Abstract

Projection data obtained through optical techniques for tomographic measurements, such as interferometry for refractive-index-based measurements, are often incomplete. This is due to limitations in the optical system, data storage, and alignment and vignette issues. Algebraic iterative reconstruction techniques are usually favored for such incomplete projections. A number of iterative algorithms, based on additive and multiplicative corrections, are used with a known simulated phantom and noise source to assess the reconstruction performance of incomplete data sets. In addition, we present reconstructions using experimental data obtained from a coherent gradient sensing interferometer for a steady temperature field in a fluid medium. We tested the algorithms using the simulated data set for incompleteness conditions similar to those found in the experimental data, and the best-performing algorithm is identified.

© 2004 Optical Society of America

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  1. D. Mishra, K. Muralidhar, P. Munshi, “Experimental study of Rayleigh-Benard convection at intermediate Rayleigh numbers using interferometric tomography,” Fluid Dyn. Res. 25, 231–255 (1999).
    [CrossRef]
  2. 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]
  3. R. J. Goldstein, ed., Fluid Mechanics Measurements (Hemisphere, New York, 1983).
  4. P. Munshi, P. Jayakumar, P. Sthyamurthy, T. K. Thiyagarajan, N. S. Dixit, N. Venkatramani, “Void-fraction measurements in a steady-state mercury-nitrogen flow loop,” Exp. Fluids 24, 424–430 (1998).
    [CrossRef]
  5. F. Motoo, Z. Xing, “Noncontact measurement of internal temperature distribution in a solid material using ultrasonic computed tomography,” in Proceedings of the Fourth World Conference on Experimental Heat Transfer, Fluid Mechanics and Thermodynamics, M. Giot, F. Mayinger, G. P. Celata, eds. (Edizioni ETS, Pisa, Italy, 1997), Vol. 1, pp. 169–176.
  6. P. Munshi, “X-ray and ultrasonic tomography,” Insight 45, 47–50 (2003).
    [CrossRef]
  7. G. T. Herman, Image Reconstruction from Projections (Academic, New York, 1980).
  8. Y. Censor, “Finite series expansion reconstruction methods,” Proc. IEEE 71, 409–419 (1983).
    [CrossRef]
  9. S. F. Gull, T. J. Newton, “Maximum entropy tomography,” Appl. Opt. 25, 156–160 (1986).
    [CrossRef] [PubMed]
  10. D. Mishra, S. L. Wong, J. P. Longtin, R. Singh, V. Prasad, “Development of a coherent gradient-sensing tomographic interferometer for three-dimensional refractive-index based measurements,” Opt. Commun. 212, 17–27 (2002).
    [CrossRef]
  11. D. Mishra, J. P. Longtin, R. P. Singh, “Coherent gradient sensing interferometry: application in convective fluid medium for tomographic measurements,” submitted to Commun. Exp. Fluids.
  12. D. Verhoeven, “Multiplicative algebraic computed tomography algorithms for the reconstruction of multidirectional interferometric data,” Opt. Eng. 32, 410–419 (1993).
    [CrossRef]
  13. P. M. V. Subbarao, P. Munshi, K. Muralidhar, “Performance evaluation of iterative tomographic algorithms applied to reconstruction of a three-dimensional temperature field,” Numer. Heat Transfer Part B 31, 347–372 (1997).
    [CrossRef]
  14. R. Gordon, G. T. Herman, “Three-dimensional reconstructions from projections—review of algorithms,” Int. Rev. Cytol. 38, 111–151 (1974).
    [CrossRef]
  15. D. Mishra, K. Muralidhar, P. Munshi, “A robust MART algorithm for tomographic applications,” Numer. Heat Transfer Part B 35, 485–506 (1999).
    [CrossRef]
  16. F. Mayinger, ed., Optical Measurements: Techniques and Applications (Springer-Verlag, Berlin, Germany, 1994).
    [CrossRef]
  17. R. Gordon, R. Bender, G. T. Herman, “Algebraic reconstruction techniques (ART) for three-dimensional electron microscopy and x-ray photography,” J. Theor. Biol. 29, 471–481 (1970).
    [CrossRef] [PubMed]
  18. P. F. C. Gilbert, “Iterative methods for three-dimensional reconstruction of an object from its projections,” J. Theor. Biol. 36, 105–117 (1972).
    [CrossRef] [PubMed]
  19. A. H. Andersen, A. C. Kak, “Simultaneous algebraic reconstruction technique (SART)—a superior implementation of the ART algorithm,” Ultrason. Imaging 6, 81–94 (1984).
    [PubMed]

