Abstract

A modified iterative conjugate gradient technique is developed and applied to limited-view interferometric tomography for reconstructing flow fields of simple structure. The approach is also effective in dealing with significant noise present in the data. From a computer simulation of experiments, a multipath interferometric system is built, from which multidirectional projection interferograms can be captured in real time for fast transient phenomena. The system is employed to measure an asymmetric temperature field. The three-dimensional field is then reconstructed with the developed experimental and computational techniques. The results are satisfactory when compared with thermocouple measurements.

© 1998 Optical Society of America

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References

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  1. D. J. Cha, S. S. Cha, “Holographic interferometric tomography for limited data reconstruction,” AIAA J. 34, 1019–1026 (1996).
    [CrossRef]
  2. H. M. Hertz, “Experiment determination of 2-D flame temperature field by interferometric tomography,” Opt. Commun. 54, 131–136 (1985).
    [CrossRef]
  3. R. Snyder, L. Hesselink, “Measurement of mixing fluid flows with optical tomography,” Opt. Lett. 13, 87–89 (1988).
    [CrossRef] [PubMed]
  4. A. He, D. Yan, “Tomography of 3-D temperature field by laser image plane holography,” in Proceedings of Laser Interferometry, R. J. Pryputniewicz, ed., Proc. SPIE1162, 411–418 (1989).
  5. D. W. Sweeney, R. Goulard, Combustion Measurement (Hemisphere, New York, 1976).
  6. S. Bahl, J. A. Liburdy, “Measurement of local convective heat transfer coefficient using three-dimensional interferometry,” Int. J. Heat Mass Transfer 34, 949–960 (1991).
    [CrossRef]
  7. E. Yu, S. S. Cha, A. W. Burner, “Interferometric tomographic measurement of an instantaneous flow field under adverse environments,” in Proceedings of Optical Techniques in Fluid, Thermal and Combustion Flow, S. S. Cha, J. D. Trolinger, eds., Proc. SPIE2546, 33–44 (1995).
    [CrossRef]
  8. S. S. Cha, “Interferometric tomography for three-dimensional flow fields via envelope function and orthogonal series decomposition,” Opt. Eng. 27, 557–563 (1988).
    [CrossRef]
  9. S. S. Cha, “Holographic interferometric tomography for reconstructing flow fields: a review,” presented at the 17th Aerospace Ground Testing Conference, Nashville, Tenn., 1992, AIAA paper 92-3934.
  10. D. W. Sweeney, C. M. Vest, “Reconstruction of three dimensional refractive-index field from multidirectional interferometric data,” Appl. Opt. 12, 2649–2664 (1973).
    [CrossRef] [PubMed]
  11. S. Kawata, O. Nalcioglu, “Constrained iterative reconstruction by the conjugate gradient method,” IEEE Trans. Med. Imaging M1-4, 65–71 (1985).
    [CrossRef]
  12. A. He, D. Yan, X. Ni, “Large-aperture and long-path interferometer for measuring transient thermal field,” Opt. Eng. 27, 841–844 (1988).
    [CrossRef]
  13. D. Yan, J. Zhang, A. He, X. Mi, Y. Ge, “Automatic extraction and registration of shock wave fronts from series interferograms of a flow field,” Appl. Opt. 33, 2121–2124 (1994).
    [CrossRef] [PubMed]
  14. C. M. Vest, Holographic Interferometry (Wiley, New York, 1979).

1996

D. J. Cha, S. S. Cha, “Holographic interferometric tomography for limited data reconstruction,” AIAA J. 34, 1019–1026 (1996).
[CrossRef]

1994

1991

S. Bahl, J. A. Liburdy, “Measurement of local convective heat transfer coefficient using three-dimensional interferometry,” Int. J. Heat Mass Transfer 34, 949–960 (1991).
[CrossRef]

1988

S. S. Cha, “Interferometric tomography for three-dimensional flow fields via envelope function and orthogonal series decomposition,” Opt. Eng. 27, 557–563 (1988).
[CrossRef]

A. He, D. Yan, X. Ni, “Large-aperture and long-path interferometer for measuring transient thermal field,” Opt. Eng. 27, 841–844 (1988).
[CrossRef]

R. Snyder, L. Hesselink, “Measurement of mixing fluid flows with optical tomography,” Opt. Lett. 13, 87–89 (1988).
[CrossRef] [PubMed]

1985

S. Kawata, O. Nalcioglu, “Constrained iterative reconstruction by the conjugate gradient method,” IEEE Trans. Med. Imaging M1-4, 65–71 (1985).
[CrossRef]

H. M. Hertz, “Experiment determination of 2-D flame temperature field by interferometric tomography,” Opt. Commun. 54, 131–136 (1985).
[CrossRef]

1973

Bahl, S.

