Abstract

Most interferogram demodulation techniques give the detected phase wrapped owing to the arctangent function involved in the final step of the demodulation process. To obtain a continuous detected phase, an unwrapping process must be performed. Here we propose a phase-unwrapping technique based on a regularized phase-tracking (RPT) system. Phase unwrapping is achieved in two steps. First, we obtain two phase-shifted fringe patterns from the demodulated wrapped phase (the sine and the cosine), then demodulate them by using the RPT technique. In the RPT technique the unwrapping process is achieved simultaneously with the demodulation process so that the final goal of unwrapping is therefore achieved. The RPT method for unwrapping the phase is compared with the technique of least-squares integration of wrapped phase differences to outline the substantial noise robustness of the RPT technique.

© 1999 Optical Society of America

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References

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    [CrossRef]

1998 (2)

M. Servin, R. Rodriguez-Vera, J. L. Marroquin, D. Malacara, “Phase-shifting interferometry using a two-dimensional regularized phase-tracking technique,” J. Mod. Opt. 45, 1809–1820 (1998).
[CrossRef]

M. Servin, J. L. Marroquin, D. Malacara, F. J. Cuevas, “Phase unwrapping with a regularized phase-tracking system,” Appl. Opt. 37, 1917–1923 (1998).
[CrossRef]

1997 (1)

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)

1988 (1)

1987 (1)

D. C. Ghiglia, G. A. Mastin, L. A. Romero, “Cellular automata method for phase unwrapping,” J. Opt. Soc. Am. 4, 267–280 (1987).
[CrossRef]

1986 (1)

1983 (2)

1982 (2)

1979 (1)

1978 (1)

1977 (2)

1974 (1)

1972 (1)

Bone, D. J.

Brangaccio, D. J.

Bruning, J. H.

Cuevas, F. J.

M. Servin, J. L. Marroquin, D. Malacara, F. J. Cuevas, “Phase unwrapping with a regularized phase-tracking system,” Appl. Opt. 37, 1917–1923 (1998).
[CrossRef]

M. Servin, J. L. Marroquin, F. J. Cuevas, “Demodulation of a single interferogram by use of a two-dimensional regularized phase-tracking technique,” Appl. Opt. 36, 4540–4548 (1997).
[CrossRef] [PubMed]

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.

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.

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 Eart’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 Eart’s Atmosphere, P. B. Ulrich, L. E. Wilson, eds., Proc. SPIE1221, 394–403 (1990).
[CrossRef]

Macy, W.

Malacara, D.

M. Servin, J. L. Marroquin, D. Malacara, F. J. Cuevas, “Phase unwrapping with a regularized phase-tracking system,” Appl. Opt. 37, 1917–1923 (1998).
[CrossRef]

M. Servin, R. Rodriguez-Vera, J. L. Marroquin, D. Malacara, “Phase-shifting interferometry using a two-dimensional regularized phase-tracking technique,” J. Mod. Opt. 45, 1809–1820 (1998).
[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]

Marroquin, J. L.

Mastin, G. A.

D. C. Ghiglia, G. A. Mastin, L. A. Romero, “Cellular automata method for phase unwrapping,” J. Opt. Soc. Am. 4, 267–280 (1987).
[CrossRef]

Mertz, L.

Noll, R. J.

Ransom, P. L.

Rivera, M.

Rodriguez-Vera, R.

M. Servin, R. Rodriguez-Vera, J. L. Marroquin, D. Malacara, “Phase-shifting interferometry using a two-dimensional regularized phase-tracking technique,” J. Mod. Opt. 45, 1809–1820 (1998).
[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, 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, J. L. Marroquin, D. Malacara, F. J. Cuevas, “Phase unwrapping with a regularized phase-tracking system,” Appl. Opt. 37, 1917–1923 (1998).
[CrossRef]

M. Servin, R. Rodriguez-Vera, J. L. Marroquin, D. Malacara, “Phase-shifting interferometry using a two-dimensional regularized phase-tracking technique,” J. Mod. Opt. 45, 1809–1820 (1998).
[CrossRef]

M. Servin, J. L. Marroquin, F. J. Cuevas, “Demodulation of a single interferogram by use of a two-dimensional regularized phase-tracking technique,” Appl. Opt. 36, 4540–4548 (1997).
[CrossRef] [PubMed]

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 Eart’s Atmosphere, P. B. Ulrich, L. E. Wilson, eds., Proc. SPIE1221, 394–403 (1990).
[CrossRef]

Takahashi,

Takajo, H.