2003 (1)

P. Munshi, “X-ray and ultrasonic tomography,” Insight 45, 47–50 (2003).
[CrossRef]

2002 (1)

D. Mishra, S. L. Wong, J. P. Longtin, R. Singh, V. Prasad, “Development of a coherent gradient-sensing tomographic interferometer for three-dimensional refractive-index based measurements,” Opt. Commun. 212, 17–27 (2002).
[CrossRef]

1999 (2)

D. Mishra, K. Muralidhar, P. Munshi, “Experimental study of Rayleigh-Benard convection at intermediate Rayleigh numbers using interferometric tomography,” Fluid Dyn. Res. 25, 231–255 (1999).
[CrossRef]

D. Mishra, K. Muralidhar, P. Munshi, “A robust MART algorithm for tomographic applications,” Numer. Heat Transfer Part B 35, 485–506 (1999).
[CrossRef]

1998 (1)

P. Munshi, P. Jayakumar, P. Sthyamurthy, T. K. Thiyagarajan, N. S. Dixit, N. Venkatramani, “Void-fraction measurements in a steady-state mercury-nitrogen flow loop,” Exp. Fluids 24, 424–430 (1998).
[CrossRef]

1997 (1)

P. M. V. Subbarao, P. Munshi, K. Muralidhar, “Performance evaluation of iterative tomographic algorithms applied to reconstruction of a three-dimensional temperature field,” Numer. Heat Transfer Part B 31, 347–372 (1997).
[CrossRef]

1993 (1)

D. Verhoeven, “Multiplicative algebraic computed tomography algorithms for the reconstruction of multidirectional interferometric data,” Opt. Eng. 32, 410–419 (1993).
[CrossRef]

1991 (1)

1986 (1)

1984 (1)

A. H. Andersen, A. C. Kak, “Simultaneous algebraic reconstruction technique (SART)—a superior implementation of the ART algorithm,” Ultrason. Imaging 6, 81–94 (1984).
[PubMed]

1983 (1)

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

1974 (1)

R. Gordon, G. T. Herman, “Three-dimensional reconstructions from projections—review of algorithms,” Int. Rev. Cytol. 38, 111–151 (1974).
[CrossRef]

1972 (1)

P. F. C. Gilbert, “Iterative methods for three-dimensional reconstruction of an object from its projections,” J. Theor. Biol. 36, 105–117 (1972).
[CrossRef] [PubMed]

1970 (1)

R. Gordon, R. Bender, G. T. Herman, “Algebraic reconstruction techniques (ART) for three-dimensional electron microscopy and x-ray photography,” J. Theor. Biol. 29, 471–481 (1970).
[CrossRef] [PubMed]

Andersen, A. H.

A. H. Andersen, A. C. Kak, “Simultaneous algebraic reconstruction technique (SART)—a superior implementation of the ART algorithm,” Ultrason. Imaging 6, 81–94 (1984).
[PubMed]

Bahl, S.

Bender, R.

R. Gordon, R. Bender, G. T. Herman, “Algebraic reconstruction techniques (ART) for three-dimensional electron microscopy and x-ray photography,” J. Theor. Biol. 29, 471–481 (1970).
[CrossRef] [PubMed]

Censor, Y.

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

Dixit, N. S.

P. Munshi, P. Jayakumar, P. Sthyamurthy, T. K. Thiyagarajan, N. S. Dixit, N. Venkatramani, “Void-fraction measurements in a steady-state mercury-nitrogen flow loop,” Exp. Fluids 24, 424–430 (1998).
[CrossRef]

Gilbert, P. F. C.

