Abstract

We present a phase shifting robust method for irregular and unknown phase steps. The method is formulated as the minimization of a half-quadratic (robust) regularized cost function for simultaneously computing phase maps and arbitrary phase shifts. The convergence to, at least, a local minimum is guaranteed. The algorithm can be understood as a phase refinement strategy that uses as initial guess a coarsely computed phase and coarsely estimated phase shifts. Such a coarse phase is assumed to be corrupted with artifacts produced by the use of a phase shifting algorithm but with imprecise phase steps. The refinement is achieved by iterating alternated minimization of the cost function for computing the phase map correction, an outliers rejection map and the phase shifts correction, respectively. The method performance is demonstrated by comparison with standard filtering and arbitrary phase steps detecting algorithms.

© 2006 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  27. K. Okada, A. Sato, and J. Tsujiuchi, "Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry," Opt. Commun. 84, 118-124 (1991).
    [CrossRef]
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    [CrossRef]
  29. D. Geman and G. Reynolds, "Constrained restoration and the recovery of discontinuities," IEEE Trans. Image Process. 14, 367-383 (1992).
  30. M.J. Black and A. Rangarajan, "Unification of line process, outlier rejection, and robust statistics with application in early vision," Int. J. Comput. Vis. 19, 57-91 (1996).
    [CrossRef]
  31. P. Charbonnier, L. Blanc-F´eraud, G. Aubert and M. Barlaud, "Deterministic edge-preserving regularization in computer imaging," IEEE Trans. Image Process. 6, 298-311 (1997).
    [CrossRef] [PubMed]
  32. M. Rivera and J.L. Marroquin, "Adaptive rest condition potentials: Second order edge-preserving regularization," Comput. Vision Image Understand. 88, 76-93 (2002).
    [CrossRef]
  33. M. Rivera, and J.L. Marroquin, "Efficient half-quadratic regularization with granularity control," Image and Vision Computing 21, 345—357 (2003).
    [CrossRef]

2005 (5)

2004 (3)

2003 (2)

L. Z. Cai, Q. Liu, and X. L. Yang, "Phase-shift extraction and wave-front reconstruction in phase-shifting interferometry with arbitrary phase steps," Opt. Lett. 28, 1808-1810 (2003).
[CrossRef] [PubMed]

M. Rivera, and J.L. Marroquin, "Efficient half-quadratic regularization with granularity control," Image and Vision Computing 21, 345—357 (2003).
[CrossRef]

2002 (1)

M. Rivera and J.L. Marroquin, "Adaptive rest condition potentials: Second order edge-preserving regularization," Comput. Vision Image Understand. 88, 76-93 (2002).
[CrossRef]

2001 (3)

2000 (2)

1999 (2)

H. van Brug, "Phase-step calibration for phase-stepped interferometry," Appl. Opt. 383549-3555 (1999).
[CrossRef]

C. Wei, M. Chen, Z. Wang, "General phase-stepping algorithm with automatic calibration of phase steps," Opt. Eng. 38, 1357-1360 (1999).
[CrossRef]

1998 (1)

1997 (2)

B. Zhao and Y. Surrel, "Effect of quantization error on the computed phase of phase-shifting measurements," Appl. Opt. 36, 2070-2075 (1997).
[CrossRef] [PubMed]

P. Charbonnier, L. Blanc-F´eraud, G. Aubert and M. Barlaud, "Deterministic edge-preserving regularization in computer imaging," IEEE Trans. Image Process. 6, 298-311 (1997).
[CrossRef] [PubMed]

1996 (2)

M.J. Black and A. Rangarajan, "Unification of line process, outlier rejection, and robust statistics with application in early vision," Int. J. Comput. Vis. 19, 57-91 (1996).
[CrossRef]

Y. Surrel, "Design of algorithms for phase measurements by the use of phase stepping," Appl. Opt. 35, 51-60 (1996).
[CrossRef] [PubMed]

1995 (2)

1993 (1)

1992 (1)

D. Geman and G. Reynolds, "Constrained restoration and the recovery of discontinuities," IEEE Trans. Image Process. 14, 367-383 (1992).

