Abstract

We propose a phase measurement technique to retrieve optical phase distributions coded in noisy temporal speckle pattern interferometry signals presenting regions of adjacent low-modulated pixels, which is based on the bidimensional empirical mode decomposition and the Hilbert transform. It is shown that this approach can effectively remove noise and minimize the influence of large sets of adjacent nonmodulated pixels located in the time series of speckle interferograms. The performance of the phase retrieval approach is analyzed using computer-simulated speckle interferograms modulated with a temporal carrier. The results are also compared with those given by a technique based on the one-dimensional empirical mode decomposition. The advantages and limitations of the proposed approach are finally discussed.

© 2011 Optical Society of America

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References

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    [CrossRef]
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2009

2008

2007

L. R. Watkins, “Phase recovery from fringe patterns using the continuous wavelet transform,” Opt. Lasers Eng. 45, 298–303(2007).
[CrossRef]

2006

2005

C. Damerval, S. Meignen, and V. Perrier, “A fast algorithm for bidimensional EMD,” IEEE Signal Process. Lett. 12, 701–704(2005).
[CrossRef]

2003

G. H. Kaufmann, “Phase measurement in temporal speckle pattern interferometry using the Fourier transform method with and without a temporal carrier,” Opt. Commun. 217, 141–149 (2003).
[CrossRef]

2002

1998

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. A 454, 903–995 (1998).
[CrossRef]

1982

Bernini, M. B.

Damerval, C.

C. Damerval, S. Meignen, and V. Perrier, “A fast algorithm for bidimensional EMD,” IEEE Signal Process. Lett. 12, 701–704(2005).
[CrossRef]

Equis, S.

S. Equis and P. Jacquot, “Phase extraction in dynamic speckle interferometry with empirical mode decomposition and Hilbert transform,” Strain 46, 550–558 (2008).
[CrossRef]

S. Equis and P. Jacquot, “A new application of the Delaunay triangulation: the processing of speckle interferometry signals,” in Proceedings of Fringe, the International Workshop on Automatic Processing of Fringe Patterns, W.Osten and M. Kujawinska, eds. (Springer, 2009), pp 123–131.

Federico, A.

Flannery, B.

W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge University, 1992), Chap. 13.

Galizzi, G. E.

Huang, N. E.

Z. Wu and N. E. Huang, “Ensemble empirical mode decomposition: a noise-assisted data analysis method,” Advances Adapt. Data Anal. 1, 1–41 (2009).
[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. A 454, 903–995 (1998).
[CrossRef]

Huntley, J. M.

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

Ina, H.

Jacquot, P.

S. Equis and P. Jacquot, “Phase extraction in dynamic speckle interferometry with empirical mode decomposition and Hilbert transform,” Strain 46, 550–558 (2008).
[CrossRef]

S. Equis and P. Jacquot, “A new application of the Delaunay triangulation: the processing of speckle interferometry signals,” in Proceedings of Fringe, the International Workshop on Automatic Processing of Fringe Patterns, W.Osten and M. Kujawinska, eds. (Springer, 2009), pp 123–131.

Kadono, H.

Kaufmann, G. H.

Kobayashi, S.

Kujawinska, M.

S. Equis and P. Jacquot, “A new application of the Delaunay triangulation: the processing of speckle interferometry signals,” in Proceedings of Fringe, the International Workshop on Automatic Processing of Fringe Patterns, W.Osten and M. Kujawinska, eds. (Springer, 2009), pp 123–131.

Liu, H. H.

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. A 454, 903–995 (1998).
[CrossRef]

Long, S. R.

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. A 454, 903–995 (1998).
[CrossRef]

Madjarova, V. D.

Marengo Rodriguez, F. A.

Meignen, S.

C. Damerval, S. Meignen, and V. Perrier, “A fast algorithm for bidimensional EMD,” IEEE Signal Process. Lett. 12, 701–704(2005).
[CrossRef]

Perrier, V.

C. Damerval, S. Meignen, and V. Perrier, “A fast algorithm for bidimensional EMD,” IEEE Signal Process. Lett. 12, 701–704(2005).
[CrossRef]

Press, W.

W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge University, 1992), Chap. 13.

Shen, Z.

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. A 454, 903–995 (1998).
[CrossRef]

Shih, H. H.

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. A 454, 903–995 (1998).
[CrossRef]

Takeda, M.

Teukolsky, S.

W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge University, 1992), Chap. 13.

Toyooka, S.

Tung, C. C.

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. A 454, 903–995 (1998).
[CrossRef]

Vetterling, W.

W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge University, 1992), Chap. 13.

Watkins, L. R.

L. R. Watkins, “Phase recovery from fringe patterns using the continuous wavelet transform,” Opt. Lasers Eng. 45, 298–303(2007).
[CrossRef]

Wu, M. C.

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. A 454, 903–995 (1998).
[CrossRef]

Wu, Z.

