Abstract

It is well known that spatial phase shifting interferometry (SPSI) may be used to demodulate two- dimensional (2D) spatial-carrier interferograms. In these cases the application of SPSI is straightforward because the modulating phase is a monotonic increasing function of space. However, this is not true when we apply SPSI to demodulate a single-image interferogram containing closed fringes. This is because using these algorithms, one would obtain a wrongly demodulated monotonic phase all over the 2D space. We present a technique to overcome this drawback and to allow any SPSI algorithm to be used as a single-image fringe pattern demodulator containing closed fringes. We make use of the 2D spatial orientation direction of the fringes to steer (orient) the one-dimensional SPSI algorithm in order to correctly demodulate the nonmonotonic 2D phase all over the interferogram.

© 2009 Optical Society of America

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References

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2008

2007

X. Yang, Q. Yu, and S. Fu, “An algorithm for estimating both fringe orientation and fringe density,” Opt. Commun. 274, 286-292 (2007).
[CrossRef]

Q. Kemao, “Two-dimensional windowed Fourier transform for fringe pattern analysis: principles, applications and implementations,” Opt. Lasers Eng. 45, 304-317 (2007).
[CrossRef]

2005

2004

J. Zhong and J. Weng “Spatial carrier-fringe pattern analysis by means of wavelet transform: wavelet transform profilometry,” Appl. Opt. 43, 4993-4998 (2004).
[CrossRef] [PubMed]

J. A. Gomez Pedrero, J. A. Quiroga, and M. Servín, “Asynchronous phase demodulation algorithm for temporal evaluation of fringe patterns with spatial carrier,” J. Mod. Opt. 51, 97-109 (2004).
[CrossRef]

2003

2001

1999

1998

1996

1995

M. Servin, R. Rodriguez-Vera, and D. Malacara, “Noisy fringe pattern demodulation by an iterative phase locked loop,” Opt. Lasers Eng. 23, 355-365 (1995).
[CrossRef]

1987

1983

1982

Bone, D. J.

Botello, S.

Cuevas, F.

De la Rosa, I.

Eiju, T.

Fu, S.

X. Yang, Q. Yu, and S. Fu, “An algorithm for estimating both fringe orientation and fringe density,” Opt. Commun. 274, 286-292 (2007).
[CrossRef]

Gomez Pedrero, J. A.

J. A. Gomez Pedrero, J. A. Quiroga, and M. Servín, “Asynchronous phase demodulation algorithm for temporal evaluation of fringe patterns with spatial carrier,” J. Mod. Opt. 51, 97-109 (2004).
[CrossRef]

Gómez-Pedrero, J. A.

Hariharan, P.

Ishii, Y.

R. Onodera, Y. Yamamoto, and Y. Ishii, “Signal processing of interferogram using a two-dimensional discrete Hilbert transform,” in Proceedings of Fringe 2005, Fifth International Workshop on Automatic Processing of Fringe Patterns, W. Osten, ed. (European Space Agency, 2005), pp. 82-89.

Kemao, Q.

Q. Kemao, “Two-dimensional windowed Fourier transform for fringe pattern analysis: principles, applications and implementations,” Opt. Lasers Eng. 45, 304-317 (2007).
[CrossRef]

Larkin, K.

Malacara, D.

M. Servin, R. Rodriguez-Vera, and D. Malacara, “Noisy fringe pattern demodulation by an iterative phase locked loop,” Opt. Lasers Eng. 23, 355-365 (1995).
[CrossRef]

Marroquin, J. L.

Marroquín, J. L.

Marroquin,, J. L.

Miramontes, G.

Morgan, C. J.

Mutoh, K.

Oldfield, M. A.

Onodera, R.

R. Onodera, Y. Yamamoto, and Y. Ishii, “Signal processing of interferogram using a two-dimensional discrete Hilbert transform,” in Proceedings of Fringe 2005, Fifth International Workshop on Automatic Processing of Fringe Patterns, W. Osten, ed. (European Space Agency, 2005), pp. 82-89.

Oreb, B. F.

Quiroga, J. A.

Rivera, M.

Rodriguez-Vera, R.

Servin, M.

Servín, M.

J. A. Gomez Pedrero, J. A. Quiroga, and M. Servín, “Asynchronous phase demodulation algorithm for temporal evaluation of fringe patterns with spatial carrier,” J. Mod. Opt. 51, 97-109 (2004).
[CrossRef]

Strobel, B.

Takeda, M.

Villa, J.

Weng, J.

Yamamoto, Y.

