Abstract

Tomographic methods are used for the investigation of three-dimensional compressible flow fields by means of interferometric methods. A modified algebraic reconstruction technique algorithm is applied. The algorithm proved to give reliable reconstructions from experimentally measured projection data in the case of an unrestricted angular view. The method was used for the reconstruction of density distributions of weakly perturbed supersonic free jets exiting from a deformed Laval nozzle. Even small perturbations of the jet resulted in significant three-dimensional effects. Reconstruction of a multiple system of jets emanating from a sievelike nozzle showed mutual interactions between the constituent jets. For the investigation of unsteady flows a setup for the recording of holographic interferograms was designed. Here, because of experimental restrictions, only a limited angular range of views was accessible. In the context of this limited-view geometry, reconstructions revealed considerable distortions for objects containing steep gradients.

© 1994 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. C. M. Vest, Holographic Interferometry (Wiley, New York, 1979), Chap. 6.3.
  2. L. T. Clark, D. C. Koepp, J. J. Thykkuttathil, “Three-dimensional density field measurement of a transonic flow from a square nozzle using holographic interferometry,” J. Fluids Eng. 99, 737–744 (1977).
    [CrossRef]
  3. G. W. Faris, R. L. Byer, “Three-dimensional beam-deflection optical tomography of a supersonic jet,” Appl. Opt. 27, 5202–5212 (1988).
    [CrossRef] [PubMed]
  4. R. D. Matulka, D. J. Collins, “Determination of three-dimensional density fields from holographic interferometry,” J. Appl. Phys. 42, 1109–1119 (1971).
    [CrossRef]
  5. R. Rangayyan, A. P. Dhawan, R. Gordon, “Algorithms for limited-view tomography: an annoted bibliography and a challenge,” Appl. Opt. 24, 4000–4020 (1985).
    [CrossRef] [PubMed]
  6. D. D. Verhoeven, “Applications of computed tomography and holographic interferometry to the study of transparent media,” Rep. 38501 (Institut Français du Pétrole, B.P. 311, 92500 Rueil-Malmaisons, France, 1990).
  7. H.-H. Bartels-Lehnhoff, P. H. Baumann, B. Bretthauer, G. E. A. Meier, “Computer-aided evaluation of interferograms,” Exp. Fluids (to be published).
  8. G. T. Herman, Image Reconstruction from Projections (Academic, New York, 1980), Chap. 6, pp. 90–95.
  9. S. R. Deans, The Radon Transform and Some of Its Applications (Wiley, New York, 1983), Chap. 2, pp. 55–59.
  10. R. Wenskus, “Tomographische Untersuchugen an schwach gestörten Überschallfreistrahlen,” Ph.D. dissertation (Georg-August-Universität, Göttingen, Germany, 1988).
  11. G. Grabitz, “Formen dreidimensionaler Überschallfreistrahlen in linearer Näherung,” Z. Angew. Math. Mech. 61, T155–T156 (1981).
  12. A. Dillmann, “Analytische Theorie zylindrischer dreidimensionaler Überschallfreistrahlen,” Rep. DLR-1B222-92A24 (Deutsche Forschungsanstalt für Luft- und Raumfahrt, Institut für Experimentelle Strömungsmechanik, Göttingen, Germany, 1992).
  13. B. P. Medoff, “Image reconstruction from limited data: theory and applications in computerized tomography,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, Orlando, Fla., 1987), pp. 321–368.
  14. Ref. 9, Chap. 6.2, pp. 131–142.
  15. F. Natterer, The Mathematics of Computerized Tomography (Wiley, New York, 1986), Chap. V.2, pp. 119–128.
  16. Y. Censor, “Finite series-expansion methods,” Proc. IEEE 71, 409–419 (1983).
    [CrossRef]
  17. A. K. Louis, F. Natterer, “Mathematical problems of computerized tomography,” Proc. IEEE 71, 379–389 (1983).
    [CrossRef]
  18. M. B. Katz, Questions of Uniqueness and Resolution in Reconstruction from Projections (Springer, New York, 1978), Chap. 6, pp. 62–90.
    [CrossRef]
  19. A. van der Sluis, A. van der Vorst, “SIRT- and CG-type methods for the iterative solution of sparse linear least-squares problems,” Linear Algebra Appl. 130, 257–302 (1990).
    [CrossRef]
  20. S. Bahl, J. A. Liburdy, “Three-dimensional image reconstruction using interferometric data from a limited field of view with noise,” Appl. Opt. 30, 4218–4226 (1991).
    [CrossRef] [PubMed]
  21. N. J. Dausaussoy, I. E. Abdou, “Some new multiplicative algorithms for image reconstruction from projections,” Linear Algebra Appl. 130, 111–132 (1990)
    [CrossRef]
  22. F. Natterer, The Mathematics of Computerized Tomography (Wiley, New York, 1986), Chap. VI.2, pp. 160–163.
  23. Ref. 22, pp. 164–166.
  24. W. Hiller, H. M. Lent, G. E. A. Meier, B. Stasicki, “A pulsed light generator for high speed photography,” Exp. Fluids 5, 141–144 (1987).
    [CrossRef]
  25. D. W. Watt, C. M. Vest, “Turbulent flow vizualization by interferometric integral imaging and computed tomography,” Exp. Fluids 8, 301–311 (1990).
    [CrossRef]

