Abstract

We propose phase retrieval from three or more interferograms corresponding to different tilts of an object wavefront. The algorithm uses the information contained in the interferogram differences to reduce the problem to phase shifting. Three interferograms is the minimum for restoring the phase over most of the image. Four or more interferograms are needed to restore the phase over the whole image. The method works with images including open and closed fringes in any combination.

© 2005 Optical Society of America

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References

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  1. D. Malacara, M. Servín, and Z. Malacara, Interferogram analysis for optical shop testing (Marcel Dekker, Inc., New York, Basel, Hong Kong, 1998).
  2. Y. Surrel, “Fringe analysis,” in Photomechanics, P. K. Rastogi, ed. (Springer, 1999).
  3. Y. Surrel, “Design of algorithms for phase measurements by the use of phase stepping,” Appl. Opt. 35(1), 51–60 (1996).
    [Crossref]
  4. D. Malacara, ed., Optical Shop Testing, 2nd ed. (John Wiley & Sons, Inc., 1992).
  5. G. Lai and T. Yatagai, “Generalized phase-shifting interferometry,” J. Opt. Soc. Am. A 8(5), 822–827 (1991).
    [Crossref]
  6. K. Okada, A. Sato, and J. Tujiuchi, “Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry,” Opt. Commun. 84(3,4), 118–124 (1991).
    [Crossref]
  7. I.-B. Kong and S.-W. Kim, “General algorithm of phase-shifting interferometry by iterative least-squares fitting,” Opt. Eng. 34, 183–187 (1995).
    [Crossref]
  8. M. Chen, H. Guo, and C. Wei, “Algorithm immune to tilt phase-shifting error for phase-shifting interferometers,” Appl. Opt. 39(22), 3894–3898 (2000).
    [Crossref]
  9. A. Dobroiu, D. Apostol, V. Nascov, and V. Damian, “Tilt-compensating algorithm for phase-shift interferometry,” Appl. Opt. 41(13), 2435–2439 (2002).
    [Crossref]
  10. T. Kreis, “Digital holographic interference-phase measurement using the Fourier-transform method,” J. Opt. Soc. Am. A 3(6), 847–855 (1986).
    [Crossref]
  11. J. A. Quiroga, J. A. Gómez-Pedrero, and á. García-Botella, “Algorithm for fringe pattern normalisation,” Opt. Commun. 197, 43–51 (2001).
    [Crossref]
  12. C. Roddier and F. Roddier, “Interferogram Analysis Using Fourier Transform Techniques,” Appl. Opt. 26, 1668–1673 (1987).
    [Crossref] [PubMed]
  13. Z. Ge, F. Kobayashi, S. Matsuda, and M. Takeda, “Coordinate-transform technique for closed-fringe analysis by the Fourier-transform method,” Appl. Opt. 40(10), 1649–1657 (2001).
    [Crossref]
  14. K. A. Goldberg and J. Bokor, “Fourier-transform method of phase-shift determination,” Appl. Opt. 40(17), 2886–2893 (2001).
    [Crossref]
  15. F. Cuevas, J. Sossa-Azuela, and M. Servin, “A parametric method applied to phase recovery from a fringe pattern based on a genetic algorithm,” Opt. Commun. 203, 213–223 (2002).
    [Crossref]
  16. M. Servin, J. A. Quiroga, and J. L. Marroquin, “General n-dimensional quadrature transform and its application to interferogram demodulation,” J. Opt. Soc. Am. A 20(5), 925–934 (2003).
    [Crossref]
  17. E. Yu and S. S. Cha, “Two-dimensional regression for interferometric phase extraction,” Appl. Opt. 37(8), 1370–1376 (1998).
    [Crossref]
  18. G. X. Ritter and J. N. Wilson, Handbook of computer vision algoritm in image algebra (CRC Press, Boca Raton, New York, London, Tokyo, 1996).

2003 (1)

M. Servin, J. A. Quiroga, and J. L. Marroquin, “General n-dimensional quadrature transform and its application to interferogram demodulation,” J. Opt. Soc. Am. A 20(5), 925–934 (2003).
[Crossref]

2002 (2)

F. Cuevas, J. Sossa-Azuela, and M. Servin, “A parametric method applied to phase recovery from a fringe pattern based on a genetic algorithm,” Opt. Commun. 203, 213–223 (2002).
[Crossref]

A. Dobroiu, D. Apostol, V. Nascov, and V. Damian, “Tilt-compensating algorithm for phase-shift interferometry,” Appl. Opt. 41(13), 2435–2439 (2002).
[Crossref]

