal Mass Difussion,” Int. J. Heat Mass Transfer 14, 2025 (1971).
[CrossRef]

Radcliff, R. D.

R. D. Radcliff, C. A. Balanis, “Electromagnetic Geophysical Imaging Incorporating Refraction and Reflection,” IEEE Trans. Antennas Propag. AP-29, 288 (1981).
[CrossRef]

Ramachandran, G. N.

G. N. Ramachandran, A. V. Lakshminarayanan, “Three Dimensional Reconstruction from Radiographs and Electron Micrographs: Application of Convolution instead of Fourier Transforms,” Proc. Natl. Acad. Sci. U.S.A. 68, 2236 (1971).
[CrossRef] [PubMed]

Reuss, D. L.

Schmidt, F. W.

F. W. Schmidt, M. E. Newell, “Evaluation of Refraction Errors in Interferometric Heat Transfer Studies,” Rev. Sci. Instrum. 39, 592 (1968).
[CrossRef]

Schomberg, H.

H. Schomberg, “An Improved Approach to Reconstructive Ultrasound Tomography,” J. Phys. D 11, L181 (1978).
[CrossRef]

Schreiber, P. W.

Shepp, L. A.

L. A. Shepp, B. F. Logan, “The Fourier Reconstruction of a Head Section,” IEEE Trans. Nucl. Sci. NS-21, 21 (1974).

Sjostrand, J.

S. A. Johnson, J. F. Greenleaf, A. Chu, J. Sjostrand, B. K. Gilbert, E. H. Wood, “Reconstruction of Material Characteristics from Highly Refraction Distorted Projections by Ray Tracing,” in Technical Digest of Topical Meeting on Image Processing for 2-D and 3-D Reconstruction from Projections: Theory and Practice in Medicine and the Physical Sciences (Optical Society of America, Washington, DC, 1975), paper TuB2.

Stevenson, W. H.

Svensson, H.

H. Svensson, “The Second-Order Aberrations in the Interferometric Measurement of Concentration Gradients,” Opt. Acta 1, 25 (1954).
[CrossRef]

Sweeney, D. W.

Tobias, C. W.

Vest, C. M.

Viskanta, R.

Wachtel, G. P.

G. P. Wachtel, “Refraction Error in Interferometry of Boundary Layer in Supersonic Flow Along a Flat Plate,” Ph.D. Thesis, Princeton U. (1951).

Wolf, E.

M. Born, E. Wolf, Principles of Optics(Pergamon, New York, 1975).

Wood, E. H.

S. A. Johnson, J. F. Greenleaf, A. Chu, J. Sjostrand, B. K. Gilbert, E. H. Wood, “Reconstruction of Material Characteristics from Highly Refraction Distorted Projections by Ray Tracing,” in Technical Digest of Topical Meeting on Image Processing for 2-D and 3-D Reconstruction from Projections: Theory and Practice in Medicine and the Physical Sciences (Optical Society of America, Washington, DC, 1975), paper TuB2.

Worek, W. M.

Zimin, V. D.

V. D. Zimin, P. G. Frik, “Strong Refraction by Axisymmetric Optical Inhomogeneities,” Sov. Phys. Tech. Phys. 21, 233 (1976).

Appl. Opt. (12)

C. M. Vest “Tomography for Properties of Materials that Bend Rays: a Tutorial,” Appl. Opt. 24, 4089 (1985).
[CrossRef] [PubMed]

E. E. Anderson, W. H. Stevenson, R. Viskanta, “Estimating the Refractive Error in Optical Measurements of Transport Phenomena,” Appl. Opt. 14, 185 (1975).
[PubMed]

J. M. Mehta, W. Z. Black, “Errors Associated with Interferometric Measurement of Convective Heat Transfer Coefficients,” Appl. Opt. 16, 1720 (1977).
[CrossRef] [PubMed]

J. M. Mehta, W. M. Worek, “Analysis of Refraction Errors for Interferometric Measurements in Multicomponent Systems,” Appl. Opt. 23, 928 (1984).
[CrossRef] [PubMed]

W. L. Howes, “Rainbow Schlieren vs Mach-Zehnder Interferometer: a Comparison,” Appl. Opt. 24, 816 (1985).
[CrossRef] [PubMed]

C. M. Vest, “Interferometry of Strongly Refracting Axisymmetric Phase Objects,” Appl. Opt. 14, 1601 (1975).
[CrossRef] [PubMed]

G. P. Montogomery, D. L. Reuss, “Effects of Refraction on Axisymmetric Flame Temperatures Measured by Holographic Interferometry,” Appl. Opt. 21, 1373 (1982).
[CrossRef]

