Abstract

We evaluate a data-driven technique to perform bias suppression and modulation normalization of fringe patterns. The proposed technique uses a bidimensional empirical mode decomposition method to decompose a fringe pattern in a set of intrinsic frequency modes and the partial Hilbert transform to characterize the local amplitude of the modes in order to perform the normalization. The performance of the technique is tested using computer simulated fringe patterns of different fringe densities and illu mination defects with high local variations of the modulation, and its advantages and limitations are discussed. Finally, the performance of the normalization approach in processing real data is also illustrated.

© 2009 Optical Society of America

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References

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  1. D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Testing (Marcel Dekker, 1998).
  2. J. A. Quiroga, J. A. Gómez-Pedrero, and A. García-Botella, “Algorithm for fringe pattern normalization,” Opt. Commun. 197, 43-51 (2001).
    [CrossRef]
  3. J. A. Quiroga and M. Servin, “Isotropic n-dimensional fringe pattern normalization,” Opt. Commun. 224, 221-227 (2003).
    [CrossRef]
  4. R. Legarda-Sáenz, W. Osten, and W. Jüptner, “Improvement of the regularized phase tracking technique for the processing of nonnormalized fringe patterns,” Appl. Opt. 41, 5519-5526(2002).
    [CrossRef] [PubMed]
  5. P. K. Rastogi, Digital Speckle Pattern Interferometry and Related Techniques (Wiley, 2001).
  6. L. Watkins, S. Tan, and T. Barnes, “Determination of interferometer phase distributions by use of wavelets,” Opt. Lett. 24, 905-907 (1999).
    [CrossRef]
  7. C. A. Sciammarella and T. Kim, “Determination of strains from fringe patterns using space-frequency representations,” Opt. Eng. 42, 3182-3193 (2003).
    [CrossRef]
  8. A. Federico and G. H. Kaufmann, “Phase retrieval in digital speckle pattern interferometry using a smoothed time-frequency distribution,” Appl. Opt. 42, 7066-7071 (2003).
    [CrossRef] [PubMed]
  9. A. Federico and G. H. Kaufmann, “Phase retrieval in digital speckle pattern interferometry by application of two-dimensional active contours called snakes,” Appl. Opt. 45, 1909-1916 (2006).
    [CrossRef] [PubMed]
  10. A. Federico and G. H. Kaufmann, “Local denoising of digital speckle pattern interferometry fringes using multiplicative correlation and weighted smoothing splines,” Appl. Opt. 44, 2728-2735 (2005).
    [CrossRef] [PubMed]
  11. A. Federico and G. H. Kaufmann, “Denoising in digital speckle pattern interferometry using wave atoms,” Opt. Lett. 32, 1232-1234 (2007).
    [CrossRef] [PubMed]
  12. J. A. Guerrero, J. L. Marroquin, M. Rivera, and J. A. Quiroga, “Adaptive monogenic filtering and normalization of ESPI fringe patterns,” Opt. Lett. 30, 3018-3020 (2005).
    [CrossRef] [PubMed]
  13. N. A. Ochoa and A. A. Silva-Moreno, “Normalization and noise-reduction algorithm for fringe patterns,” Opt. Commun. 270, 161-168 (2007).
    [CrossRef]
  14. M. B. Bernini, A. Federico, and G. H. Kaufmann, “Noise reduction in digital speckle pattern interferometry using bidimensional empirical mode decomposition,” Appl. Opt. 47, 2592-2598 (2008).
    [CrossRef] [PubMed]
  15. N. E. Huang, Z. Sheng, 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 non-linear and non-stationary time series analysis,” Proc. R. Soc. London Ser. A 454, 903-995 (1998).
    [CrossRef]
  16. C. Damerval, S. Meignen, and V. Perrier, “A fast algorithm for bidimensional EMD,” IEEE Signal Process Lett. 12, 701-704(2005).
    [CrossRef]
  17. H. T. Yang, Finite Element Structural Analysis (Prentice-Hall, 1986).
  18. Y. Tian, Y. Huang, and Y. Li, “Image zooming method using 2D EMD technique,” in Proceedings of IEEE 6th World Congress on Intelligent Control and Automation (IEEE, 2006), pp. 10036-10040.
    [CrossRef]
  19. K. Zeng and M. He, “A simple boundary process technique for empirical mode decomposition,” in Proceedings of IEEE Geoscience and Remote Sensing Symposium (IEEE, 2004), Vol. 6, pp. 4258-4261.
  20. G. Rilling, P. Flandrin, and P. Gonçalves, “On empirical mode decomposition and its algorithms,” in Proceedings of IEEE-EURASIP Workshop on Nonlinear Signal and Image Processing (IEEE, 2003) (http://perso.ens-lyon.fr/paulo.goncalves/index.php?page=Publications=emd-eurasip03#Communications).
  21. Z. Liu and S. Peng, “Boundary processing of bidimensional EMD using texture synthesis,” IEEE Signal Process Lett. 12, 33-36 (2005).
  22. M. Shen, H. Tang, and B. Li, “The modified bidimensional empirical mode decomposition for image denoising,” in Proceedings of IEEE 8th International Conference on Signal Processing (IEEE, 2006), pp. 16-20.
  23. W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge Univ. Press, 1992), Chap. 13.
  24. S. L. Hahn, Hilbert Transforms in Signal Processing (Artech House, 1996).
  25. Z. Wang and A. C. Bovik, “A universal quality index,” IEEE Signal Process Lett. 9, 81-84 (2002).
    [CrossRef]
  26. P. D. Ruiz and G. H. Kaufmann, “Evaluation of a scale-space filter for speckle noise reduction in electronic speckle pattern interferometry,” Opt. Eng. 37, 2395-2401(1998).
    [CrossRef]
  27. G. H. Kaufmann, “Nondestructive testing with thermal waves using phase shifted temporal speckle pattern interferometry,” Opt. Eng. 42, 2010-2014 (2003).
    [CrossRef]

