Abstract

A method for automated phase reconstruction from holographic interferograms of nonideal phase objects based on a two-dimensional Fourier transform is described. In particular, the problem of phase unwrapping is solved because earlier techniques are inappropriate for the phase unwrapping from interferograms of partially absorbent objects. A noise-level-dependent criterion for the binary mask that defines the unwrapping path for the flood algorithm is derived. The method shows high noise immunity, and the result is reliable provided that the true phase is free of discontinuities. The phase distribution in the outmasked regions is estimated by a linear least-squares fit to the surrounding unwrapped pixels.

© 1995 Optical Society of America

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References

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  1. K. Creath, “Phase-measurement interferometry techniques,” Prog. Opt. 26, 349–393 (1988).
    [CrossRef]
  2. M. Takeda, H. Ina, S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72, 156–160 (1982).
    [CrossRef]
  3. D. J. Bone, H.-A. Bachor, R. J. Sandeman, “Fringe- pattern analysis using a two-dimensional Fourier transform,” Appl. Opt. 25, 165–1660 (1986).
    [CrossRef]
  4. W. W. Macy, “Two-dimensional fringe-pattern analysis,” Appl. Opt. 22, 3898–3901 (1983).
    [CrossRef] [PubMed]
  5. D. C. Ghiglia, G. A. Mastin, L. A. Romero, “Cellular-automata method for phase unwrapping,” J. Opt. Soc. Am. A 4, 267–278 (1987).
    [CrossRef]
  6. J. J. Gierloff, “Phase unwrapping by regions,” in Current Developments in Optical Engineering II, R. E. Fischer, W. G. Smith, eds., Proc. Soc. Photo-Opt. Instrum. Eng.818, 2–9 (1987).
  7. R. M. Goldstein, H. A. Zebker, C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23, 713–720 (1988).
    [CrossRef]
  8. J. M. Huntley, “Noise-immune phase unwrapping algorithm,” Appl. Opt. 28, 3268–3270 (1989).
    [CrossRef] [PubMed]
  9. D. J. Bone, “Fourier fringe analysis: the two-dimensional phase unwrapping problem,” Appl. Opt. 30, 3627–3632 (1991).
    [CrossRef] [PubMed]
  10. H. A. Vrooman, A. A. M. Maas, “Image-processing algorithms for the analysis of phase-shifted speckle interference patterns,” Appl. Opt. 30, 1636–1641 (1991).
    [CrossRef] [PubMed]
  11. D. P. Towers, T. R. Judge, P. J. Bryanston-Cross, “A quasi heterodyne holographic technique and automatic algorithms for phase unwrapping,” in Fringe Pattern Analysis, G. T. Reid, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1163, 95–119 (1989).
  12. T. R. Judge, C. Quan, P. J. Bryanston-Cross, “Holographic deformation measurements by Fourier transform technique with automatic phase unwrapping,” Opt. Eng. 31, 533–543 (1992).
    [CrossRef]

1992 (1)

T. R. Judge, C. Quan, P. J. Bryanston-Cross, “Holographic deformation measurements by Fourier transform technique with automatic phase unwrapping,” Opt. Eng. 31, 533–543 (1992).
[CrossRef]

1991 (2)

D. J. Bone, “Fourier fringe analysis: the two-dimensional phase unwrapping problem,” Appl. Opt. 30, 3627–3632 (1991).
[CrossRef] [PubMed]

H. A. Vrooman, A. A. M. Maas, “Image-processing algorithms for the analysis of phase-shifted speckle interference patterns,” Appl. Opt. 30, 1636–1641 (1991).
[CrossRef] [PubMed]

1989 (1)

1988 (2)

R. M. Goldstein, H. A. Zebker, C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23, 713–720 (1988).
[CrossRef]

K. Creath, “Phase-measurement interferometry techniques,” Prog. Opt. 26, 349–393 (1988).
[CrossRef]

1987 (1)

1986 (1)

D. J. Bone, H.-A. Bachor, R. J. Sandeman, “Fringe- pattern analysis using a two-dimensional Fourier transform,” Appl. Opt. 25, 165–1660 (1986).
[CrossRef]

1983 (1)

1982 (1)

M. Takeda, H. Ina, S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72, 156–160 (1982).
[CrossRef]

Bachor, H.-A.