D. W. Sweeney, D. T. Attwood, L. W. Coleman, “Interferometric Probing of Laser Produced Plasmas,” Appl. Opt. 15, 1126 (1976).
[CrossRef] [PubMed]

Y. Maruyama, K. Iwata, R. Nagata, “Determination of Axially Symmetrical Refractive Index Distribution from Directions of Emerging Rays,” Appl. Opt. 16, 2500 (1977).
[CrossRef] [PubMed]

A. M. Hunter, P. W. Schreiber, “Mach-Zehnder Interferometer Data Reduction Method for Refractively Inhomogeneous Test Objects,” Appl. Opt. 14, 634 (1975).
[CrossRef]

S. Cha, C. M. Vest, “Tomographic Reconstruction of Strongly Refracting Fields and Its Application to Interferometric Measurement of Boundary Layers,” Appl. Opt. 20, 2787 (1981).
[CrossRef] [PubMed]

I. H. Lira, C. M. Vest, “Perturbation Correction for Refraction in Interferometric Tomography,” Appl. Opt. 26, 774 (1987).
[CrossRef] [PubMed]

Australas. Phys. Sci. Med. (1)

R. H. T. Bates, G. C. McKinnon, “Towards Improving Images in Ultrasonic Transmission Tomography,” Australas. Phys. Sci. Med. 2–3, 134 (1979).

IEEE Proc. (1)

Y. Censor, “Finite Series-Expansion Reconstruction Methods,” IEEE Proc. 71, 409 (1983).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

R. D. Radcliff, C. A. Balanis, “Electromagnetic Geophysical Imaging Incorporating Refraction and Reflection,” IEEE Trans. Antennas Propag. AP-29, 288 (1981).
[CrossRef]

IEEE Trans. Geosci. Remote Sensing (1)

R. J. Lytle, K. A. Dines, “Iterative Ray Tracing Between Boreholes for Underground Image Reconstruction,” IEEE Trans. Geosci. Remote Sensing GE-18, 234 (1980).
[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 (1974).

Int. J. Heat Mass Transfer (1)

B. Gebhart, L. Pera, “The Nature of Vertical Natural Convection Flows Resulting from the Combined Buoyancy Effects on Thermal Mass Difussion,” Int. J. Heat Mass Transfer 14, 2025 (1971).
[CrossRef]

J. Acoust. Soc. Am. (1)

A. H. Anderson, A. C. Kak, “Digital Ray Tracing in Two-Dimensional Refractive Fields,” J. Acoust. Soc. Am. 72, 1593 (1982).
[CrossRef]

J. Opt. Soc. Am. (4)

J. Phys. D (1)

H. Schomberg, “An Improved Approach to Reconstructive Ultrasound Tomography,” J. Phys. D 11, L181 (1978).
[CrossRef]

Jpn. J. Appl. Phys. (1)

Y. Maruyama, K. Iwata, R. Nagata, “A Method for Measuring Axially Symmetrical Refractive Index Distribution Using Eikonal Approximation,” Jpn. J. Appl. Phys. 15, 1921 (1976).
[CrossRef]

NACA TN-2693 (1)

W. L. Howes, D. R. Buchele, “A Theory and Method for Applying Interferometry to the Measurement of Certain Two-Dimensional Density Fields,” NACA TN-2693 (1952).

NACA TN-3340 (1)

W. L. Howes, D. R. Buchele, “Generalization of Gas-Flow Interferometry Theory and Interferogram Evaluation Equations for One-Dimensional Density Fields,” NACA TN-3340 (1955).

NACA TN-3507 (1)

W. L. Howes, D. R. Buchele, “Practical Considerations in Specific Applications of Gas-Flow Interferometry,” NACA TN-3507 (1955).

Opt. Acta (1)

H. Svensson, “The Second-Order Aberrations in the Interferometric Measurement of Concentration Gradients,” Opt. Acta 1, 25 (1954).
[CrossRef]

Opt. Commun. (1)

G. Gillman, “Interferometer Focussing Accuracy and the Effect on Interferograms for Density Measurements of Laser Produced Plasmas,” Opt. Commun. 35, 127 (1980).
[CrossRef]

Proc. Natl. Acad. Sci. U.S.A. (1)

G. N. Ramachandran, A. V. Lakshminarayanan, “Three Dimensional Reconstruction from Radiographs and Electron Micrographs: Application of Convolution instead of Fourier Transforms,” Proc. Natl. Acad. Sci. U.S.A. 68, 2236 (1971).
[CrossRef] [PubMed]

Rev. Sci. Instrum. (1)

F. W. Schmidt, M. E. Newell, “Evaluation of Refraction Errors in Interferometric Heat Transfer Studies,” Rev. Sci. Instrum. 39, 592 (1968).
[CrossRef]

Sov. Phys. Tech. Phys. (1)

V. D. Zimin, P. G. Frik, “Strong Refraction by Axisymmetric Optical Inhomogeneities,” Sov. Phys. Tech. Phys. 21, 233 (1976).

