Abstract

We describe a new algorithm for phase determination from a single interferogram with closed fringes based on an unwrapping procedure. Here we use bandpass filtering in the Fourier domain, obtaining two wrapped phases with sign changes corresponding to the orientation of the applied filters. An unwrapping scheme that corrects the sign ambiguities by comparing the local derivatives is then proposed. This can be done, assuming that the phase derivatives do not change abruptly among adjacent areas as occurs with smooth continuous phase maps. The proposed algorithm works fast and is robust against noise, as demonstrated in experimental and simulated data.

© 2010 Optical Society of America

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References

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    [CrossRef]
  5. M. Pirga and M. Kujawinska, “Two-directional spatial-carrier phase-shifting method for analysis of crossed and closed fringe patterns,” Opt. Eng. 34, 2459–2466 (1995).
    [CrossRef]
  6. M. Servin and R. Rodriguez-Vera, “Two dimensional phase locked loop demodulation of carrier frequency interferograms,” J. Mod. Opt. 40, 2087–2094 (1993).
    [CrossRef]
  7. J. Kozlowski and G. Serra, “New modified phase locked loop method for fringe pattern demodulation,” Opt. Eng. 36, 2025–2030 (1997).
    [CrossRef]
  8. J. E. Greivenkamp, “Sub-Nyquist interferometry,” Appl. Opt. 26, 5245–5257 (1987).
    [CrossRef] [PubMed]
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    [CrossRef]
  10. K. H. Womack, “Interferometric phase measurement using spatial synchronous detection,” Opt. Eng. 23, 391–395 (1984).
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    [CrossRef] [PubMed]
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    [CrossRef]
  13. J. A. Quiroga, M. Servin, and F. J. Cuevas, “Modulo 2π fringe orientation angle estimation by phase unwrapping with a regularized phase tracking algorithm,” J. Opt. Soc. Am. A 19, 1524–1531 (2002).
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  17. J. Muñoz, F. Lecona, C. Castillo, F. Casillas, V. Duran, and M. Mora, “Phase recovery from a single interferogram using multiple Fourier transforms,” Proc. SPIE 6292, 62920D (2006).
    [CrossRef]
  18. Q. Kemao and H. Seah, “Sequential demodulation of a single fringe pattern guided by local frequencies,” Opt. Lett. 32, 127–129 (2007).
    [CrossRef]
  19. J. L. Marroquin and M. Rivera, “Quadratic regularization functionals for phase unwrapping,” J. Opt. Soc. Am. A 12, 2393–2400 (1995).
    [CrossRef]
  20. D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, 1998).

2007 (1)

2006 (1)

J. Muñoz, F. Lecona, C. Castillo, F. Casillas, V. Duran, and M. Mora, “Phase recovery from a single interferogram using multiple Fourier transforms,” Proc. SPIE 6292, 62920D (2006).
[CrossRef]

2002 (2)

2001 (3)

1997 (2)

1995 (2)

M. Pirga and M. Kujawinska, “Two-directional spatial-carrier phase-shifting method for analysis of crossed and closed fringe patterns,” Opt. Eng. 34, 2459–2466 (1995).
[CrossRef]

J. L. Marroquin and M. Rivera, “Quadratic regularization functionals for phase unwrapping,” J. Opt. Soc. Am. A 12, 2393–2400 (1995).
[CrossRef]

1993 (1)

M. Servin and R. Rodriguez-Vera, “Two dimensional phase locked loop demodulation of carrier frequency interferograms,” J. Mod. Opt. 40, 2087–2094 (1993).
[CrossRef]

1991 (1)

M. Kujawinska and J. Schmidt, “Spatial-carrier phase-shifting technique of fringe pattern analysis,” Proc. SPIE 1508, 61–67 (1991).
[CrossRef]

1987 (1)

1986 (1)

1984 (1)

K. H. Womack, “Interferometric phase measurement using spatial synchronous detection,” Opt. Eng. 23, 391–395 (1984).

1982 (1)

1974 (1)

Bone, D.

Brangaccio, D. J.

Bruning, J. H.

Casillas, F.

J. Muñoz, F. Lecona, C. Castillo, F. Casillas, V. Duran, and M. Mora, “Phase recovery from a single interferogram using multiple Fourier transforms,” Proc. SPIE 6292, 62920D (2006).
[CrossRef]

Castillo, C.

J. Muñoz, F. Lecona, C. Castillo, F. Casillas, V. Duran, and M. Mora, “Phase recovery from a single interferogram using multiple Fourier transforms,” Proc. SPIE 6292, 62920D (2006).
[CrossRef]

Cuevas, F. J.

Duran, V.

J. Muñoz, F. Lecona, C. Castillo, F. Casillas, V. Duran, and M. Mora, “Phase recovery from a single interferogram using multiple Fourier transforms,” Proc. SPIE 6292, 62920D (2006).
[CrossRef]

Gallagher, J. E.