S. Bahl, J. A. Liburdy, “Measurement of local convective heat transfer coefficient using three-dimensional interferometry,” Int. J. Heat Mass Transfer 34, 949–960 (1991).
[CrossRef]

Burner, A. W.

E. Yu, S. S. Cha, A. W. Burner, “Interferometric tomographic measurement of an instantaneous flow field under adverse environments,” in Proceedings of Optical Techniques in Fluid, Thermal and Combustion Flow, S. S. Cha, J. D. Trolinger, eds., Proc. SPIE2546, 33–44 (1995).
[CrossRef]

Cha, D. J.

D. J. Cha, S. S. Cha, “Holographic interferometric tomography for limited data reconstruction,” AIAA J. 34, 1019–1026 (1996).
[CrossRef]

Cha, S. S.

D. J. Cha, S. S. Cha, “Holographic interferometric tomography for limited data reconstruction,” AIAA J. 34, 1019–1026 (1996).
[CrossRef]

S. S. Cha, “Interferometric tomography for three-dimensional flow fields via envelope function and orthogonal series decomposition,” Opt. Eng. 27, 557–563 (1988).
[CrossRef]

E. Yu, S. S. Cha, A. W. Burner, “Interferometric tomographic measurement of an instantaneous flow field under adverse environments,” in Proceedings of Optical Techniques in Fluid, Thermal and Combustion Flow, S. S. Cha, J. D. Trolinger, eds., Proc. SPIE2546, 33–44 (1995).
[CrossRef]

S. S. Cha, “Holographic interferometric tomography for reconstructing flow fields: a review,” presented at the 17th Aerospace Ground Testing Conference, Nashville, Tenn., 1992, AIAA paper 92-3934.

Ge, Y.

Goulard, R.

D. W. Sweeney, R. Goulard, Combustion Measurement (Hemisphere, New York, 1976).

He, A.

D. Yan, J. Zhang, A. He, X. Mi, Y. Ge, “Automatic extraction and registration of shock wave fronts from series interferograms of a flow field,” Appl. Opt. 33, 2121–2124 (1994).
[CrossRef] [PubMed]

A. He, D. Yan, X. Ni, “Large-aperture and long-path interferometer for measuring transient thermal field,” Opt. Eng. 27, 841–844 (1988).
[CrossRef]

A. He, D. Yan, “Tomography of 3-D temperature field by laser image plane holography,” in Proceedings of Laser Interferometry, R. J. Pryputniewicz, ed., Proc. SPIE1162, 411–418 (1989).

Hertz, H. M.

H. M. Hertz, “Experiment determination of 2-D flame temperature field by interferometric tomography,” Opt. Commun. 54, 131–136 (1985).
[CrossRef]

Hesselink, L.

Kawata, S.

S. Kawata, O. Nalcioglu, “Constrained iterative reconstruction by the conjugate gradient method,” IEEE Trans. Med. Imaging M1-4, 65–71 (1985).
[CrossRef]

Liburdy, J. A.

S. Bahl, J. A. Liburdy, “Measurement of local convective heat transfer coefficient using three-dimensional interferometry,” Int. J. Heat Mass Transfer 34, 949–960 (1991).
[CrossRef]

Mi, X.

Nalcioglu, O.

S. Kawata, O. Nalcioglu, “Constrained iterative reconstruction by the conjugate gradient method,” IEEE Trans. Med. Imaging M1-4, 65–71 (1985).
[CrossRef]

Ni, X.

A. He, D. Yan, X. Ni, “Large-aperture and long-path interferometer for measuring transient thermal field,” Opt. Eng. 27, 841–844 (1988).
[CrossRef]

Snyder, R.

Sweeney, D. W.

Vest, C. M.

Yan, D.

D. Yan, J. Zhang, A. He, X. Mi, Y. Ge, “Automatic extraction and registration of shock wave fronts from series interferograms of a flow field,” Appl. Opt. 33, 2121–2124 (1994).
[CrossRef] [PubMed]

A. He, D. Yan, X. Ni, “Large-aperture and long-path interferometer for measuring transient thermal field,” Opt. Eng. 27, 841–844 (1988).
[CrossRef]

A. He, D. Yan, “Tomography of 3-D temperature field by laser image plane holography,” in Proceedings of Laser Interferometry, R. J. Pryputniewicz, ed., Proc. SPIE1162, 411–418 (1989).

Yu, E.

E. Yu, S. S. Cha, A. W. Burner, “Interferometric tomographic measurement of an instantaneous flow field under adverse environments,” in Proceedings of Optical Techniques in Fluid, Thermal and Combustion Flow, S. S. Cha, J. D. Trolinger, eds., Proc. SPIE2546, 33–44 (1995).
[CrossRef]

Zhang, J.