Ultrasonic Imaging (3)

S. J. Norton, M. Linzer, “Correcting for Ray Refraction in Velocity and Attenuation Tomography: a Perturbation Approach,” Ultrasonic Imaging 4, 201 (1982).
[CrossRef] [PubMed]

G. C. McKinnon, R. H. T. Bates, “A Limitation on Ultrasonic Transmission Tomography,” Ultrasonic Imaging 2, 48 (1980).
[CrossRef] [PubMed]

A. H. Andersen, A. C. Kak, “Simultaneous Algebraic Reconstruction Technique (SART): a Superior Implementation of the ART Algorithm,” Ultrasonic Imaging 6, 81 (1984).
[CrossRef] [PubMed]

Other (6)

I. H. Lira, “Correcting for Refraction Effects in Holographic Interferometry of Transparent Objects,” Ph.D. Thesis, U. Michigan (1987).

C. M. Vest, Holographic Interferometry (Wiley, New York, 1979).

A. H. Andersen, A. C. Kak, “The Application of Ray Tracing Towards a Correction for Refracting Effects in Computed Tomography with Diffracting Sources,” TR-EE 84-14, School of Electrical Engineering, Purdue U. (1984).

S. A. Johnson, J. F. Greenleaf, A. Chu, J. Sjostrand, B. K. Gilbert, E. H. Wood, “Reconstruction of Material Characteristics from Highly Refraction Distorted Projections by Ray Tracing,” in Technical Digest of Topical Meeting on Image Processing for 2-D and 3-D Reconstruction from Projections: Theory and Practice in Medicine and the Physical Sciences (Optical Society of America, Washington, DC, 1975), paper TuB2.

M. Born, E. Wolf, Principles of Optics(Pergamon, New York, 1975).

G. P. Wachtel, “Refraction Error in Interferometry of Boundary Layer in Supersonic Flow Along a Flat Plate,” Ph.D. Thesis, Princeton U. (1951).

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Figures (11)

Fig. 1
Fig. 1

Interferometric analysis of a 1-D boundary layer-type refractive-index field.

Fig. 2
Fig. 2

Schematic representation of the formation of an interferogram in a 2-D axisymmetric refractive-index field.

Fig. 3
Fig. 3

Schematic representation of the formation of an interferogram in a 2-D asymmetric refractive-index field.

Fig. 4
Fig. 4

Straight line reconstructed profile (dashed line) of the double Gaussian distribution compared to the actual profile (solid line). The focusing plane is at xP = 2.0. In obtaining this figure, twenty viewing angles were considered, with fifty rays per view. Resolution is 40 × 40 pixels. The root-mean square error of this reconstruction is 1.76% and the maximum error is 8.74%.

Fig. 5
Fig. 5

Perturbation-corrected reconstruction (dashed line) of the double Gaussian distribution compared to the actual profile (solid line). Root-mean square error is 0.35%, maximum error is 2.97%.

Fig. 6
Fig. 6

Reconstruction error as a function of the object plane selected for the straight line (solid lines) and perturbation-corrected (dashed lines) reconstructions of the double Gaussian function. Plain line give rms errors, while lines with asterisks give maximum errors.

Fig. 7
Fig. 7

Reconstruction error as a function of inversion number for the reconstruction of the double Gaussian function using the SLIM iterative approach. The solid line gives the rms error, while the dashed line shows the maximum error. For comparison, the perturbation-corrected errors for the second inversion are shown as white and black symbols for the maximum and rms, respectively. The focusing plane is xP = 2.0.

Fig. 8
Fig. 8

Reconstruction error as a function of inversion number for the reconstruction of the double Gaussian function using the CRAI. The solid line gives the rms error, while the dashed line shows the maximum error. For comparison, the perturbation-corrected errors for the second inversion are shown as white and black symbols for the maximum and rms, respectively. The focusing plane is xP =2.0.

Fig. 9
Fig. 9

Temperature distribution in the boundary layer surrounding a heated vertical flat plate submerged in water. Plate temperature is 29° C and ambient water temperature is 20° C. Plate width is 10 cm and the results shown are for a plane 1 cm above the leading edge. In this figure the object plane is at the point where the actual and apparent plate boundaries coincide. The solid curve shows the temperature as given in Ref. 42, using which the interferometric data was generated. The dashed curve shows the temperature obtained with the conventional straight line inversion of the data. The curve with asterisks gives the perturbation-corrected temperature.

