Abstract

A computational tomographic technique has been developed to accurately reconstruct continuous flow fields of a simple shape from severely limited interferometric data. The algorithm is based on iterative reconstruction of the complementary field, the difference between the field to be reconstructed and its estimate. Its advantages lie in the treatment of various ill-posed problems in a unified manner and ease of incorporation of a priori information, even an approximate field shape. In principle it can utilize only available data. Test results demonstrated stable convergence and potential for substantial error reduction with a proper field estimate.

© 1990 Optical Society of America

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References

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  1. H. Tan, D. Modarress, “Algebraic Reconstruction Technique Code for Tomographic Interferometry,” Opt. Eng. 24, 435–440 (1985).
    [CrossRef]
  2. T. F. Zien, W. C. Ragsdale, W. C. Spring, “Quantitative Determination of Three-Dimensional Density Field by Holographic Interferometry,” AIAA J. 13, 841–842 (1975).
    [CrossRef]
  3. D. W. Sweeney, C. M. Vest, “Measurement of Three-Dimensional Temperature Fields Above Heated Sources by Holographic Interferometry,” Int. J. Heat Mass Transfer 17, 1443–1454 (1974).
    [CrossRef]
  4. S. Cha, C. M. Vest, “Tomographic Reconstruction of Strongly Refracting Fields and Its Application to Interferometric Measurements of Boundary Layers,” Appl. Opt. 20, 2787–2794 (1981).
    [CrossRef] [PubMed]
  5. S. R. Deans, The Radon Transform and Some of Its Applications (Wiley, New York, 1983).
  6. S. S. Cha, H. Sun, “Interferometric Tomography of Continuous Fields with Incomplete Projections,” Opt. Lett.14, 299–301 (1989), in press.
    [CrossRef] [PubMed]
  7. R. Rangayyan, A. P. Dhawan, R. Gordon, “Algorithms for Limited-View Computed Tomography: an Automated Bibliography and a Challenge,” Appl. Opt. 24, 4000–4012 (1985).
    [CrossRef] [PubMed]
  8. S. Cha, “Interferometric Tomography for Three-Dimensional Flow Fields via Envelope Function and Orthogonal Series Decomposition,” Opt. Eng. 27, 557–563 (1988).
    [CrossRef]
  9. A. M. Darling, T. J. Hall, M. A. Fiddy, “Stable Noniterative Object Reconstruction from Incomplete Data Using a Priori Knowledge,” J. Opt. Soc. Am. 73, 1466–1469 (1983).
    [CrossRef]
  10. R. N. Bracewell, S. J. Wernecke, “Image Reconstruction Over a Finite Field of View,” J. Opt. Soc. Am. 65, 1342–1346 (1975).
    [CrossRef]
  11. R. M. Lewitt, R. H. T. Bates, “Image Reconstruction from Projections IV: Projection Completion Methods,” Optik 50, 269–278 (1978).
  12. R. A. Brooks, G. H. Weiss, A. J. Talbert, “A New Approach to Interpolation in Computed Tomography,” J. Comput. Assisted Tomogr. 2, 577–585 (1978).
    [CrossRef]
  13. B. E. Oppenheim, “Reconstruction Tomography from Incomplete Projections,” in Reconstruction Tomography in Diagnostic Radiology and Nuclear Medicine, M. M. Ter-Pogossian et al., Eds. (University Park Press, Baltimore, 1977), pp. 155–183.
  14. T. Inouye, “Image Reconstruction with Limited Angle Projection Data,” IEEE Trans. Nucl. Sci. NS-26, 2666–2669 (1979).
  15. B. P. Medoff, W. R. Brody, M. Nassi, A. Macovski, “Iterative Convolution Backprojection Algorithms for Image Reconstruction from Limited Data,” J. Opt. Soc. Am. 73, 1493–1500 (1983).
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  16. C. M. Vest, I. Prikryl, “Tomography by Iterative Convolution: Empirical Study and Application to Interferometry,” Appl. Opt. 23, 2433–2440 (1984).
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    [CrossRef] [PubMed]
  20. R. Gordon, R. M. Rangayyan, “Geometric Deconvolution: a Mega-Algorithm for Limited View Computed Tomography,” IEEE Trans. Biomed. Eng. BME-30, 806–810 (1983).
    [CrossRef]
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  22. T. Sato, S. J. Norton, M. Linzer, O. Ikeda, M. Hirama, “Tomographic Image Reconstruction from Limited Projections Using Iterative Revisions in Image and Transform Spaces,” Appl. Opt. 20, 395–399 (1981).
    [CrossRef] [PubMed]
  23. L. A. Shepp, B. F. Logan, “The Fourier Reconstruction of a Head Section,” IEEE Trans. Nucl. Sci. NS-21, 21–43 (1974).