1991

1990

A. van der Sluis, A. van der Vorst, “SIRT- and CG-type methods for the iterative solution of sparse linear least-squares problems,” Linear Algebra Appl. 130, 257–302 (1990).
[CrossRef]

N. J. Dausaussoy, I. E. Abdou, “Some new multiplicative algorithms for image reconstruction from projections,” Linear Algebra Appl. 130, 111–132 (1990)
[CrossRef]

D. W. Watt, C. M. Vest, “Turbulent flow vizualization by interferometric integral imaging and computed tomography,” Exp. Fluids 8, 301–311 (1990).
[CrossRef]

1988

1987

W. Hiller, H. M. Lent, G. E. A. Meier, B. Stasicki, “A pulsed light generator for high speed photography,” Exp. Fluids 5, 141–144 (1987).
[CrossRef]

1985

1983

Y. Censor, “Finite series-expansion methods,” Proc. IEEE 71, 409–419 (1983).
[CrossRef]

A. K. Louis, F. Natterer, “Mathematical problems of computerized tomography,” Proc. IEEE 71, 379–389 (1983).
[CrossRef]

1981

G. Grabitz, “Formen dreidimensionaler Überschallfreistrahlen in linearer Näherung,” Z. Angew. Math. Mech. 61, T155–T156 (1981).

1977

L. T. Clark, D. C. Koepp, J. J. Thykkuttathil, “Three-dimensional density field measurement of a transonic flow from a square nozzle using holographic interferometry,” J. Fluids Eng. 99, 737–744 (1977).
[CrossRef]

1971

R. D. Matulka, D. J. Collins, “Determination of three-dimensional density fields from holographic interferometry,” J. Appl. Phys. 42, 1109–1119 (1971).
[CrossRef]

Abdou, I. E.

N. J. Dausaussoy, I. E. Abdou, “Some new multiplicative algorithms for image reconstruction from projections,” Linear Algebra Appl. 130, 111–132 (1990)
[CrossRef]

Bahl, S.

Bartels-Lehnhoff, H.-H.

H.-H. Bartels-Lehnhoff, P. H. Baumann, B. Bretthauer, G. E. A. Meier, “Computer-aided evaluation of interferograms,” Exp. Fluids (to be published).

Baumann, P. H.

H.-H. Bartels-Lehnhoff, P. H. Baumann, B. Bretthauer, G. E. A. Meier, “Computer-aided evaluation of interferograms,” Exp. Fluids (to be published).

Bretthauer, B.

H.-H. Bartels-Lehnhoff, P. H. Baumann, B. Bretthauer, G. E. A. Meier, “Computer-aided evaluation of interferograms,” Exp. Fluids (to be published).

Byer, R. L.

Censor, Y.

Y. Censor, “Finite series-expansion methods,” Proc. IEEE 71, 409–419 (1983).
[CrossRef]

Clark, L. T.

L. T. Clark, D. C. Koepp, J. J. Thykkuttathil, “Three-dimensional density field measurement of a transonic flow from a square nozzle using holographic interferometry,” J. Fluids Eng. 99, 737–744 (1977).
[CrossRef]

Collins, D. J.

R. D. Matulka, D. J. Collins, “Determination of three-dimensional density fields from holographic interferometry,” J. Appl. Phys. 42, 1109–1119 (1971).
[CrossRef]

Dausaussoy, N. J.

N. J. Dausaussoy, I. E. Abdou, “Some new multiplicative algorithms for image reconstruction from projections,” Linear Algebra Appl. 130, 111–132 (1990)
[CrossRef]

Deans, S. R.

S. R. Deans, The Radon Transform and Some of Its Applications (Wiley, New York, 1983), Chap. 2, pp. 55–59.

Dhawan, A. P.