1991 (1)

K. Okada, A. Sato, and J. Tsujiuchi, "Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry," Opt. Commun. 84, 118-124 (1991).
[CrossRef]

1984 (1)

J.E. Grievenkamp, "Generalized data reduction for heterodyne interferometry," Opt. Eng. 23, 350-352 (1984).

1982 (1)

1974 (1)

Aubert, G.

P. Charbonnier, L. Blanc-F´eraud, G. Aubert and M. Barlaud, "Deterministic edge-preserving regularization in computer imaging," IEEE Trans. Image Process. 6, 298-311 (1997).
[CrossRef] [PubMed]

Barlaud, M.

P. Charbonnier, L. Blanc-F´eraud, G. Aubert and M. Barlaud, "Deterministic edge-preserving regularization in computer imaging," IEEE Trans. Image Process. 6, 298-311 (1997).
[CrossRef] [PubMed]

Black, M.J.

M.J. Black and A. Rangarajan, "Unification of line process, outlier rejection, and robust statistics with application in early vision," Int. J. Comput. Vis. 19, 57-91 (1996).
[CrossRef]

Blanc-F´eraud, L.

P. Charbonnier, L. Blanc-F´eraud, G. Aubert and M. Barlaud, "Deterministic edge-preserving regularization in computer imaging," IEEE Trans. Image Process. 6, 298-311 (1997).
[CrossRef] [PubMed]

Bokor, J.

Botello, S.

Brangaccio, D.J.

Bruning, J.H.

Cai, L. Z.

Charbonnier, P.

P. Charbonnier, L. Blanc-F´eraud, G. Aubert and M. Barlaud, "Deterministic edge-preserving regularization in computer imaging," IEEE Trans. Image Process. 6, 298-311 (1997).
[CrossRef] [PubMed]

Chen, , L.

Chen, M.

M. Chen, H. Guo, and C. Wei,"Algorithm immune to tilt phase-shifting error for phase-shifting interferometers," Appl. Opt. 39, 3894-3898 (2000).
[CrossRef]

C. Wei, M. Chen, Z. Wang, "General phase-stepping algorithm with automatic calibration of phase steps," Opt. Eng. 38, 1357-1360 (1999).
[CrossRef]

Farrant, D.I.

Gallagher, J.E.

Geman, D.

D. Geman and G. Reynolds, "Constrained restoration and the recovery of discontinuities," IEEE Trans. Image Process. 14, 367-383 (1992).

Goldberg, K.A.

Grievenkamp, J.E.

J.E. Grievenkamp, "Generalized data reduction for heterodyne interferometry," Opt. Eng. 23, 350-352 (1984).

Guerrero, J. A.

Guo, H.

Herriot, D.R.

Hibino, K.

Langoju, R.

Larkin, K.

Larkin, K.G.

Li, W.

W. Li and X. Su, "Real-time calibration algorithm for phase shifting in phase-measuring profilometry," Opt. Commun. 40, 761-766 (2001).

Liu, Q.

Marroquin, J. L.

Marroquin, J.L.

M. Rivera and J.L. Marroquin, "Half-quadratic cost functions for phase unwrapping," Opt. Lett. 29, 504-506 (2004).
[CrossRef] [PubMed]

M. Rivera, and J.L. Marroquin, "Efficient half-quadratic regularization with granularity control," Image and Vision Computing 21, 345—357 (2003).
[CrossRef]

M. Rivera and J.L. Marroquin, "Adaptive rest condition potentials: Second order edge-preserving regularization," Comput. Vision Image Understand. 88, 76-93 (2002).
[CrossRef]

M. Rivera, J.L. Marroquin, S. Botello and M. Servin, "A robust spatio-temporal quadrature filter for multi-phase stepping," Appl. Opt. 39, 284-292 (2000).
[CrossRef]

Morgan, C.J.

Okada, K.

