Abstract

Tomographic reconstruction techniques used to obtain three-dimensional refractive index fields from interferometric data all assume that the correct fringe number has been assigned. In confined fields or where the imaging beam is smaller than the object field, only relative fringe numbers can be determined from the interferogram because no reference (undisturbed) region is visible. An algorithm that can be applied to iterative reconstruction techniques is described that permits the quantitative reconstruction of three-dimensional fields by using only relative fringe numbers provided that a priori data are supplied. This method is applied to an existing reconstruction technique and is demonstrated by using simulated interferometric data.

© 1993 Optical Society of America

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References

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  1. Y. H. Yu, J. K. Kittleson, “Reconstruction of a three-dimensional, transonic rotor flowfield from holographic interferograms,” AIAA J.25, 300–305 (1987).
    [CrossRef]
  2. S. Bahl, J. A. Liburdy, “Measurements of local convective heat transfer coefficients using three-dimensional interferometry,” Int. J. Heat Mass Transfer 34, 949–960 (1991).
    [CrossRef]
  3. 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]
  4. S. S. Cha, H. Sun, “Tomography for reconstructing continuous fields from ill-posed multidirectional interferometric data,” Appl. Opt. 29, 251–258 (1990).
    [CrossRef] [PubMed]
  5. 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]
  6. S. S. Cha, H. Sun, “Complementary field method for interferometric tomographic reconstruction of high-speed aerodynamic flows,” Opt. Eng. 28, 1241–1246 (1989).
  7. D. W. Sweeney, C. M. Vest, “Reconstruction of three-dimensional refractive index fields from multidirectional interferometric data,” Appl. Opt. 12, 2649–2664 (1973).
    [CrossRef] [PubMed]
  8. C. M. Vest, Holographic Interferometry (Wiley, New York, 1979), pp. 315–329.

1991 (2)

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

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]

1990 (1)

1989 (1)

S. S. Cha, H. Sun, “Complementary field method for interferometric tomographic reconstruction of high-speed aerodynamic flows,” Opt. Eng. 28, 1241–1246 (1989).

1985 (1)

1973 (1)

Bahl, S.

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

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]

Cha, S. S.

S. S. Cha, H. Sun, “Tomography for reconstructing continuous fields from ill-posed multidirectional interferometric data,” Appl. Opt. 29, 251–258 (1990).
[CrossRef] [PubMed]

S. S. Cha, H. Sun, “Complementary field method for interferometric tomographic reconstruction of high-speed aerodynamic flows,” Opt. Eng. 28, 1241–1246 (1989).

Dhawan, A. P.

Gordon, R.

Kittleson, J. K.

Y. H. Yu, J. K. Kittleson, “Reconstruction of a three-dimensional, transonic rotor flowfield from holographic interferograms,” AIAA J.25, 300–305 (1987).
[CrossRef]

Liburdy, J. A.

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

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]

Rangayyan, R.

Sun, H.

S. S. Cha, H. Sun, “Tomography for reconstructing continuous fields from ill-posed multidirectional interferometric data,” Appl. Opt. 29, 251–258 (1990).
[CrossRef] [PubMed]

S. S. Cha, H. Sun, “Complementary field method for interferometric tomographic reconstruction of high-speed aerodynamic flows,” Opt. Eng. 28, 1241–1246 (1989).

Sweeney, D. W.

Vest, C. M.

Yu, Y. H.

Y. H. Yu, J. K. Kittleson, “Reconstruction of a three-dimensional, transonic rotor flowfield from holographic interferograms,” AIAA J.25, 300–305 (1987).
[CrossRef]

Appl. Opt. (4)

Int. J. Heat Mass Transfer (1)

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

Opt. Eng. (1)

S. S. Cha, H. Sun, “Complementary field method for interferometric tomographic reconstruction of high-speed aerodynamic flows,” Opt. Eng. 28, 1241–1246 (1989).

Other (2)

C. M. Vest, Holographic Interferometry (Wiley, New York, 1979), pp. 315–329.

Y. H. Yu, J. K. Kittleson, “Reconstruction of a three-dimensional, transonic rotor flowfield from holographic interferograms,” AIAA J.25, 300–305 (1987).
[CrossRef]

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Figures (10)

Fig. 1
Fig. 1

Nomenclature for the projection of a refractive index field in multidirectional interferometry.

Fig. 2
Fig. 2

Analytic one-hump function used to evaluate the MCFM.

Fig. 3
Fig. 3

Convergence of the phase shift Ns as a function of the number of iterations for the one-hump distribution.

Fig. 4
Fig. 4

Iterative reduction of the average error of reconstruction for the one-hump distribution.

Fig. 5
Fig. 5

Effect of the placement of the a priori data on the iterative reduction of the average error of reconstruction for the one-hump distribution.

Fig. 6
Fig. 6

Reconstruction of the one-hump field obtained by the MCFM (36 points specified, 350 iterations).

Fig. 7
Fig. 7

Analytic two-hump function used to evaluate the MCFM.

Fig. 8
Fig. 8

Convergence of the phase shift Ns as a function of the number of iterations for the two-hump distribution.

Fig. 9
Fig. 9

Iterative reduction of the average error of reconstruction for the two-hump distribution.

Fig. 10
Fig. 10

Reconstruction of the two-hump field by the MCFM (36 points specified, 350 iterations).

Equations (14)

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g e ( ρ , θ ) = L { f e ( x , y ) } ,
g ( ρ , θ ) = { N ¯ ( ρ , θ ) + N s ( θ ) } λ / 2 ,
g ¯ ( ρ , θ ) = N ¯ ( ρ , θ ) λ / 2.
N s , i = [ 2 g e ( ρ i , θ i ) / λ ] - N ¯ i .
β = abs ( i = 1 n N s , i ) / i = 1 n abs ( N s , i ) ,
N s = { 1 n i = 1 n N s , i } β .
g ( ρ , θ ) = ( N s + N ¯ ) λ / 2.
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 ) + α f c ( x , y ) .
f r new ( x , y ) = C { f r ( x , y ) } ,
f ( x , y ) = exp { 1.0 - [ ( x - 0.2 ) 2 + ( y - 0.2 ) 2 ] } ,
Average error = { i = 1 I j = 1 J abs [ f r ( x i , y j ) - f ( x i , y j ) ] } { ( I × J ) max [ f ( x i , y j ) ] } ,
f ( x , y ) = exp { 1.0 = [ 3.5 ( x - 0.6 ) 2 + y 2 ] } + 0.7 exp { 1.0 - [ 3.5 ( x + 0.6 ) 2 + y 2 ] }

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