Abstract

We develop a regularized phase-tracking (RPT) technique to unwrap phase maps. The phase maps that can be unwrapped with this new method may be bounded by arbitrarily shaped boundaries. Moreover, the RPT unwrapper has a higher noise robustness than previously reported phase-unwrapping schemes.

© 1998 Optical Society of America

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References

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    [CrossRef]
  11. M. Servin, F. J. Cuevas, “A novel technique for spatial phase-shifting interferometry,” J. Mod. Opt. 42, 1853–1862 (1995).
    [CrossRef]
  12. M. Servin, D. Malacara, F. J. Cuevas, “Direct phase detection of modulated Ronchi rulings using a phase locked loop,” Opt. Eng. 33, 1193–1199 (1994).
    [CrossRef]
  13. M. Servin, R. Rodriguez-Vera, “Two dimensional phase locked loop demodulation of carrier frequency interferograms,” J. Mod. Opt. 40, 2087–2094 (1993).
    [CrossRef]
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    [CrossRef]
  18. D. C. Ghiglia, L. A. Romero, “Robust two-dimensional weighted and unweighted phase unwrapping that uses fast transforms and iterative methods,” J. Opt. Soc. Am. A 11, 107–117 (1994).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  26. J. L. Marroquin, R. Rodriguez-Vera, M. Servin, M. Tapia, “Parallel phase-unwrapping algorithms based on Markov random field models,” J. Opt. Soc. Am. A 12, 2578–2585 (1995).
    [CrossRef]
  27. J. L. Marroquin, M. Rivera, “Quadratic regularization functionals for phase unwrapping,” J. Opt. Soc. Am. A 12, 2393–2400 (1995).
    [CrossRef]
  28. J. Besag, “On the statistical analysis of dirty pictures,” J. R. Stat. Soc. B 48, 259–302 (1986).
  29. A. J. Moore, C. Perez-Lopez, “Fringe visibility enhancement and phase calculation in double addition ESPI,” J. Mod. Opt. 43, 1829–1844 (1996).
    [CrossRef]

1996 (1)

A. J. Moore, C. Perez-Lopez, “Fringe visibility enhancement and phase calculation in double addition ESPI,” J. Mod. Opt. 43, 1829–1844 (1996).
[CrossRef]

1995 (3)

1994 (2)

M. Servin, D. Malacara, F. J. Cuevas, “Direct phase detection of modulated Ronchi rulings using a phase locked loop,” Opt. Eng. 33, 1193–1199 (1994).
[CrossRef]

D. C. Ghiglia, L. A. Romero, “Robust two-dimensional weighted and unweighted phase unwrapping that uses fast transforms and iterative methods,” J. Opt. Soc. Am. A 11, 107–117 (1994).
[CrossRef]

1993 (1)

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

1991 (1)

1989 (1)

1988 (2)

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

H. Takajo, T. Takahashi, “Least-squares phase estimation from phase difference,” J. Opt. Soc. Am. A 5, 416–425 (1988).
[CrossRef]

1987 (1)

1986 (1)

J. Besag, “On the statistical analysis of dirty pictures,” J. R. Stat. Soc. B 48, 259–302 (1986).

1984 (1)

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

1983 (2)

1982 (2)

1979 (1)

1978 (1)

1977 (2)

1974 (1)

1972 (1)

Besag, J.

J. Besag, “On the statistical analysis of dirty pictures,” J. R. Stat. Soc. B 48, 259–302 (1986).

Bone, D. J.

Brangaccio, D. J.

Bruning, J. H.

Cuevas, F. J.

M. Servin, F. J. Cuevas, “A novel technique for spatial phase-shifting interferometry,” J. Mod. Opt. 42, 1853–1862 (1995).
[CrossRef]

M. Servin, D. Malacara, F. J. Cuevas, “Direct phase detection of modulated Ronchi rulings using a phase locked loop,” Opt. Eng. 33, 1193–1199 (1994).
[CrossRef]

Fried, D. L.

