Abstract

A cascade least-squares scheme for wrapped phase extraction using two or more phase-shifted fringe-patterns with unknown and inhomogeneous surface phase shift is proposed. This algorithm is based on the parameter estimation approach to process fringe-patterns where, except for the interest phase distribution that is a function of the space only, all other parameters are functions of both space and time. Computer simulations and experimental results show that phase computing is possible even when an inhomogeneous phase shift is induced by nonlinearity of the piezoelectric materials or miscalibrated phase shifters. The algorithm’s features and its operating conditions will been discussed. Due to the useful properties of this algorithm such as the robustness, computational efficiency, and user-free execution, this proposal could be used in automatic applications.

© 2014 Optical Society of America

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References

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    [CrossRef]

2013 (5)

2012 (1)

G. Rajshekhar, P. Rastogi, “Fringe analysis: Premise and perspectives,” Opt. Lasers Eng. 50, iii–x (2012).
[CrossRef]

2011 (2)

2008 (2)

J. Xu, Q. Xu, L. Chai, “Tilt-shift determination and compensation in phase-shifting interferometry,” J. Opt. A Pure Appl. Opt. 10(7), 075011 (2008).
[CrossRef]

J. Xu, Q. Xu, L. Chai, “Iterative algorithm for phase extraction from interferograms with random and spatially nonuniform phase shifts,” Appl. Opt. 47, 480–485 (2008).
[CrossRef] [PubMed]

2007 (1)

A. Patil, P. Rastogi, “Moving ahead with phase,” Opt. Lasers Eng. 45, 253–257 (2007).
[CrossRef]

2006 (1)

2005 (2)

2004 (1)

2002 (1)

2001 (3)

K. Larkin, “A self-calibrating phase-shifting algorithm based on the natural demodulation of two-dimensional fringe patterns,” Opt. Express 9, 236–253 (2001).
[CrossRef] [PubMed]

M. Afifi, K. Nassim, S. Rachafi, “Five-frame phase-shifting algorithm insensitive to diode laser power variation,” Opt. Commun. 197, 37–42 (2001).
[CrossRef]

W. Li, X. Su, “Real-time calibration algorithm for phase shifting in phase-measuring profilometry,” Opt. Eng. 40, 761–766 (2001).
[CrossRef]

2000 (3)

1998 (1)

M. Costantini, “A novel phase unwrapping method based on network programming,” IEEE Trans. Geosci. Remote Sens. 36(3), 813–821 (1998).
[CrossRef]

1997 (1)

1994 (1)

C. T. Farrell, M. A. Player, “Phase-step insensitive algorithms for phase-shifting interferometry,” Meas. Sci. Technol. 5(6), 648 (1994).
[CrossRef]

1991 (1)

1987 (1)

1985 (1)

1984 (1)

J. E. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23350–352 (1984).
[CrossRef]

1982 (1)

1974 (1)

1966 (1)

P. Carre, “Installation et utilisation du comparateur photohlectrique et interfhrentiel du bureau international des poids et mesures,” Metrologia 2, 13–23 (1966).
[CrossRef]

Afifi, M.

M. Afifi, K. Nassim, S. Rachafi, “Five-frame phase-shifting algorithm insensitive to diode laser power variation,” Opt. Commun. 197, 37–42 (2001).
[CrossRef]

Aguilar, L. A.

R. Juarez-Salazar, C. Robledo-Sanchez, C. Meneses-Fabian, F. Guerrero-Sanchez, L. A. Aguilar, “Generalized phase-shifting interferometry by parameter estimation with the least squares method,” Opt. Lasers Eng. 51(5), 626–632 (2013).
[CrossRef]

Aguilar, L. M. A.

Apostol, D.

Brangaccio, D. J.

Bruning, J. H.

Bruno, L.

Carre, P.

P. Carre, “Installation et utilisation du comparateur photohlectrique et interfhrentiel du bureau international des poids et mesures,” Metrologia 2, 13–23 (1966).
[CrossRef]

Chai, L.

J. Xu, Q. Xu, L. Chai, “Tilt-shift determination and compensation in phase-shifting interferometry,” J. Opt. A Pure Appl. Opt. 10(7), 075011 (2008).
[CrossRef]

J. Xu, Q. Xu, L. Chai, “Iterative algorithm for phase extraction from interferograms with random and spatially nonuniform phase shifts,” Appl. Opt. 47, 480–485 (2008).
[CrossRef] [PubMed]

Chen, M.