2001 (3)

J. A. Quiroga, J. A. Gómez-Pedrero, and á. García-Botella, “Algorithm for fringe pattern normalisation,” Opt. Commun. 197, 43–51 (2001).
[Crossref]

Z. Ge, F. Kobayashi, S. Matsuda, and M. Takeda, “Coordinate-transform technique for closed-fringe analysis by the Fourier-transform method,” Appl. Opt. 40(10), 1649–1657 (2001).
[Crossref]

K. A. Goldberg and J. Bokor, “Fourier-transform method of phase-shift determination,” Appl. Opt. 40(17), 2886–2893 (2001).
[Crossref]

2000 (1)

M. Chen, H. Guo, and C. Wei, “Algorithm immune to tilt phase-shifting error for phase-shifting interferometers,” Appl. Opt. 39(22), 3894–3898 (2000).
[Crossref]

1998 (1)

E. Yu and S. S. Cha, “Two-dimensional regression for interferometric phase extraction,” Appl. Opt. 37(8), 1370–1376 (1998).
[Crossref]

1996 (1)

Y. Surrel, “Design of algorithms for phase measurements by the use of phase stepping,” Appl. Opt. 35(1), 51–60 (1996).
[Crossref]

1995 (1)

I.-B. Kong and S.-W. Kim, “General algorithm of phase-shifting interferometry by iterative least-squares fitting,” Opt. Eng. 34, 183–187 (1995).
[Crossref]

1991 (2)

G. Lai and T. Yatagai, “Generalized phase-shifting interferometry,” J. Opt. Soc. Am. A 8(5), 822–827 (1991).
[Crossref]

K. Okada, A. Sato, and J. Tujiuchi, “Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry,” Opt. Commun. 84(3,4), 118–124 (1991).
[Crossref]

1987 (1)

1986 (1)

T. Kreis, “Digital holographic interference-phase measurement using the Fourier-transform method,” J. Opt. Soc. Am. A 3(6), 847–855 (1986).
[Crossref]

Apostol, D.

A. Dobroiu, D. Apostol, V. Nascov, and V. Damian, “Tilt-compensating algorithm for phase-shift interferometry,” Appl. Opt. 41(13), 2435–2439 (2002).
[Crossref]

Bokor, J.

K. A. Goldberg and J. Bokor, “Fourier-transform method of phase-shift determination,” Appl. Opt. 40(17), 2886–2893 (2001).
[Crossref]

Cha, S. S.

E. Yu and S. S. Cha, “Two-dimensional regression for interferometric phase extraction,” Appl. Opt. 37(8), 1370–1376 (1998).
[Crossref]

Chen, M.

M. Chen, H. Guo, and C. Wei, “Algorithm immune to tilt phase-shifting error for phase-shifting interferometers,” Appl. Opt. 39(22), 3894–3898 (2000).
[Crossref]

Cuevas, F.

F. Cuevas, J. Sossa-Azuela, and M. Servin, “A parametric method applied to phase recovery from a fringe pattern based on a genetic algorithm,” Opt. Commun. 203, 213–223 (2002).
[Crossref]

Damian, V.

A. Dobroiu, D. Apostol, V. Nascov, and V. Damian, “Tilt-compensating algorithm for phase-shift interferometry,” Appl. Opt. 41(13), 2435–2439 (2002).
[Crossref]

Dobroiu, A.

A. Dobroiu, D. Apostol, V. Nascov, and V. Damian, “Tilt-compensating algorithm for phase-shift interferometry,” Appl. Opt. 41(13), 2435–2439 (2002).
[Crossref]

García-Botella, á.

J. A. Quiroga, J. A. Gómez-Pedrero, and á. García-Botella, “Algorithm for fringe pattern normalisation,” Opt. Commun. 197, 43–51 (2001).
[Crossref]

Ge, Z.

Z. Ge, F. Kobayashi, S. Matsuda, and M. Takeda, “Coordinate-transform technique for closed-fringe analysis by the Fourier-transform method,” Appl. Opt. 40(10), 1649–1657 (2001).
[Crossref]

Goldberg, K. A.

K. A. Goldberg and J. Bokor, “Fourier-transform method of phase-shift determination,” Appl. Opt. 40(17), 2886–2893 (2001).
[Crossref]

Gómez-Pedrero, J. A.

J. A. Quiroga, J. A. Gómez-Pedrero, and á. García-Botella, “Algorithm for fringe pattern normalisation,” Opt. Commun. 197, 43–51 (2001).
[Crossref]

Guo, H.

