Abstract

The paper introduces an adaptive-window-spectrogram–based method to directly estimate the phase derivative from a single fringe pattern. The proposed method relies on estimating the phase derivative using spectrogram peak detection for a set of different window lengths. Then the optimal window length is selected from the set by resolving the estimator’s bias variance trade-off using the intersection of confidence intervals rule. Finally, the phase derivative estimate corresponding to the optimum window is selected. The method’s applicability to phase derivative estimation is demonstrated using simulation and experimental results.

© 2009 Optical Society of America

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  1. Q. Kemao, S. H. Soon, and A. Asundi, “Instantaneous frequency and its application to strain extraction in moiré interferometry,” Appl. Opt. 42, 6504-6513 (2003).
    [CrossRef] [PubMed]
  2. K. Qian, S. H. Soon, and A. Asundi, “Phase-shifting windowed Fourier ridges for determination of phase derivatives,” Opt. Lett. 28, 1657-1659 (2003).
    [CrossRef] [PubMed]
  3. C. J. Tay, C. Quan, W. Sun, and X. Y. He, “Demodulation of a single interferogram based on continuous wavelet transform and phase derivative,” Opt. Commun. 280, 327-336 (2007).
    [CrossRef]
  4. C. A. Sciammarella and T. Kim, “Frequency modulation interpretation of fringes and computation of strains,” Exp. Mech. 45, 393-403 (2005).
    [CrossRef]
  5. U. Schnars and W. P. O. Juptner, “Digital recording and reconstruction of holograms in hologram interferometry and shearography,” Appl. Opt. 33, 4373-4377 (1994).
    [CrossRef] [PubMed]
  6. B. Boashash, “Estimating and interpreting the instantaneous frequency of a signal--part 1: fundamentals,” Proc. IEEE 80, 519-538 (1992).
  7. L. Cohen, Time Frequency Analysis (Prentice Hall, 1995).
  8. L. Stankovic and V. Katkovnik, “Algorithm for the instantaneous frequency estimation using time-frequency distributions with adaptive window width,” IEEE Signal Process. Lett. 5, 224-227 (1998).
    [CrossRef]
  9. V. Katkovnik and L. J. Stankovic, “Periodogram with varying and data-driven window length,” Signal Process. 67, 345-358 (1998).
    [CrossRef]
  10. V. Katkovnik and L. Stankovic, “Instantaneous frequency estimation using the Wigner distribution with varying and data-driven window length,” IEEE Trans. Signal Process. 45, 2147 (1997).
  11. D. R. Pauluzzi and N. C. Beaulieu, “A comparison of snr estimation techniques for the awgn channel,” IEEE Trans. Commun. 48, 1601 (2000).
    [CrossRef]
  12. R. Matzner and F. Englberger, “Snr estimation algorithm using fourth-order moments,” Proceedings of the IEEE International Symposium on Information Theory (IEEE, 1994).
  13. S. C. Sekhar and T. V. Sreenivas, “Signal-to-noise ratio estimation using higher-order moments,” Signal Process. 86, 716-732 (2006).
    [CrossRef]
  14. J. A. Quiroga, J. Antonio Gomez-Pedrero, and A. Garcia-Botella, “Algorithm for fringe pattern normalization,” Opt. Commun. 197, 43-51 (2001).
    [CrossRef]
  15. S. Lawrence Marple Jr., “Computing the discrete-time analytic signal via fft,” IEEE Trans. Signal Process. 47, 2600-2603 (1999).
    [CrossRef]

2007

C. J. Tay, C. Quan, W. Sun, and X. Y. He, “Demodulation of a single interferogram based on continuous wavelet transform and phase derivative,” Opt. Commun. 280, 327-336 (2007).
[CrossRef]

2006

S. C. Sekhar and T. V. Sreenivas, “Signal-to-noise ratio estimation using higher-order moments,” Signal Process. 86, 716-732 (2006).
[CrossRef]

2005

C. A. Sciammarella and T. Kim, “Frequency modulation interpretation of fringes and computation of strains,” Exp. Mech. 45, 393-403 (2005).
[CrossRef]

2003

2001

J. A. Quiroga, J. Antonio Gomez-Pedrero, and A. Garcia-Botella, “Algorithm for fringe pattern normalization,” Opt. Commun. 197, 43-51 (2001).
[CrossRef]

2000

D. R. Pauluzzi and N. C. Beaulieu, “A comparison of snr estimation techniques for the awgn channel,” IEEE Trans. Commun. 48, 1601 (2000).
[CrossRef]

1999

S. Lawrence Marple Jr., “Computing the discrete-time analytic signal via fft,” IEEE Trans. Signal Process. 47, 2600-2603 (1999).
[CrossRef]

1998

L. Stankovic and V. Katkovnik, “Algorithm for the instantaneous frequency estimation using time-frequency distributions with adaptive window width,” IEEE Signal Process. Lett. 5, 224-227 (1998).
[CrossRef]

V. Katkovnik and L. J. Stankovic, “Periodogram with varying and data-driven window length,” Signal Process. 67, 345-358 (1998).
[CrossRef]

1997

V. Katkovnik and L. Stankovic, “Instantaneous frequency estimation using the Wigner distribution with varying and data-driven window length,” IEEE Trans. Signal Process. 45, 2147 (1997).