Ghiglia, D. C.

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, 1998).

Greivenkamp, J. E.

J. E. Greivenkamp, “Sub-Nyquist interferometry,” Appl. Opt. 26, 5245–5257 (1987).
[CrossRef] [PubMed]

J. E. Greivenkamp and J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, D.Malacara, ed. (Wiley, 1992), pp. 501–598.

Herriott, D. R.

Ina, H.

Kemao, Q.

Kobayashi, S.

Kozlowski, J.

J. Kozlowski and G. Serra, “New modified phase locked loop method for fringe pattern demodulation,” Opt. Eng. 36, 2025–2030 (1997).
[CrossRef]

Kreis, T.

Kujawinska, M.

M. Pirga and M. Kujawinska, “Two-directional spatial-carrier phase-shifting method for analysis of crossed and closed fringe patterns,” Opt. Eng. 34, 2459–2466 (1995).
[CrossRef]

M. Kujawinska and J. Schmidt, “Spatial-carrier phase-shifting technique of fringe pattern analysis,” Proc. SPIE 1508, 61–67 (1991).
[CrossRef]

Larkin, K. G.

Lecona, F.

J. Muñoz, F. Lecona, C. Castillo, F. Casillas, V. Duran, and M. Mora, “Phase recovery from a single interferogram using multiple Fourier transforms,” Proc. SPIE 6292, 62920D (2006).
[CrossRef]

Marroquin, J. L.

Mora, M.

J. Muñoz, F. Lecona, C. Castillo, F. Casillas, V. Duran, and M. Mora, “Phase recovery from a single interferogram using multiple Fourier transforms,” Proc. SPIE 6292, 62920D (2006).
[CrossRef]

Muñoz, J.

J. Muñoz, F. Lecona, C. Castillo, F. Casillas, V. Duran, and M. Mora, “Phase recovery from a single interferogram using multiple Fourier transforms,” Proc. SPIE 6292, 62920D (2006).
[CrossRef]

J. Muñoz, M. Strojnik, and G. Páez, “Detection and interpretation of high-frequency spatial interferograms,” Proc. SPIE 4486, 523–532 (2002).
[CrossRef]

Oldfield, M. A.

Páez, G.

J. Muñoz, M. Strojnik, and G. Páez, “Detection and interpretation of high-frequency spatial interferograms,” Proc. SPIE 4486, 523–532 (2002).
[CrossRef]

Pirga, M.

M. Pirga and M. Kujawinska, “Two-directional spatial-carrier phase-shifting method for analysis of crossed and closed fringe patterns,” Opt. Eng. 34, 2459–2466 (1995).
[CrossRef]

Pritt, M. D.

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, 1998).

Quiroga, J. A.

Rivera, M.

Rodriguez-Vera, R.

M. Servin and R. Rodriguez-Vera, “Two dimensional phase locked loop demodulation of carrier frequency interferograms,” J. Mod. Opt. 40, 2087–2094 (1993).
[CrossRef]

Rosenfeld, D. P.

Schmidt, J.

M. Kujawinska and J. Schmidt, “Spatial-carrier phase-shifting technique of fringe pattern analysis,” Proc. SPIE 1508, 61–67 (1991).
[CrossRef]

Seah, H.

Serra, G.

J. Kozlowski and G. Serra, “New modified phase locked loop method for fringe pattern demodulation,” Opt. Eng. 36, 2025–2030 (1997).
[CrossRef]

Servin, M.

Strojnik, M.

J. Muñoz, M. Strojnik, and G. Páez, “Detection and interpretation of high-frequency spatial interferograms,” Proc. SPIE 4486, 523–532 (2002).
[CrossRef]

Takeda, M.

White, A. D.

Womack, K. H.

K. H. Womack, “Interferometric phase measurement using spatial synchronous detection,” Opt. Eng. 23, 391–395 (1984).

Appl. Opt. (3)

J. Mod. Opt. (1)

M. Servin and R. Rodriguez-Vera, “Two dimensional phase locked loop demodulation of carrier frequency interferograms,” J. Mod. Opt. 40, 2087–2094 (1993).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (6)

Opt. Eng. (3)

J. Kozlowski and G. Serra, “New modified phase locked loop method for fringe pattern demodulation,” Opt. Eng. 36, 2025–2030 (1997).
[CrossRef]

M. Pirga and M. Kujawinska, “Two-directional spatial-carrier phase-shifting method for analysis of crossed and closed fringe patterns,” Opt. Eng. 34, 2459–2466 (1995).
[CrossRef]

K. H. Womack, “Interferometric phase measurement using spatial synchronous detection,” Opt. Eng. 23, 391–395 (1984).

Opt. Lett. (1)

Proc. SPIE (3)

J. Muñoz, F. Lecona, C. Castillo, F. Casillas, V. Duran, and M. Mora, “Phase recovery from a single interferogram using multiple Fourier transforms,” Proc. SPIE 6292, 62920D (2006).
[CrossRef]

M. Kujawinska and J. Schmidt, “Spatial-carrier phase-shifting technique of fringe pattern analysis,” Proc. SPIE 1508, 61–67 (1991).
[CrossRef]

J. Muñoz, M. Strojnik, and G. Páez, “Detection and interpretation of high-frequency spatial interferograms,” Proc. SPIE 4486, 523–532 (2002).
[CrossRef]

Other (2)

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, 1998).