2008 (1)

2007 (2)

N. A. Ochoa and A. A. Silva-Moreno, “Normalization and noise-reduction algorithm for fringe patterns,” Opt. Commun. 270, 161-168 (2007).
[CrossRef]

A. Federico and G. H. Kaufmann, “Denoising in digital speckle pattern interferometry using wave atoms,” Opt. Lett. 32, 1232-1234 (2007).
[CrossRef] [PubMed]

2006 (1)

2005 (4)

2003 (4)

J. A. Quiroga and M. Servin, “Isotropic n-dimensional fringe pattern normalization,” Opt. Commun. 224, 221-227 (2003).
[CrossRef]

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

G. H. Kaufmann, “Nondestructive testing with thermal waves using phase shifted temporal speckle pattern interferometry,” Opt. Eng. 42, 2010-2014 (2003).
[CrossRef]

A. Federico and G. H. Kaufmann, “Phase retrieval in digital speckle pattern interferometry using a smoothed time-frequency distribution,” Appl. Opt. 42, 7066-7071 (2003).
[CrossRef] [PubMed]

2002 (2)

2001 (1)

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

1999 (1)

1998 (2)

N. E. Huang, Z. Sheng, 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 non-linear and non-stationary time series analysis,” Proc. R. Soc. London Ser. A 454, 903-995 (1998).
[CrossRef]

P. D. Ruiz and G. H. Kaufmann, “Evaluation of a scale-space filter for speckle noise reduction in electronic speckle pattern interferometry,” Opt. Eng. 37, 2395-2401(1998).
[CrossRef]

Barnes, T.

Bernini, M. B.

Bovik, A. C.

Z. Wang and A. C. Bovik, “A universal quality index,” IEEE Signal Process Lett. 9, 81-84 (2002).
[CrossRef]

Damerval, C.

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

Federico, A.

Flandrin, P.

G. Rilling, P. Flandrin, and P. Gonçalves, “On empirical mode decomposition and its algorithms,” in Proceedings of IEEE-EURASIP Workshop on Nonlinear Signal and Image Processing (IEEE, 2003) (http://perso.ens-lyon.fr/paulo.goncalves/index.php?page=Publications=emd-eurasip03#Communications).

Flannery, B.

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

García-Botella, A.

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

Gómez-Pedrero, J. A.

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

Gonçalves, P.

