Abstract

Fringe patterns with a multiplicative phase shift among them appear in experimental techniques as photoelasticity and RGB shadow moiré, among others. These patterns cannot be processed using standard phase-shifting demodulation techniques. In this work, we propose to use a multiframe regularized optical flow algorithm to obtain the interesting modulating phase. The proposed technique has been applied to simulated and experimental interferograms obtaining satisfactory results.

© 2012 Optical Society of America

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References

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  1. D. Malacara, M. Servín, and Z. Malacara, Interferogram Analysis for Optical Testing (Cambridge University, 2004).
  2. Z. Wang and B. Han, “Advanced iterative algorithm for phase extraction of randomly phase-shifted interferograms,” Opt. Lett. 29, 1671–1673 (2004).
    [CrossRef]
  3. J. Vargas, J. Antonio Quiroga, and T. Belenguer, “Phase-shifting interferometry based on principal component analysis,” Opt. Lett. 36, 1326–1328 (2011).
    [CrossRef]
  4. J. A. Quiroga, J. A. Gómez-Pedrero, M. J. Terrón-López, and M. Servín, “Temporal demodulation of fringe patterns with sensitivity change,” Opt. Commun. 253, 266–275 (2005).
    [CrossRef]
  5. J. A. Gómez-Pedrero, J. A. Quiroga, M. J. Terrón-López, and D. Crespo, “Measurement of surface topography by RGB shadow-moiré with direct phase demodulation,” Opt. Lasers Eng. 44, 1297–1310 (2006).
    [CrossRef]
  6. J. Vargas, J. A. Quiroga, C. O. S. Sorzano, J. C. Estrada, and J. M. Carazo, “Two-step interferometry by a regularized optical flow algorithm,” Opt. Lett. 36, 3485–3487 (2011).
    [CrossRef]
  7. K. G. Larkin, D. J. Bone, and 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]
  8. J. Villa, I. De la Rosa, G. Miramontes, and J. A. Quiroga, “Phase recovery from a single fringe pattern using an orientational vector-field-regularized estimator,” J. Opt. Soc. Am. A 22, 2766–2773 (2005).
    [CrossRef]
  9. B. K. P. Horn and B. G. Schunck, “Determining optical flow,” Artif. Intell. 17, 185–203 (1981).
    [CrossRef]
  10. B. Ströbel, “Processing of interferometric phase maps as complex-valued phasor images,” Appl. Opt. 35, 2192–2198 (1996).
    [CrossRef]
  11. M. Arevallilo-Herráez, D. R. Burton, M. J. Lalor, and M. A. Gdeisat, “Fast two-dimensional phase-unwrapping algorithm based on sorting by reliability following a noncontinuous path,” Appl. Opt. 41, 7437–7444 (2002).
    [CrossRef]

2011 (2)

2006 (1)

J. A. Gómez-Pedrero, J. A. Quiroga, M. J. Terrón-López, and D. Crespo, “Measurement of surface topography by RGB shadow-moiré with direct phase demodulation,” Opt. Lasers Eng. 44, 1297–1310 (2006).
[CrossRef]

2005 (2)

J. A. Quiroga, J. A. Gómez-Pedrero, M. J. Terrón-López, and M. Servín, “Temporal demodulation of fringe patterns with sensitivity change,” Opt. Commun. 253, 266–275 (2005).
[CrossRef]

J. Villa, I. De la Rosa, G. Miramontes, and J. A. Quiroga, “Phase recovery from a single fringe pattern using an orientational vector-field-regularized estimator,” J. Opt. Soc. Am. A 22, 2766–2773 (2005).
[CrossRef]

2004 (1)

2002 (1)

2001 (1)

1996 (1)

1981 (1)

B. K. P. Horn and B. G. Schunck, “Determining optical flow,” Artif. Intell. 17, 185–203 (1981).
[CrossRef]

Antonio Quiroga, J.

Arevallilo-Herráez, M.

Belenguer, T.

Bone, D. J.

Burton, D. R.

Carazo, J. M.

Crespo, D.

J. A. Gómez-Pedrero, J. A. Quiroga, M. J. Terrón-López, and D. Crespo, “Measurement of surface topography by RGB shadow-moiré with direct phase demodulation,” Opt. Lasers Eng. 44, 1297–1310 (2006).
[CrossRef]

De la Rosa, I.

Estrada, J. C.

Gdeisat, M. A.

Gómez-Pedrero, J. A.

J. A. Gómez-Pedrero, J. A. Quiroga, M. J. Terrón-López, and D. Crespo, “Measurement of surface topography by RGB shadow-moiré with direct phase demodulation,” Opt. Lasers Eng. 44, 1297–1310 (2006).
[CrossRef]

J. A. Quiroga, J. A. Gómez-Pedrero, M. J. Terrón-López, and M. Servín, “Temporal demodulation of fringe patterns with sensitivity change,” Opt. Commun. 253, 266–275 (2005).
[CrossRef]

Han, B.

Horn, B. K. P.

B. K. P. Horn and B. G. Schunck, “Determining optical flow,” Artif. Intell. 17, 185–203 (1981).
[CrossRef]

Lalor, M. J.

Larkin, K. G.

Malacara, D.

D. Malacara, M. Servín, and Z. Malacara, Interferogram Analysis for Optical Testing (Cambridge University, 2004).

Malacara, Z.

D. Malacara, M. Servín, and Z. Malacara, Interferogram Analysis for Optical Testing (Cambridge University, 2004).

Miramontes, G.

Oldfield, M. A.

Quiroga, J. A.