D. J. Bone, H.-A. Bachor, R. J. Sandeman, “Fringe- pattern analysis using a two-dimensional Fourier transform,” Appl. Opt. 25, 165–1660 (1986).
[CrossRef]

Bone, D. J.

D. J. Bone, “Fourier fringe analysis: the two-dimensional phase unwrapping problem,” Appl. Opt. 30, 3627–3632 (1991).
[CrossRef] [PubMed]

D. J. Bone, H.-A. Bachor, R. J. Sandeman, “Fringe- pattern analysis using a two-dimensional Fourier transform,” Appl. Opt. 25, 165–1660 (1986).
[CrossRef]

Bryanston-Cross, P. J.

T. R. Judge, C. Quan, P. J. Bryanston-Cross, “Holographic deformation measurements by Fourier transform technique with automatic phase unwrapping,” Opt. Eng. 31, 533–543 (1992).
[CrossRef]

D. P. Towers, T. R. Judge, P. J. Bryanston-Cross, “A quasi heterodyne holographic technique and automatic algorithms for phase unwrapping,” in Fringe Pattern Analysis, G. T. Reid, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1163, 95–119 (1989).

Creath, K.

K. Creath, “Phase-measurement interferometry techniques,” Prog. Opt. 26, 349–393 (1988).
[CrossRef]

Ghiglia, D. C.

Gierloff, J. J.

J. J. Gierloff, “Phase unwrapping by regions,” in Current Developments in Optical Engineering II, R. E. Fischer, W. G. Smith, eds., Proc. Soc. Photo-Opt. Instrum. Eng.818, 2–9 (1987).

Goldstein, R. M.

R. M. Goldstein, H. A. Zebker, C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23, 713–720 (1988).
[CrossRef]

Huntley, J. M.

Ina, H.

M. Takeda, H. Ina, S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72, 156–160 (1982).
[CrossRef]

Judge, T. R.

T. R. Judge, C. Quan, P. J. Bryanston-Cross, “Holographic deformation measurements by Fourier transform technique with automatic phase unwrapping,” Opt. Eng. 31, 533–543 (1992).
[CrossRef]

D. P. Towers, T. R. Judge, P. J. Bryanston-Cross, “A quasi heterodyne holographic technique and automatic algorithms for phase unwrapping,” in Fringe Pattern Analysis, G. T. Reid, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1163, 95–119 (1989).

Kobayashi, S.

M. Takeda, H. Ina, S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72, 156–160 (1982).
[CrossRef]

Maas, A. A. M.

H. A. Vrooman, A. A. M. Maas, “Image-processing algorithms for the analysis of phase-shifted speckle interference patterns,” Appl. Opt. 30, 1636–1641 (1991).
[CrossRef] [PubMed]

Macy, W. W.

Mastin, G. A.

Quan, C.

T. R. Judge, C. Quan, P. J. Bryanston-Cross, “Holographic deformation measurements by Fourier transform technique with automatic phase unwrapping,” Opt. Eng. 31, 533–543 (1992).
[CrossRef]

Romero, L. A.

Sandeman, R. J.

D. J. Bone, H.-A. Bachor, R. J. Sandeman, “Fringe- pattern analysis using a two-dimensional Fourier transform,” Appl. Opt. 25, 165–1660 (1986).
[CrossRef]

Takeda, M.

M. Takeda, H. Ina, S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72, 156–160 (1982).
[CrossRef]

Towers, D. P.

D. P. Towers, T. R. Judge, P. J. Bryanston-Cross, “A quasi heterodyne holographic technique and automatic algorithms for phase unwrapping,” in Fringe Pattern Analysis, G. T. Reid, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1163, 95–119 (1989).

Vrooman, H. A.

H. A. Vrooman, A. A. M. Maas, “Image-processing algorithms for the analysis of phase-shifted speckle interference patterns,” Appl. Opt. 30, 1636–1641 (1991).
[CrossRef] [PubMed]

Werner, C. L.

R. M. Goldstein, H. A. Zebker, C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23, 713–720 (1988).
[CrossRef]

Zebker, H. A.