G. Rilling, P. Flandrin, and P. Gonçalves, “On empirical mode decomposition and its algorithms,” in Proceedings of IEEE-EURASIP Workshop on Nonlinear Signal and Image Processing (IEEE, 2003) (http://perso.ens-lyon.fr/paulo.goncalves/index.php?page=Publications=emd-eurasip03#Communications).

Guerrero, J. A.

Hahn, S. L.

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

He, M.

K. Zeng and M. He, “A simple boundary process technique for empirical mode decomposition,” in Proceedings of IEEE Geoscience and Remote Sensing Symposium (IEEE, 2004), Vol. 6, pp. 4258-4261.

Huang, N. E.

N. E. Huang, Z. Sheng, 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 non-linear and non-stationary time series analysis,” Proc. R. Soc. London Ser. A 454, 903-995 (1998).
[CrossRef]

Huang, Y.

Y. Tian, Y. Huang, and Y. Li, “Image zooming method using 2D EMD technique,” in Proceedings of IEEE 6th World Congress on Intelligent Control and Automation (IEEE, 2006), pp. 10036-10040.
[CrossRef]

Jüptner, W.

Kaufmann, G. H.

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]

Legarda-Sáenz, R.

Li, B.

M. Shen, H. Tang, and B. Li, “The modified bidimensional empirical mode decomposition for image denoising,” in Proceedings of IEEE 8th International Conference on Signal Processing (IEEE, 2006), pp. 16-20.

Li, Y.

Y. Tian, Y. Huang, and Y. Li, “Image zooming method using 2D EMD technique,” in Proceedings of IEEE 6th World Congress on Intelligent Control and Automation (IEEE, 2006), pp. 10036-10040.
[CrossRef]

Liu, H. H.

N. E. Huang, Z. Sheng, 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 non-linear and non-stationary time series analysis,” Proc. R. Soc. London Ser. A 454, 903-995 (1998).
[CrossRef]

Liu, Z.

Z. Liu and S. Peng, “Boundary processing of bidimensional EMD using texture synthesis,” IEEE Signal Process Lett. 12, 33-36 (2005).

Long, S. R.

N. E. Huang, Z. Sheng, 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 non-linear and non-stationary time series analysis,” Proc. R. Soc. London Ser. A 454, 903-995 (1998).
[CrossRef]

Malacara, D.

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

Malacara, Z.

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

Marroquin, J. L.

Meignen, S.

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

Ochoa, N. A.

N. A. Ochoa and A. A. Silva-Moreno, “Normalization and noise-reduction algorithm for fringe patterns,” Opt. Commun. 270, 161-168 (2007).
[CrossRef]

Osten, W.

Peng, S.

Z. Liu and S. Peng, “Boundary processing of bidimensional EMD using texture synthesis,” IEEE Signal Process Lett. 12, 33-36 (2005).

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 Univ. Press, 1992), Chap. 13.

Quiroga, J. A.

J. A. Guerrero, J. L. Marroquin, M. Rivera, and J. A. Quiroga, “Adaptive monogenic filtering and normalization of ESPI fringe patterns,” Opt. Lett. 30, 3018-3020 (2005).
[CrossRef] [PubMed]

J. A. Quiroga and M. Servin, “Isotropic n-dimensional fringe pattern normalization,” Opt. Commun. 224, 221-227 (2003).
[CrossRef]

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

Rastogi, P. K.

P. K. Rastogi, Digital Speckle Pattern Interferometry and Related Techniques (Wiley, 2001).

Rilling, G.

G. Rilling, P. Flandrin, and P. Gonçalves, “On empirical mode decomposition and its algorithms,” in Proceedings of IEEE-EURASIP Workshop on Nonlinear Signal and Image Processing (IEEE, 2003) (http://perso.ens-lyon.fr/paulo.goncalves/index.php?page=Publications=emd-eurasip03#Communications).

Rivera, M.

Ruiz, P. D.

P. D. Ruiz and G. H. Kaufmann, “Evaluation of a scale-space filter for speckle noise reduction in electronic speckle pattern interferometry,” Opt. Eng. 37, 2395-2401(1998).
[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. Quiroga and M. Servin, “Isotropic n-dimensional fringe pattern normalization,” Opt. Commun. 224, 221-227 (2003).
[CrossRef]

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

Shen, M.