1994

1992

B. Boashash, “Estimating and interpreting the instantaneous frequency of a signal--part 1: fundamentals,” Proc. IEEE 80, 519-538 (1992).

Asundi, A.

Beaulieu, N. C.

D. R. Pauluzzi and N. C. Beaulieu, “A comparison of snr estimation techniques for the awgn channel,” IEEE Trans. Commun. 48, 1601 (2000).
[CrossRef]

Boashash, B.

B. Boashash, “Estimating and interpreting the instantaneous frequency of a signal--part 1: fundamentals,” Proc. IEEE 80, 519-538 (1992).

Cohen, L.

L. Cohen, Time Frequency Analysis (Prentice Hall, 1995).

Englberger, F.

R. Matzner and F. Englberger, “Snr estimation algorithm using fourth-order moments,” Proceedings of the IEEE International Symposium on Information Theory (IEEE, 1994).

Garcia-Botella, A.

J. A. Quiroga, J. Antonio Gomez-Pedrero, and A. Garcia-Botella, “Algorithm for fringe pattern normalization,” Opt. Commun. 197, 43-51 (2001).
[CrossRef]

Gomez-Pedrero, J. Antonio

J. A. Quiroga, J. Antonio Gomez-Pedrero, and A. Garcia-Botella, “Algorithm for fringe pattern normalization,” Opt. Commun. 197, 43-51 (2001).
[CrossRef]

He, X. Y.

C. J. Tay, C. Quan, W. Sun, and X. Y. He, “Demodulation of a single interferogram based on continuous wavelet transform and phase derivative,” Opt. Commun. 280, 327-336 (2007).
[CrossRef]

Juptner, W. P. O.

Katkovnik, V.

L. Stankovic and V. Katkovnik, “Algorithm for the instantaneous frequency estimation using time-frequency distributions with adaptive window width,” IEEE Signal Process. Lett. 5, 224-227 (1998).
[CrossRef]

V. Katkovnik and L. J. Stankovic, “Periodogram with varying and data-driven window length,” Signal Process. 67, 345-358 (1998).
[CrossRef]

V. Katkovnik and L. Stankovic, “Instantaneous frequency estimation using the Wigner distribution with varying and data-driven window length,” IEEE Trans. Signal Process. 45, 2147 (1997).

Kemao, Q.

Kim, T.

C. A. Sciammarella and T. Kim, “Frequency modulation interpretation of fringes and computation of strains,” Exp. Mech. 45, 393-403 (2005).
[CrossRef]

Lawrence Marple, S.

S. Lawrence Marple Jr., “Computing the discrete-time analytic signal via fft,” IEEE Trans. Signal Process. 47, 2600-2603 (1999).
[CrossRef]

Matzner, R.

R. Matzner and F. Englberger, “Snr estimation algorithm using fourth-order moments,” Proceedings of the IEEE International Symposium on Information Theory (IEEE, 1994).

Pauluzzi, D. R.

D. R. Pauluzzi and N. C. Beaulieu, “A comparison of snr estimation techniques for the awgn channel,” IEEE Trans. Commun. 48, 1601 (2000).
[CrossRef]

Qian, K.

Quan, C.

C. J. Tay, C. Quan, W. Sun, and X. Y. He, “Demodulation of a single interferogram based on continuous wavelet transform and phase derivative,” Opt. Commun. 280, 327-336 (2007).
[CrossRef]

Quiroga, J. A.

J. A. Quiroga, J. Antonio Gomez-Pedrero, and A. Garcia-Botella, “Algorithm for fringe pattern normalization,” Opt. Commun. 197, 43-51 (2001).
[CrossRef]

Schnars, U.

Sciammarella, C. A.

C. A. Sciammarella and T. Kim, “Frequency modulation interpretation of fringes and computation of strains,” Exp. Mech. 45, 393-403 (2005).
[CrossRef]

Sekhar, S. C.

S. C. Sekhar and T. V. Sreenivas, “Signal-to-noise ratio estimation using higher-order moments,” Signal Process. 86, 716-732 (2006).
[CrossRef]

Soon, S. H.

Sreenivas, T. V.

