Abstract

In many fringe pattern processing applications the local phase has to be obtained from a sinusoidal irradiance signal with unknown local frequency. This process is called asynchronous phase demodulation. Existing algorithms for asynchronous phase detection, or asynchronous algorithms, have been designed to yield no algebraic error in the recovered value of the phase for any signal frequency. However, each asynchronous algorithm has a characteristic frequency response curve. Existing asynchronous algorithms present a range of frequencies with low response, reaching zero for particular values of the signal frequency. For real noisy signals, low response implies a low signal-to-noise ratio in the recovered phase and therefore unreliable results. We present a new Fourier-based methodology for designing asynchronous algorithms with any user-defined frequency response curve and known limit of algebraic error. We show how asynchronous algorithms designed with this method can have better properties for real conditions of noise and signal frequency variation.

© 2006 Optical Society of America

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References

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    [CrossRef]
  2. D. Malacara, M. Servin, and Z. Malcacara, Interferogram Analysis for Optical Testing (Marcel Dekker, 1998).
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  6. K. G. Larkin, "Efficient nonlinear algorithm for envelope detection in white light interferometry," J. Opt. Soc. Am. A 13, 832-843 (1996).
    [CrossRef]
  7. J. A. Gomez-Pedrero, J. A. Quiroga, and M. Servin, "Asynchronous phase demodulation algorithm for temporal evaluation of fringe patterns with spatial carrier," J. Mod. Opt. 51, 97-109 (2004).
    [CrossRef]
  8. B. Ströbel, "Processing of interferometric phase maps as complex-valued phasor images," Appl. Opt. 35, 2192-2198 (1996).
    [CrossRef] [PubMed]
  9. M. Cherbuliez and P. Jacquot, "Phase computation through wavelet analysis: yesterday and nowadays," in Proceedings of Fringe'01: The Fourth International Workshop on Automatic Processing of Fringe Patterns, W.Ostes and W.Juptner, eds. (Elsevier, 2001), pp. 154-162.
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    [CrossRef] [PubMed]
  11. C. A. Sciammarella and T. Kim, "Determination of strains from fringe patterns using space-frequency representations," Opt. Eng. 42, 3182-3193 (2003).
    [CrossRef]
  12. Q. Kemao, "Windowed Fourier transform for fringe pattern analysis," Appl. Opt. 43, 2695-2702 (2004).
    [CrossRef] [PubMed]
  13. A. Asundi and W. Jun, "Strain contouring using Gabor filters: principle and algorithm," Opt. Eng. 41, 1400-1405 (2002).
    [CrossRef]
  14. J. W. Harris and H. Stocker, Handbook of Mathematics and Computational Science (Springer, 1998).
    [CrossRef]

2004 (3)

2003 (1)

C. A. Sciammarella and T. Kim, "Determination of strains from fringe patterns using space-frequency representations," Opt. Eng. 42, 3182-3193 (2003).
[CrossRef]

2002 (1)

A. Asundi and W. Jun, "Strain contouring using Gabor filters: principle and algorithm," Opt. Eng. 41, 1400-1405 (2002).
[CrossRef]

1996 (2)

1995 (1)

M. Servin and F. J. Cuevas, "A novel technique for spatial phase-shifting interferometry," J. Mod. Opt. 42, 1853-1862 (1995).
[CrossRef]

1990 (1)

1987 (1)

1966 (1)

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

Asundi, A.

A. Asundi and W. Jun, "Strain contouring using Gabor filters: principle and algorithm," Opt. Eng. 41, 1400-1405 (2002).
[CrossRef]

Carré, P.

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

Cherbuliez, M.

M. Cherbuliez and P. Jacquot, "Phase computation through wavelet analysis: yesterday and nowadays," in Proceedings of Fringe'01: The Fourth International Workshop on Automatic Processing of Fringe Patterns, W.Ostes and W.Juptner, eds. (Elsevier, 2001), pp. 154-162.

Cuevas, F. J.

M. Servin and F. J. Cuevas, "A novel technique for spatial phase-shifting interferometry," J. Mod. Opt. 42, 1853-1862 (1995).
[CrossRef]

Eiju, T.

Freischlad, K.

Gomez-Pedrero, J. A.

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

Hariharan, P.

Harris, J. W.