J. Vargas, J. A. Quiroga, C. O. S. Sorzano, J. C. Estrada, and J. M. Carazo, “Two-step interferometry by a regularized optical flow algorithm,” Opt. Lett. 36, 3485–3487 (2011).
[CrossRef]

J. A. Gómez-Pedrero, J. A. Quiroga, M. J. Terrón-López, and D. Crespo, “Measurement of surface topography by RGB shadow-moiré with direct phase demodulation,” Opt. Lasers Eng. 44, 1297–1310 (2006).
[CrossRef]

J. Villa, I. De la Rosa, G. Miramontes, and J. A. Quiroga, “Phase recovery from a single fringe pattern using an orientational vector-field-regularized estimator,” J. Opt. Soc. Am. A 22, 2766–2773 (2005).
[CrossRef]

J. A. Quiroga, J. A. Gómez-Pedrero, M. J. Terrón-López, and M. Servín, “Temporal demodulation of fringe patterns with sensitivity change,” Opt. Commun. 253, 266–275 (2005).
[CrossRef]

Schunck, B. G.

B. K. P. Horn and B. G. Schunck, “Determining optical flow,” Artif. Intell. 17, 185–203 (1981).
[CrossRef]

Servín, M.

J. A. Quiroga, J. A. Gómez-Pedrero, M. J. Terrón-López, and M. Servín, “Temporal demodulation of fringe patterns with sensitivity change,” Opt. Commun. 253, 266–275 (2005).
[CrossRef]

D. Malacara, M. Servín, and Z. Malacara, Interferogram Analysis for Optical Testing (Cambridge University, 2004).

Sorzano, C. O. S.

Ströbel, B.

Terrón-López, M. J.

J. A. Gómez-Pedrero, J. A. Quiroga, M. J. Terrón-López, and D. Crespo, “Measurement of surface topography by RGB shadow-moiré with direct phase demodulation,” Opt. Lasers Eng. 44, 1297–1310 (2006).
[CrossRef]

J. A. Quiroga, J. A. Gómez-Pedrero, M. J. Terrón-López, and M. Servín, “Temporal demodulation of fringe patterns with sensitivity change,” Opt. Commun. 253, 266–275 (2005).
[CrossRef]

Vargas, J.

Villa, J.

Wang, Z.

Appl. Opt. (2)

Artif. Intell. (1)

B. K. P. Horn and B. G. Schunck, “Determining optical flow,” Artif. Intell. 17, 185–203 (1981).
[CrossRef]

J. Opt. Soc. Am. A (2)

Opt. Commun. (1)

J. A. Quiroga, J. A. Gómez-Pedrero, M. J. Terrón-López, and M. Servín, “Temporal demodulation of fringe patterns with sensitivity change,” Opt. Commun. 253, 266–275 (2005).
[CrossRef]

Opt. Lasers Eng. (1)

J. A. Gómez-Pedrero, J. A. Quiroga, M. J. Terrón-López, and D. Crespo, “Measurement of surface topography by RGB shadow-moiré with direct phase demodulation,” Opt. Lasers Eng. 44, 1297–1310 (2006).
[CrossRef]

Opt. Lett. (3)

Other (1)

D. Malacara, M. Servín, and Z. Malacara, Interferogram Analysis for Optical Testing (Cambridge University, 2004).

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Figures (7)

Fig. 1.
Fig. 1.

Simulated fringe patterns used in the numerical analysis section.

Fig. 2.
Fig. 2.

(a) Ground truth modulating phase and retrieved phases using (b) the proposed MOF method, by the (c) TDM and by the (d) OVFR method.

Fig. 3.
Fig. 3.

Root mean square errors (RMS) obtained by the proposed method (solid black curve with dark squares), the TDM (solid black curve with gray triangles), and the OVFR method (solid black curve with gray circles) for (a) different levels of noise, (b) multiplicative phase-shifting, and (c) number of fringe patterns.

Fig. 4.
Fig. 4.

Two experimental fringe patterns obtained in a loading-stepping photoelastic experiment.

Fig. 5.
Fig. 5.

Obtained phases using (a) the proposed MOF method, (b) the TDM, and (c) the OVFR methods.

Fig. 6.
Fig. 6.

Experimental fringe patterns obtained from an aeronautical surface with an indentation in an RGB shadow-moiré experiment.

Fig. 7.
Fig. 7.

Obtained phases using (a) the proposed MOF method, (b) the TDM, and (c) the OVFR methods.

Equations (13)

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In(x,y)=a(x,y)+b(x,y)cos[αnΦ(x,y)],n[1,N],
I(x+Δx,y+Δy,t+Δt)I(x,y,t)+IxΔx+IyΔy+ItΔt.
Ixu+Iyv+It=0,
E2=(Ixu+Iyv+It)2+μ(ux2+uy2+vx2+vy2),
uk+1=u¯k-Ix[Ixu¯k+Iyv¯k+It]/(μ2+Ix2+Iy2)vk+1=v¯kIy[Ixu¯k+Iyv¯k+It]/(μ2+Ix2+Iy2),
θ=arctan(vu).
Qn(x,y)=un(x,y)2+vn(x,y)2.
cos(θ)=ncos(θn)Qn,sin(θ)=nsin(θn)Qn,θ=arctan(sin(θ)cos(θ)).
SPT{I˜n}=iexp(iθ)bsin(αnΦ),
bsin(αnΦ)=-iexp(iθ)SPT{I˜n},
W{Φn}=W{αnΦ}=arctan(bsin(αnΦ)I˜n),
α˜Φ=nΦn/N,
α˜=(xyα˜Φ/(NxNyα1Φ)),Φ=α˜Φ/α˜,

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