S. C. Sekhar and T. V. Sreenivas, “Signal-to-noise ratio estimation using higher-order moments,” Signal Process. 86, 716-732 (2006).
[CrossRef]

Stankovic, L.

L. Stankovic and V. Katkovnik, “Algorithm for the instantaneous frequency estimation using time-frequency distributions with adaptive window width,” IEEE Signal Process. Lett. 5, 224-227 (1998).
[CrossRef]

V. Katkovnik and L. Stankovic, “Instantaneous frequency estimation using the Wigner distribution with varying and data-driven window length,” IEEE Trans. Signal Process. 45, 2147 (1997).

Stankovic, L. J.

V. Katkovnik and L. J. Stankovic, “Periodogram with varying and data-driven window length,” Signal Process. 67, 345-358 (1998).
[CrossRef]

Sun, W.

C. J. Tay, C. Quan, W. Sun, and X. Y. He, “Demodulation of a single interferogram based on continuous wavelet transform and phase derivative,” Opt. Commun. 280, 327-336 (2007).
[CrossRef]

Tay, C. J.

C. J. Tay, C. Quan, W. Sun, and X. Y. He, “Demodulation of a single interferogram based on continuous wavelet transform and phase derivative,” Opt. Commun. 280, 327-336 (2007).
[CrossRef]

Appl. Opt.

Exp. Mech.

C. A. Sciammarella and T. Kim, “Frequency modulation interpretation of fringes and computation of strains,” Exp. Mech. 45, 393-403 (2005).
[CrossRef]

IEEE Signal Process. Lett.

L. Stankovic and V. Katkovnik, “Algorithm for the instantaneous frequency estimation using time-frequency distributions with adaptive window width,” IEEE Signal Process. Lett. 5, 224-227 (1998).
[CrossRef]

IEEE Trans. Commun.

D. R. Pauluzzi and N. C. Beaulieu, “A comparison of snr estimation techniques for the awgn channel,” IEEE Trans. Commun. 48, 1601 (2000).
[CrossRef]

IEEE Trans. Signal Process.

V. Katkovnik and L. Stankovic, “Instantaneous frequency estimation using the Wigner distribution with varying and data-driven window length,” IEEE Trans. Signal Process. 45, 2147 (1997).

S. Lawrence Marple Jr., “Computing the discrete-time analytic signal via fft,” IEEE Trans. Signal Process. 47, 2600-2603 (1999).
[CrossRef]

Opt. Commun.

C. J. Tay, C. Quan, W. Sun, and X. Y. He, “Demodulation of a single interferogram based on continuous wavelet transform and phase derivative,” Opt. Commun. 280, 327-336 (2007).
[CrossRef]

J. A. Quiroga, J. Antonio Gomez-Pedrero, and A. Garcia-Botella, “Algorithm for fringe pattern normalization,” Opt. Commun. 197, 43-51 (2001).
[CrossRef]

Opt. Lett.

Proc. IEEE

B. Boashash, “Estimating and interpreting the instantaneous frequency of a signal--part 1: fundamentals,” Proc. IEEE 80, 519-538 (1992).

Signal Process.

V. Katkovnik and L. J. Stankovic, “Periodogram with varying and data-driven window length,” Signal Process. 67, 345-358 (1998).
[CrossRef]

S. C. Sekhar and T. V. Sreenivas, “Signal-to-noise ratio estimation using higher-order moments,” Signal Process. 86, 716-732 (2006).
[CrossRef]

Other

R. Matzner and F. Englberger, “Snr estimation algorithm using fourth-order moments,” Proceedings of the IEEE International Symposium on Information Theory (IEEE, 1994).

L. Cohen, Time Frequency Analysis (Prentice Hall, 1995).

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Figures (3)

Fig. 1
Fig. 1

(a) Probability density function for IF estimate. Original and estimated phase derivatives in radians/pixel for different window sizes: (b) small window, (c) large window, (d) adaptive window.

Fig. 2
Fig. 2

(a) Simulated fringe pattern ( 256 × 256 ) at SNR of 30 dB , (b) original phase derivative along the x direction in radians/pixel, (c) original phase derivative and estimated phase derivative in radians/pixel for row y = 129 , (d) optimum window length h opt , (e) estimated phase derivative in radians/pixel for the entire fringe pattern, (f) cosine fringes of the estimated phase derivative.

Fig. 3
Fig. 3

(a) Experimental fringe pattern obtained in DHI ( 256 × 256 ) , (b) estimated phase derivative by the proposed method in radians/pixel, (c) wrapped phase derivative estimate, (d) wrapped phase derivative obtained by the pixel-shearing approach.