J. W. Harris and H. Stocker, Handbook of Mathematics and Computational Science (Springer, 1998).
[CrossRef]

Jacquot, P.

M. Cherbuliez and P. Jacquot, "Phase computation through wavelet analysis: yesterday and nowadays," in Proceedings of Fringe'01: The Fourth International Workshop on Automatic Processing of Fringe Patterns, W.Ostes and W.Juptner, eds. (Elsevier, 2001), pp. 154-162.

Jun, W.

A. Asundi and W. Jun, "Strain contouring using Gabor filters: principle and algorithm," Opt. Eng. 41, 1400-1405 (2002).
[CrossRef]

Kemao, Q.

Kim, T.

C. A. Sciammarella and T. Kim, "Determination of strains from fringe patterns using space-frequency representations," Opt. Eng. 42, 3182-3193 (2003).
[CrossRef]

Koliopoulos, C. L.

Larkin, K. G.

Malacara, D.

D. Malacara, M. Servin, and Z. Malcacara, Interferogram Analysis for Optical Testing (Marcel Dekker, 1998).

Malcacara, Z.

D. Malacara, M. Servin, and Z. Malcacara, Interferogram Analysis for Optical Testing (Marcel Dekker, 1998).

Oreb, B. F.

Quiroga, J. A.

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

Sciammarella, C. A.

C. A. Sciammarella and T. Kim, "Determination of strains from fringe patterns using space-frequency representations," Opt. Eng. 42, 3182-3193 (2003).
[CrossRef]

Servin, M.

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

M. Servin and F. J. Cuevas, "A novel technique for spatial phase-shifting interferometry," J. Mod. Opt. 42, 1853-1862 (1995).
[CrossRef]

D. Malacara, M. Servin, and Z. Malcacara, Interferogram Analysis for Optical Testing (Marcel Dekker, 1998).

Stocker, H.

J. W. Harris and H. Stocker, Handbook of Mathematics and Computational Science (Springer, 1998).
[CrossRef]

Ströbel, B.

Weng, J.

Zhong, J.

Opt. Eng. (1)

A. Asundi and W. Jun, "Strain contouring using Gabor filters: principle and algorithm," Opt. Eng. 41, 1400-1405 (2002).
[CrossRef]

Appl. Opt. (4)

J. Mod. Opt. (1)

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

J. Mod. Opt. (1)

M. Servin and F. J. Cuevas, "A novel technique for spatial phase-shifting interferometry," J. Mod. Opt. 42, 1853-1862 (1995).
[CrossRef]

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

Metrologia (1)

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

Opt. Eng. (1)

C. A. Sciammarella and T. Kim, "Determination of strains from fringe patterns using space-frequency representations," Opt. Eng. 42, 3182-3193 (2003).
[CrossRef]

Other (3)

J. W. Harris and H. Stocker, Handbook of Mathematics and Computational Science (Springer, 1998).
[CrossRef]

M. Cherbuliez and P. Jacquot, "Phase computation through wavelet analysis: yesterday and nowadays," in Proceedings of Fringe'01: The Fourth International Workshop on Automatic Processing of Fringe Patterns, W.Ostes and W.Juptner, eds. (Elsevier, 2001), pp. 154-162.

D. Malacara, M. Servin, and Z. Malcacara, Interferogram Analysis for Optical Testing (Marcel Dekker, 1998).

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Figures (9)

Fig. 1
Fig. 1

Response as a function of frequency ω in radians per sample for the Carré[3] algorithm (solid curve) and the Servin and Cuevas[5] and the Gomez-Pedrero et al.[7] algorithms (dotted curve).

Fig. 2
Fig. 2

Functions that define the optimum asynchronous algorithm for all values of σ ∈ (0, 0.5): (a) symmetric MC (σ) and (b) antisymmetric MS (σ).

Fig. 3
Fig. 3

Symmetric function MC (σ) that defines the optimum background-insensitive algorithm for all values of σ ∈ (ε, 0.5). The antisymmetric function MS (σ) is the same as depicted in Fig. 2(b).

Fig. 4
Fig. 4

Functions (a) MC (σ) and (b) MS (σ) that define the algorithm used in the experiment. These are the functions obtained from the optimum background-insensitive algorithm when μ = 0.025 and for different values of N.