M. Shen, H. Tang, and B. Li, “The modified bidimensional empirical mode decomposition for image denoising,” in Proceedings of IEEE 8th International Conference on Signal Processing (IEEE, 2006), pp. 16-20.

Sheng, Z.

N. E. Huang, Z. Sheng, 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 non-linear and non-stationary time series analysis,” Proc. R. Soc. London Ser. A 454, 903-995 (1998).
[CrossRef]

Shih, H. H.

N. E. Huang, Z. Sheng, 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 non-linear and non-stationary time series analysis,” Proc. R. Soc. London Ser. A 454, 903-995 (1998).
[CrossRef]

Silva-Moreno, A. A.

N. A. Ochoa and A. A. Silva-Moreno, “Normalization and noise-reduction algorithm for fringe patterns,” Opt. Commun. 270, 161-168 (2007).
[CrossRef]

Tan, S.

Tang, H.

M. Shen, H. Tang, and B. Li, “The modified bidimensional empirical mode decomposition for image denoising,” in Proceedings of IEEE 8th International Conference on Signal Processing (IEEE, 2006), pp. 16-20.

Teukolsky, S.

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

Tian, Y.

Y. Tian, Y. Huang, and Y. Li, “Image zooming method using 2D EMD technique,” in Proceedings of IEEE 6th World Congress on Intelligent Control and Automation (IEEE, 2006), pp. 10036-10040.
[CrossRef]

Tung, C. C.

N. E. Huang, Z. Sheng, 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 non-linear and non-stationary time series analysis,” Proc. R. Soc. London Ser. A 454, 903-995 (1998).
[CrossRef]

Vetterling, W.

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

Wang, Z.

Z. Wang and A. C. Bovik, “A universal quality index,” IEEE Signal Process Lett. 9, 81-84 (2002).
[CrossRef]

Watkins, L.

Wu, M. C.

N. E. Huang, Z. Sheng, 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 non-linear and non-stationary time series analysis,” Proc. R. Soc. London Ser. A 454, 903-995 (1998).
[CrossRef]

Yang, H. T.

H. T. Yang, Finite Element Structural Analysis (Prentice-Hall, 1986).

Yen, N. C.

N. E. Huang, Z. Sheng, 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 non-linear and non-stationary time series analysis,” Proc. R. Soc. London Ser. A 454, 903-995 (1998).
[CrossRef]

Zeng, K.

K. Zeng and M. He, “A simple boundary process technique for empirical mode decomposition,” in Proceedings of IEEE Geoscience and Remote Sensing Symposium (IEEE, 2004), Vol. 6, pp. 4258-4261.

Zheng, Q.

N. E. Huang, Z. Sheng, 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 non-linear and non-stationary time series analysis,” Proc. R. Soc. London Ser. A 454, 903-995 (1998).
[CrossRef]

Appl. Opt. (5)

IEEE Signal Process Lett. (3)

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

Z. Liu and S. Peng, “Boundary processing of bidimensional EMD using texture synthesis,” IEEE Signal Process Lett. 12, 33-36 (2005).

Z. Wang and A. C. Bovik, “A universal quality index,” IEEE Signal Process Lett. 9, 81-84 (2002).
[CrossRef]

Opt. Commun. (3)

N. A. Ochoa and A. A. Silva-Moreno, “Normalization and noise-reduction algorithm for fringe patterns,” Opt. Commun. 270, 161-168 (2007).
[CrossRef]

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

J. A. Quiroga and M. Servin, “Isotropic n-dimensional fringe pattern normalization,” Opt. Commun. 224, 221-227 (2003).
[CrossRef]

Opt. Eng. (3)

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

P. D. Ruiz and G. H. Kaufmann, “Evaluation of a scale-space filter for speckle noise reduction in electronic speckle pattern interferometry,” Opt. Eng. 37, 2395-2401(1998).
[CrossRef]

G. H. Kaufmann, “Nondestructive testing with thermal waves using phase shifted temporal speckle pattern interferometry,” Opt. Eng. 42, 2010-2014 (2003).
[CrossRef]

Opt. Lett. (3)

Proc. R. Soc. London Ser. A (1)