1988 (1)

S. Cha, “Interferometric Tomography for Three-Dimensional Flow Fields via Envelope Function and Orthogonal Series Decomposition,” Opt. Eng. 27, 557–563 (1988).
[CrossRef]

1985 (3)

1984 (1)

1983 (4)

1982 (1)

1981 (2)

1979 (1)

T. Inouye, “Image Reconstruction with Limited Angle Projection Data,” IEEE Trans. Nucl. Sci. NS-26, 2666–2669 (1979).

1978 (2)

R. M. Lewitt, R. H. T. Bates, “Image Reconstruction from Projections IV: Projection Completion Methods,” Optik 50, 269–278 (1978).

R. A. Brooks, G. H. Weiss, A. J. Talbert, “A New Approach to Interpolation in Computed Tomography,” J. Comput. Assisted Tomogr. 2, 577–585 (1978).
[CrossRef]

1975 (2)

T. F. Zien, W. C. Ragsdale, W. C. Spring, “Quantitative Determination of Three-Dimensional Density Field by Holographic Interferometry,” AIAA J. 13, 841–842 (1975).
[CrossRef]

R. N. Bracewell, S. J. Wernecke, “Image Reconstruction Over a Finite Field of View,” J. Opt. Soc. Am. 65, 1342–1346 (1975).
[CrossRef]

1974 (2)

D. W. Sweeney, C. M. Vest, “Measurement of Three-Dimensional Temperature Fields Above Heated Sources by Holographic Interferometry,” Int. J. Heat Mass Transfer 17, 1443–1454 (1974).
[CrossRef]

L. A. Shepp, B. F. Logan, “The Fourier Reconstruction of a Head Section,” IEEE Trans. Nucl. Sci. NS-21, 21–43 (1974).

Bates, R. H. T.

R. M. Lewitt, R. H. T. Bates, “Image Reconstruction from Projections IV: Projection Completion Methods,” Optik 50, 269–278 (1978).

Bracewell, R.

R. Bracewell, The Fourier Transform and Its Application (McGraw-Hill, New York, 1965).

Bracewell, R. N.

Brody, W. R.

Brooks, R. A.

R. A. Brooks, G. H. Weiss, A. J. Talbert, “A New Approach to Interpolation in Computed Tomography,” J. Comput. Assisted Tomogr. 2, 577–585 (1978).
[CrossRef]

Cha, S.

S. Cha, “Interferometric Tomography for Three-Dimensional Flow Fields via Envelope Function and Orthogonal Series Decomposition,” Opt. Eng. 27, 557–563 (1988).
[CrossRef]

S. Cha, C. M. Vest, “Tomographic Reconstruction of Strongly Refracting Fields and Its Application to Interferometric Measurements of Boundary Layers,” Appl. Opt. 20, 2787–2794 (1981).
[CrossRef] [PubMed]

Cha, S. S.

S. S. Cha, H. Sun, “Interferometric Tomography of Continuous Fields with Incomplete Projections,” Opt. Lett.14, 299–301 (1989), in press.
[CrossRef] [PubMed]

Darling, A. M.

Deans, S. R.

S. R. Deans, The Radon Transform and Some of Its Applications (Wiley, New York, 1983).

Dhawan, A. P.

Fiddy, M. A.

Fienup, J. R.

Gordon, R.

Hall, T. J.

Hanson, K. M.

Hirama, M.

Ikeda, O.

Inouye, T.

T. Inouye, “Image Reconstruction with Limited Angle Projection Data,” IEEE Trans. Nucl. Sci. NS-26, 2666–2669 (1979).

Lewitt, R. M.

R. M. Lewitt, R. H. T. Bates, “Image Reconstruction from Projections IV: Projection Completion Methods,” Optik 50, 269–278 (1978).

Linzer, M.

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).

Macovski, A.

Medoff, B. P.

Modarress, D.

H. Tan, D. Modarress, “Algebraic Reconstruction Technique Code for Tomographic Interferometry,” Opt. Eng. 24, 435–440 (1985).
[CrossRef]

Nassi, M.

Norton, S. J.

Oppenheim, B. E.

B. E. Oppenheim, “Reconstruction Tomography from Incomplete Projections,” in Reconstruction Tomography in Diagnostic Radiology and Nuclear Medicine, M. M. Ter-Pogossian et al., Eds. (University Park Press, Baltimore, 1977), pp. 155–183.

