Abstract

Some effective filtering methods for wrapped phase maps, a regularized phase-tracking method (RPT) without the regularization term, a multiple-parameter least-square method (MPLS), a windowed Fourier ridges method (WFR), an autocorrelation function method (ACF), and a sine∕cosine average filter (SCAF), are analyzed in order to establish their transversal relationship. The analysis shows that principles of the RPT, MPLS, WFR, and ACF are equivalent and the SCAF also leads to the WFR by some extension, which elegantly unifies all these methods for filtering unwrapped phase maps.

© 2007 Optical Society of America

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  1. D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithm, and Software (Wiley, 1998).
  2. C. M. Loeffler and R. E. Leonard, Jr., "Phase unwrapping via media filtering," in IEEE International Conference Acoustics, Speech, and Signal Process. (ICASSP) (IEEE, 1984), pp. 48.6.1-48.6.3.
  3. J. Lee, "Speckle suppression and analysis for synthetic aperture radar images," Opt. Eng. 25, 636-643 (1986).
  4. G. Bo, S. Dellepiane, and G. Beneventano, "A locally adaptive noise filtering approach for phase-unwrapping improvement," in SPIE Conference on SAR Image Analysis, Modeling, and Techniques, Florence, Italy, September 1999, Proc. SPIE 3869, 116-125 (1999).
    [CrossRef]
  5. A. Capanni, L. Pezzati, D. Bertani, M. Cetica, and F. Francini, "Phase-shifting speckle interferometry: a noise reduction filter for phase unwrapping," Opt. Eng. 36, 2466-2472 (1997).
    [CrossRef]
  6. F. Qian, X. Wang, X. Wang, and Y. Bu, "Adaptive filter for unwrapping noisy phase image in phase-stepping interferometry," Opt. Laser Technol. 33, 479-486 (2001).
    [CrossRef]
  7. M. J. Huang and W. Sheu, "Histogram-data-orientated filter for inconsistency removal of interferometric phase maps," Opt. Eng. 44, 045602 (2005).
    [CrossRef]
  8. M. Servin, F. J. Cuevas, D. Malacara, J. L. Marroguin, and R. Rodriguez-Vera, "Phase unwrapping through demodulation by use of the regularized phase-tracking technique," Appl. Opt. 38, 1934-1941 (1999).
    [CrossRef]
  9. M. Servin and M. Kujawinska, "Modern fringe pattern analysis in interferometry," Handbook of Optical Engineering, D. Malacara and B. J. Thompson, eds. (Dekker, 2001), Chap. 12, pp. 373-426.
  10. H. Y. Yun, C. K. Hong, and S. W. Chang, "Least-square phase estimation with multiple parameters in phase-shifting electronic speckle pattern interferometry," J. Opt. Soc. Am. A 20, 240-247 (2003).
    [CrossRef]
  11. Q. Kemao, "Two-dimensional windowed Fourier transform for fringe pattern analysis: principles, applications and implementations," Opt. Lasers Eng. 45, 304-317 (2007).
    [CrossRef]
  12. R. Kramer and O. Loffeld, "Presentation of an improved phase unwrapping algorithm based on Kalman filters combined with local slope estimation," in Proceedings of Fringe'96 ESA Workshop Applications of ERS SAR Interferometry (ESA, 1996).
  13. E. Trouve, M. Caramma, and H. Maitre, "Fringe detection in noisy complex interferograms," Appl. Opt. 35, 3799-3806 (1996).
    [CrossRef] [PubMed]
  14. E. Trouve, J. Nicolas, and H. Maitre, "Improving phase unwrapping techniques by the use of local frequency estimates," IEEE Trans. Geosci. Remote Sens. 36, 1963-1972 (1998).
    [CrossRef]
  15. J. M. Huntley, "Random phase measurement errors in digital speckle pattern interferometry," Opt. Lasers Eng. 26, 131-150 (1997).
    [CrossRef]
  16. H. A. Aebischer and S. Waldner, "A simple and effective method for filtering speckle-interferometric phase fringe patterns," Opt. Commun. 162, 205-210 (1999).
    [CrossRef]
  17. F. Palacios, E. Goncalves, J. Ricardo, and J. L. Valin, "Adaptive filter to improve the performance of phase-unwrapping in digital holography," Opt. Commun. 238, 245-251 (2004).
    [CrossRef]
  18. V. Rosso, F. Michel, V. Moreau, Y. Renotte, B. Tilkens, and Y. Lion, "Highlighting properties for filters for their application in temporal phase shifting interferometry," in Photonic Applications in Biosensing and Imaging, C. W. Chan, K. Yu, U. J. Krull, R. I. Hornsey, B. C. Wilson, and R. A. Weersink, eds., Proc. SPIE 5969, 59692M (2005).
    [CrossRef]
  19. S. Fu, H. Lin, J. Chen, and Q. Yu, "Influence of window size on the fringe orientation estimation," Opt. Commun. 272, 73-80 (2007).
    [CrossRef]
  20. C. Tang, W. Wang, H. Yan, and X. Gu, "Tangent least-square fitting filtering method for electrical speckle pattern interferometry phase fringe patterns," Appl. Opt. 46, 2907-2913 (2007).
    [CrossRef] [PubMed]
  21. S. Mallat, A Wavelet Tour of Signal Processing, 2nd ed. (Academic, 1999).
  22. G. Chunsheng, Z. Zhaoda, and Z. Daiyin, "InSAR interferogram filtering based on PD operator," J. Nanjing University of Aeronautics and Astronautics 35, 72-76 (2003).
  23. R. M. Goldstein and C. L. Werner, "Radar interferogram filtering for geophysical applications," Geophys. Res. Lett. 25, 4035-4038 (1998).
    [CrossRef]
  24. M. Servin, J. L. Marroguin, and F. J. Cucvas, "Demodulation of a single interferogram by use of a two-dimensional regularized phase-tracking technique," Appl. Opt. 36, 4540-4548 (1997).
    [CrossRef] [PubMed]
  25. M. Servin, J. L. Marroquin, and F. J. Cuevas, "Fringe-follower regularized phase tracker for demodulation of closed-fringe interferograms," J. Opt. Soc. Am. A 18, 689-695 (2001).
    [CrossRef]
  26. K. Qian, S. H. Soon, and A. Asundi, "A simple phase unwrapping approach based on filtering by windowed Fourier transform," Opt. Laser Technol. 37, 458-462 (2005).
    [CrossRef]
  27. Q. Kemao and S. H. Soon, "A simple phase unwrapping approach based on filtering by windowed Fourier transform (II)," Key Eng. Mater. 326-328, 67-70 (2006).
    [CrossRef]
  28. Q. Kemao, "A simple phase unwrapping approach based on filtering by windowed Fourier transform: the phase near edges," Opt. Laser Technol. 39, 1364-1369 (2007).
    [CrossRef]

2007

S. Fu, H. Lin, J. Chen, and Q. Yu, "Influence of window size on the fringe orientation estimation," Opt. Commun. 272, 73-80 (2007).
[CrossRef]

Q. