Abstract

We present a spatial adaptive asynchronous algorithm for fringe pattern demodulation. The proposed algorithm is based on the standard five-step asynchronous method with the one modification that we select the best sample spacing for each point of the fringe pattern. As we show, the frequency response of any asynchronous method depends on the sample spacing. This interesting behavior is used to select the best sample spacing as the one that gives the biggest response for each location. The overall result is a spatial demodulation algorithm with an improved frequency response compared to the existing ones. We show the feasibility of the proposed method with theoretical analysis as well as experimental results.

© 2008 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  6. P. Carré, “Installation et utilisation du comparateur photoelectrique et interferentiel du Buerau International des Poids et Mesures,” Metrologia 2, 13-23 (1966).
    [CrossRef]
  7. D. Crespo, J. A. Quiroga, and J. A. Gómez-Pedrero, “Design of asynchronous phase detection algorithms optimized for wide frequency response,” Appl. Opt. 45, 4037-4045 (2006).
    [CrossRef] [PubMed]
  8. Q. Kemao Quian, “Windowed Fourier transform for fringe pattern analysis,” Appl. Opt. 43, 2695-2702 (2004).
    [CrossRef]
  9. E. Berger, W. von der Linden, V. Dose, M. W. Ruprecht, and A. W. Koch “Approach for the evaluation of speckle deformation measurements by application of the wavelet transformation,” Appl. Opt. 36, 7455-7460 (1997).
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  10. J. Zhong and J. Weng, “Spatial carrier-fringe pattern analysis by means of wavelet transform: wavelet transform profilometry,” Appl. Opt. 43, 4993-4998 (2004).
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  11. 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]

2006 (1)

2005 (1)

2004 (3)

1999 (1)

B. V. Dorrío and J. L. Fernández, “Phase-evaluation methods in whole-field optical measurement techniques,” Meas. Sci. Technol. 10, R33-R55 (1999).
[CrossRef]

1997 (1)

1996 (1)

1966 (1)

P. Carré, “Installation et utilisation du comparateur photoelectrique et interferentiel du Buerau International des Poids et Mesures,” Metrologia 2, 13-23 (1966).
[CrossRef]

Berger, E.

Carré, P.

P. Carré, “Installation et utilisation du comparateur photoelectrique et interferentiel du Buerau International des Poids et Mesures,” Metrologia 2, 13-23 (1966).
[CrossRef]

Crespo, D.

De la Rosa, I.

Dorrío, B. V.

B. V. Dorrío and J. L. Fernández, “Phase-evaluation methods in whole-field optical measurement techniques,” Meas. Sci. Technol. 10, R33-R55 (1999).
[CrossRef]

Dose, V.

Fernández, J. L.

B. V. Dorrío and J. L. Fernández, “Phase-evaluation methods in whole-field optical measurement techniques,” Meas. Sci. Technol. 10, R33-R55 (1999).
[CrossRef]

Gómez-Pedrero, J. A.

D. Crespo, J. A. Quiroga, and J. A. Gómez-Pedrero, “Design of asynchronous phase detection algorithms optimized for wide frequency response,” Appl. Opt. 45, 4037-4045 (2006).
[CrossRef] [PubMed]

J. A. Gómez-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]

Koch, A. W.

Kreiss, T.

T. Kreiss, Holographic Interferometry (Akademie Verlag, 1996).

Larkin, K. G.

Malacara, D.

D. Malacara, M. Servín, and Z. Malacara, Interferogram Analysis for Optical Testing, 2nd ed. (CRC Press, 2005).
[CrossRef]

Malacara, Z.

D. Malacara, M. Servín, and Z. Malacara, Interferogram Analysis for Optical Testing, 2nd ed. (CRC Press, 2005).
[CrossRef]

Miramontes, G.

Quian, Q. Kemao

Quiroga, J. A.

Ruprecht, M. W.

Servín, M.

J. A. Gómez-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]

D. Malacara, M. Servín, and Z. Malacara, Interferogram Analysis for Optical Testing, 2nd ed. (CRC Press, 2005).
[CrossRef]

Villa, J.

von der Linden, W.

Weng, J.

Zhong, J.