P. F. C. Gilbert, “Iterative methods for three-dimensional reconstruction of an object from its projections,” J. Theor. Biol. 36, 105–117 (1972).
[CrossRef] [PubMed]

Gordon, R.

R. Gordon, G. T. Herman, “Three-dimensional reconstructions from projections—review of algorithms,” Int. Rev. Cytol. 38, 111–151 (1974).
[CrossRef]

R. Gordon, R. Bender, G. T. Herman, “Algebraic reconstruction techniques (ART) for three-dimensional electron microscopy and x-ray photography,” J. Theor. Biol. 29, 471–481 (1970).
[CrossRef] [PubMed]

Gull, S. F.

Herman, G. T.

R. Gordon, G. T. Herman, “Three-dimensional reconstructions from projections—review of algorithms,” Int. Rev. Cytol. 38, 111–151 (1974).
[CrossRef]

R. Gordon, R. Bender, G. T. Herman, “Algebraic reconstruction techniques (ART) for three-dimensional electron microscopy and x-ray photography,” J. Theor. Biol. 29, 471–481 (1970).
[CrossRef] [PubMed]

G. T. Herman, Image Reconstruction from Projections (Academic, New York, 1980).

Jayakumar, P.

P. Munshi, P. Jayakumar, P. Sthyamurthy, T. K. Thiyagarajan, N. S. Dixit, N. Venkatramani, “Void-fraction measurements in a steady-state mercury-nitrogen flow loop,” Exp. Fluids 24, 424–430 (1998).
[CrossRef]

Kak, A. C.

A. H. Andersen, A. C. Kak, “Simultaneous algebraic reconstruction technique (SART)—a superior implementation of the ART algorithm,” Ultrason. Imaging 6, 81–94 (1984).
[PubMed]

Liburdy, J. A.

Longtin, J. P.

D. Mishra, S. L. Wong, J. P. Longtin, R. Singh, V. Prasad, “Development of a coherent gradient-sensing tomographic interferometer for three-dimensional refractive-index based measurements,” Opt. Commun. 212, 17–27 (2002).
[CrossRef]

D. Mishra, J. P. Longtin, R. P. Singh, “Coherent gradient sensing interferometry: application in convective fluid medium for tomographic measurements,” submitted to Commun. Exp. Fluids.

Mishra, D.

D. Mishra, S. L. Wong, J. P. Longtin, R. Singh, V. Prasad, “Development of a coherent gradient-sensing tomographic interferometer for three-dimensional refractive-index based measurements,” Opt. Commun. 212, 17–27 (2002).
[CrossRef]

D. Mishra, K. Muralidhar, P. Munshi, “Experimental study of Rayleigh-Benard convection at intermediate Rayleigh numbers using interferometric tomography,” Fluid Dyn. Res. 25, 231–255 (1999).
[CrossRef]

D. Mishra, K. Muralidhar, P. Munshi, “A robust MART algorithm for tomographic applications,” Numer. Heat Transfer Part B 35, 485–506 (1999).
[CrossRef]

D. Mishra, J. P. Longtin, R. P. Singh, “Coherent gradient sensing interferometry: application in convective fluid medium for tomographic measurements,” submitted to Commun. Exp. Fluids.

Motoo, F.

F. Motoo, Z. Xing, “Noncontact measurement of internal temperature distribution in a solid material using ultrasonic computed tomography,” in Proceedings of the Fourth World Conference on Experimental Heat Transfer, Fluid Mechanics and Thermodynamics, M. Giot, F. Mayinger, G. P. Celata, eds. (Edizioni ETS, Pisa, Italy, 1997), Vol. 1, pp. 169–176.

Munshi, P.

