Abstract

A new sequential phase demodulator based on a regularized quadrature and phase tracker system (RQPT) is applied to demodulate two-dimensional fringe patterns. This RQPT system tracks the fringe pattern’s quadrature and phase in a sequential way by following the path of the fringes. To make the RQPT system more robust to noise, the modulating phase around a small neighborhood is modeled as a plane and the quadrature of the signal is estimated simultaneously with the fringe’s modulating phase. By sequentially calculating the quadrature of the fringe pattern, one obtains a more robust sequential demodulator than was previously possible. This system may be applied to the demodulation of a single interferogram having closed fringes.

© 2004 Optical Society of America

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References

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  1. D. Malacara, M. Servin, Z. Malacara, Interferogram Analysis for Optical Testing (Marcel Dekker, New York, 1998), Chap. 1.
  2. M. Takeda, H. Ina, S. Kobayashi, “Fourier-transform method of fringe pattern analysis,” J. Opt. Soc. Am. 72, 156–160 (1982).
    [CrossRef]
  3. D. Malacara, M. Servin, Z. Malacara, Interferogram Analysis for Optical Testing (Marcel Dekker, New York, 1998), Chap. 6.
  4. G. Cloud, Optical Methods of Engineering Analysis (Cambridge U. Press, Cambridge, UK, 1995).
  5. K. G. Larkin, D. J. Bone, M. A. Oldfield, “Natural demodulation of two-dimensional fringe patterns. I. General background of the spiral phase quadrature transform,” J. Opt. Soc. Am. A 18, 1862–1870 (2001).
    [CrossRef]
  6. M. Servin, J. A. Quiroga, J. L. Marroquin, “General n-dimensional quadrature transform and its application to interferogram demodulation,” J. Opt. Soc. Am. A 20, 925–934 (2003).
    [CrossRef]
  7. J. A. Quiroga, M. Servin, 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).
    [CrossRef]
  8. M. Servin, J. L. Marroquin, F. J. Cuevas, “Fringe-follower regularized phase tracker for demodulation of closed-fringe interferograms,” J. Opt. Soc. Am. A 18, 689–695 (2001).
    [CrossRef]
  9. M. Servin, R. Rodriguez-Vera, “Two dimensional phase locked loop demodulation of carrier frequency interferograms,” J. Mod. Opt. 40, 2087–2094 (1993).
    [CrossRef]
  10. J. Kozlowski, G. Serra, “New modified phase locked loop method for fringe pattern demodulation,” Opt. Eng. 36, 2025–2030 (1997).
    [CrossRef]
  11. J. Kozlowski, G. Serra, “Complex phase tracing method for fringe pattern demodulation,” Appl. Opt. 38, 2256–2262 (1999).
    [CrossRef]
  12. B. Ströbel, “Processing of interferometric phase maps as complex-valued phasor images,” Appl. Opt. 35, 2192–2198 (1996).
    [CrossRef] [PubMed]
  13. C. W. Groetsch, Inverse Problems in the Mathematical Sciences (Vieweg, Braunschweig, Germany, 1993).

2003 (1)

2002 (1)

2001 (2)

1999 (1)

1997 (1)

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

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]

1982 (1)

Bone, D. J.

Cloud, G.

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

Cuevas, F. J.

Groetsch, C. W.

C. W. Groetsch, Inverse Problems in the Mathematical Sciences (Vieweg, Braunschweig, Germany, 1993).

Ina, H.

Kobayashi, S.

Kozlowski, J.

J. Kozlowski, G. Serra, “Complex phase tracing method for fringe pattern demodulation,” Appl. Opt. 38, 2256–2262 (1999).
[CrossRef]

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

Larkin, K. G.

Malacara, D.

D. Malacara, M. Servin, Z. Malacara, Interferogram Analysis for Optical Testing (Marcel Dekker, New York, 1998), Chap. 6.

D. Malacara, M. Servin, Z. Malacara, Interferogram Analysis for Optical Testing (Marcel Dekker, New York, 1998), Chap. 1.

