Abstract

We present an optical phase measurement method based on the Hilbert transform for the analysis of a time series of speckle interferograms modulated by a temporal carrier. We discuss the influence of nonmodulating pixels, modulation loss, and noise that affect the bias and modulation intensities of the interferometric signal and propose the application of the empirical mode decomposition method for its minimization. We also show the equivalence between the phase recovery approaches that are based on the Hilbert and the Fourier transforms. Finally, we present a numerical comparison between these methods using computer-simulated speckle interferograms modulated with a temporal carrier.

© 2008 Optical Society of America

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References

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  1. J. M. Huntley, “Automated analysis of speckle interferograms,” in Digital Speckle Pattern Interferometry and Related Techniques, P. K. Rastogi, ed. (Wiley, 2001), Chap. 2, pp. 59-139.
  2. M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72, 156-160(1982).
    [CrossRef]
  3. L. H. Hahn, Hilbert Transforms in Signal Processing (Artech House, 1996).
  4. V. D. Madjarova, H. Kadono, and S. Toyooka, “Dynamic electronic speckle pattern interferometry (DESPI) phase analyses with temporal Hilbert transform,” Opt. Express 11, 617-623(2003).
    [CrossRef] [PubMed]
  5. V. D. Madjarova, H. Kadono, and S. Toyooka, “Use of dynamic electronic speckle pattern interferometry with the Hilbert transform method to investigate thermal expansion of a joint material,” Appl. Opt. 45, 7590-7596 (2006).
    [CrossRef]
  6. A. Baldi, S. Equis, and P. Jacquot, “Phase extraction in dynamic speckle interferometry by empirical mode decomposition,” in Experimental Analysis of Nano and Engineering Materials and Structures, E. E. Gdoutos, ed. (Springer, 2007), pp. 719-720.
    [CrossRef]
  7. F. A. Marengo Rodriguez, A. Federico, and G. H. Kaufmann, “Phase measurement improvement in temporal speckle pattern interferometry using empirical mode decomposition,” Opt. Commun. 275, 38-41 (2007).
    [CrossRef]
  8. N. E. Huang, Z. Shen, S. R. Long, M. C. Wu, H. H. Shih, Q. Zheng, N.-C. Yen, C. C. Tung, and H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis,” Proc. R. Soc. London, Ser. A 454, 903-995 (1998).
    [CrossRef]
  9. B. Boashash, “Estimating and interpreting the instantaneous frequency of a signal, ” Proc. IEEE 80, 520-568 (1992).
    [CrossRef]
  10. E. Bedrosian, “A product theorem for Hilbert transforms,” Proc. IEEE 51, 868-869 (1963).
    [CrossRef]
  11. Z. Wu and N. E. Huang, “A study of the characteristics of white noise using the empirical mode decomposition method,” Proc. R. Soc. London, Ser. A 460, 1597-1611 (2004).
    [CrossRef]
  12. G. H. Kaufmann and G. E. Galizzi, “Phase measurement in temporal speckle pattern interferometry: comparison between the phase-shifting and the Fourier transform methods,” Appl. Opt. 41, 7254-7263 (2002).
    [CrossRef] [PubMed]

2007 (1)

F. A. Marengo Rodriguez, A. Federico, and G. H. Kaufmann, “Phase measurement improvement in temporal speckle pattern interferometry using empirical mode decomposition,” Opt. Commun. 275, 38-41 (2007).
[CrossRef]

2006 (1)

2004 (1)

Z. Wu and N. E. Huang, “A study of the characteristics of white noise using the empirical mode decomposition method,” Proc. R. Soc. London, Ser. A 460, 1597-1611 (2004).
[CrossRef]

2003 (1)

2002 (1)

1998 (1)

N. E. Huang, Z. Shen, S. R. Long, M. C. Wu, H. H. Shih, Q. Zheng, N.-C. Yen, C. C. Tung, and H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis,” Proc. R. Soc. London, Ser. A 454, 903-995 (1998).
[CrossRef]

1992 (1)