Fig. 5
Fig. 5

(a) Response of the asynchronous algorithm defined by the functions shown in Fig. 4 for N = 3, 7, and 15. (b) Maximum algebraic error for the algorithm with N = 7.

Fig. 6
Fig. 6

Error in the recovered phase using (a) the new design algorithm and (b) the Gomez-Pedrero et al.[7] algorithm.

Fig. 7
Fig. 7

(a) Comparison between the response obtained using the new design algorithm and the Gomez-Pedrero et al. algorithm. (b) Comparison between the experimental and the theoretical response of the algorithm.

Fig. 8
Fig. 8

(a) Experimental fringe pattern. (b) Profile of Fig. 7 along row 100, (c) phase recovered with the Gomez-Pedrero et al. algorithm.

Fig. 9
Fig. 9

Phase recovered from the image in Fig. 8(a) with the proposed algorithm for (a) N = 3, (b) N = 7, and (c) N = 15.

Equations (35)

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I ( x ) = b + m cos ( ω x + ϕ ) ,
W ( ϕ ) = tan 1 ( S C ) .
m ϕ ( ω ) = m m D ( ω ) = S 2 ( ω ) + C 2 ( ω ) ,
I ( x ) = m cos ( ω x + ϕ ) ,
I s ( x ) = n = δ ( x n p s ) I ( x ) = m n = δ ( x n p s ) cos ( 2 π n σ + ϕ ) ,
C ( ϕ ) = d x n = c ( n ) δ ( x n p s ) I ( x ) = n = c ( n ) I ( n p s ) ,
S ( ϕ ) = d x n = s ( n ) δ ( x n p s ) I ( x ) = n = s ( n ) I ( n p s ) .
C ( ϕ ) = m n = c ( n ) cos ( 2 π n σ + ϕ ) ,
S ( ϕ ) = m n = s ( n ) cos ( 2 π n σ + ϕ ) .
n = c ( n ) cos ( 2 π n σ + ϕ ) = cos ( ϕ ) ,
n = s ( n ) cos ( 2 π n σ + ϕ ) = sin ( ϕ ) .
cos ( ϕ ) n = c ( n ) cos ( 2 π n σ ) sin ( ϕ ) n = c ( n ) sin ( 2 π n σ )
= cos ( ϕ ) ,
cos ( ϕ ) n = s ( n ) cos ( 2 π n σ ) sin ( ϕ ) n = s ( n ) sin ( 2 π n σ )
= sin ( ϕ ) .
n = c ( n ) sin ( 2 π n σ ) = 0 ,
n = s ( n ) cos ( 2 π n σ ) = 0 .
c ( n ) = c ( n ) ,
s ( n ) = s ( n ) .
C ( ϕ ) = cos ( ϕ ) m M C ( σ ) ,
S ( ϕ ) = sin ( ϕ ) m M S ( σ ) ,
M C ( σ ) = n = c ( n ) cos ( 2 π n σ ) ,
M S ( σ ) = n = s ( n ) sin ( 2 π n σ ) .
M C ( σ ) = n = c ( n ) exp ( i 2 π n σ ) ,
M S ( σ ) = i n = s ( n ) exp ( i 2 π n σ ) .
ϕ e = arctan [ S ( ϕ ) C ( ϕ ) ] = arctan [ M S ( σ ) sin ( ϕ ) M C ( σ ) cos ( ϕ ) ] ,
m D ( σ ) = M S ( σ ) 2 + M C ( σ ) 2 .
ε ( σ ) = | ϕ e ϕ | 1 2 | M S ( σ ) M C ( σ ) 1 | .
tan ϕ = n = s ( n ) I ( n ) n = c ( n ) I ( n ) ,
tan ϕ = n = 0 x 2 π ( 2 n + 1 ) [ I ( 2 n + 1 ) I ( 2 n 1 ) ] I ( 0 ) .
n = c ( n ) = 0 ,
n = s ( n ) = 0.
tan ϕ =
n = 0 2 π ( 2 n + 1 ) [ I ( 2 n + 1 ) I ( 2 n 1 ) ] ( 1 2 μ ) I ( 0 ) n = 1 sin ( 2 n π μ ) n π [ I ( n ) + I ( n ) ] .
U N [ s ( n ) , c ( n ) ] = 0 0.5 { [ 1 m D ( σ ) ] 2 + β ε 2 ( σ ) } d σ ,

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