Malacara, Z.

D. Malacara, M. Servin, Z. Malacara, Interferogram Analysis for Optical Testing (Marcel Dekker, New York, 1998), Chap. 1.

D. Malacara, M. Servin, Z. Malacara, Interferogram Analysis for Optical Testing (Marcel Dekker, New York, 1998), Chap. 6.

Marroquin, J. L.

Oldfield, M. A.

Quiroga, J. A.

Rodriguez-Vera, R.

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

Serra, G.

J. Kozlowski, G. Serra, “Complex phase tracing method for fringe pattern demodulation,” Appl. Opt. 38, 2256–2262 (1999).
[CrossRef]

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

Servin, M.

M. Servin, J. A. Quiroga, J. L. Marroquin, “General n-dimensional quadrature transform and its application to interferogram demodulation,” J. Opt. Soc. Am. A 20, 925–934 (2003).
[CrossRef]

J. A. Quiroga, M. Servin, 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).
[CrossRef]

M. Servin, J. L. Marroquin, F. J. Cuevas, “Fringe-follower regularized phase tracker for demodulation of closed-fringe interferograms,” J. Opt. Soc. Am. A 18, 689–695 (2001).
[CrossRef]

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

D. Malacara, M. Servin, Z. Malacara, Interferogram Analysis for Optical Testing (Marcel Dekker, New York, 1998), Chap. 1.

D. Malacara, M. Servin, Z. Malacara, Interferogram Analysis for Optical Testing (Marcel Dekker, New York, 1998), Chap. 6.

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 (4)

Opt. Eng. (1)

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

Other (4)

D. Malacara, M. Servin, Z. Malacara, Interferogram Analysis for Optical Testing (Marcel Dekker, New York, 1998), Chap. 6.

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

C. W. Groetsch, Inverse Problems in the Mathematical Sciences (Vieweg, Braunschweig, Germany, 1993).

D. Malacara, M. Servin, Z. Malacara, Interferogram Analysis for Optical Testing (Marcel Dekker, New York, 1998), Chap. 1.

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Figures (6)

Fig. 1
Fig. 1

Phase demodulation of a sinusoidal signal. (a) Estimated phase given by ϕˆ=cos-1(ϕ), which is the minimum of U in Eq. (10); (b) minimizing function of Eq. (12) along with zero initial conditions [Eqs. (15)]; (c) function that minimizes Eq. (12) but now using as initial conditions the previously found estimates [Eqs. (16)].

Fig. 2
Fig. 2

Demodulation of a 2D fringe pattern using the nonregularized quadrature and phase tracker (QPT) estimator. (a) Given noiseless computer-generated fringe pattern. (b) Path followed by the sequential demodulating system. Whiter paths are preferably followed as being the sites with “higher quality.” (c)–(e) Path demodulation progress of the QPT system. (f) Demodulated phase. Although the QPT demodulator finds the phase unwrapped, the phase was rewrapped to compare it with (a).

Fig. 3
Fig. 3

Phase demodulation of an experimentally obtained speckle interferometric pattern. (a) Speckle pattern. (b) Traced path that the regularized QPT (RQPT) demodulating system follows. Whiter zones are preferably first demodulated. (c), (d) Two “moments” on the path followed by the RQPT demodulating system. (e) Quadrature of the original fringe pattern. (f) Demodulated phase shown rewrapped.

Fig. 4
Fig. 4

Phase demodulation of a noisy computer-generated fringe pattern. (a) Noisy fringe pattern. (b) Traced path that the sequential RQPT demodulating system will follow. Whiter zones are preferably first demodulated. (c) Quadrature of the original fringe pattern. (d) Demodulated phase shown rewrapped.

Fig. 5
Fig. 5

Phase demodulation of a fringe pattern with less than expected fringe contrast of 1.0. (a) Fringe pattern I(x)=0.8 cos(ω0x). (b) Wrong demodulated phase found by the RPT [Eq. (28)], which is close to ϕˆ(x)=cos-1[I(x)]. (c) Correctly demodulated phase using the RQPT system.