Z. Wu and N. E. Huang, “Ensemble empirical mode decomposition: a noise-assisted data analysis method,” Advances Adapt. Data Anal. 1, 1–41 (2009).
[CrossRef]

Yen, N. C.

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. A 454, 903–995 (1998).
[CrossRef]

Zheng, Q.

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. A 454, 903–995 (1998).
[CrossRef]

Advances Adapt. Data Anal.

Z. Wu and N. E. Huang, “Ensemble empirical mode decomposition: a noise-assisted data analysis method,” Advances Adapt. Data Anal. 1, 1–41 (2009).
[CrossRef]

Appl. Opt.

IEEE Signal Process. Lett.

C. Damerval, S. Meignen, and V. Perrier, “A fast algorithm for bidimensional EMD,” IEEE Signal Process. Lett. 12, 701–704(2005).
[CrossRef]

J. Opt. Soc. Am.

Opt. Commun.

G. H. Kaufmann, “Phase measurement in temporal speckle pattern interferometry using the Fourier transform method with and without a temporal carrier,” Opt. Commun. 217, 141–149 (2003).
[CrossRef]

Opt. Lasers Eng.

L. R. Watkins, “Phase recovery from fringe patterns using the continuous wavelet transform,” Opt. Lasers Eng. 45, 298–303(2007).
[CrossRef]

Opt. Lett.

Proc. R. Soc. A

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. A 454, 903–995 (1998).
[CrossRef]

Strain

S. Equis and P. Jacquot, “Phase extraction in dynamic speckle interferometry with empirical mode decomposition and Hilbert transform,” Strain 46, 550–558 (2008).
[CrossRef]

Other

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

S. Equis and P. Jacquot, “A new application of the Delaunay triangulation: the processing of speckle interferometry signals,” in Proceedings of Fringe, the International Workshop on Automatic Processing of Fringe Patterns, W.Osten and M. Kujawinska, eds. (Springer, 2009), pp 123–131.

W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge University, 1992), Chap. 13.

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Figures (7)

Fig. 1
Fig. 1

(a) 3D matrix produced by a series of speckle interferograms showing some frames and column slices, (b) column slice k extracted for processing.

Fig. 2
Fig. 2

Simulated phase evolution used for the numerical evaluation and produced by a pixel at coordinates x = 64 and y = 64 .

Fig. 3
Fig. 3

Time-varying intensity at a pixel belonging to a region of nonmodulated pixels of size ( 4 × 3 × 25 ) .

Fig. 4
Fig. 4

Simulated (thin curve) and retrieved (bold curve) phase distributions obtained using the BEMD-HT method for the pixel shown in Fig. 3.

Fig. 5
Fig. 5

Time-varying intensity at a pixel belonging to a region of nonmodulated pixels of size ( 6 × 6 × 30 ) .

Fig. 6
Fig. 6

Column slice located at the coordinate y = 61 at four different stages of the phase measurement technique: (a) original column slice with noise and a region of nonmodulated pixels, (b) slice after the application of the mean filter, (c) denoised slice, (d) column slice of the reconstructed 3D matrix after the application of the median filter.

Fig. 7
Fig. 7

Comparison of the simulated phase distribution (thin curve) and the retrieved maps obtained using the BEMD-HT (bold curve) and the EMD-HT (triangles) methods for the pixel shown in Fig. 5.

Tables (1)

Tables Icon

Table 1 RMS Phase Error σ Obtained Using the BEMD-HT and the EMD-HT Methods as a Function of the Size of the Nonmodulated Regions

Equations (8)

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z ( t ) = s ( t ) + i H T [ s ( t ) ] = a ( t ) exp [ i φ ( t ) ] ,
φ w ( t ) = arctan { H T [ s ( t ) ] s ( t ) } ,
I ( x , y , t ) = I 0 ( x , y , t ) + I M ( x , y , t ) × cos [ ϕ r ( x , y ) + ϕ ( x , y , t ) + ψ ( x , y , t ) ] ,
SD = x = 0 x = n x 1 y = 0 y = n y 1 | h i k 1 ( x , y ) h i k ( x , y ) | 2 x = 0 x = n x 1 y = 0 y = n y 1 h i k 1 2 ( x , y ) < ε ,
I ( x , y , t ) = | R exp ( i α ) + F 1 [ H ( u , v ) F ( exp { i [ ϕ r ( x , y ) + ϕ ( x , y , t ) + ψ ( x , y , t ) ] } ) ] | 2 ,
σ = { 1 K Σ t = 0 t = n t 1 Σ x = 0 x = n 1 Σ y = 0 y = n 1 [ Δ ϕ ( x , y , t ) Δ ϕ 0 ( x , y , t ) ] 2 } 1 / 2 ,
ϕ ( x , y , t ) = n × exp [ ( y n / 2 ) 2 + ( x n / 2 ) 2 4 n 2 ] g ( t ) ,
g ( t ) = exp ( 1 / 2 3 ) exp [ ( 1 / 2 t / n t ) 3 ] ,

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