R. Onodera, Y. Yamamoto, and Y. Ishii, “Signal processing of interferogram using a two-dimensional discrete Hilbert transform,” in Proceedings of Fringe 2005, Fifth International Workshop on Automatic Processing of Fringe Patterns, W. Osten, ed. (European Space Agency, 2005), pp. 82-89.

Yang, X.

X. Yang, Q. Yu, and S. Fu, “An algorithm for estimating both fringe orientation and fringe density,” Opt. Commun. 274, 286-292 (2007).
[CrossRef]

Yu, Q.

X. Yang, Q. Yu, and S. Fu, “An algorithm for estimating both fringe orientation and fringe density,” Opt. Commun. 274, 286-292 (2007).
[CrossRef]

Zhong, J.

Appl. Opt.

J. Mod. Opt.

J. A. Gomez Pedrero, J. A. Quiroga, and M. Servín, “Asynchronous phase demodulation algorithm for temporal evaluation of fringe patterns with spatial carrier,” J. Mod. Opt. 51, 97-109 (2004).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Commun.

X. Yang, Q. Yu, and S. Fu, “An algorithm for estimating both fringe orientation and fringe density,” Opt. Commun. 274, 286-292 (2007).
[CrossRef]

Opt. Lasers Eng.

Q. Kemao, “Two-dimensional windowed Fourier transform for fringe pattern analysis: principles, applications and implementations,” Opt. Lasers Eng. 45, 304-317 (2007).
[CrossRef]

M. Servin, R. Rodriguez-Vera, and D. Malacara, “Noisy fringe pattern demodulation by an iterative phase locked loop,” Opt. Lasers Eng. 23, 355-365 (1995).
[CrossRef]

Opt. Lett.

Other

R. Onodera, Y. Yamamoto, and Y. Ishii, “Signal processing of interferogram using a two-dimensional discrete Hilbert transform,” in Proceedings of Fringe 2005, Fifth International Workshop on Automatic Processing of Fringe Patterns, W. Osten, ed. (European Space Agency, 2005), pp. 82-89.

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Figures (7)

Fig. 1
Fig. 1

Computer generated closed-fringe interferogram.

Fig. 2
Fig. 2

Phasor terms obtained from the processing of the interferogram of Fig. 1 by the five-step SPSI asynchronous method applied in the horizontal direction: (a) sine term before the sign correction, (b) cosine term. Note in both images the fringe amplitude toward the central vertical line predicted by the sin 2 ω x term.

Fig. 3
Fig. 3

Demodulated phase from the interferogram shown in Fig. 1 using the steered five-step SPSI asynchronous technique: (a) using the orientation, note the incorrect demodulation due to the nonmonotonic phase, and (b) using the direction angle.

Fig. 4
Fig. 4

Demodulated phase from the interferogram of Fig. 1 using two steered SPSI methods: (a) the five-step Hariharan and (b) the standard three-step SPSI method.

Fig. 5
Fig. 5

(a) Experimental closed-fringe Fizeau interferogram and (b) demodulated phase using the steered five-step asynchronous SPSI method.

Fig. 6
Fig. 6

(a) Experimental shadow moiré topography interferogram, (b) demodulated phase using the steered five-step asynchronous SPSI method, (c) phasor amplitude used as quality map for unwrapping the phase depicted in Fig. 6b, and (d) unwrapped phase using a quality-guided phase unwrapping method.

Fig. 7
Fig. 7

(a) Experimental out-of-plane ESPI interferogram, (b) demodulated phase using a noise-rejecting SPSI method [17], (c)  3 × 3 averaging sine-cosine low-pass filtered phase map, (d) phase map obtained using four temporal phase-shifted images, (e) cosine of the phase depicted in Fig. 7c; the comparison with Fig. 7a shows the correctness of the demodulated phase.

Equations (38)