P. Munshi, “X-ray and ultrasonic tomography,” Insight 45, 47–50 (2003).
[CrossRef]

D. Mishra, K. Muralidhar, P. Munshi, “Experimental study of Rayleigh-Benard convection at intermediate Rayleigh numbers using interferometric tomography,” Fluid Dyn. Res. 25, 231–255 (1999).
[CrossRef]

D. Mishra, K. Muralidhar, P. Munshi, “A robust MART algorithm for tomographic applications,” Numer. Heat Transfer Part B 35, 485–506 (1999).
[CrossRef]

P. Munshi, P. Jayakumar, P. Sthyamurthy, T. K. Thiyagarajan, N. S. Dixit, N. Venkatramani, “Void-fraction measurements in a steady-state mercury-nitrogen flow loop,” Exp. Fluids 24, 424–430 (1998).
[CrossRef]

P. M. V. Subbarao, P. Munshi, K. Muralidhar, “Performance evaluation of iterative tomographic algorithms applied to reconstruction of a three-dimensional temperature field,” Numer. Heat Transfer Part B 31, 347–372 (1997).
[CrossRef]

Muralidhar, K.

D. Mishra, K. Muralidhar, P. Munshi, “A robust MART algorithm for tomographic applications,” Numer. Heat Transfer Part B 35, 485–506 (1999).
[CrossRef]

D. Mishra, K. Muralidhar, P. Munshi, “Experimental study of Rayleigh-Benard convection at intermediate Rayleigh numbers using interferometric tomography,” Fluid Dyn. Res. 25, 231–255 (1999).
[CrossRef]

P. M. V. Subbarao, P. Munshi, K. Muralidhar, “Performance evaluation of iterative tomographic algorithms applied to reconstruction of a three-dimensional temperature field,” Numer. Heat Transfer Part B 31, 347–372 (1997).
[CrossRef]

Newton, T. J.

Prasad, V.

D. Mishra, S. L. Wong, J. P. Longtin, R. Singh, V. Prasad, “Development of a coherent gradient-sensing tomographic interferometer for three-dimensional refractive-index based measurements,” Opt. Commun. 212, 17–27 (2002).
[CrossRef]

Singh, R.

D. Mishra, S. L. Wong, J. P. Longtin, R. Singh, V. Prasad, “Development of a coherent gradient-sensing tomographic interferometer for three-dimensional refractive-index based measurements,” Opt. Commun. 212, 17–27 (2002).
[CrossRef]

Singh, R. P.

D. Mishra, J. P. Longtin, R. P. Singh, “Coherent gradient sensing interferometry: application in convective fluid medium for tomographic measurements,” submitted to Commun. Exp. Fluids.

Sthyamurthy, P.

P. Munshi, P. Jayakumar, P. Sthyamurthy, T. K. Thiyagarajan, N. S. Dixit, N. Venkatramani, “Void-fraction measurements in a steady-state mercury-nitrogen flow loop,” Exp. Fluids 24, 424–430 (1998).
[CrossRef]

Subbarao, P. M. V.

P. M. V. Subbarao, P. Munshi, K. Muralidhar, “Performance evaluation of iterative tomographic algorithms applied to reconstruction of a three-dimensional temperature field,” Numer. Heat Transfer Part B 31, 347–372 (1997).
[CrossRef]

Thiyagarajan, T. K.

P. Munshi, P. Jayakumar, P. Sthyamurthy, T. K. Thiyagarajan, N. S. Dixit, N. Venkatramani, “Void-fraction measurements in a steady-state mercury-nitrogen flow loop,” Exp. Fluids 24, 424–430 (1998).
[CrossRef]

Venkatramani, N.

P. Munshi, P. Jayakumar, P. Sthyamurthy, T. K. Thiyagarajan, N. S. Dixit, N. Venkatramani, “Void-fraction measurements in a steady-state mercury-nitrogen flow loop,” Exp. Fluids 24, 424–430 (1998).
[CrossRef]

Verhoeven, D.

D. Verhoeven, “Multiplicative algebraic computed tomography algorithms for the reconstruction of multidirectional interferometric data,” Opt. Eng. 32, 410–419 (1993).
[CrossRef]

Wong, S. L.

D. Mishra, S. L. Wong, J. P. Longtin, R. Singh, V. Prasad, “Development of a coherent gradient-sensing tomographic interferometer for three-dimensional refractive-index based measurements,” Opt. Commun. 212, 17–27 (2002).
[CrossRef]

Xing, Z.

