Abstract

A reconstruction of phase objects using an algebraic reconstruction technique in an unconfined environment from multidirectional interferometric data is presented. The effect of noise on the data from the interference patterns is studied. It is shown that in the presence of noise the number of iterations need to be critically evaluated; otherwise the solution tends to diverge. Criteria used to quantify the noise are presented. Also shown is a relationship between noise and the required number of iterations, which yields the least error in reconstruction. This procedure is applied to experimentally obtained interferometric data.

© 1991 Optical Society of America

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References

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  1. R. W. Gerchberg, “Super resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
    [CrossRef]
  2. 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]
  3. N. Baba, K. Murata, “Image reconstruction from limited-angle projections,” Optik 60, 327–332 (1982).
  4. R. D. Matulka, D. J. Collins, “Determination of three-dimensional density fields from holographic interferograms,” J. Appl. Phys. 42, 1109–1119 (1971).
    [CrossRef]
  5. J. D. Trollinger, “Flow visualization holography,” Opt. Eng. 14, 470–481 (1975).
    [CrossRef]
  6. C. M. Vest, Holographic Interferometry (Wiley, New York, 1979).
  7. S. Cha, C. M. Vest, “Use of series expansion and envelope function for interferometric reconstruction of continuous flow fields,” in Flow Visualization and Aero-Optics in Simulated Environments, H. T. Bentley, ed., Proc. Soc. Photo-Opt. Instrum. Eng.788, 73–80 (1987).
  8. D. W. Sweeney, C. M. Vest, “Reconstruction of 3-D refractive index fields from multidirectional interferometric data,” Appl. Opt. 12, 2649–2664 (1973).
    [CrossRef] [PubMed]
  9. D. W. Sweeny, C. M. Vest, “Measurement of three dimensional temperature fields above heated surfaces by holographic interferometry,” Int. J. Heat Mass Trans. 17, 1443–1457 (1974).
    [CrossRef]
  10. S. Bahl, “Studies of three dimensional natural convection above horizontal heated disks using laser holographic interferometry,” Ph.D. dissertation (Clemson University; Clemson, S.C., 1988).
  11. B. E. Oppenheim, “More accurate algorithms for iterative 3-dimensional reconstruction,” IEEE Trans. Nucl. Sci. NS-21, 72–77 (1974).
  12. R. Gordon, “A tutorial on ART,” IEEE Trans. Nucl. Sci. NS-21 (3), 78–93 (1974).
  13. A. Rosenfeld, A. C. Kak, Digital Picture Processing, (Academic, New York, 1982), Vol. 1, pp. 353–430.
  14. S. Bahl, J. A. Liburdy, “Measurement of the three dimensional convective heat transfer coefficient on a heated disk,” accepted for publication in Int. J. Heat Mass Transfer 34, 949–960 (1991).
    [CrossRef]

1991

S. Bahl, J. A. Liburdy, “Measurement of the three dimensional convective heat transfer coefficient on a heated disk,” accepted for publication in Int. J. Heat Mass Transfer 34, 949–960 (1991).
[CrossRef]

1982

N. Baba, K. Murata, “Image reconstruction from limited-angle projections,” Optik 60, 327–332 (1982).

1981

1975

J. D. Trollinger, “Flow visualization holography,” Opt. Eng. 14, 470–481 (1975).
[CrossRef]

1974

D. W. Sweeny, C. M. Vest, “Measurement of three dimensional temperature fields above heated surfaces by holographic interferometry,” Int. J. Heat Mass Trans. 17, 1443–1457 (1974).
[CrossRef]

B. E. Oppenheim, “More accurate algorithms for iterative 3-dimensional reconstruction,” IEEE Trans. Nucl. Sci. NS-21, 72–77 (1974).

R. Gordon, “A tutorial on ART,” IEEE Trans. Nucl. Sci. NS-21 (3), 78–93 (1974).

R. W. Gerchberg, “Super resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
[CrossRef]

1973

1971

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

Baba, N.

N. Baba, K. Murata, “Image reconstruction from limited-angle projections,” Optik 60, 327–332 (1982).

Bahl, S.

S. Bahl, J. A. Liburdy, “Measurement of the three dimensional convective heat transfer coefficient on a heated disk,” accepted for publication in Int. J. Heat Mass Transfer 34, 949–960 (1991).
[CrossRef]

S. Bahl, “Studies of three dimensional natural convection above horizontal heated disks using laser holographic interferometry,” Ph.D. dissertation (Clemson University; Clemson, S.C., 1988).

Cha, S.

S. Cha, C. M. Vest, “Use of series expansion and envelope function for interferometric reconstruction of continuous flow fields,” in Flow Visualization and Aero-Optics in Simulated Environments, H. T. Bentley, ed., Proc. Soc. Photo-Opt. Instrum. Eng.788, 73–80 (1987).

Collins, D. J.

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

Gerchberg, R. W.

R. W. Gerchberg, “Super resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
[CrossRef]

Gordon, R.

R. Gordon, “A tutorial on ART,” IEEE Trans. Nucl. Sci. NS-21 (3), 78–93 (1974).

Hirama, M.

Ikeda, O.

Kak, A. C.

A. Rosenfeld, A. C. Kak, Digital Picture Processing, (Academic, New York, 1982), Vol. 1, pp. 353–430.

Liburdy, J. A.

S. Bahl, J. A. Liburdy, “Measurement of the three dimensional convective heat transfer coefficient on a heated disk,” accepted for publication in Int. J. Heat Mass Transfer 34, 949–960 (1991).
[CrossRef]

Linzer, M.

Matulka, R. D.

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

Murata, K.

N. Baba, K. Murata, “Image reconstruction from limited-angle projections,” Optik 60, 327–332 (1982).

Norton, S. J.