Chen, X.

Cheng, Y.-Y.

Costantini, M.

M. Costantini, “A novel phase unwrapping method based on network programming,” IEEE Trans. Geosci. Remote Sens. 36(3), 813–821 (1998).
[CrossRef]

Damian, V.

Dobroiu, A.

Eiju, T.

Farrant, D. I.

Farrell, C. T.

C. T. Farrell, M. A. Player, “Phase-step insensitive algorithms for phase-shifting interferometry,” Meas. Sci. Technol. 5(6), 648 (1994).
[CrossRef]

Gallagher, J. E.

Gao, F.

Gramaglia, M.

Greivenkamp, J. E.

J. E. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23350–352 (1984).
[CrossRef]

Gu, H.

Guerrero-Sanchez, F.

R. Juarez-Salazar, C. Robledo-Sanchez, C. Meneses-Fabian, F. Guerrero-Sanchez, L. A. Aguilar, “Generalized phase-shifting interferometry by parameter estimation with the least squares method,” Opt. Lasers Eng. 51(5), 626–632 (2013).
[CrossRef]

Guerrero-Sánchez, F.

Guo, H.

Han, B.

Han, G.

Hariharan, P.

Herriott, D. R.

Hibino, K.

Ixba-Santos, V.

Jiao, Z.

Jin, G.

Juarez-Salazar, R.

C. Robledo-Sanchez, R. Juarez-Salazar, C. Meneses-Fabian, F. Guerrero-Sánchez, L. M. A. Aguilar, G. Rodriguez-Zurita, V. Ixba-Santos, “Phase-shifting interferometry based on the lateral displacement of the light source,” Opt. Express 21, 17228–17233 (2013).
[CrossRef] [PubMed]

R. Juarez-Salazar, C. Robledo-Sanchez, C. Meneses-Fabian, F. Guerrero-Sanchez, L. A. Aguilar, “Generalized phase-shifting interferometry by parameter estimation with the least squares method,” Opt. Lasers Eng. 51(5), 626–632 (2013).
[CrossRef]

Lai, G.

Larkin, K.

Larkin, K. G.

Li, D.

Li, J.

J. Li, Y. Wang, X. Meng, X. Yang, Q. Wang, “An evaluation method for phase shift extraction algorithms in generalized phase-shifting interferometry,” J. Opt. 15(10), 105408 (2013).
[CrossRef]

Li, W.

W. Li, X. Su, “Real-time calibration algorithm for phase shifting in phase-measuring profilometry,” Opt. Eng. 40, 761–766 (2001).
[CrossRef]

Lu, G.

Malacara, D.

D. Malacara, M. Servin, Z. Malacara, Interferogram Analysis for Optical Testing, 2nd ed. (Taylor and Francis, 2005).
[CrossRef]

Malacara, Z.

D. Malacara, M. Servin, Z. Malacara, Interferogram Analysis for Optical Testing, 2nd ed. (Taylor and Francis, 2005).
[CrossRef]

Meneses-Fabian, C.

Meng, X.

J. Li, Y. Wang, X. Meng, X. Yang, Q. Wang, “An evaluation method for phase shift extraction algorithms in generalized phase-shifting interferometry,” J. Opt. 15(10), 105408 (2013).
[CrossRef]

Morgan, C. J.

Nascov, V.

Nassim, K.

M. Afifi, K. Nassim, S. Rachafi, “Five-frame phase-shifting algorithm insensitive to diode laser power variation,” Opt. Commun. 197, 37–42 (2001).
[CrossRef]

Oreb, B. F.

Patil, A.

A. Patil, P. Rastogi, “Moving ahead with phase,” Opt. Lasers Eng. 45, 253–257 (2007).
[CrossRef]

A. Patil, P. Rastogi, “Approaches in generalized phase shifting interferometry,” Opt. Lasers Eng. 43, 475–490 (2005).
[CrossRef]

Patorski, K.

Player, M. A.

C. T. Farrell, M. A. Player, “Phase-step insensitive algorithms for phase-shifting interferometry,” Meas. Sci. Technol. 5(6), 648 (1994).
[CrossRef]

Rachafi, S.