K. Okada, A. Sato, and J. Tsujiuchi, "Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry," Opt. Commun. 84, 118-124 (1991).
[CrossRef]

Oreb, B.F.

Patil, A.

Quan, C.

Quiroga, J. A.

Rangarajan, A.

M.J. Black and A. Rangarajan, "Unification of line process, outlier rejection, and robust statistics with application in early vision," Int. J. Comput. Vis. 19, 57-91 (1996).
[CrossRef]

Raphael, B.

Rastogi, P.

Rathjen, C.

Reynolds, G.

D. Geman and G. Reynolds, "Constrained restoration and the recovery of discontinuities," IEEE Trans. Image Process. 14, 367-383 (1992).

Rivera, M.

Rodriguez-Vera, R.

Rosenfeld, D.P.

Sato, A.

K. Okada, A. Sato, and J. Tsujiuchi, "Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry," Opt. Commun. 84, 118-124 (1991).
[CrossRef]

Servin, M.

Soloviev, O.

Su, X.

W. Li and X. Su, "Real-time calibration algorithm for phase shifting in phase-measuring profilometry," Opt. Commun. 40, 761-766 (2001).

Surrel, Y.

Tay, C.J.

Trolinger, J. D.

Tsujiuchi, J.

K. Okada, A. Sato, and J. Tsujiuchi, "Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry," Opt. Commun. 84, 118-124 (1991).
[CrossRef]

van Brug, H.

Vdovin, G.

Vikram, C. S.

Wang, Z.

C. Wei, M. Chen, Z. Wang, "General phase-stepping algorithm with automatic calibration of phase steps," Opt. Eng. 38, 1357-1360 (1999).
[CrossRef]

Wei, C.

M. Chen, H. Guo, and C. Wei,"Algorithm immune to tilt phase-shifting error for phase-shifting interferometers," Appl. Opt. 39, 3894-3898 (2000).
[CrossRef]

C. Wei, M. Chen, Z. Wang, "General phase-stepping algorithm with automatic calibration of phase steps," Opt. Eng. 38, 1357-1360 (1999).
[CrossRef]

White, A.D.

Witherow, W. K.

Yang, X. L.

Zhao, B.

Appl. Opt. (9)

Comput. Vision Image Understand. (1)

M. Rivera and J.L. Marroquin, "Adaptive rest condition potentials: Second order edge-preserving regularization," Comput. Vision Image Understand. 88, 76-93 (2002).
[CrossRef]

IEEE Trans. Image Process. (2)

P. Charbonnier, L. Blanc-F´eraud, G. Aubert and M. Barlaud, "Deterministic edge-preserving regularization in computer imaging," IEEE Trans. Image Process. 6, 298-311 (1997).
[CrossRef] [PubMed]

D. Geman and G. Reynolds, "Constrained restoration and the recovery of discontinuities," IEEE Trans. Image Process. 14, 367-383 (1992).

Image and Vision Computing (1)

M. Rivera, and J.L. Marroquin, "Efficient half-quadratic regularization with granularity control," Image and Vision Computing 21, 345—357 (2003).
[CrossRef]

Int. J. Comput. Vis. (1)

M.J. Black and A. Rangarajan, "Unification of line process, outlier rejection, and robust statistics with application in early vision," Int. J. Comput. Vis. 19, 57-91 (1996).
[CrossRef]

J. Mod. Opt. (1)

A. Patil and P. Rastogi, "Nonlinear regression technique applied to generalized phase-shifting interferometry," J. Mod. Opt. 52, 573 - 582 (2005).
[CrossRef]

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

Opt. Commun. (2)

K. Okada, A. Sato, and J. Tsujiuchi, "Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry," Opt. Commun. 84, 118-124 (1991).
[CrossRef]

W. Li and X. Su, "Real-time calibration algorithm for phase shifting in phase-measuring profilometry," Opt. Commun. 40, 761-766 (2001).

