Abstract

We present a two-dimensional regularized phase-tracking technique that is capable of demodulating a single fringe pattern with either open or closed fringes. The proposed regularized phase-tracking system gives the detected phase continuously so that no further unwrapping is needed over the detected phase.

© 1997 Optical Society of America

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References

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  1. K. J. Gasvik, Optical Metrology (Wiley, New York, 1987).
  2. D. Malacara, ed., Optical Shop Testing (Wiley, New York, 1992).
  3. J. H. Bruninig, D. R. Herriott, J. E. Gallager, D. P. Rosenfel, A. D. White, D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. 13, 2693–2703 (1974).
    [CrossRef]
  4. J. E. Greivenkamp, J. H. Bruning, “Phase shifting interferometry,” in Ref. 2, pp. 501–598.
  5. M. Takeda, H. Ina, S. Kobayashi, “Fourier transform methods of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72, 156–160 (1982).
    [CrossRef]
  6. Y. Ichioka, M. Inuiya, “Direct phase detecting system,” Appl. Opt. 11, 1507–1514 (1972).
    [CrossRef] [PubMed]
  7. K. H. Womack, “Interferometric phase measurement using spatial synchronous detection,” Opt. Eng. 23, 391–395 (1984).
    [CrossRef]
  8. L. Mertz, “Real time fringe pattern analysis,” Appl. Opt. 22, 1535–1539 (1983).
    [CrossRef]
  9. P. L. Ransom, J. V. Kokal, “Interferogram analysis by a modified sinusoidal fitting technique,” Appl. Opt. 25, 4199–4204 (1986).
    [CrossRef]
  10. 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, E. Wilson, ed., Proc. SPIE1221, 394–403.
  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, R. Rodriguez-Vera, “Two dimensional phase locked loop demodulation of carrier frequency interferograms,” J. Mod. Opt. 40, 2087–2094 (1993).
    [CrossRef]
  13. 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]
  14. T. Kreis, “Digital holographic interference-phase measurement using the Fourier transform method,” J. Opt. Soc. Am. A 3, 847–855 (1986).
    [CrossRef]
  15. A. Spik, D. W. Robinson, “Investigation of the cellular automata method for phase unwrapping and its implementation on an array processor,” Opt. Laser Eng. 14, 25–37 (1991).
    [CrossRef]
  16. D. C. Ghiglia, G. A. Mastin, L. A. Romero, “Cellular automata method for phase unwrapping,” J. Opt. Soc. Am. 4, 267–280 (1987).
    [CrossRef]
  17. 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]
  18. 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]
  19. J. L. Marroquin, M. Rivera, “Quadratic regularization functionals for phase unwrapping,” J. Opt. Soc. Am. A 12, 2393–2400 (1995).
    [CrossRef]
  20. J. L. Marroquin, M. Servin, “Adaptive quadrature filters and the recovery of phase from fringe pattern images,” J. Opt. Soc. Am.
  21. J. Besag, “On the statistical analysis of dirty pictures,” J. R. Stat. Soc. B 48, 259–302 (1986).
  22. R. Jaffe, E. Rechtin, “Design and performance of phase-lock circuits capable of near-optimum performance over a wide range of input signal and noise levels,” in Phase Locked Loops and Their Applications, C. W. Lindsey, K. M. Simon, eds. (IEEE, New York, 1978), pp. 20–30.

1995 (3)

1994 (2)

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]

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]

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)

A. Spik, D. W. Robinson, “Investigation of the cellular automata method for phase unwrapping and its implementation on an array processor,” Opt. Laser Eng. 14, 25–37 (1991).
[CrossRef]

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 (3)

1984 (1)

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

1983 (1)

1982 (1)

1974 (1)

1972 (1)

Besag, J.

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

Brangaccio, D. J.

Bruninig, 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]

Gallager, J. E.

Gasvik, K. J.

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

Ghiglia, D. C.

Herriott, D. R.

Ichioka, Y.

Ina, H.

Inuiya, M.

Jaffe, R.

R. Jaffe, E. Rechtin, “Design and performance of phase-lock circuits capable of near-optimum performance over a wide range of input signal and noise levels,” in Phase Locked Loops and Their Applications, C. W. Lindsey, K. M. Simon, eds. (IEEE, New York, 1978), pp. 20–30.

Kobayashi, S.

Kokal, J. V.

Kreis, T.

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, E. Wilson, ed., Proc. SPIE1221, 394–403.

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, E. Wilson, ed., Proc. SPIE1221, 394–403.

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.

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.

Ransom, P. L.

Rechtin, E.

R. Jaffe, E. Rechtin, “Design and performance of phase-lock circuits capable of near-optimum performance over a wide range of input signal and noise levels,” in Phase Locked Loops and Their Applications, C. W. Lindsey, K. M. Simon, eds. (IEEE, New York, 1978), pp. 20–30.

Rivera, M.

Robinson, D. W.

A. Spik, D. W. Robinson, “Investigation of the cellular automata method for phase unwrapping and its implementation on an array processor,” Opt. Laser Eng. 14, 25–37 (1991).
[CrossRef]

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.

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, 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]

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

J. L. Marroquin, M. Servin, “Adaptive quadrature filters and the recovery of phase from fringe pattern images,” J. Opt. Soc. Am.

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, E. Wilson, ed., Proc. SPIE1221, 394–403.

Spik, A.