Prikryl, I.

Ragsdale, W. C.

T. F. Zien, W. C. Ragsdale, W. C. Spring, “Quantitative Determination of Three-Dimensional Density Field by Holographic Interferometry,” AIAA J. 13, 841–842 (1975).
[CrossRef]

Rangayyan, R.

Rangayyan, R. M.

A. P. Dhawan, R. M. Rangayyan, R. Gordon, “Image Restoration by Wiener Deconvolution in Limited-View Computed Tomography,” Appl. Opt. 24, 4013–4020 (1985).
[CrossRef] [PubMed]

R. Gordon, R. M. Rangayyan, “Geometric Deconvolution: a Mega-Algorithm for Limited View Computed Tomography,” IEEE Trans. Biomed. Eng. BME-30, 806–810 (1983).
[CrossRef]

Sato, T.

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).

Spring, W. C.

T. F. Zien, W. C. Ragsdale, W. C. Spring, “Quantitative Determination of Three-Dimensional Density Field by Holographic Interferometry,” AIAA J. 13, 841–842 (1975).
[CrossRef]

Sun, H.

S. S. Cha, H. Sun, “Interferometric Tomography of Continuous Fields with Incomplete Projections,” Opt. Lett.14, 299–301 (1989), in press.
[CrossRef] [PubMed]

Sweeney, D. W.

D. W. Sweeney, C. M. Vest, “Measurement of Three-Dimensional Temperature Fields Above Heated Sources by Holographic Interferometry,” Int. J. Heat Mass Transfer 17, 1443–1454 (1974).
[CrossRef]

Talbert, A. J.

R. A. Brooks, G. H. Weiss, A. J. Talbert, “A New Approach to Interpolation in Computed Tomography,” J. Comput. Assisted Tomogr. 2, 577–585 (1978).
[CrossRef]

Tan, H.

H. Tan, D. Modarress, “Algebraic Reconstruction Technique Code for Tomographic Interferometry,” Opt. Eng. 24, 435–440 (1985).
[CrossRef]

Vest, C. M.

Wecksung, G. W.

Weiss, G. H.

R. A. Brooks, G. H. Weiss, A. J. Talbert, “A New Approach to Interpolation in Computed Tomography,” J. Comput. Assisted Tomogr. 2, 577–585 (1978).
[CrossRef]

Wernecke, S. J.

Zien, T. F.

T. F. Zien, W. C. Ragsdale, W. C. Spring, “Quantitative Determination of Three-Dimensional Density Field by Holographic Interferometry,” AIAA J. 13, 841–842 (1975).
[CrossRef]

AIAA J. (1)

T. F. Zien, W. C. Ragsdale, W. C. Spring, “Quantitative Determination of Three-Dimensional Density Field by Holographic Interferometry,” AIAA J. 13, 841–842 (1975).
[CrossRef]

Appl. Opt. (6)

IEEE Trans. Biomed. Eng. (1)

R. Gordon, R. M. Rangayyan, “Geometric Deconvolution: a Mega-Algorithm for Limited View Computed Tomography,” IEEE Trans. Biomed. Eng. BME-30, 806–810 (1983).
[CrossRef]

IEEE Trans. Nucl. Sci. (2)

T. Inouye, “Image Reconstruction with Limited Angle Projection Data,” IEEE Trans. Nucl. Sci. NS-26, 2666–2669 (1979).

L. A. Shepp, B. F. Logan, “The Fourier Reconstruction of a Head Section,” IEEE Trans. Nucl. Sci. NS-21, 21–43 (1974).

Int. J. Heat Mass Transfer (1)

D. W. Sweeney, C. M. Vest, “Measurement of Three-Dimensional Temperature Fields Above Heated Sources by Holographic Interferometry,” Int. J. Heat Mass Transfer 17, 1443–1454 (1974).
[CrossRef]

J. Comput. Assisted Tomogr. (1)

R. A. Brooks, G. H. Weiss, A. J. Talbert, “A New Approach to Interpolation in Computed Tomography,” J. Comput. Assisted Tomogr. 2, 577–585 (1978).
[CrossRef]

J. Opt. Soc. Am. (4)

Opt. Eng. (2)

S. Cha, “Interferometric Tomography for Three-Dimensional Flow Fields via Envelope Function and Orthogonal Series Decomposition,” Opt. Eng. 27, 557–563 (1988).
[CrossRef]

H. Tan, D. Modarress, “Algebraic Reconstruction Technique Code for Tomographic Interferometry,” Opt. Eng. 24, 435–440 (1985).
[CrossRef]

Optik (1)

R. M. Lewitt, R. H. T. Bates, “Image Reconstruction from Projections IV: Projection Completion Methods,” Optik 50, 269–278 (1978).