M. Chen, H. Guo, and C. Wei, “Algorithm immune to tilt phase-shifting error for phase-shifting interferometers,” Appl. Opt. 39(22), 3894–3898 (2000).
[Crossref]

Kim, S.-W.

I.-B. Kong and S.-W. Kim, “General algorithm of phase-shifting interferometry by iterative least-squares fitting,” Opt. Eng. 34, 183–187 (1995).
[Crossref]

Kobayashi, F.

Z. Ge, F. Kobayashi, S. Matsuda, and M. Takeda, “Coordinate-transform technique for closed-fringe analysis by the Fourier-transform method,” Appl. Opt. 40(10), 1649–1657 (2001).
[Crossref]

Kong, I.-B.

I.-B. Kong and S.-W. Kim, “General algorithm of phase-shifting interferometry by iterative least-squares fitting,” Opt. Eng. 34, 183–187 (1995).
[Crossref]

Kreis, T.

T. Kreis, “Digital holographic interference-phase measurement using the Fourier-transform method,” J. Opt. Soc. Am. A 3(6), 847–855 (1986).
[Crossref]

Lai, G.

G. Lai and T. Yatagai, “Generalized phase-shifting interferometry,” J. Opt. Soc. Am. A 8(5), 822–827 (1991).
[Crossref]

Malacara, D.

D. Malacara, M. Servín, and Z. Malacara, Interferogram analysis for optical shop testing (Marcel Dekker, Inc., New York, Basel, Hong Kong, 1998).

Malacara, Z.

D. Malacara, M. Servín, and Z. Malacara, Interferogram analysis for optical shop testing (Marcel Dekker, Inc., New York, Basel, Hong Kong, 1998).

Marroquin, J. L.

M. Servin, J. A. Quiroga, and J. L. Marroquin, “General n-dimensional quadrature transform and its application to interferogram demodulation,” J. Opt. Soc. Am. A 20(5), 925–934 (2003).
[Crossref]

Matsuda, S.

Z. Ge, F. Kobayashi, S. Matsuda, and M. Takeda, “Coordinate-transform technique for closed-fringe analysis by the Fourier-transform method,” Appl. Opt. 40(10), 1649–1657 (2001).
[Crossref]

Nascov, V.

A. Dobroiu, D. Apostol, V. Nascov, and V. Damian, “Tilt-compensating algorithm for phase-shift interferometry,” Appl. Opt. 41(13), 2435–2439 (2002).
[Crossref]

Okada, K.

K. Okada, A. Sato, and J. Tujiuchi, “Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry,” Opt. Commun. 84(3,4), 118–124 (1991).
[Crossref]

Quiroga, J. A.

M. Servin, J. A. Quiroga, and J. L. Marroquin, “General n-dimensional quadrature transform and its application to interferogram demodulation,” J. Opt. Soc. Am. A 20(5), 925–934 (2003).
[Crossref]

J. A. Quiroga, J. A. Gómez-Pedrero, and á. García-Botella, “Algorithm for fringe pattern normalisation,” Opt. Commun. 197, 43–51 (2001).
[Crossref]

Ritter, G. X.

G. X. Ritter and J. N. Wilson, Handbook of computer vision algoritm in image algebra (CRC Press, Boca Raton, New York, London, Tokyo, 1996).

Roddier, C.

Roddier, F.

Sato, A.

K. Okada, A. Sato, and J. Tujiuchi, “Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry,” Opt. Commun. 84(3,4), 118–124 (1991).
[Crossref]

Servin, M.

M. Servin, J. A. Quiroga, and J. L. Marroquin, “General n-dimensional quadrature transform and its application to interferogram demodulation,” J. Opt. Soc. Am. A 20(5), 925–934 (2003).
[Crossref]

F. Cuevas, J. Sossa-Azuela, and M. Servin, “A parametric method applied to phase recovery from a fringe pattern based on a genetic algorithm,” Opt. Commun. 203, 213–223 (2002).
[Crossref]

Servín, M.

D. Malacara, M. Servín, and Z. Malacara, Interferogram analysis for optical shop testing (Marcel Dekker, Inc., New York, Basel, Hong Kong, 1998).

Sossa-Azuela, J.

F. Cuevas, J. Sossa-Azuela, and M. Servin, “A parametric method applied to phase recovery from a fringe pattern based on a genetic algorithm,” Opt. Commun. 203, 213–223 (2002).
[Crossref]

Surrel, Y.

