Abstract

An algorithm for phase demodulation of a single interferogram that may contain closed fringes is presented. This algorithm uses the regularized phase-tracker system as a robust phase estimator, together with a new scanning technique that estimates the phase that initially follows the bright zones of the interferogram. The combination of these two elements constitutes a powerful new method, the fringe-follower regularized phase tracker, that makes it possible to correctly demodulate complex, single-image interferograms for which traditional methods fail.

© 2001 Optical Society of America

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References

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  1. G. Cloud, Optical Methods of Engineering Analysis (Cambridge U. Press, Cambridge, UK, 1995).
  2. 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]
  3. T. Kreis, “Digital holographic interference phase measurement using the Fourier transform method,” J. Opt. Soc. Am. A 3, 847–855 (1986).
    [CrossRef]
  4. D. W. Robinson, “Phase unwrapping methods,” in Interferogram Analysis, D. W. Rabinson, G. T. Reid, eds. (Institute of Physics, Bristol, UK, 1993), Chap. 6.
  5. J. L. Marroquin, M. Servin, R. Rodriguez-Vera, “Adaptive quadrature filters and the recovery of phase from fringe pattern images,” J. Opt. Soc. Am. A 14, 1742–1753 (1997).
    [CrossRef]
  6. J. L. Marroquin, R. Rodriguez-Vera, M. Servin, “Local phase from local orientation by solution of a sequence of linear systems,” J. Opt. Soc. Am. A 15, 1536–1543 (1998).
    [CrossRef]
  7. M. Servin, R. Rodriguez-Vera, “Two dimensional phase locked loop demodulation of carrier frequency interferograms,” J. Mod. Opt. 40, 2087–2094 (1993).
    [CrossRef]
  8. 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. (Institute of Electrical and Electronics Engineers, Piscataway, N. J., 1978), pp. 20–30.
  9. 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]
  10. B. Ströbel, “Processing of interferometric phase maps as complex-valued phasor images,” Appl. Opt. 35, 2192–2198 (1996).
    [CrossRef] [PubMed]

1998 (1)

1997 (2)

1996 (1)

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]

1986 (1)

1982 (1)

Cloud, G.

G. Cloud, Optical Methods of Engineering Analysis (Cambridge U. Press, Cambridge, UK, 1995).

Cuevas, F. J.

Ina, H.

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. (Institute of Electrical and Electronics Engineers, Piscataway, N. J., 1978), pp. 20–30.

Kobayashi, S.

Kreis, T.

Marroquin, J. 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. (Institute of Electrical and Electronics Engineers, Piscataway, N. J., 1978), pp. 20–30.

Robinson, D. W.

D. W. Robinson, “Phase unwrapping methods,” in Interferogram Analysis, D. W. Rabinson, G. T. Reid, eds. (Institute of Physics, Bristol, UK, 1993), Chap. 6.

Rodriguez-Vera, R.

Servin, M.

Ströbel, B.

Takeda, M.

Appl. Opt. (2)

J. Mod. Opt. (1)

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. (1)

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

Other (3)

D. W. Robinson, “Phase unwrapping methods,” in Interferogram Analysis, D. W. Rabinson, G. T. Reid, eds. (Institute of Physics, Bristol, UK, 1993), Chap. 6.

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. (Institute of Electrical and Electronics Engineers, Piscataway, N. J., 1978), pp. 20–30.

G. Cloud, Optical Methods of Engineering Analysis (Cambridge U. Press, Cambridge, UK, 1995).

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Figures (7)

Fig. 1
Fig. 1

Computer-generated subtraction specklegram. (a) Fringe pattern being phase demodulated, (b) demodulated phase map obtained by the Fourier method.

Fig. 2
Fig. 2

Computer-generated subtraction specklegram. (a) Fringe pattern being phase demodulated; (b), (c) isotropic demodulation path followed by the RPT; (d) wrongly demodulated phase.

Fig. 3
Fig. 3

Computer-generated subtraction specklegram. (a) Fringe pattern being phase demodulated, (b) binarized fringe pattern, (c)–(e) sequence followed by the FFRPT to demodulate the fringe pattern, (f) fully demodulated phase. The estimated phase was rewrapped for comparison with the source fringe pattern.

Fig. 4
Fig. 4

(a) Computer-generated subtraction specklegram, (b) binarized fringe pattern, (c)–(e) path followed by the FFRPT, (f) fully demodulated phase. The estimated phase was rewrapped for comparison with the interferogram.

Fig. 5
Fig. 5

(a) Experimentally obtained static subtraction ESPI pattern, (b) binarized fringe pattern used to guide the FFPTR within the fringe pattern, (c)–(e) three moments in the demodulation process followed by the FFRPT, (f) rewrapped demodulated phase.

Fig. 6
Fig. 6

(a) Experimentally obtained static subtraction ESPI pattern, (b) binarized fringe pattern used to guide the FFPTR within the fringe pattern, (c)–(e) three moments in the demodulation process followed by the FFRPT, (f) rewrapped demodulated phase.

Fig. 7
Fig. 7

(a) Computer-generated fringe pattern, (b) wrongly demodulated computer-generated noiseless interferogram. The reason for this was the fact that the stationary points at the boundary were not completely surrounded by a demodulating path.

Equations (11)

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I(x, y)=a(x, y)+b(x, y)cos[ϕ(x, y)],
ϕ1(x, y)=-ϕ(x, y)+2kπ(x, y)Sϕ(x, y)(x, y)S,
cos[ϕ(x, y)+α]cos[ϕ1(x, y)+α].
I(x, y)cos[ϕ(x, y)].
U(x, y)=(,η)(Nx,yP){[I(, η)-cos p(x, y, , η)]2+β[ϕ0(, η)-p(x, y, , η)]2m(, η)},
p(x, y, , η)=ϕ0(x, y)+ωx(x, y)(x-)+ωy(x, y)(y-η),
rk+1(x, y)=rk(x, y)-μrU(x, y),
rU(x, y)=U(x, y)ϕ0(x, y), U(x, y)ωx(x, y), U(x, y)ωy(x, y).
U(x, y)=(,η)(Nx,yS){[I(, η)-cos p(x, y, , η)]2+[I(, η)-cos p1(x, y, , η)]2+β[ϕ0(, η)-p(x, y, , η)]2m(, η)},
p1(x, y, , η)=p(x, y, , η)+α,
Ib(x, y)=Thresh[I(x, y)**LPF(x, y)],

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