Other (4)

B. E. Oppenheim, “Reconstruction Tomography from Incomplete Projections,” in Reconstruction Tomography in Diagnostic Radiology and Nuclear Medicine, M. M. Ter-Pogossian et al., Eds. (University Park Press, Baltimore, 1977), pp. 155–183.

S. R. Deans, The Radon Transform and Some of Its Applications (Wiley, New York, 1983).

S. S. Cha, H. Sun, “Interferometric Tomography of Continuous Fields with Incomplete Projections,” Opt. Lett.14, 299–301 (1989), in press.
[CrossRef] [PubMed]

R. Bracewell, The Fourier Transform and Its Application (McGraw-Hill, New York, 1965).

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Figures (8)

Fig. 1
Fig. 1

Projection of a refractive index field in multidirectional interferometry.

Fig. 2
Fig. 2

Plots of the test fields: (a) four-hump field and (b) one-hump field.

Fig. 3
Fig. 3

Iterative reduction of percent errors of incomplete projection reconstruction: (a) four-hump field and (b) one-hump field.

Fig. 4
Fig. 4

Plots of the CFM reconstructed four-hump field from incomplete projections: (a) after one iteration and (b) after twelve iterations.

Fig. 5
Fig. 5

Plots of the CFM reconstructed one-hump field from incomplete projections: (a) after one iteration and (b) after twelve iterations.

Fig. 6
Fig. 6

Iterative reduction of percent errors of limited view angle reconstruction: (a) four-hump field and (b) one-hump field.

Fig. 7
Fig. 7

Plots of the CFM reconstructed four-hump field from limited view angle reconstruction: (a) after one iteration and (b) after twelve iterations.

Fig. 8
Fig. 8

Plots of the CFM reconstructed one-hump field from limited view angle reconstruction: (a) after one iteration and (b) after twelve iterations.

Equations (16)

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g e ( ρ , θ ) = L { f e ( x , y ) } ,
g c ( ρ , θ ) = g ( ρ , θ ) - g e ( ρ , θ ) .
f c ( x , y ) = R { g c ( ρ , θ } ,
f r ( x , y ) = f e ( x , y ) + f c ( x , y ) .
f e new ( x , y ) = C { f r ( x , y ) } ,
f o ( x , y ) = RL { f ( x , y ) } = f ( x , y ) * * h ( x , y ) ,
e o ( x , y ) = f ( x , y ) - f o ( x , y ) = f ( x , y ) * * [ δ ( x , y ) - h ( x , y ) ] ,
f r ( x , y ) = RL { f ( x , y ) - f e ( x , y ) } + f e ( x , y ) ,
e r ( x , y ) = f ( x , y ) - f r ( x , y ) = [ f ( x , y ) - f e ( x , y ) ] * * [ δ ( x , y ) - h ( x , y ) ] .
ɛ o = e o ( x , y ) 2 d x d y = F ( u , v ) [ H ( u , v ) - 1 ] 2 d u d v ,
ɛ r = e r ( x , y ) 2 d x d y = [ F ( u , v ) - F e ( u , v ) ] [ H ( u , v ) - 1 ] 2 d u d v ,
F ( u , v ) - F e i + 1 ( u , v ) 2 d u d v < F ( u , v ) - F e i ( u , v ) 2 d u d v .
F ( u , v ) - F e i + 1 ( u , v ) 2 H ( u , v ) - 1 2 d u d v < F ( u , v ) - F e i ( u , v ) 2 H ( u , v ) - 1 2 d u d v .
ɛ r = f ( x , y ) - f r ( x , y ) 2 d x d y f ( x , y ) - f e ( x , y ) 2 d x d y = ɛ e .
f ( x , y ) = 3.0 exp { - 6 [ ( x - 0.6 ) 2 + y 2 ] 1 - ( x 2 + y 2 ) } + 1.5 exp { - 6 [ ( x - 0.6 ) 2 + y 2 ] 1 - ( x 2 + y 2 ) } + 3.0 exp { - 6 [ x 2 ( y - 0.6 ) 2 ] 1 - ( x 2 + y 2 ) } + 1.5 exp { - 6 [ x 2 ( y - 0.6 ) 2 ] 1 - ( x 2 + y 2 ) } .
f ( x , y ) = 1.0 ( 1 - x 2 - y 2 ) 1 / 2 exp { 1.0 - 3.0 [ ( x - 0.6 ) 2 + y 2 ] } ,

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