Appl. Opt. (4)

J. Mod. Opt. (1)

J. A. Gómez-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 (2)

Meas. Sci. Technol. (1)

B. V. Dorrío and J. L. Fernández, “Phase-evaluation methods in whole-field optical measurement techniques,” Meas. Sci. Technol. 10, R33-R55 (1999).
[CrossRef]

Metrologia (1)

P. Carré, “Installation et utilisation du comparateur photoelectrique et interferentiel du Buerau International des Poids et Mesures,” Metrologia 2, 13-23 (1966).
[CrossRef]

Other (2)

D. Malacara, M. Servín, and Z. Malacara, Interferogram Analysis for Optical Testing, 2nd ed. (CRC Press, 2005).
[CrossRef]

T. Kreiss, Holographic Interferometry (Akademie Verlag, 1996).

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Figures (9)

Fig. 1
Fig. 1

Theoretical response function for the adaptive five-step asynchronous algorithm as function of the local frequency (continuous line) compared with the response of the standard five-step asynchronous algorithm decimated by 0 ( Δ = 1 ), 1 ( Δ = 2 ), and 2 ( Δ = 3 ) pixels.

Fig. 2
Fig. 2

Flow chart of the proposed algorithm.

Fig. 3
Fig. 3

Response function for the adaptive five-step asynchronous algorithm measured from the noisy fringe pattern described in the text compared to that of the standard five-step asynchronous algorithm. The adaptive algorithm presents higher response for the low and high frequencies, which will lead to a more reliable phase demodulation for these frequencies.

Fig. 4
Fig. 4

Comparison of the phase recovery error for the adaptive five-step asynchronous algorithm and the standard five-step asynchronous algorithm. As it could be expected from the response function of Fig. 3, the standard five-step algorithm behaves worse than the adaptive algorithm for low and high frequencies.

Fig. 5
Fig. 5

Shadow-moiré fringe pattern of a smooth surface with a central indentation.

Fig. 6
Fig. 6

(a) Wrapped phase of the fringe pattern of Fig. 5 recovered with the proposed method, (b) wrapped phase recovered with the standard five-step algorithm, (c) wrapped phase corrected through the fringe orientation, (d) unwrapped phase of the fringe pattern of Fig. 5, and (e) horizontal profile of the unwrapped phase along the 210th row of Fig. 6d.

Fig. 7
Fig. 7

(a) Fringe pattern projected over a human back, (b) wrapped phase recovered with the adaptive algorithm, and (c) wrapped phase recovered with the standard five-step algorithm.

Fig. 8
Fig. 8

Histogram of the fit error corresponding to the region labeled A in Fig. 7a.

Fig. 9
Fig. 9

(a) Experimental fringe pattern obtained in a photoelastic experiment, (b) distribution of the decimation factor over the fringe pattern, (c) wrapped phase recovered with the adaptive algorithm, (d) corrected wrapped phase, (e) unwrapped phase, and (f) horizontal profile of the continuous phase along points A and B of 9(e).

Tables (1)

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Table 1 Standard Deviation of the Fit Error Obtained When Adjusting the Unwrapped Phase to a 3rd Degree Polynomial for the Adaptive and Standard Algorithms

Equations (14)

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I j = a + b cos ( φ + α j ) , j = 1 , 2 , ... , N ,
W [ φ ] = arctan [ 3 ( I 2 I 3 ) ( I 1 I 4 ) ] [ ( I 1 I 4 ) + ( I 2 I 3 ) ] ( I 2 + I 3 ) ( I 1 + I 4 ) ,
tan 2 ω = 3 ( I 2 I 3 ) ( I 1 I 4 ) ( I 1 I 4 ) + ( I 2 I 3 ) .
m ( ω ) = u 2 + v 2 .
m ( ω ) = | 2 cos 3 ω cos ω | .
I j = a + b cos [ φ + ω ( j 3 ) ] , j = 1. .. 5 ,
tan φ = u v = sign ( I 2 I 4 ) 4 ( I 2 I 4 ) 2 ( I 1 I 5 ) 2 2 I 3 I 1 I 5 ,
u = 4 b sin φ sin 2 ω , v = 4 b cos φ sin 2 ω .
m ( ω ) = 4 b sin 2 ( ω ) .
I 1 = a + b cos ( φ 2 ω Δ ) , I 2 = a + b cos ( φ ω Δ ) , I 3 = a + b cos ( φ ) , I 4 = a + b cos ( φ + ω Δ ) , I 5 = a + b cos ( φ + 2 ω Δ ) .
m ( Δ , ω ) = 4 b sin 2 ( ω Δ ) .
δ = { Δ | m ( Δ ) = max [ m ( 1 ) , m ( 2 ) , m ( N ) ] } .
I ( x , y ) = 100 + 60 cos [ φ ( x ) + r ( x , y ) ] ,
W [ φ ] ¯ = sign [ cos ( θ ) ] W [ φ ] ,

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