Fig. 10
Fig. 10

Reconstruction error as a function of inversion number for the reconstruction of the 1-D temperature profile using the SLIM approach. The solid line gives the rms error, while the dashed line shows the maximum error. For comparison, the perturbation corrected errors for the second inversion are shown as white and black symbols for the maximum and rms, respectively. Percentages are based on 50 pixel values. All other parameters are as in Fig. 9.

Fig. 11
Fig. 11

Reconstruction error as a function of inversion number for the reconstruction of the 1-D temperature profile using the CRAI. The solid line gives the rms error, while the dashed line shows the maximum error. For comparison, the perturbation-corrected errors for the second inversion are shown as white and black symbols for the maximum and rms, respectively. Percentages are based on 50 pixel values. All other parameters are as in Fig. 9.

Equations (47)

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Δ Φ P = Φ P Φ 0 ,
Φ P = A B n ( y ) d s , Φ 0 = n 0 x C ,
d d s ( n d r d s ) = n ,
n y = ( n y n x y ) [ 1 + y 2 ] ,
1 + y 2 = ( n n A ) 2 ,
Φ P = n A 0 L ( 1 + y 2 ) d x ,
Φ P = 1 n A 0 L n 2 d x .
n = n A + i = 1 b i ( y y A ) i .
n = n A + n A ( y y A ) ,
y = y A + l A [ cosh ( x l A ) 1 ] ,
y = y A + 1 2 x 2 l A .
Φ P n A L [ 1 + 1 3 ( L l A ) 2 ] .
Φ 0 = n 0 x C = n 0 [ x P + ( L x P ) 1 + tan 2 ϕ ] ,
tan ϕ L l A .
n A n 0 Δ Φ P L n A 6 [ 2 3 K ( n 0 n A ) ] ( L l A ) 2 ,
n A 1 L d Δ Φ d y y = y P ,
tan ϕ = y B y P x B x P .
y P y A L 2 l A ( 1 2 K ) ,
Δ Φ ¯ P = ( n P n 0 ) L ,
n P n A + n A ( y P y A ) .
Δ Φ ¯ P Δ Φ P = n A L [ 1 6 K ( 1 n 0 2 n A ) ] ( L l A ) 2 .
Δ Φ P = Φ P + 2 n 0 x A + n 0 y A tan ϕ + n 0 x P ( sec ϕ 1 ) ,
y P = y A sec ϕ + x P tan ϕ .
Φ P = 2 r p r 0 η 2 r η 2 p 2 d r ,
θ = 2 r p r 0 p 2 r η 2 p 2 d r .
r ( η ) = r 0 exp [ 1 π η η 0 cosh 1 ( p η ) 1 p d Φ P d p d p ] ,
r ( η ) = r 0 exp [ 1 π η η 0 cosh 1 ( p η ) d θ d p d p ] ,
r ( η ) = r 0 exp [ 1 π η η 0 ϴ p 2 η 2 d p ] .
sin ϕ = d Δ Φ P d y P .
r 0 sin i 0 = x P 2 + y P 2 sin ( ϕ + tan 1 y P x P ) .
n ( η ) = n 0 exp [ 1 π η η 0 ϕ p 2 η 2 d p ] .
n ( η ) = n 0 exp [ 1 π η η 0 cosh 1 ( p η ) 1 p d Δ Φ P d p d P ] .
Δ Φ ¯ k + 1 = Δ Φ ¯ k + ( Δ Φ Δ Φ k ) with k = 1 , 2 , ,
Δ Φ ¯ Δ Φ c ,
c = 1 2 [ ( x C x p ) H ( x C ) x E x C H ( x ) d x ] ,
H ( x ) 1 n 0 [ x E x n P y ( μ ) d μ ] 2 .
c = ( n P y x B ) 2 1 n 0 [ 1 3 x B 1 2 x P ] .
Φ i k h 2 n k ( r i 1 ) + h l = 2 K i 1 n k ( r i l ) + h i n k ( r i K i ) ,
n k ( r i l ) = j = 1 m c ilj k n j k l = 1 , 2 , , K i ,
Φ i k j = 1 m a i j k n j k ,
a i j k h 2 c i 1 j k + h l = 2 K i 1 c ilj k + h i c i K i j k .
Φ 0 i k = n 0 l i k .
Δ Φ k = A k n k n 0 l k ,
Δ Φ = A n n 0 l ,
A k δ n k = Δ Φ Δ Φ k .
n ( x , y ) = n 0 0.01 n 0 { exp [ x 2 + ( y 0.1 ) 2 ] 0.09 + exp [ x 2 + ( y + 0.5 ) 2 ] 0.04 } .
e ( rms ) = 100 max | n i e n 0 | 1 m i = 1 m ( n i e n i r ) 2 , e ( max ) = max | n i e n i r | max | n i e n 0 | × 100 ,

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