M. Afifi, K. Nassim, S. Rachafi, “Five-frame phase-shifting algorithm insensitive to diode laser power variation,” Opt. Commun. 197, 37–42 (2001).
[CrossRef]

Rajshekhar, G.

G. Rajshekhar, P. Rastogi, “Fringe analysis: Premise and perspectives,” Opt. Lasers Eng. 50, iii–x (2012).
[CrossRef]

Rastogi, P.

G. Rajshekhar, P. Rastogi, “Fringe analysis: Premise and perspectives,” Opt. Lasers Eng. 50, iii–x (2012).
[CrossRef]

A. Patil, P. Rastogi, “Moving ahead with phase,” Opt. Lasers Eng. 45, 253–257 (2007).
[CrossRef]

A. Patil, P. Rastogi, “Approaches in generalized phase shifting interferometry,” Opt. Lasers Eng. 43, 475–490 (2005).
[CrossRef]

Rivera-Ortega, U.

Robledo-Sanchez, C.

C. Robledo-Sanchez, R. Juarez-Salazar, C. Meneses-Fabian, F. Guerrero-Sánchez, L. M. A. Aguilar, G. Rodriguez-Zurita, V. Ixba-Santos, “Phase-shifting interferometry based on the lateral displacement of the light source,” Opt. Express 21, 17228–17233 (2013).
[CrossRef] [PubMed]

R. Juarez-Salazar, C. Robledo-Sanchez, C. Meneses-Fabian, F. Guerrero-Sanchez, L. A. Aguilar, “Generalized phase-shifting interferometry by parameter estimation with the least squares method,” Opt. Lasers Eng. 51(5), 626–632 (2013).
[CrossRef]

Rodriguez-Zurita, G.

Rosenfeld, D. P.

Servin, M.

D. Malacara, M. Servin, Z. Malacara, Interferogram Analysis for Optical Testing, 2nd ed. (Taylor and Francis, 2005).
[CrossRef]

Soloviev, O.

Styk, A.

Su, X.

W. Li, X. Su, “Real-time calibration algorithm for phase shifting in phase-measuring profilometry,” Opt. Eng. 40, 761–766 (2001).
[CrossRef]

Szwaykowski, P.

Tan, Q.

Vdovin, G.

Wang, Q.

J. Li, Y. Wang, X. Meng, X. Yang, Q. Wang, “An evaluation method for phase shift extraction algorithms in generalized phase-shifting interferometry,” J. Opt. 15(10), 105408 (2013).
[CrossRef]

Wang, Y.

J. Li, Y. Wang, X. Meng, X. Yang, Q. Wang, “An evaluation method for phase shift extraction algorithms in generalized phase-shifting interferometry,” J. Opt. 15(10), 105408 (2013).
[CrossRef]

Wang, Z.

Wei, C.

White, A. D.

Wyant, J. C.

Xu, J.

J. Xu, Q. Xu, L. Chai, “Iterative algorithm for phase extraction from interferograms with random and spatially nonuniform phase shifts,” Appl. Opt. 47, 480–485 (2008).
[CrossRef] [PubMed]

J. Xu, Q. Xu, L. Chai, “Tilt-shift determination and compensation in phase-shifting interferometry,” J. Opt. A Pure Appl. Opt. 10(7), 075011 (2008).
[CrossRef]

Xu, Q.

J. Xu, Q. Xu, L. Chai, “Tilt-shift determination and compensation in phase-shifting interferometry,” J. Opt. A Pure Appl. Opt. 10(7), 075011 (2008).
[CrossRef]

J. Xu, Q. Xu, L. Chai, “Iterative algorithm for phase extraction from interferograms with random and spatially nonuniform phase shifts,” Appl. Opt. 47, 480–485 (2008).
[CrossRef] [PubMed]

Xu, X.

Yang, X.

J. Li, Y. Wang, X. Meng, X. Yang, Q. Wang, “An evaluation method for phase shift extraction algorithms in generalized phase-shifting interferometry,” J. Opt. 15(10), 105408 (2013).
[CrossRef]

Yatagai, T.

Yeazell, J. A.

Zeng, F.