F. Motoo, Z. Xing, “Noncontact measurement of internal temperature distribution in a solid material using ultrasonic computed tomography,” in Proceedings of the Fourth World Conference on Experimental Heat Transfer, Fluid Mechanics and Thermodynamics, M. Giot, F. Mayinger, G. P. Celata, eds. (Edizioni ETS, Pisa, Italy, 1997), Vol. 1, pp. 169–176.

Appl. Opt. (2)

Exp. Fluids (1)

P. Munshi, P. Jayakumar, P. Sthyamurthy, T. K. Thiyagarajan, N. S. Dixit, N. Venkatramani, “Void-fraction measurements in a steady-state mercury-nitrogen flow loop,” Exp. Fluids 24, 424–430 (1998).
[CrossRef]

Fluid Dyn. Res. (1)

D. Mishra, K. Muralidhar, P. Munshi, “Experimental study of Rayleigh-Benard convection at intermediate Rayleigh numbers using interferometric tomography,” Fluid Dyn. Res. 25, 231–255 (1999).
[CrossRef]

Insight (1)

P. Munshi, “X-ray and ultrasonic tomography,” Insight 45, 47–50 (2003).
[CrossRef]

Int. Rev. Cytol. (1)

R. Gordon, G. T. Herman, “Three-dimensional reconstructions from projections—review of algorithms,” Int. Rev. Cytol. 38, 111–151 (1974).
[CrossRef]

J. Theor. Biol. (2)

R. Gordon, R. Bender, G. T. Herman, “Algebraic reconstruction techniques (ART) for three-dimensional electron microscopy and x-ray photography,” J. Theor. Biol. 29, 471–481 (1970).
[CrossRef] [PubMed]

P. F. C. Gilbert, “Iterative methods for three-dimensional reconstruction of an object from its projections,” J. Theor. Biol. 36, 105–117 (1972).
[CrossRef] [PubMed]

Numer. Heat Transfer Part B (2)

D. Mishra, K. Muralidhar, P. Munshi, “A robust MART algorithm for tomographic applications,” Numer. Heat Transfer Part B 35, 485–506 (1999).
[CrossRef]

P. M. V. Subbarao, P. Munshi, K. Muralidhar, “Performance evaluation of iterative tomographic algorithms applied to reconstruction of a three-dimensional temperature field,” Numer. Heat Transfer Part B 31, 347–372 (1997).
[CrossRef]

Opt. Commun. (1)

D. Mishra, S. L. Wong, J. P. Longtin, R. Singh, V. Prasad, “Development of a coherent gradient-sensing tomographic interferometer for three-dimensional refractive-index based measurements,” Opt. Commun. 212, 17–27 (2002).
[CrossRef]

Opt. Eng. (1)

D. Verhoeven, “Multiplicative algebraic computed tomography algorithms for the reconstruction of multidirectional interferometric data,” Opt. Eng. 32, 410–419 (1993).
[CrossRef]

Proc. IEEE (1)

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

Ultrason. Imaging (1)

A. H. Andersen, A. C. Kak, “Simultaneous algebraic reconstruction technique (SART)—a superior implementation of the ART algorithm,” Ultrason. Imaging 6, 81–94 (1984).
[PubMed]

Other (5)

F. Mayinger, ed., Optical Measurements: Techniques and Applications (Springer-Verlag, Berlin, Germany, 1994).
[CrossRef]

D. Mishra, J. P. Longtin, R. P. Singh, “Coherent gradient sensing interferometry: application in convective fluid medium for tomographic measurements,” submitted to Commun. Exp. Fluids.

G. T. Herman, Image Reconstruction from Projections (Academic, New York, 1980).

F. Motoo, Z. Xing, “Noncontact measurement of internal temperature distribution in a solid material using ultrasonic computed tomography,” in Proceedings of the Fourth World Conference on Experimental Heat Transfer, Fluid Mechanics and Thermodynamics, M. Giot, F. Mayinger, G. P. Celata, eds. (Edizioni ETS, Pisa, Italy, 1997), Vol. 1, pp. 169–176.