A. Spik, D. W. Robinson, “Investigation of the cellular automata method for phase unwrapping and its implementation on an array processor,” Opt. Laser Eng. 14, 25–37 (1991).
[CrossRef]

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 (1984).
[CrossRef]

Appl. Opt. (4)

J. Mod. Opt. (2)

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. (2)

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

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

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

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)

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

Opt. Laser Eng. (1)

A. Spik, D. W. Robinson, “Investigation of the cellular automata method for phase unwrapping and its implementation on an array processor,” Opt. Laser Eng. 14, 25–37 (1991).
[CrossRef]

Other (6)

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

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

J. E. Greivenkamp, J. H. Bruning, “Phase shifting interferometry,” in Ref. 2, pp. 501–598.

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, E. Wilson, ed., Proc. SPIE1221, 394–403.

J. L. Marroquin, M. Servin, “Adaptive quadrature filters and the recovery of phase from fringe pattern images,” J. Opt. Soc. Am.

R. Jaffe, E. Rechtin, “Design and performance of phase-lock circuits capable of near-optimum performance over a wide range of input signal and noise levels,” in Phase Locked Loops and Their Applications, C. W. Lindsey, K. M. Simon, eds. (IEEE, New York, 1978), pp. 20–30.

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Figures (7)

Fig. 1
Fig. 1

Demodulation of a computer-generated interferogram: (a) wideband noisy carrier-frequency interferogram [the amount of phase noise introduced was uniformly distributed in the range of (-π/2, +π/2)]; (b) the best we could obtain by demodulating the fringe pattern shown in (a) by using the Fourier technique; (c) the estimated phase obtained with the proposed RPT technique (shown wrapped for comparison purposes).

Fig. 2
Fig. 2

Demodulation of the experimentally obtained carrier-frequency fringe pattern: (a) fringe pattern, (b) the estimated phase obtained with the proposed RPT technique (note the high noise rejection and immunity to the boundary effects), (c) the best phase estimation we could obtain by using the Fourier technique.

Fig. 3
Fig. 3

Effect of phase shifting the indistinguishable fringe patterns described by Eq. (3): (a) fringe pattern of cos[ϕ1(x, y) - 1.0], (b) fringe pattern that corresponds to cos[ϕ2(x, y) - 1.0], (c) fringe pattern that corresponds to cos[ϕ3(x, y) - 1.0]. As we can see from this figure, their phase-shifted versions are now distinguishable. The searched demodulating phase (a) is the one that has the minimum frequency content.

Fig. 4
Fig. 4

(a) 128 × 128 pixel computer-generated interferogram that has a uniformly distributed phase noise between -1.25 and 1.25 rad, (b) detected continuous phase (shown wrapped for comparison purposes).

Fig. 5
Fig. 5

(a) Same fringe pattern as in Fig. 4(a), but in this case the amount of phase noise (also uniformly distributed) was increased to the range -1.5–1.5 rad; (b) misdetected continuous phase (shown wrapped for comparison purposes).

Fig. 6
Fig. 6

Demodulation of a single closed-fringe pattern by use of the presented RPT technique: (a) experimentally obtained fringe pattern, (b) continuous estimated phase (shown wrapped for comparison purposes). We can see that no noticeable edge effects are presented.

Fig. 7
Fig. 7

Demodulation of a single ESPI closed-fringe pattern by use of the presented RPT technique: (a) subtraction ESPI fringe pattern of the fast loading of an aluminum clamped plate, (b) the continuous estimated phase (shown wrapped for comparison purposes).

Equations (12)

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Ix, y=ax, y+bx, ycosϕx, y,
ϕ1x, y=x2+y2,  ϕ2x, y=Wx2+y2,  ϕ3x, y=x2+y2,  for x0,=-x2+y2,  for x>0,
cosϕ1x, y=cosϕ2x, y=cosϕ3x, y,
UT=x,yL Ux,yϕ0, ωx, ωy,
Ux,yϕ0, ωx, ωy=ϵ,ηNx,yL Iϵ, η-cosϕex, y, ϵ, η2+λϕ0ϵ, η -ϕex, y, ϵ, η2mϵ, η,
ϕex, y, ϵ, η=ϕ0x, y+ωxx, yx-ϵ+ωyx, yy-η,
ϕ0k+1x, y=ϕ0kx, y-τUx,yϕ0, ωx, ωyϕ0x, y,  ωxk+1x, y=ωxkx, y-τ Ux,yϕ0, ωx, ωyωxx, y,  ωyk+1x, y=ωykx, y-τUx,yϕ0, ωx, ωyωyx, y,
Pθ=1Z exp-Uθ
Uθ=C VCθ,
Ux,yϕ0, ωx, ωy=ϵ,ηNx,yL Iϵ, η-cosϕex, y, ϵ, η2+ϵ,ηNx,yL λϕ0ϵ, η-ϕex, y, ϵ, η2+ϕ0x, y-ϕeϵ, η, x, y2,
ϕ0k+1x, y=ϕ0kx, y-τ Ux,yϕ0, ωx, ωyϕ0x, y,  ωxk+1x, y=ωxkx, y-τUx,yϕ0, ωx, ωyωxx, y,  ωyk+1x, y=ωykx, y-τUx,yϕ0, ωx, ωyωyx, y,
Ux,yϕ0, ωx, ωy=ϵ,ηNx,yL Iϵ, η-cosϕex, y, ϵ, η2+ Iϵ, η-cosϕex, y, ϵ, η-δ2+Iϵ, η -cosϕex, y, ϵ, η+δ2+λϕ0ϵ, η-ϕex, y, ϵ, η2mϵ, η,

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