Appl. Opt. (8)

IEEE Trans. Geosci. Remote Sens. (1)

M. Costantini, “A novel phase unwrapping method based on network programming,” IEEE Trans. Geosci. Remote Sens. 36(3), 813–821 (1998).
[CrossRef]

J. Opt. (1)

J. Li, Y. Wang, X. Meng, X. Yang, Q. Wang, “An evaluation method for phase shift extraction algorithms in generalized phase-shifting interferometry,” J. Opt. 15(10), 105408 (2013).
[CrossRef]

J. Opt. A Pure Appl. Opt. (1)

J. Xu, Q. Xu, L. Chai, “Tilt-shift determination and compensation in phase-shifting interferometry,” J. Opt. A Pure Appl. Opt. 10(7), 075011 (2008).
[CrossRef]

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

Meas. Sci. Technol. (1)

C. T. Farrell, M. A. Player, “Phase-step insensitive algorithms for phase-shifting interferometry,” Meas. Sci. Technol. 5(6), 648 (1994).
[CrossRef]

Metrologia (1)

P. Carre, “Installation et utilisation du comparateur photohlectrique et interfhrentiel du bureau international des poids et mesures,” Metrologia 2, 13–23 (1966).
[CrossRef]

Opt. Commun. (1)

M. Afifi, K. Nassim, S. Rachafi, “Five-frame phase-shifting algorithm insensitive to diode laser power variation,” Opt. Commun. 197, 37–42 (2001).
[CrossRef]

Opt. Eng. (2)

J. E. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23350–352 (1984).
[CrossRef]

W. Li, X. Su, “Real-time calibration algorithm for phase shifting in phase-measuring profilometry,” Opt. Eng. 40, 761–766 (2001).
[CrossRef]

Opt. Express (6)

Opt. Lasers Eng. (4)

R. Juarez-Salazar, C. Robledo-Sanchez, C. Meneses-Fabian, F. Guerrero-Sanchez, L. A. Aguilar, “Generalized phase-shifting interferometry by parameter estimation with the least squares method,” Opt. Lasers Eng. 51(5), 626–632 (2013).
[CrossRef]

A. Patil, P. Rastogi, “Approaches in generalized phase shifting interferometry,” Opt. Lasers Eng. 43, 475–490 (2005).
[CrossRef]

G. Rajshekhar, P. Rastogi, “Fringe analysis: Premise and perspectives,” Opt. Lasers Eng. 50, iii–x (2012).
[CrossRef]

A. Patil, P. Rastogi, “Moving ahead with phase,” Opt. Lasers Eng. 45, 253–257 (2007).
[CrossRef]

Opt. Lett. (3)

Other (1)

D. Malacara, M. Servin, Z. Malacara, Interferogram Analysis for Optical Testing, 2nd ed. (Taylor and Francis, 2005).
[CrossRef]

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Figures (10)

Fig. 1
Fig. 1

Phase shift functions in the generalized phase-shifting approach. Homogeneous phase shift: (a) linear, and (b) nonlinear on k. Inhomogeneous phase shift: (c) nonlinear on k but linear on p, and (d) nonlinear on both k and p.

Fig. 2
Fig. 2

From a set of phase-shifted fringe-patterns modelled by Eq. (1), the interest phase distribution ϕ(p) and the phase shift δk(p) can be determined, respectively, as the dynamic and static parts of the encoded phase Φk(p) = ϕ(p) + δk(p).

Fig. 3
Fig. 3

(a) The data A k 2 1 and 1 S k 2, given by Eqs. (7)(a) and (7)(b), for the estimation of the term cosαk. For αk ∈ [0, π/2] it is convenient to choose 1 S k 2 because the noise amplitude is lower, and vice versa for αk ∈ [π/2, π]. (b) Switch functions by hyperbolic tangent approximation of unit step functions. The constant ω sets the smoothness of the transition at ck = 0. In this plot, ω = 15. (c) Equivalent data to polynomial fitting by the procedure described with Eq. (9).

Fig. 4
Fig. 4

Proposed inhomogeneous generalized phase-shifting algorithm by a cascade least-squares scheme. (a) Normalization stage. (b) Phase shift estimation stage. (c) Wrapped phase extraction stage.

Fig. 5
Fig. 5

Computer simulation. (1st column) Phase step α1, phase distribution ϕ to be recovered and its wrapped version. (2nd column) Background a0 and modulation b0 lights for the first fringe-pattern I0. Similarly, the 3rd column shows a1, b1 and I1. (4th column) Normalized fringe-patterns, computed phase step, and recovered wrapped phase.