Gallager, J. E.

Gasvik, K. J.

K. J. Gasvik, Optical Metrology (Wiley, New York, 1987).

Ghiglia, D. C.

Goldstein, R. M.

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

Greivenkamp, J. E.

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

Herriott, D. R.

Hudgin, R. H.

Hunt, B. R.

Huntley, J. M.

Ichioka, Y.

Ina, H.

Inuiya, M.

Itoh, K.

Kobayashi, S.

Kokal, J. V.

Kwon, O. Y.

D. W. Shough, O. Y. Kwon, D. F. Leavy, “High speed interferometric measurements of aerodynamic phenomena,” in Propagation of High-Energy Laser Beams through the Earth’s Atmosphere, P. B. Ulrich, L. E. Wilson, eds., Proc. SPIE1221, 394–403 (1990).
[CrossRef]

Leavy, D. F.

D. W. Shough, O. Y. Kwon, D. F. Leavy, “High speed interferometric measurements of aerodynamic phenomena,” in Propagation of High-Energy Laser Beams through the Earth’s Atmosphere, P. B. Ulrich, L. E. Wilson, eds., Proc. SPIE1221, 394–403 (1990).
[CrossRef]

Macy, W. W.

Malacara, D.

M. Servin, D. Malacara, F. J. Cuevas, “Direct phase detection of modulated Ronchi rulings using a phase locked loop,” Opt. Eng. 33, 1193–1199 (1994).
[CrossRef]

Marroquin, J. L.

Mastin, G. A.

Mertz, L.

Moore, A. J.

A. J. Moore, C. Perez-Lopez, “Fringe visibility enhancement and phase calculation in double addition ESPI,” J. Mod. Opt. 43, 1829–1844 (1996).
[CrossRef]

Noll, R. J.

Perez-Lopez, C.

A. J. Moore, C. Perez-Lopez, “Fringe visibility enhancement and phase calculation in double addition ESPI,” J. Mod. Opt. 43, 1829–1844 (1996).
[CrossRef]

Ransom, P. L.

Rivera, M.

Rodriguez-Vera, R.

J. L. Marroquin, R. Rodriguez-Vera, M. Servin, M. Tapia, “Parallel phase-unwrapping algorithms based on Markov random field models,” J. Opt. Soc. Am. A 12, 2578–2585 (1995).
[CrossRef]

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

Romero, L. A.

Rosenfel, D. P.

Servin, M.

M. Servin, F. J. Cuevas, “A novel technique for spatial phase-shifting interferometry,” J. Mod. Opt. 42, 1853–1862 (1995).
[CrossRef]

J. L. Marroquin, R. Rodriguez-Vera, M. Servin, M. Tapia, “Parallel phase-unwrapping algorithms based on Markov random field models,” J. Opt. Soc. Am. A 12, 2578–2585 (1995).
[CrossRef]

M. Servin, D. Malacara, F. J. Cuevas, “Direct phase detection of modulated Ronchi rulings using a phase locked loop,” Opt. Eng. 33, 1193–1199 (1994).
[CrossRef]

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

Shough, D. W.

D. W. Shough, O. Y. Kwon, D. F. Leavy, “High speed interferometric measurements of aerodynamic phenomena,” in Propagation of High-Energy Laser Beams through the Earth’s Atmosphere, P. B. Ulrich, L. E. Wilson, eds., Proc. SPIE1221, 394–403 (1990).
[CrossRef]

Takahashi, T.

Takajo, H.

Takeda, M.

Tapia, M.

Werner, C. L.

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

White, A. D.

Womack, K. H.

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

Zebker, H. A.