N. E. Huang, Z. Sheng, 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 non-linear and non-stationary time series analysis,” Proc. R. Soc. London Ser. A 454, 903-995 (1998).
[CrossRef]

Other (9)

P. K. Rastogi, Digital Speckle Pattern Interferometry and Related Techniques (Wiley, 2001).

H. T. Yang, Finite Element Structural Analysis (Prentice-Hall, 1986).

Y. Tian, Y. Huang, and Y. Li, “Image zooming method using 2D EMD technique,” in Proceedings of IEEE 6th World Congress on Intelligent Control and Automation (IEEE, 2006), pp. 10036-10040.
[CrossRef]

K. Zeng and M. He, “A simple boundary process technique for empirical mode decomposition,” in Proceedings of IEEE Geoscience and Remote Sensing Symposium (IEEE, 2004), Vol. 6, pp. 4258-4261.

G. Rilling, P. Flandrin, and P. Gonçalves, “On empirical mode decomposition and its algorithms,” in Proceedings of IEEE-EURASIP Workshop on Nonlinear Signal and Image Processing (IEEE, 2003) (http://perso.ens-lyon.fr/paulo.goncalves/index.php?page=Publications=emd-eurasip03#Communications).

M. Shen, H. Tang, and B. Li, “The modified bidimensional empirical mode decomposition for image denoising,” in Proceedings of IEEE 8th International Conference on Signal Processing (IEEE, 2006), pp. 16-20.

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

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

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

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Figures (6)

Fig. 1
Fig. 1

Normalization of a low frequency computer-generated fringe pattern: (a) fringe pattern containing a bias component and a modulation defect, (b) normalized fringe pattern obtained using the first 2D IMF ( Q = 0.764 ).

Fig. 2
Fig. 2

Normalization of a more complex computer-generated fringe pattern: (a) fringe pattern containing a bias component and a modulation defect, (b) normalized fringe pattern obtained using the first 2D IMF ( Q = 0.898 ).

Fig. 3
Fig. 3

Comparison between the proposed technique and another normalization method: (a) fringe pattern containing a bias component and a modulation defect, (b) normalized fringe pattern obtained using the first 2D IMF ( Q = 0.845 ), (c) fringe pattern normalized using the two orthogonal bandpass filter method of Ref. [2] ( Q = 0.160 ).

Fig. 4
Fig. 4

Intensity profiles along the middle row of the fringe patterns depicted in Fig. 3: (a) original fringe pattern, (b) fringe pattern normalized with the proposed method, (c) fringe pattern normalized with the two orthogonal bandpass filter technique of Ref. [2].

Fig. 5
Fig. 5

Computer-generated DSPI fringe pattern: (a) original fringe pattern, (b) smoothed fringe pattern obtained using the wave atoms technique of Ref. [11] containing modulation defects introduced by the denoising procedure, (c) fringe pattern normalized with the proposed method using the first 2D IMF ( Q = 0.842 ).

Fig. 6
Fig. 6

Experimental DSPI fringe pattern containing an illumination defect: (a) original fringe pattern, (b) fringe pattern denoised using the BEMD method, (c) fringe pattern normalized with the proposed technique using the first two 2D IMFs, (d) fringe pattern normalized with the two orthogonal bandpass filter technique of Ref. [2].

Equations (6)

Equations on this page are rendered with MathJax. Learn more.

SD = x = 1 x = X y = 1 y = Y | h i k 1 ( x , y ) h i k ( x , y ) | 2 x = 1 x = X y = 1 y = Y h i k 1 2 ( x , y ) < ε ,
H x ( x , y ) = 1 π PV + d ν x I IMF ( ν x , y ) ( x - ν x ) ,
H y ( x , y ) = 1 π PV + d ν y I IMF ( x , ν y ) ( y ν y ) .
A ( x , y ) = | I IMF ( x , y ) + i H x ( x , y ) | + | I IMF ( x , y ) + i H y ( x , y ) | 2 .
Q 4 σ E O E ¯ O ¯ ( σ E 2 + σ O 2 ) [ E ¯ 2 + O ¯ 2 ] ,
Q = σ E O σ E σ O 2 E ¯ O ¯ ( E ¯ ) 2 + ( O ¯ ) 2 2 σ E σ O σ E 2 + σ O 2 .

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