AIAA J.

D. J. Cha, S. S. Cha, “Holographic interferometric tomography for limited data reconstruction,” AIAA J. 34, 1019–1026 (1996).
[CrossRef]

Appl. Opt.

IEEE Trans. Med. Imaging

S. Kawata, O. Nalcioglu, “Constrained iterative reconstruction by the conjugate gradient method,” IEEE Trans. Med. Imaging M1-4, 65–71 (1985).
[CrossRef]

Int. J. Heat Mass Transfer

S. Bahl, J. A. Liburdy, “Measurement of local convective heat transfer coefficient using three-dimensional interferometry,” Int. J. Heat Mass Transfer 34, 949–960 (1991).
[CrossRef]

Opt. Commun.

H. M. Hertz, “Experiment determination of 2-D flame temperature field by interferometric tomography,” Opt. Commun. 54, 131–136 (1985).
[CrossRef]

Opt. Eng.

S. S. Cha, “Interferometric tomography for three-dimensional flow fields via envelope function and orthogonal series decomposition,” Opt. Eng. 27, 557–563 (1988).
[CrossRef]

A. He, D. Yan, X. Ni, “Large-aperture and long-path interferometer for measuring transient thermal field,” Opt. Eng. 27, 841–844 (1988).
[CrossRef]

Opt. Lett.

Other

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

S. S. Cha, “Holographic interferometric tomography for reconstructing flow fields: a review,” presented at the 17th Aerospace Ground Testing Conference, Nashville, Tenn., 1992, AIAA paper 92-3934.

E. Yu, S. S. Cha, A. W. Burner, “Interferometric tomographic measurement of an instantaneous flow field under adverse environments,” in Proceedings of Optical Techniques in Fluid, Thermal and Combustion Flow, S. S. Cha, J. D. Trolinger, eds., Proc. SPIE2546, 33–44 (1995).
[CrossRef]

A. He, D. Yan, “Tomography of 3-D temperature field by laser image plane holography,” in Proceedings of Laser Interferometry, R. J. Pryputniewicz, ed., Proc. SPIE1162, 411–418 (1989).

D. W. Sweeney, R. Goulard, Combustion Measurement (Hemisphere, New York, 1976).

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Figures (7)

Fig. 1
Fig. 1

Effects of the number of projections on reconstruction: (a) average errors, (b) maximum errors.

Fig. 2
Fig. 2

Iterative behavior of the relative differences between two peak values of the original and reconstructed fields.

Fig. 3
Fig. 3

Reconstruction results of the limited-projection data with high-level noise.

Fig. 4
Fig. 4

Plot of the exact and reconstructed fields: (a) exact; (b) reconstructed.

Fig. 5
Fig. 5

Multipath interferometric system for simultaneously capturing multidirectional projections in real time: O, object; L’s, lenses; D’s, diaphragms; P’s, plate mirrors; R’s, recording systems; M’s, mirrors; BS’s, beam splitters.

Fig. 6
Fig. 6

Interferograms captured at different scanning angles of (a) 0°, (b) 30°, (c) 60°, (d) 90°.

Fig. 7
Fig. 7

Plot of a cross section of the reconstructed temperature field.

Equations (14)

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f x ,   y = m = 1 M n = 1 N   f mn b mn x ,   y ,
b mn x ,   y = sinc 2 l x x - ml x sinc 2 l y y - nl y
f mn = f ml x ,   nl y .
Δ Φ p ,   θ = m = 1 M n = 1 N   f mn a mn p ,   θ ,
a mn p ,   θ = ray   b mn x ,   y d s .
Φ = AF ,
R = AF a - Φ .
A T A - σ I F = A T Φ ,
f min f x ,   y 0 .
k = k + 1 ; Q k = P k - 1 T G k - 1 P k - 1 T A T A + σ I P k - 1 ;   G k = G k - 1 - Q k A T A + σ I P k - 1 ; E k = G k A T A + σ I P k - 1 P k - 1 T A T A + σ I P k - 1 ; P k = G k - E k P k - 1 ;   and F k = F k - 1 + Q k P k - 1 .
k = k + 1 ; G k = A T R k 1 ; E k = G k G k - 1 ;   P k = G k   for   k = 1   and   G k - E k P k - 1   for   k 2 ; Q k = G k AP k ;   and F k = F k - 1 + Q k P k .
f x ,   y = exp - 12 x - 0.5 2 + y 2 + exp - 12 x + 0.5 2 + y 2 .
e i = f e - f r f max ,
e av = 1 N s i = 1 N s   | e i | ,

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