Fig. 6
Fig. 6

Phase demodulation of a noisy fringe pattern with the right (expected) contrast equal to 1.0. (a) Noisy fringe pattern I(x)=cos[ϕ(x)+noise(x)]. (b) Wrong demodulated phase using the RPT system given by Eq. (28), which is close to ϕˆ(x)=cos-1[I(x)]. (c) Correctly demodulated phase obtained by the RQPT system.

Equations (35)

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I(r)=a(r)+b(r)cos[ϕ(r)],
I1(x)=a(x)+b(x)cos[ω0x+ϕ(x)],
I(x)=b(x)cos[ω0x+ϕ(x)].
dϕˆ(x)dx=τI(x)sin[ω0x+ϕˆ(x)],
ϕˆ(x)=τ-xcos[ω0ξ+ϕ(ξ)]sin[ω0ξ+ϕˆ(ξ)]dξ,
ϕˆ(x)=τ2-x{sin[ϕ(ξ)-ϕˆ(ξ)]+sin[2ω0ξ+ϕˆ(ξ)+ϕ(ξ)]}dξ.
dϕˆ(x)dx=τ sin[ϕ(x)-ϕˆ(x)].
dϕˆ(x)dx=τ[ϕ(x)-ϕˆ(x)].
ϕˆ(x+1)=ϕˆ(x)+τI(x)sin[ω0x+ϕˆ(x)].
U={I(x)-cos[ϕˆ(x)]}2,
ϕˆ(x)=arccos[I(x)];
U={I(x)-cos[ϕˆ(x)]}2+{Ix(x)+ωˆ(x)sin[ϕˆ(x)]}2,
ϕˆ(x)k+1=ϕˆ(x)k-μUϕˆ(x),
ωˆ(x)k+1=ωˆ(x)k-μUωˆ(x),
ϕˆ(x)0=0,    ωˆ(x)0=0,
ϕˆ(x)0=ϕˆ(x-1),    ωˆ(x)0=ωˆ(x-1)
U=[I-cos(ϕˆ)]2+[Ix+ωˆx sin(ϕˆ)]2+[Iy+ωˆy sin(ϕˆ)]2,
ϕˆ(ri)k+1=ϕˆ(ri)k-μUϕˆ(ri),
ωˆx(ri)k+1=ωˆx(ri)k-μUωˆx(ri),
ωˆy(ri)k+1=ωˆy(ri)k-μUωˆy(ri),
ϕˆ(ri)0=ϕˆ(ri-1),    ωˆx(ri)0=ωˆx(ri-1),ωˆy(ri)0=ωˆy(ri-1),
ωˆ(r)=[ωˆx2(r)+ωˆy2(r)]1/2>0;
if I(x)>0,  good data,
if I(x)0,  bad data.
p(, η)=ϕˆ(x, y)+ωˆx(x, y)(x-)+ωˆy(x, y)(y-η).
U=(,η)Nx,y{[I-cos(p)]2+[Ix+ωˆx sin(p)]2+[Iy+ωˆy sin(p)]2},
U=(,η)Nx,y{[I-cos(p)]2+[Ix+ωˆx sin(p)]2+[Iy+ωˆy sin(p)]2+λ(ϕˆ-p)2m},
URPT=(,η)Nx,y{[I-cos(p)]2+λ(ϕˆ-p)2m},
URQPT=(,η)Nx,y{[I-cos(p)]2+[Ix+ωˆx sin(p)]2+[Iy+ωˆy sin(p)]2+λ(ϕˆ-p)2m},
p(, η)=ϕˆ(x)+ωˆx(x, y)(x-)+ωˆx(x, y)(y-η).
ϕˆ1(x)ϕ(x, y),
ϕˆ2(x)cos-1[I(x, y)],
I(x, y)=0.8 cos(ω0x).
I(x, y)=cos[ω0x+n(x, y)].
URPT=(,η)Nx,y{[I-cos(p)]2+[I-cos(p+α)]2+λ(ϕˆ-p)2m},

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