Opt. Eng. (2)

C. Wei, M. Chen, Z. Wang, "General phase-stepping algorithm with automatic calibration of phase steps," Opt. Eng. 38, 1357-1360 (1999).
[CrossRef]

J.E. Grievenkamp, "Generalized data reduction for heterodyne interferometry," Opt. Eng. 23, 350-352 (1984).

Opt. Express (3)

Opt. Lett. (6)

Other (2)

J.E. Grievenkamp and J.H. Bruning, "Phase shifting interferometry," in Optical Shop Testing, D. Malacara ed. (John Wiley & Sons, Inc. New York, 1992) pp. 501-598.

G, Lai and T. Yatagai, "Generalized phase-shifting interferometry," J. Opt. Soc. Am. A 8, 822- (1991)
[CrossRef]

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Figures (5)

Fig. 1.
Fig. 1.

First row: Synthetic fringe pattern set. Second row: Reconstruction using the computed phase map and the computed phase steps.

Fig. 2.
Fig. 2.

Coarse solution computed with Eq. (3) by assuming ideal steps (π/2). From left to right: Wrapped phase, unwrapped phase computed with the convex algorithm in [4] and its cosine (reconstructed fringe pattern).

Fig. 3.
Fig. 3.

From left to right: Rewrapped phase (for illustration purposes), unwrapped phase and its cosine: Results computed with the proposed method (first row) and by smoothing, with a thin plate regularization filter [32], the coarsely computed phase in Fig. 2

Fig. 4.
Fig. 4.

Phaser plots: (a) Ideal phase shifts (with steps equal to π/2), (b) real phase shifts, (c) phase shifts computed with the proposed method and (d) phase shift computed with the method in [18].

Fig. 5.
Fig. 5.

Real data experiment: (a) An original ESPI fringe pattern, (b) computed wrapped phase with a standard four steps (assuming phase steps equal to π/2), (c) refined phase computed with the proposed algorithm (rewrapped for illustration purposes) and (d) computed phase shifts.

Equations (18)

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

I kr = a kr + b kr cos ( f r + Δ k ) + η kr ,
I kr = b k cos ( f r + Δ k ) + η ̂ kr ;
f ̂ r = tan 1 ( I 4 r I 2 r I 1 r I 3 r ) .
min f k = 1 K r L H kr 2 ( f ; Δ ) ,
H kr ( f ; Δ ) = I kr cos ( f r ) cos ( Δ k ) + sin ( f r ) sin ( Δ k ) ;
cos ( δ IJ b ) = corr I J I . * J I J ( I . * I I 2 ) 1 2 ( J . * J J 2 ) 1 2 ,
δ IJ c = 2 sin 1 ( π 2 I J 2 b )
Δ k = δ k + α k ,
f r = ϕ r + ψ r .
I kr cos ( ψ r + δ k + ϕ r + α k ) .
E kr ϕ α I kr cos ( ψ r + δ k ) + ( ϕ r + α k ) sin ( ψ r + δ k ) 0 .
U ( ϕ , α , ω ) = k = 1 K r L [ ω r 2 E kr 2 ϕ α + μ ( 1 ω r ) 2 ] + γ [ k = 1 K α k 2 + r L ϕ r 2 ]
+ λ q , r , s L [ ψ q + ϕ q 2 ( ψ r + ϕ r ) + ψ s + ϕ s ] 2 ,
δ dc = arg min d D I 0 cos ( ψ ̂ + d ) 2 2 ,
U ( ϕ = 0 , ω , α ) = r [ ω r 2 k ( g kr cos ψ ̂ kr + α k sin ψ kr ) 2 + γα k 2 ] + Q ( ϕ = 0 , ω ) ,
α k = r ω r 2 sin ψ ̂ kr ( cos ψ ̂ kr g kr ) γ + r ω r 2 sin 2 ψ ̂ kr .
ω r = μ μ + k ( g kr cos ψ ̂ kr ) 2 .
U p ( ϕ ) = r L ( ψ r ϕ ) 2 + λ q r s L [ ϕ q 2 ϕ r + ϕ s ] 2 ,

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