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s ( x ) = b ( x ) + m ( x ) cos ( ϕ 0 + ω 0 x ) ,
G ( 0 ) = 0 , G ( ω 0 ) = 0 , G ( ω 0 ) 0.
A ( x ) = s ( x ) * g ( x ) = m ( x ) G ( ω 0 ) e i ( ϕ 0 + ω 0 x ) ,
g ( x ) = [ 2 δ ( x ) δ ( x 1 ) δ ( x + 1 ) ] ( 1 cos ω 0 ) + i [ δ ( x 1 ) δ ( x + 1 ) ] sin ω 0 ,
G ( ω ) = 4 sin ( ω ω 0 2 ) sin ω .
s ( x ) = b ( x ) + m ( x ) cos [ ϕ ( x ) + ω 0 x ] ,
G ( 0 ) = 0 , G ( ω ω 0 ) = 0 , G ( ω ω 0 ) 0.
A ( x ) = s ( x ) * g ( ω ) = m ( x ) G ( ω ) e i ( ϕ ( x ) + ω 0 x ) for     ω ω 0 .
g ( x ) = [ 2 δ ( x ) δ ( x 2 ) δ ( x + 2 ) ] + 2 i [ δ ( x 1 ) δ ( x + 1 ) ] ,
G ( ω ) = 2 sin ω ( sin ω 1 ) .
s ( x ) = b ( x ) + m ( x ) cos [ ϕ ( x ) ] .
G ( ω > 0 ) = 0 , G ( ω 0 ) 0.
A ( x ) = m ( x ) G ( ω ) e i ϕ ( x ) .
tan ϕ = sign ( s 2 s 4 ) 4 ( s 2 s 4 ) 2 ( s 1 s 5 ) 2 2 s 3 ( s 1 + s 5 ) .
G ( ω ) = 4 sin 2 ( ω ) , ω > 0.
g ( s ) = f ( s ) + i h 1 ( s ) ,
G ( ω ) = F ( ω ) + i H 1 ( ω ) .
h 1 ( s ) = sign ( s 2 s 4 ) 4 ( s 2 s 4 ) 2 ( s 1 s 5 ) 2 = 4 m sin 2 ω sin ϕ , f ( s ) = 2 s 3 ( s 1 + s 5 ) = 4 m sin 2 ω cos ϕ ,
h N ( s ) = s | ϕ | = m sin ϕ ω | ω | = m sin ϕ · n ,
h 1 ( s ) = d s / d x d ϕ / d x = m sin ϕ ω | ω | = m sin ϕ · sign ( ω ) .
q N ( s ) = h N ( s ) · n = m sin ϕ ,
q 1 ( s ) = h 1 ( s ) · sign ( ω ) = m sin ϕ .
A ( x ) = m ( x ) G ( ω ) [ cos ϕ + i sign ( ω ) sin ϕ ] .
s ( x , y ) = b ( x , y ) + m ( x , y ) cos [ ϕ ( x , y ) ]
h 2 x [ s ( x , y ) ] = d s / d x | ϕ | = m sin ϕ ω x | ω | = m sin ϕ cos β , h 2 y [ s ( x , y ) ] = d s / d y | ϕ | = m sin ϕ ω y | ω | = m sin ϕ sin β ,
A x ( x , y ) = m ( x , y ) G ( ω x ) [ cos ϕ + i cos β sin ϕ ] .
A y ( x , y ) = m ( x , y ) G ( ω y ) [ cos ϕ + i sin β sin ϕ ] ,
q 2 ( s ) = [ h 2 x ( s ) , h 2 y ( s ) ] · n = h 2 x ( s ) cos β + h 2 y ( s ) sin β = m [ G ( ω x ) + G ( ω y ) ] sin ϕ ,
tan ϕ = h 2 x ( s ) cos β + h 2 y ( s ) sin β f x ( s ) + f y ( s ) ,
s ( k , l ) = b + m cos [ ϕ + ( k 2 ) ω x + ( l 2 ) ω y ] , k , l = 1 , , 5 ,
h 2 x ( s ) = sign ( s 2 , l s 4 , l ) 4 ( s 2 , l s 4 , l ) 2 ( s 1 , l s 5 , l ) 2 = 4 m sin 2 ω x cos β sin ϕ , f x ( s ) = 2 s 3 , l ( s 1 , l + s 5 , l ) = 4 m sin 2 ω x cos ϕ ,
h 2 y ( s ) = sign ( s k , 2 s k , 4 ) 4 ( s k , 2 s k , 4 ) 2 ( s k , 1 s k , 5 ) 2 = 4 m sin 2 ω x sin β sin ϕ , f y ( s ) = 2 s k , 3 ( s k , 1 + s k , 5 ) = 4 m sin 2 ω y cos ϕ .
h 2 x ( s ) = 2 ( s 2 , l s 4 , l ) = 4 m sin 2 ω x cos β sin ϕ , f x ( s ) = 2 s 3 , l ( s 1 , l + s 5 , l ) = 4 m sin 2 ω x cos ϕ ,
h 2 y ( s ) = 2 ( s k , 2 s k , 4 ) = 4 m sin 2 ω y sin β sin ϕ , f y ( s ) = 2 s k , 3 ( s k , 1 + s k , 5 ) = 4 m sin 2 ω y cos ϕ .
θ = β ± K π ,
cos θ = ( 1 ) K cos β , sin θ = ( 1 ) K sin β .
tan ϕ θ = ( 1 ) K ( x , y ) tan ϕ ,
s = 128 + 128 cos [ 0.5 π ( x 2 + y 2 ) / 256 ] , x , y = 128 , , 128.

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