Takeda, M.

Tapia, M.

White, A. D.

Womack, K. H.

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

Appl. Opt. (9)

J. Mod. Opt. (3)

M. Servin, R. Rodriguez-Vera, J. L. Marroquin, D. Malacara, “Phase-shifting interferometry using a two-dimensional regularized phase-tracking technique,” J. Mod. Opt. 45, 1809–1820 (1998).
[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. (6)

J. Opt. Soc. Am. A (4)

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 (1984a).
[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 Eart’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 (10)

Fig. 1
Fig. 1

Possible sequence of the demodulation process followed by the RPT, shown to have a graphic reference to the different domains involved.

Fig. 2
Fig. 2

Noiseless wrapped phase used in the numerical experiments. This image has 128 × 128 pixels with 256 gray levels.

Fig. 3
Fig. 3

Three phase-shifted interferograms (2π/3 phase shift) obtained when a white and uniformly distributed phase noise in the range of (0.0, 1.55π) rad was added to the phase shown in Fig. 2. Then the interferograms were convolved three times with an average of 3 × 3 pixels. The low-pass-filtered interferograms are shown.

Fig. 4
Fig. 4

Unwrapped phase (shown rewrapped for comparison with the noiseless phase) obtained from the interferograms shown in Fig. 3 by using the least-squares unwrapper. The phase map was detected by using the well-known three-step phase-shifted formula.

Fig. 5
Fig. 5

Unwrapped phase (also shown rewrapped) obtained by using the RPT unwrapping proposed here. The input phase map in this case was obtained directly from the noisy phase-shifted interferograms, that is, the low pass filtering used in the least-squares unwrapping was not required. This method was used because the RPT unwrapped works better this way.

Fig. 6
Fig. 6

Three phase-shifted ESPI patterns of a clamped metallic plate heated by a soldering tip. The phase shift among these interferograms is π/2 rad.

Fig. 7
Fig. 7

Wrapped phase estimated directly (no low pass filtering) from the three phase-shifted interferograms shown in Fig. 6.

Fig. 8
Fig. 8

(a) Cosine of the wrapped phase; (b) sine of the wrapped phase.

Fig. 9
Fig. 9

Unwrapped phase (wrapped again for comparison with the interferograms) obtained by using the RPT unwrapped proposed here.

Fig. 10
Fig. 10

Wire mesh of the unwrapped phase shown in Fig. 9.

Equations (11)

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

Ix, y=ax, y+bx, ycosϕx, y+nx, y,
ϕx, y=ϕWx, y+2πkx, y,
ICx, y=cosϕWx, y, ISx, y=sinϕWx, y,
Ux, y=,ηNx,yLIC, η-cos px, y, , η2+IS, η-sin px, y, , η2+λϕ0, η-px, y, , η2m, η,
px, y, , η=ϕ0x, y+ωxx, yx-+ωyx, yy-η,
ϕ0k+1x, y=ϕ0kx, y-τ Ux, yϕ0x, y, ωxk+1x, y=ωxkx, y-τ Ux, yωxx, y, ωyk+1x, y=ωykx, y-τ Ux, yωyx, y,
Ux, y=,ηNx,yLVϕw, η-px, y, , η+λϕ0, η-px, y, , η2m, η,
I1x, y=1+cosϕx, y+noisex, y-2π3, I2x, y=1+cosϕx, y+noisex, y, I3x, y=1+cosϕx, y+noisex, y+2π3.
ϕWx, y=arctan1-cosαsin α×LPFI1x, y-LPFI3x, y2LPFI2x, y-LPFI1x, y-LPFI3x, y,
ϕWx, y=arctan1-cosαsin α×I1x, y-I3x, y2I2x, y-I1x, y-I3x, y.
ϕWx, y=arctanI1x, y-I3x, y2I2x, y-I1x, y-I3x, y,

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