Y. Surrel, “Design of algorithms for phase measurements by the use of phase stepping,” Appl. Opt. 35(1), 51–60 (1996).
[Crossref]

Y. Surrel, “Fringe analysis,” in Photomechanics, P. K. Rastogi, ed. (Springer, 1999).

Takeda, M.

Z. Ge, F. Kobayashi, S. Matsuda, and M. Takeda, “Coordinate-transform technique for closed-fringe analysis by the Fourier-transform method,” Appl. Opt. 40(10), 1649–1657 (2001).
[Crossref]

Tujiuchi, J.

K. Okada, A. Sato, and J. Tujiuchi, “Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry,” Opt. Commun. 84(3,4), 118–124 (1991).
[Crossref]

Wei, C.

M. Chen, H. Guo, and C. Wei, “Algorithm immune to tilt phase-shifting error for phase-shifting interferometers,” Appl. Opt. 39(22), 3894–3898 (2000).
[Crossref]

Wilson, J. N.

G. X. Ritter and J. N. Wilson, Handbook of computer vision algoritm in image algebra (CRC Press, Boca Raton, New York, London, Tokyo, 1996).

Yatagai, T.

G. Lai and T. Yatagai, “Generalized phase-shifting interferometry,” J. Opt. Soc. Am. A 8(5), 822–827 (1991).
[Crossref]

Yu, E.

E. Yu and S. S. Cha, “Two-dimensional regression for interferometric phase extraction,” Appl. Opt. 37(8), 1370–1376 (1998).
[Crossref]

Appl. Opt. (7)

Y. Surrel, “Design of algorithms for phase measurements by the use of phase stepping,” Appl. Opt. 35(1), 51–60 (1996).
[Crossref]

M. Chen, H. Guo, and C. Wei, “Algorithm immune to tilt phase-shifting error for phase-shifting interferometers,” Appl. Opt. 39(22), 3894–3898 (2000).
[Crossref]

A. Dobroiu, D. Apostol, V. Nascov, and V. Damian, “Tilt-compensating algorithm for phase-shift interferometry,” Appl. Opt. 41(13), 2435–2439 (2002).
[Crossref]

C. Roddier and F. Roddier, “Interferogram Analysis Using Fourier Transform Techniques,” Appl. Opt. 26, 1668–1673 (1987).
[Crossref] [PubMed]

Z. Ge, F. Kobayashi, S. Matsuda, and M. Takeda, “Coordinate-transform technique for closed-fringe analysis by the Fourier-transform method,” Appl. Opt. 40(10), 1649–1657 (2001).
[Crossref]

K. A. Goldberg and J. Bokor, “Fourier-transform method of phase-shift determination,” Appl. Opt. 40(17), 2886–2893 (2001).
[Crossref]

E. Yu and S. S. Cha, “Two-dimensional regression for interferometric phase extraction,” Appl. Opt. 37(8), 1370–1376 (1998).
[Crossref]

J. Opt. Soc. Am. A (3)

G. Lai and T. Yatagai, “Generalized phase-shifting interferometry,” J. Opt. Soc. Am. A 8(5), 822–827 (1991).
[Crossref]

T. Kreis, “Digital holographic interference-phase measurement using the Fourier-transform method,” J. Opt. Soc. Am. A 3(6), 847–855 (1986).
[Crossref]

M. Servin, J. A. Quiroga, and J. L. Marroquin, “General n-dimensional quadrature transform and its application to interferogram demodulation,” J. Opt. Soc. Am. A 20(5), 925–934 (2003).
[Crossref]

Opt. Commun. (3)

J. A. Quiroga, J. A. Gómez-Pedrero, and á. García-Botella, “Algorithm for fringe pattern normalisation,” Opt. Commun. 197, 43–51 (2001).
[Crossref]

K. Okada, A. Sato, and J. Tujiuchi, “Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry,” Opt. Commun. 84(3,4), 118–124 (1991).
[Crossref]

F. Cuevas, J. Sossa-Azuela, and M. Servin, “A parametric method applied to phase recovery from a fringe pattern based on a genetic algorithm,” Opt. Commun. 203, 213–223 (2002).
[Crossref]

Opt. Eng. (1)

I.-B. Kong and S.-W. Kim, “General algorithm of phase-shifting interferometry by iterative least-squares fitting,” Opt. Eng. 34, 183–187 (1995).
[Crossref]

Other (4)

G. X. Ritter and J. N. Wilson, Handbook of computer vision algoritm in image algebra (CRC Press, Boca Raton, New York, London, Tokyo, 1996).