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

Appl. Opt. (8)

J. Mod. Opt. (3)

A. J. Moore, C. Perez-Lopez, “Fringe visibility enhancement and phase calculation in double addition ESPI,” J. Mod. Opt. 43, 1829–1844 (1996).
[CrossRef]

M. Servin, F. J. Cuevas, “A novel technique for spatial phase-shifting interferometry,” J. Mod. Opt. 42, 1853–1862 (1995).
[CrossRef]

M. Servin, 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. (5)

J. Opt. Soc. Am. A (5)

J. R. Stat. Soc. B (1)

J. Besag, “On the statistical analysis of dirty pictures,” J. R. Stat. Soc. B 48, 259–302 (1986).

Opt. Eng. (2)

M. Servin, D. Malacara, F. J. Cuevas, “Direct phase detection of modulated Ronchi rulings using a phase locked loop,” Opt. Eng. 33, 1193–1199 (1994).
[CrossRef]

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

Radio Sci. (1)

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

Other (4)

D. W. Shough, O. Y. Kwon, D. F. Leavy, “High speed interferometric measurements of aerodynamic phenomena,” in Propagation of High-Energy Laser Beams through the Earth’s Atmosphere, P. B. Ulrich, L. E. Wilson, eds., Proc. SPIE1221, 394–403 (1990).
[CrossRef]

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

K. J. Gasvik, Optical Metrology (Wiley, New York, 1987).

D. Malacara, ed., Optical Shop Testing (Wiley, New York, 1992).

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Figures (6)

Fig. 1
Fig. 1

(a) Potential corresponding to the first term in Eq. (4). (b) Derivative of the potential shown in Fig. 1(a) [Eq. (8)].

Fig. 2
Fig. 2

(a) Computer-generated phase map of two Gaussians. (b) Same phase map, but now a uniformly distributed phase noise was added.

Fig. 3
Fig. 3

(a) Solution for which the least-squares integration of the phase gradient as reported in Ref. 18 was used. (b) Solution of the regularized system as reported in Ref. 25.

Fig. 4
Fig. 4

(a) Resulting unwrapped phase taking as input the noisy phase map shown in Fig. 2(b). (The unwrapped phase is shown wrapped for comparison purpose.) (b) Wire mesh of the unwrapped phase.

Fig. 5
Fig. 5

Phase map obtained experimentally from an electronic speckle interferogram of the addition fringes reported in Ref. 27.

Fig. 6
Fig. 6

(a) Unwrapped phase (shown wrapped) obtained with the least-squares integration proposed in Ref. 18. (b) Unwrapped phase (also shown wrapped) according to the technique herein proposed.

Equations (8)

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I x ,   y = a x ,   y + b x ,   y cos ϕ x ,   y   noise ,
ϕ W x ,   y = ϕ x ,   y + 2 π k x ,   y ,
U T ϕ 0 ,   ω x ,   ω y = x , y L   U x , y ϕ 0 ,   ω x ,   ω y ,
U x , y ϕ 0 ,   ω x ,   ω y = , η N x , y L V ϕ W ,   η - ϕ e x ,   y ,   ,   η + λ ϕ 0 ,   η - ϕ e x ,   y ,   ,   η 2 m ,   η ,
ϕ e x ,   y ,   ,   η = ϕ 0 x ,   y + ω x x ,   y x - + ω y x ,   y y - η ,
ϕ 0 k + 1 x ,   y = ϕ 0 k x ,   y - τ   U x , y ϕ 0 ,   ω x ,   ω y ϕ 0 x ,   y , ω x k + 1 x ,   y = ω x k x ,   y - τ   U x , y ϕ 0 ,   ω x ,   ω y ω x x ,   y , ω y k + 1 x ,   y = ω y k x ,   y - τ   U x , y ϕ 0 ,   ω x ,   ω y ω y x ,   y ,
ϕ 0 k + 1 x ,   y = ϕ 0 k x ,   y - τ   U T ϕ 0 ,   ω x ,   ω y ϕ 0 x ,   y , ω x k + 1 x ,   y = ω x k x ,   y - τ   U T ϕ 0 ,   ω x ,   ω y ω x x ,   y , ω y k + 1 x ,   y = ω y k x ,   y - τ   U T ϕ 0 ,   ω x ,   ω y ω y x ,   y .
V x = 2 x - 2 π   int x / π .

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