R. M. Goldstein, H. A. Zebker, C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23, 713–720 (1988).
[CrossRef]

Appl. Opt. (1)

H. A. Vrooman, A. A. M. Maas, “Image-processing algorithms for the analysis of phase-shifted speckle interference patterns,” Appl. Opt. 30, 1636–1641 (1991).
[CrossRef] [PubMed]

Appl. Opt. (4)

J. Opt. Soc. Am. (1)

M. Takeda, H. Ina, S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72, 156–160 (1982).
[CrossRef]

J. Opt. Soc. Am. A (1)

Opt. Eng. (1)

T. R. Judge, C. Quan, P. J. Bryanston-Cross, “Holographic deformation measurements by Fourier transform technique with automatic phase unwrapping,” Opt. Eng. 31, 533–543 (1992).
[CrossRef]

Prog. Opt. (1)

K. Creath, “Phase-measurement interferometry techniques,” Prog. Opt. 26, 349–393 (1988).
[CrossRef]

Radio Sci. (1)

R. M. Goldstein, H. A. Zebker, C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23, 713–720 (1988).
[CrossRef]

Other (2)

D. P. Towers, T. R. Judge, P. J. Bryanston-Cross, “A quasi heterodyne holographic technique and automatic algorithms for phase unwrapping,” in Fringe Pattern Analysis, G. T. Reid, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1163, 95–119 (1989).

J. J. Gierloff, “Phase unwrapping by regions,” in Current Developments in Optical Engineering II, R. E. Fischer, W. G. Smith, eds., Proc. Soc. Photo-Opt. Instrum. Eng.818, 2–9 (1987).

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Figures (4)

Fig. 1
Fig. 1

(a) Processing steps for the reconstruction of the object wave from the interferogram, (b) the subsequent calculation and unwrapping of the phase distribution.

Fig. 2
Fig. 2

Phase error δ can be estimated from the complex error Δ u and the amplitude A(x, y) (see Appendix A).

Fig. 3
Fig. 3

(a) Holographical interferogram of a microvessel (venule), (b) the wrapped phase, (c) the amplitude distribution of the object wave, and (d) the binary mask obtained by thresholding of the amplitude image.

Fig. 4
Fig. 4

Reconstructed phase distribution of the microvessel.

Equations (16)

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u ( x , y ) = A ( x , y ) exp [ i ϕ ( x , y ) ] ,
I ( x , y ) R 2 + | u ( x , y ) | 2 + h ( x , y ) u ( x , y ) * + h ( x , y ) * u ( x , y ) .
[ u ] ( k x , k y ) 0 for ( k x , k y ) > k 0 .
u ( x , y ) = Re ( x , y ) + i Im ( x , y ) ,
Δ u := E [ | u ( x , y ) u true ( x , y ) | ] .
ϕ ω ( x , y ) = { arcsin [ Im ! ( Re 2 + Im 2 ) 1 / 2 ] for Re > 0 π arcsin [ Im ! ( Re 2 + Im 2 ) 1 / 2 ] othewise .
Γ = k = 1 N { [ ϕ ω ( x k , y k ) ϕ ω ( x k 1 , y k 1 ) ] / 2 π } ,
Δ sin δ ( x , y ) E ( | sin δ | ) = 2 Δ u π A ( x , y ) .
A ( x , y ) > 8 Δ u π 3 Δ u ,
A ( x , y ) = [ Re ( x , y ) 2 + Im ( x , y ) 2 ] 1 / 2
Γ = ( k , l ) W * [ a ( k x ) + b ( l y ) + c ϕ ( k , l ) ] 2 ,
0 d r ( r ) 0 2 π d α f ( r , α ) = 1 .
Δ u = E ( r ) = 0 d r r 2 0 2 π d α f ( r ) = 2 π 0 d r r 2 f ( r ) .
sin δ = r ( r 2 + A 2 2 A r cos α ) 1 / 2 sin α .
Δ sin δ = E ( | sin δ | ) = 0 d r r f ( r ) 2 0 π d α r sin α ( r 2 + A 2 2 A r cos α ) 1 / 2 = 2 0 d r r f ( r ) 1 1 d x 1 ( 1 + ( A 2 / r 2 ) 2 ( A / r ) x ) 1 / 2 = 2 A 0 d r r 2 f ( r ) ( 1 A / r ) 2 ( 1 + A / r ) 2 d y 1 2 y = 2 A 0 d r r 2 f ( r ) ( 1 + A r | 1 A r | ) = 4 A 0 A d r r 2 f ( r ) + 4 A d r r f ( r ) ,
Δ sin δ ( x , y ) 4 A ( x , y ) 0 d r r 2 f ( r ) = 2 Δ u π A ( x , y ) .

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