D. Malacara, ed., Optical Shop Testing, 2nd ed. (John Wiley & Sons, Inc., 1992).

D. Malacara, M. Servín, and Z. Malacara, Interferogram analysis for optical shop testing (Marcel Dekker, Inc., New York, Basel, Hong Kong, 1998).

Y. Surrel, “Fringe analysis,” in Photomechanics, P. K. Rastogi, ed. (Springer, 1999).

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Figures (13)

Fig. 1.
Fig. 1.

Three original interferograms I 0, I 1, I 2 obtained with a Twyman-Green interferometer. The first interferogram corresponds to the original object; additional tilts were introduced in the second and third interferograms. Note that all three interferograms contain closed fringes.

Fig. 2.
Fig. 2.

Differences of the interferograms Id,1 =I 1-I 0 and I d,2 =I 2-I 0. Black and white correspond to minimum (-255) and maximum (255) levels, respectively. The circle marks the mask. Masking the object edges helps to avoid spurious zero level points.

Fig. 3.
Fig. 3.

Points in interferogram differences which are close to zero are used as zero levels. The threshold was chosen to be 1, the minimal possible level that is greater then 0.

Fig. 4.
Fig. 4.

The differential images I d,1 and I d,2 in the Hough space. R=354, δρ =1, δθ =π/360. One can see that zero-level points of I d,1 should contain about 8 lines at angle θ about 170π/360, and I d,2 — about 14 lines at angle approximately equal to 70π/360.

Fig. 5.
Fig. 5.

Accumulators’ rows containing the maximum elements. The local maxima corresponds to the sinusoidal intersection points. The highest maxima corresponds to parallel lines in zero-crossing curves. The second raw of pictures shows the maxima values for normalized accumulator.

Fig. 6.
Fig. 6.

Normalized accumulators’ rows containing the maximum elements and a norm function we have used. The norm function is the same for every horizontal line in the Hough space.

Fig. 7.
Fig. 7.

Lines detected by the Hough transform. The threshold α for maxima selection was set to 0.45 for the first set of zero-level points and to 0.25 for the second one.

Fig. 8.
Fig. 8.

Interferogram differences divided by 2 sin τ i + σ i 2 .

Fig. 9.
Fig. 9.

Extracted phase ϕ + τ 2 + σ 2 2 .

Fig. 10.
Fig. 10.

The fourth and fifth interferograms, the phase calculated for other tilt values, and the resulting phase obtained by median.

Fig. 11.
Fig. 11.

An interferogram with a linear carrier, the phase extracted with Fourier-transform method used as a reference, and wrapped difference between the reference and the median phase from Fig. 10.

Fig. 12.
Fig. 12.

A small tilt will give a wrong slope.

Fig. 13.
Fig. 13.

If one of the tilt signs is incorrect, formula (8) produces unrealistic phase map.

Equations (12)

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

I [ x ] = a [ x ] + b [ x ] cos ( ϕ [ x ] ) ,
I i [ x ] = a + b cos ( ϕ i [ x ] ) , i = 0 , 1 , 2 ,
ϕ 1 [ x ] = ϕ 0 [ x ] + τ 1 [ x ] + σ 1 ,
ϕ 2 [ x ] = ϕ 0 [ x ] + τ 2 [ x ] + σ 2 ,
ϕ = arctan I 2 I 1 + ( I 0 I 2 ) cos δ 1 + ( I 1 I 0 ) cos δ 2 ( I 0 I 2 ) sin δ 1 + ( I 1 I 0 ) sin δ 2 .
I d , 1 = 2 b sin τ 1 + σ 1 2 sin ( ϕ [ x ] + τ 1 + σ 1 2 ) ,
I d , 2 = 2 b sin τ 2 + σ 2 2 sin ( ϕ [ x ] + τ 2 + σ 2 2 ) .
θ , λ , s ,
I ̂ d , 1 = b sin ( ϕ [ x ] + τ 1 + σ 1 2 ) and
I ̂ d , 2 = b sin ( ϕ [ x ] + τ 2 + σ 2 2 ) ,
b cos ( ϕ [ x ] + τ 2 + σ 2 2 ) = I ̂ d , 1 cos ( τ 1 + σ 1 τ 2 σ 2 2 ) I ̂ d , 2 sin ( τ 1 + σ 1 τ 2 σ 2 2 ) .
ϕ + τ 2 + σ 2 2 = arctan ( I ̂ d , 1 cos ( τ 1 + σ 1 τ 2 σ 2 2 ) I ̂ d , 2 sin ( τ 1 + σ 1 τ 2 σ 2 2 ) , I ̂ d , 2 ) .

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