B. Boashash, “Estimating and interpreting the instantaneous frequency of a signal, ” Proc. IEEE 80, 520-568 (1992).
[CrossRef]

1982 (1)

1963 (1)

E. Bedrosian, “A product theorem for Hilbert transforms,” Proc. IEEE 51, 868-869 (1963).
[CrossRef]

Appl. Opt. (2)

J. Opt. Soc. Am. (1)

Opt. Commun. (1)

F. A. Marengo Rodriguez, A. Federico, and G. H. Kaufmann, “Phase measurement improvement in temporal speckle pattern interferometry using empirical mode decomposition,” Opt. Commun. 275, 38-41 (2007).
[CrossRef]

Opt. Express (1)

Proc. IEEE (2)

B. Boashash, “Estimating and interpreting the instantaneous frequency of a signal, ” Proc. IEEE 80, 520-568 (1992).
[CrossRef]

E. Bedrosian, “A product theorem for Hilbert transforms,” Proc. IEEE 51, 868-869 (1963).
[CrossRef]

Proc. R. Soc. London, Ser. A (2)

Z. Wu and N. E. Huang, “A study of the characteristics of white noise using the empirical mode decomposition method,” Proc. R. Soc. London, Ser. A 460, 1597-1611 (2004).
[CrossRef]

N. E. Huang, Z. Shen, S. R. Long, M. C. Wu, H. H. Shih, Q. Zheng, N.-C. Yen, C. C. Tung, and H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis,” Proc. R. Soc. London, Ser. A 454, 903-995 (1998).
[CrossRef]

Other (3)

A. Baldi, S. Equis, and P. Jacquot, “Phase extraction in dynamic speckle interferometry by empirical mode decomposition,” in Experimental Analysis of Nano and Engineering Materials and Structures, E. E. Gdoutos, ed. (Springer, 2007), pp. 719-720.
[CrossRef]

J. M. Huntley, “Automated analysis of speckle interferograms,” in Digital Speckle Pattern Interferometry and Related Techniques, P. K. Rastogi, ed. (Wiley, 2001), Chap. 2, pp. 59-139.

L. H. Hahn, Hilbert Transforms in Signal Processing (Artech House, 1996).

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Figures (3)

Fig. 1
Fig. 1

Application of the EMD method to the intensity signal produced by an arbitrary pixel of the CCD: (a) original signal, (b) processed signal after the variation of the bias intensity was removed, (c) filtered signal after the removal of the high frequency noise.

Fig. 2
Fig. 2

Simulated phase used in the numerical analysis: (a) temporal history for k = 2 (continuous curve) and k = 3 (dashed curve), (b) spatial distribution for t = 257 and k = 2 .

Fig. 3
Fig. 3

Comparison of the recuperated phases by using the HT + EMD , the HT and the FT methods for a low modulated pixel.

Tables (1)

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Table 1 RMS Phase Error σ Obtained Using the Three Phase Measurement Methods as a Function of k and the Average Speckle Size s

Equations (7)

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I ( t ) = I 0 ( t ) + I r + 2 I 0 ( t ) I r cos [ ϕ ( t ) + ϕ c ( t ) + ϕ r ( t ) ] ,
Φ w ( t ) = arctan { H [ s ( t ) ] s ( t ) } .
F { H [ s ( t ) ] } ( f ) = i sgn ( f ) F [ s ( t ) ] ( f ) ,
sgn ( f ) = { 1 , f > 0 0 , f = 0 - 1 , f < 0 .
I ( m , n , t ) = | R exp ( i α ) + F 1 { H ( u , v ) F { exp [ i ϕ ( m , n , t ) + i ϕ c ( t ) ] U ( n , m ) } } | 2 ,
ϕ ( m , n , t ) = N exp [ - ( n - N / 2 ) 2 + ( m - N / 2 ) 2 4 N 2 ] g ( t ) ,
σ = { 1 K t = 0 N t - 1 m = 0 N - 1 n = 0 N - 1 [ Δ ϕ ( m , n , t ) - Δ ϕ 0 ( m , n , t ) ] 2 } 1 / 2 ,

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