Oppenheim, B. E.

B. E. Oppenheim, “More accurate algorithms for iterative 3-dimensional reconstruction,” IEEE Trans. Nucl. Sci. NS-21, 72–77 (1974).

Rosenfeld, A.

A. Rosenfeld, A. C. Kak, Digital Picture Processing, (Academic, New York, 1982), Vol. 1, pp. 353–430.

Sato, T.

Sweeney, D. W.

Sweeny, D. W.

D. W. Sweeny, C. M. Vest, “Measurement of three dimensional temperature fields above heated surfaces by holographic interferometry,” Int. J. Heat Mass Trans. 17, 1443–1457 (1974).
[CrossRef]

Trollinger, J. D.

J. D. Trollinger, “Flow visualization holography,” Opt. Eng. 14, 470–481 (1975).
[CrossRef]

Vest, C. M.

D. W. Sweeny, C. M. Vest, “Measurement of three dimensional temperature fields above heated surfaces by holographic interferometry,” Int. J. Heat Mass Trans. 17, 1443–1457 (1974).
[CrossRef]

D. W. Sweeney, C. M. Vest, “Reconstruction of 3-D refractive index fields from multidirectional interferometric data,” Appl. Opt. 12, 2649–2664 (1973).
[CrossRef] [PubMed]

S. Cha, C. M. Vest, “Use of series expansion and envelope function for interferometric reconstruction of continuous flow fields,” in Flow Visualization and Aero-Optics in Simulated Environments, H. T. Bentley, ed., Proc. Soc. Photo-Opt. Instrum. Eng.788, 73–80 (1987).

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

Appl. Opt.

IEEE Trans. Nucl. Sci.

B. E. Oppenheim, “More accurate algorithms for iterative 3-dimensional reconstruction,” IEEE Trans. Nucl. Sci. NS-21, 72–77 (1974).

R. Gordon, “A tutorial on ART,” IEEE Trans. Nucl. Sci. NS-21 (3), 78–93 (1974).

Int. J. Heat Mass Trans.

D. W. Sweeny, C. M. Vest, “Measurement of three dimensional temperature fields above heated surfaces by holographic interferometry,” Int. J. Heat Mass Trans. 17, 1443–1457 (1974).
[CrossRef]

Int. J. Heat Mass Transfer

S. Bahl, J. A. Liburdy, “Measurement of the three dimensional convective heat transfer coefficient on a heated disk,” accepted for publication in Int. J. Heat Mass Transfer 34, 949–960 (1991).
[CrossRef]

J. Appl. Phys.

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

Opt. Acta

R. W. Gerchberg, “Super resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
[CrossRef]

Opt. Eng.

J. D. Trollinger, “Flow visualization holography,” Opt. Eng. 14, 470–481 (1975).
[CrossRef]

Optik

N. Baba, K. Murata, “Image reconstruction from limited-angle projections,” Optik 60, 327–332 (1982).

Other

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

S. Cha, C. M. Vest, “Use of series expansion and envelope function for interferometric reconstruction of continuous flow fields,” in Flow Visualization and Aero-Optics in Simulated Environments, H. T. Bentley, ed., Proc. Soc. Photo-Opt. Instrum. Eng.788, 73–80 (1987).

S. Bahl, “Studies of three dimensional natural convection above horizontal heated disks using laser holographic interferometry,” Ph.D. dissertation (Clemson University; Clemson, S.C., 1988).

A. Rosenfeld, A. C. Kak, Digital Picture Processing, (Academic, New York, 1982), Vol. 1, pp. 353–430.

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Figures (9)

Fig. 1
Fig. 1

Geometry and coordinate systems.

Fig. 2
Fig. 2

Asymmetric functions used in the verification of the reconstruction procedures: (a) f1 double-Gaussian function and (b) f2 single-peak function.

Fig. 3
Fig. 3

Effect of choosing different frequency grid sizes for the direct Fourier transform method with function f1: (a) 45° field of view; grid size, 5 × 11; (b) same field of view; grid size, 7 × 11.

Fig. 4
Fig. 4

(a) Reconstruction of f2 using the direct Fourier transform method with a 90° angle of view along the x axis; (b) same as (a) with the projection data added from one viewing direction along the y axis.

Fig. 5
Fig. 5

(a) Reconstruction of f2 using the iterative method with a 45° angle of view along the x axis; (b) same as (a) with the projection data added from one viewing direction along the y axis.

Fig. 6
Fig. 6

Term ∊i versus the number of iterations for various ∊p values using the ART with a 90° angle of view.

Fig. 7
Fig. 7

Residual ∊r versus the number of iterations for varius ∊p values using the ART with a 90° angle of view.

Fig. 8
Fig. 8

Local reconstructed function values f at several grid locations for ∊p = 0.09.

Fig. 9
Fig. 9

Residuals ∊t versus the number of iterations for three reconstructed temperature distributions obtained from interferometric fringe data.

Tables (3)

Tables Icon

Table I Errors in the Reconstruction of Functions f1 and f2a

Tables Icon

Table II Variation of Reconstruction Errors Using the ART with the Number of Iterations for Functions f1 and f2

Tables Icon

Table III Variation of Errors with the Extent of Reconstruction for Functions f1 and f2 Using the ART

Equations (5)

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

P ( ρ , θ ) = f ( x , y ) δ ( x cos θ + y sin θ ρ ) d x d y ,
i = f i f i 1 2 f 2 ;
r = f f 2 f 2 ;
t = T i T i 1 2 T i 2 .
E 1 = Σ | f f | N t | f max | , E 2 = Σ | f f | Σ | f | , E 3 = f f 2 f 2 , E 4 = | f max f max | f max ;

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