Fig. 6
Fig. 6

Computer simulation. (1st column) Absolute error from the phase step and wrapped phase estimations. (2nd column) The respective error histograms.

Fig. 7
Fig. 7

(a) Optical setup for the phase-shifting experiments. Pictographic description of the piezoelectric control signals to generate phase shifts: (a) homogeneous, (b) inhomogeneous tilted planes, and (c) inhomogeneous surfaces.

Fig. 8
Fig. 8

Experimental results. Wrapped phase computed from three phase-shifted fringe-patterns when the phase shift is: (1st row) homogeneous (non-tilted planes), (2nd row) inhomogeneous tilted planes, and (3rd row) inhomogeneous surfaces. (1–3rd columns) Fringe-patterns to be processed. (4th column) Computed phase shift. (5th column) Recovered wrapped phase. (6th column) Resulting background light.

Fig. 9
Fig. 9

(1st row) Unwrapped phases ϕ1, ϕ2, and ϕ3, respectively, of the wrapped phase maps shown in the 5th column of Fig. 8. (2nd row) The difference ϕ2ϕ1, its fitted plane, and the fitting error (4.5 × 10−15 average value). (3rd row) The difference ϕ3ϕ1, its fitted plane, and the fitting error (−3.3 × 10−14 average value).

Fig. 10
Fig. 10

Computing time required for processing: (a) Two 500 × 500 fringe-patterns with inhomogeneous nonlinear phase shift (Fig. 5). (b)–(d) Three 1824 × 1418 fringe-patterns with: homogeneous (constant), inhomogeneous linear, and inhomogeneous nonlinear phase shifts, respectively (Fig. 8). The hardware used was a 2.5 GHz laptop.

Equations (22)

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

I k ( p ) = a k ( p ) + b k ( p ) cos Φ k ( p ) , with Φ k ( p ) = ϕ ( p ) + δ k ( p ) ,
d d k Φ k ( p ) = d d k δ k ( p ) = α k ( p ) .
δ k ( p ) = δ 0 + 0 k α ( p ) d .
a ˜ k = 𝔸 a 𝔸 a I k ,
b ˜ k 2 = 2 𝔸 b 𝔸 b ( I k a ˜ k ) 2 ,
I ¯ k = sat ( I k a ˜ k b ˜ k ) cos [ ϕ + δ k ] ,
A k = I ¯ k 1 + I ¯ k = 2 cos α k 2 cos [ ϕ + δ k 1 + δ k 2 ] ,
S k = I ¯ k 1 I ¯ k = 2 sin α k 2 sin [ ϕ + δ k 1 + δ k 2 ] ,
A k 2 1 = cos α k + 2 cos 2 α k 2 η k ,
1 S k 2 = cos α k + 2 sin 2 α k 2 η k ,
c A k = 𝔸 c 𝔸 c ( A k 2 1 ) ,
c S k = 𝔸 c 𝔸 c ( 1 S k 2 ) ,
cos α k = c S k Γ ( ω c k ) + c A k Γ ( ω c k ) = c k + 1 2 ( c S k c A k ) tanh ( ω c k ) ,
δ k = δ 0 + = 1 k α , k = 1 , 2 , , K 1 .
ϕ w = arctan ( ζ / ξ ) ,
[ ξ ζ ] = 𝔸 ϕ [ I 0 I 1 I K 1 ] T ,
𝔸 ϕ = [ cos δ 0 sin δ 0 cos δ K 1 sin δ K 1 ] .
α 1 = π [ 1 2 + 1 3 ( x 2 y 2 ) ] ,
ϕ = 6 peaks ( 500 ) + 12 ( x 2 + y 2 ) + 1 ,
I k = a k + b k cos ( ϕ + δ k ) + ρ k ,
a 0 = 15 ( x 2 + y 2 + 1 ) , b 0 = x 3 + y 2 + 2 , a 1 = 20 ( y 2 x 2 + 1 ) + 5 x , b 1 = y 3 x 2 + 2.5 .
I = a + b cos ( ϕ + δ ˜ ) a ˜ + b cos ( ϕ + δ ) ,

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