Equations (29)

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I ( x , y ) = A ( x , y ) exp [ j φ ( x , y ) ] + η ( x , y ) ,
I ( x ) = A exp [ j φ ( x ) ] + η ( x )
f i ( x ) = 1 2 π φ ( x ) x .
S ( x , f ) = τ = I ( τ ) w ( x τ ) exp ( j 2 π f τ ) ,
f ̂ i ( x ) = arg max f | S ( x , f ) | ,
w h ( x ) = 1 ( 2 π h 2 ) 1 2 exp ( x 2 2 h 2 ) x [ h 2 , h 2 ] ,
S h ( x , f ) = τ = I ( τ ) w h ( x τ ) exp ( j 2 π f τ ) ,
f ̂ h ( x ) = arg max f | S h ( x , f ) | ,
B = E [ f i ( x ) f ̂ h ( x ) ] ,
σ 2 = E [ ( f ̂ h ( x ) E [ f ̂ h ( x ) ] ) 2 ] ,
MSE = E [ ( f i ( x ) f ̂ h ( x ) ) 2 ] = bias 2 + variance,
B ( h ) = 1 2 π s = 1 h 2 s b s ( 2 π f i ( x ) ) 2 s ,
σ 2 ( h ) = σ ε 2 T E 1 8 π 2 | A | 2 h 3 F 1 2 ,
E 1 = 1 2 1 2 w 2 ( x ) x 2 d x ,
F 1 = 1 2 1 2 w ( x ) x 2 d x ,
b s = 1 ( 2 s + 1 ) ! F 1 1 2 1 2 w ( x ) x 2 s + 2 d x .
A ̂ 2 = { | 2 [ 1 N x = 1 N | I ( x ) | 2 ] 2 1 N x = 1 N | I ( x ) | 4 | } 1 2 ,
σ ̂ ε 2 = | x = 1 N | I ( x ) | 2 N { | 2 [ x = 1 N | I ( x ) | 2 N ] 2 x = 1 N | I ( x ) | 4 N | } 1 2 | ,
σ 2 ( h ) = σ ̂ ε 2 T E 1 8 π 2 | A ̂ | 2 h 3 F 1 2 .
| f ̂ h ( x ) E [ f ̂ h ( x ) ] | = | f ̂ h ( x ) f i ( x ) + f i ( x ) E [ f ̂ h ( x ) ] | | f ̂ h ( x ) f i ( x ) | | f i ( x ) E [ f ̂ h ( x ) ] |
| f ̂ h ( x ) f i ( x ) | | bias | k σ ( h ) .
| bias | k σ ( h ) .
| f ̂ h ( x ) f i ( x ) | 2 k σ ( h )
f ̂ h ( x ) 2 k σ ( h ) f i ( x ) f ̂ h ( x ) + 2 k σ ( h ) .
I ( x , y ) = a ( x , y ) + b ( x , y ) cos [ φ ( x , y ) ] + η ( x , y ) ,
I ( x , y ) = cos [ φ ( x , y ) ] + η ( x , y ) .
z ( x , y ) = exp [ j φ ( x , y ) ] + η ( x , y ) .
MSE = E [ ( f i ( x ) f ̂ h ( x ) ) 2 ] = E [ ( ( f i ( x ) E [ f ̂ h ( x ) ] ) + ( E [ f ̂ h ( x ) ] f ̂ h ( x ) ) ) 2 ] = E [ ( f i ( x ) E [ f ̂ h ( x ) ] ) 2 + ( f ̂ h ( x ) E [ f ̂ h ( x ) ] ) 2 ] + 2 E [ ( f i ( x ) E [ f ̂ h ( x ) ] ) ( E [ f ̂ h ( x ) ] f ̂ h ( x ) ) ] = E [ ( f i ( x ) E [ f ̂ h ( x ) ] ) 2 ] + E [ ( f ̂ h ( x ) E [ f ̂ h ( x ) ] ) 2 ] + 2 E [ ( f i ( x ) E [ f ̂ h ( x ) ] ) ( E [ f ̂ h ( x ) ] f ̂ h ( x ) ) ] = B 2 + σ 2 + 2 E [ f i ( x ) E [ f ̂ h ( x ) ] f i ( x ) f ̂ h ( x ) ] 2 E [ ( E [ f ̂ h ( x ) ] ) 2 E [ f ̂ h ( x ) ] f ̂ h ( x ) ] ,
MSE = B 2 + σ 2 + 2 f i ( x ) E [ f ̂ h ( x ) ] 2 f i ( x ) E [ f ̂ h ( x ) ] 2 ( E [ f ̂ h ( x ) ] ) 2 + 2 E [ f ̂ h ( x ) ] E [ f ̂ h ( x ) ] = B 2 + σ 2 = bias 2 + variance.

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