Dillmann, A.

A. Dillmann, “Analytische Theorie zylindrischer dreidimensionaler Überschallfreistrahlen,” Rep. DLR-1B222-92A24 (Deutsche Forschungsanstalt für Luft- und Raumfahrt, Institut für Experimentelle Strömungsmechanik, Göttingen, Germany, 1992).

Faris, G. W.

Gordon, R.

Grabitz, G.

G. Grabitz, “Formen dreidimensionaler Überschallfreistrahlen in linearer Näherung,” Z. Angew. Math. Mech. 61, T155–T156 (1981).

Herman, G. T.

G. T. Herman, Image Reconstruction from Projections (Academic, New York, 1980), Chap. 6, pp. 90–95.

Hiller, W.

W. Hiller, H. M. Lent, G. E. A. Meier, B. Stasicki, “A pulsed light generator for high speed photography,” Exp. Fluids 5, 141–144 (1987).
[CrossRef]

Katz, M. B.

M. B. Katz, Questions of Uniqueness and Resolution in Reconstruction from Projections (Springer, New York, 1978), Chap. 6, pp. 62–90.
[CrossRef]

Koepp, D. C.

L. T. Clark, D. C. Koepp, J. J. Thykkuttathil, “Three-dimensional density field measurement of a transonic flow from a square nozzle using holographic interferometry,” J. Fluids Eng. 99, 737–744 (1977).
[CrossRef]

Lent, H. M.

W. Hiller, H. M. Lent, G. E. A. Meier, B. Stasicki, “A pulsed light generator for high speed photography,” Exp. Fluids 5, 141–144 (1987).
[CrossRef]

Liburdy, J. A.

Louis, A. K.

A. K. Louis, F. Natterer, “Mathematical problems of computerized tomography,” Proc. IEEE 71, 379–389 (1983).
[CrossRef]

Matulka, R. D.

R. D. Matulka, D. J. Collins, “Determination of three-dimensional density fields from holographic interferometry,” J. Appl. Phys. 42, 1109–1119 (1971).
[CrossRef]

Medoff, B. P.

B. P. Medoff, “Image reconstruction from limited data: theory and applications in computerized tomography,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, Orlando, Fla., 1987), pp. 321–368.

Meier, G. E. A.

W. Hiller, H. M. Lent, G. E. A. Meier, B. Stasicki, “A pulsed light generator for high speed photography,” Exp. Fluids 5, 141–144 (1987).
[CrossRef]

H.-H. Bartels-Lehnhoff, P. H. Baumann, B. Bretthauer, G. E. A. Meier, “Computer-aided evaluation of interferograms,” Exp. Fluids (to be published).

Natterer, F.

A. K. Louis, F. Natterer, “Mathematical problems of computerized tomography,” Proc. IEEE 71, 379–389 (1983).
[CrossRef]

F. Natterer, The Mathematics of Computerized Tomography (Wiley, New York, 1986), Chap. V.2, pp. 119–128.

F. Natterer, The Mathematics of Computerized Tomography (Wiley, New York, 1986), Chap. VI.2, pp. 160–163.

Rangayyan, R.

Stasicki, B.

W. Hiller, H. M. Lent, G. E. A. Meier, B. Stasicki, “A pulsed light generator for high speed photography,” Exp. Fluids 5, 141–144 (1987).
[CrossRef]

Thykkuttathil, J. J.

L. T. Clark, D. C. Koepp, J. J. Thykkuttathil, “Three-dimensional density field measurement of a transonic flow from a square nozzle using holographic interferometry,” J. Fluids Eng. 99, 737–744 (1977).
[CrossRef]

van der Sluis, A.

A. van der Sluis, A. van der Vorst, “SIRT- and CG-type methods for the iterative solution of sparse linear least-squares problems,” Linear Algebra Appl. 130, 257–302 (1990).
[CrossRef]

van der Vorst, A.

A. van der Sluis, A. van der Vorst, “SIRT- and CG-type methods for the iterative solution of sparse linear least-squares problems,” Linear Algebra Appl. 130, 257–302 (1990).
[CrossRef]

Verhoeven, D. D.

D. D. Verhoeven, “Applications of computed tomography and holographic interferometry to the study of transparent media,” Rep. 38501 (Institut Français du Pétrole, B.P. 311, 92500 Rueil-Malmaisons, France, 1990).

Vest, C. M.

D. W. Watt, C. M. Vest, “Turbulent flow vizualization by interferometric integral imaging and computed tomography,” Exp. Fluids 8, 301–311 (1990).
[CrossRef]

C. M. Vest, Holographic Interferometry (Wiley, New York, 1979), Chap. 6.3.

Watt, D. W.