J. E. Greivenkamp and J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, D.Malacara, ed. (Wiley, 1992), pp. 501–598.

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Figures (4)

Fig. 1
Fig. 1

Results of applying the Fourier method to a circular fringe pattern: (a) interferogram with closed fringes, (b) filtered Fourier spectrum and (c) the resulting wrapped phase (x direction), (d) filtered Fourier spectrum and (e) the resulting wrapped phase (y direction), and (f) the actual wrapped phase.

Fig. 2
Fig. 2

(a) Phase reconstruction after two iterations with threshold values of t = t 0 = t 1 = 0.1 π , (b) result with thresholds values of t 0 = t 1 = 0.01 π , (c) final phase estimation after extrapolation over regions with inconsistent data, and (d) wrapped final phase.

Fig. 3
Fig. 3

(a) Noisy interferogram with complex fringe distribution, (b) wrapped phases with sign changes in the x direction and (c) in the y direction. Partial unwrapped phases after (d) six and (e) eight iterations. (f) Final continuous phase recovered over the whole interferogram field, (g) wrapped final phase, and (h) wrapped noiseless phase shown for comparison purposes.

Fig. 4
Fig. 4

(a) Experimental interferogram with high fringe density and closed fringes, (b) phase solution after eight iterations, (c) extrapolated phase, and (d) cosine of the recovered phase.

Equations (12)

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I ( x , y ) = I b ( x , y ) + I m ( x , y ) cos [ ϕ ( x , y ) ] ,
I ˜ ( η , ξ ) = I { I ( x , y ) } = I ˜ b ( η , ξ ) + G ( η , ξ ) + G * ( η , ξ ) .
ϕ w ( x , y ) = arctan { Im [ I K ( x , y ) ] Re [ I K ( x , y ) ] } ,
I K ( x , y ) = I 1 { I ˜ K ( η , ξ ) } .
ϕ w k ( x , y ) = ± ϕ w ( x , y ) .
ϕ i , j x 1 = ϕ i , j x + ϕ i + 1 , j x + ϕ i , j 1 x + ϕ i + 1 , j 1 x + ϕ i , j + 1 x + ϕ i + 1 , j + 1 x 6 ,
ϕ i , j y 1 = ϕ i , j y + ϕ i , j + 1 y + ϕ i + 1 , j y + ϕ i + 1 , j + 1 y + ϕ i 1 , j y + ϕ i 1 , j + 1 y 6 ,
ϕ i , j x = arctan [ sin ( ϕ i , j w ϕ i 1 , j w ) cos ( ϕ i , j w ϕ i 1 , j w ) ] ,
ϕ i , j y = arctan [ sin ( ϕ i , j w ϕ i , j 1 w ) cos ( ϕ i , j w ϕ i , j 1 w ) ] .
if     [ | ϕ i , j w 1 ϕ i , j w 2 | t and | ϕ i , j y 1 ϕ i , j y 2 | t and | ϕ i , j x 1 ϕ i , j x 2 | t ] then ϕ i , j r = ϕ i , j w 1 + ϕ i , j w 2 2 , ϕ i , j y r = ϕ i , j y 1 + ϕ i , j y 2 2 , ϕ i , j x r = ϕ i , j x 1 + ϕ i , j x 2 2 ,
if     { [ ( | ϕ i , j y 1 | t 0 ) or ( | ϕ i , j x 1 | t 0 ) ] and | ϕ i , j y 1 ϕ ^ y r | t 1 and | ϕ i , j x 1 ϕ ^ x r | t 1 } then ϕ i , j r = ϕ ^ r + arctan [ sin ( ϕ i , j w 1 ϕ ^ r ) cos ( ϕ i , j w 1 ϕ ^ r ) ] , ϕ i , j y r = ϕ i , j y 1 ϕ i , j x r = ϕ i , j x 1 ,
if     { [ ( | ϕ i , j y 2 | t 0 ) or ( | ϕ i , j x 2 | t 0 ) ] and | ϕ i , j y 2 ϕ ^ y r | t 1 and | ϕ i , j x 2 ϕ ^ x r | t 1 } then ϕ i , j r = ϕ ^ r + arctan [ sin ( ϕ i , j w 2 ϕ ^ r ) cos ( ϕ i , j w 2 ϕ ^ r ) ] , ϕ i , j y r = ϕ i , j y 2 , ϕ i , j x r = ϕ i , j x 2 ,

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