R. J. Goldstein, ed., Fluid Mechanics Measurements (Hemisphere, New York, 1983).

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Figures (12)

Fig. 1
Fig. 1

Definition of incomplete projection data.

Fig. 2
Fig. 2

(a) Discretization of the volume into two-dimensional horizontal planes, (b) discretization of the horizontal planes and definition of weight factor.

Fig. 3
Fig. 3

(a) Simulated phantom function over a grid of 100 × 100 used to test the performance of the iterative tomographic algorithms and (b) a plan view of the function with contour lines.

Fig. 4
Fig. 4

Generated projection data from the simulated phantom function without any noise and with 10% maximum noise in the data.

Fig. 5
Fig. 5

Reconstructed function from MART2 algorithm with (a) 15, (b) 20, (c) 25 view projections.

Fig. 6
Fig. 6

Reconstructed function with the AVMART2 algorithm and 30 view projections.

Fig. 7
Fig. 7

Convergence history for MART2 and AVMART2 algorithms with 5% noise in projection data with 25 view projections.

Fig. 8
Fig. 8

E 1 error field with the MART2 algorithm. (a) No noise, (b) 5% noise, (c) 10% noise.

Fig. 9
Fig. 9

E 1 error field with the AVMART2 algorithm with 30 view projections. (a) No noise, (b) 5% noise, (c) 10% noise.

Fig. 10
Fig. 10

(a) Interferogram of the fluid layer at -46.9° angle of projection. The fringe bands represent the contours of gradient of the temperature field in the vertical direction. (b) Temperature field derived from the interferograms with isotherms at equal intervals; isotherms values are shown in degrees Celsius. HG, high-temperature, and LG, low-temperature, gradient in the temperature field.

Fig. 11
Fig. 11

Tomographically reconstructed temperature field is visualized in three different vertical planes for constant z values. Reconstructions are obtained with the AVMART2 algorithm.

Fig. 12
Fig. 12

Comparison of experimental projections with projections generated from the reconstructed field by tomography.

Tables (3)

Tables Icon

Table 1 Errors in Iterative Algorithms for Incomplete Projection Data

Tables Icon

Table 2 Errors in Iterative Algorithms for Incomplete Projection Data with 30-View Projection Data

Tables Icon

Table 3 Convergence, Relaxation Parameter, Iterations, and CPU Time for Various Iterative Algorithms with Incomplete Projection Data

Equations (18)

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

ϕi=j=1N wijfj, i=1, 2,M N>M,
wijfj=ϕi.
ϕiθ=j=1N wiθ,jfj, iθ=1, 2 , Mθ.
Δϕiθ=ϕiθ-ϕiθ
Wiθ=j=1N wiθ,j
Δϕiθ¯=ΔϕiθWiθ,
fjnew=fjold+μΔϕiθ¯,
Wiθ=j=1Nwiθ,j2,
fjnew=fjold+μ Δϕiθwiθ,jWiθ.
fjnew=fjold+1cjcj μ wiθ,jΔϕiθWiθ,
Δϕiθ=ϕiθϕiθ.
MART1: fjnew=fjold1-μΔϕiθ,MART2: fjnew=fjold1-μ wiθ,jwiθ,jmax1-Δϕiθ,MART3: fjnew=fjoldΔϕiθμwiθ,jwiθ,jmax.
AVMART1: fjnew=fjoldcj1-μΔϕiθ1cj,AVMART2: fjnew=fjoldcj1-μ wiθ,jwiθ,jmax×1-Δϕiθ1cj,AVMART3: fjnew=fjoldcjΔϕiθμwiθ,jwiθ,jmax1cj.
fx, y=f1x, yf2x, y,
f1x, y=A0 exp-x-x02Rx02exp-y-y02Ry02+A1 exp-x-x12Rx12exp-y-y12Ry12,f2x, y=sinπ30 xsinπ30 y,
ρmax=n ϕmax-ϕmin100,
rms=Nϕ-ϕnoise2N1/2.
E1=max|frecon-forig| maximum absolute difference,E2=Nfrecon-forig2N1/2rms error,E3=E2maxfrecon-minfrecon 100 normalized rms percent error,

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