D. W. Watt, C. M. Vest, “Turbulent flow vizualization by interferometric integral imaging and computed tomography,” Exp. Fluids 8, 301–311 (1990).
[CrossRef]

Wenskus, R.

R. Wenskus, “Tomographische Untersuchugen an schwach gestörten Überschallfreistrahlen,” Ph.D. dissertation (Georg-August-Universität, Göttingen, Germany, 1988).

Appl. Opt.

Exp. Fluids

W. Hiller, H. M. Lent, G. E. A. Meier, B. Stasicki, “A pulsed light generator for high speed photography,” Exp. Fluids 5, 141–144 (1987).
[CrossRef]

D. W. Watt, C. M. Vest, “Turbulent flow vizualization by interferometric integral imaging and computed tomography,” Exp. Fluids 8, 301–311 (1990).
[CrossRef]

J. Appl. Phys.

R. D. Matulka, D. J. Collins, “Determination of three-dimensional density fields from holographic interferometry,” J. Appl. Phys. 42, 1109–1119 (1971).
[CrossRef]

J. Fluids Eng.

L. T. Clark, D. C. Koepp, J. J. Thykkuttathil, “Three-dimensional density field measurement of a transonic flow from a square nozzle using holographic interferometry,” J. Fluids Eng. 99, 737–744 (1977).
[CrossRef]

Linear Algebra Appl.

N. J. Dausaussoy, I. E. Abdou, “Some new multiplicative algorithms for image reconstruction from projections,” Linear Algebra Appl. 130, 111–132 (1990)
[CrossRef]

A. van der Sluis, A. van der Vorst, “SIRT- and CG-type methods for the iterative solution of sparse linear least-squares problems,” Linear Algebra Appl. 130, 257–302 (1990).
[CrossRef]

Proc. IEEE

Y. Censor, “Finite series-expansion methods,” Proc. IEEE 71, 409–419 (1983).
[CrossRef]

A. K. Louis, F. Natterer, “Mathematical problems of computerized tomography,” Proc. IEEE 71, 379–389 (1983).
[CrossRef]

Z. Angew. Math. Mech.

G. Grabitz, “Formen dreidimensionaler Überschallfreistrahlen in linearer Näherung,” Z. Angew. Math. Mech. 61, T155–T156 (1981).

Other

A. Dillmann, “Analytische Theorie zylindrischer dreidimensionaler Überschallfreistrahlen,” Rep. DLR-1B222-92A24 (Deutsche Forschungsanstalt für Luft- und Raumfahrt, Institut für Experimentelle Strömungsmechanik, Göttingen, Germany, 1992).

B. P. Medoff, “Image reconstruction from limited data: theory and applications in computerized tomography,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, Orlando, Fla., 1987), pp. 321–368.

Ref. 9, Chap. 6.2, pp. 131–142.

F. Natterer, The Mathematics of Computerized Tomography (Wiley, New York, 1986), Chap. V.2, pp. 119–128.

D. D. Verhoeven, “Applications of computed tomography and holographic interferometry to the study of transparent media,” Rep. 38501 (Institut Français du Pétrole, B.P. 311, 92500 Rueil-Malmaisons, France, 1990).

H.-H. Bartels-Lehnhoff, P. H. Baumann, B. Bretthauer, G. E. A. Meier, “Computer-aided evaluation of interferograms,” Exp. Fluids (to be published).

G. T. Herman, Image Reconstruction from Projections (Academic, New York, 1980), Chap. 6, pp. 90–95.

S. R. Deans, The Radon Transform and Some of Its Applications (Wiley, New York, 1983), Chap. 2, pp. 55–59.

R. Wenskus, “Tomographische Untersuchugen an schwach gestörten Überschallfreistrahlen,” Ph.D. dissertation (Georg-August-Universität, Göttingen, Germany, 1988).

M. B. Katz, Questions of Uniqueness and Resolution in Reconstruction from Projections (Springer, New York, 1978), Chap. 6, pp. 62–90.
[CrossRef]

C. M. Vest, Holographic Interferometry (Wiley, New York, 1979), Chap. 6.3.

F. Natterer, The Mathematics of Computerized Tomography (Wiley, New York, 1986), Chap. VI.2, pp. 160–163.

Ref. 22, pp. 164–166.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (14)

Fig. 1
Fig. 1

Schematic drawing of the setup that was used for the tomographic investigation of supersonic free jets: D dehumidifier; C compressor; PC, pressure chamber; P, pressure probe; TE, threshold electronics; N, nozzle, CCD, CCD camera; FG-100 image processor; MZ, Mach–Zehnder interferometer; HLP, high-speed light pulser; TC, trigger conditioner; LED, light-emitting diode.

Fig. 2
Fig. 2

Front view of the sievelike nozzle. It consists of a 1-cm-thick circular plate in which seven holes are drilled. The holes have a diameter of 1 cm. Interferograms of free jets exiting from this nozzle are shown in Fig. 3.

Fig. 3
Fig. 3

Two interferograms of the free jet behind the sievelike nozzle. The Mach number at the exit is ~ 1.1. The view directions differ by 90°. The dashes marks the position of the reconstructed cross section depicted in Fig. 5. Flow direction is from left to right.

Fig. 4
Fig. 4

(Upper) Interferogram of the mean flow behind the underformed, axisymmetric Laval nozzle that is calculated from the average of eight evaluated interferograms. (Lower) Two calculated interferograms of the jet behind the slightly deformed Laval nozzle (the ratio of short semiaxis to long semiaxis is 0.94), each of which is calculated from the average FOF of three evaluated interferograms. A and B mark the positions of reconstructions shown in Figs. 7(a) and 7(b), respectively. The view directions in the left and the right interferograms are 90° and 0°, respectively. For all images the Mach number at the nozzle exit (black vertical line) is 1.65. Flow direction is as in Fig. 3.

Fig. 5
Fig. 5

Reconstructed density field in a cross section near the nozzle exit of the sievelike nozzle. Density is plotted in arbitrary units.

Fig. 6
Fig. 6

Contour plot of the reconstruction in a plane in the streamwise direction indicating the spatial oscillation of the density maxima in that direction.

Fig. 7
Fig. 7

(a) Reconstructed density field in a cross section through the deformed underexpanded free jet in position A (Fig. 4). (b) Reconstructed cross section in position B. Density is plotted in arbitrary units.

Fig. 8
Fig. 8

Reconstructed density field of the deformed under expanded jet in the streamwise direction: (a) in the plane parallel to the long semiaxis; (b) in the plane parallel to the short semiaxis. The additional disturbance is marked by dashed lines. Density is plotted in arbitrary units. Flow direction is from left to right.

Fig. 9
Fig. 9

Optical arrangement for the recording of holograms of large angular aperture: S, shutter; M’s, mirrors; VBS, variable beam splitter; BS, beam splitter; SP, spatial filter; L’s, lenses; GG, ground-glass plates; O, object; HP, holographic plate.

Fig. 10
Fig. 10

Two interferograms of the light bulb obtained from one hologram. View directions differ by 85° horizontally. The position of the cross sections shown in Figs. 1114 is indicated by dashes.

Fig. 11
Fig. 11

Full-view reconstruction of the temperature field in the light bulb immediately above the filament with AVART. Temperature is plotted in arbitrary units.

Fig. 12
Fig. 12

Limited-view reconstruction of the temperature field above the filament with AVART. Twenty-one angles over 90° are used. Temperature is plotted in arbitrary units.

Fig. 13
Fig. 13

Limited-view reconstruction of the temperature field above the filament with AVART from Fig. 12 projection data. Between iterations, temperature values were set to zero outside the bulb boundary.

Fig. 14
Fig. 14

Limited-view reconstruction of the temperature field above the filament with MART from Fig. 12 projection data. Between iterations, temperature values were set to zero outside the bulb boundary.

Tables (1)

Tables Icon

Table 1 Comparison of Limited-View Reconstructions with the Averaging Algebraic Reconstruction Technique Full-View Reconstructiona

Equations (8)

Equations on this page are rendered with MathJax. Learn more.

FOF = 1 λ ( n - n 0 ) d s .
p ( Θ , ρ ) = R [ f ( x , y ) ] = - f ( ρ cos Θ - s sin Θ , ρ sin Θ + s cos Θ ) d s .
p i = ( I - I 0 ) d s j = 1 N A i j x j ,
Ax = p .
x k + 1 = x k + ω p i - j A i j x j k j A i j 2 a i ,
x j k + 1 = x j k + ω 1 γ j i p i - j A i j x j k j A i j 2 A i j ,
x j k + 1 = x j k [ 1 + ω A i j A i max ( p i j A i j x j k - 1 ) ] ,
I ( x , y ) = I 0 ( x , y ) cos [ 2 π FOF ( x , y ) ] + I B ( x , y ) ,

Metrics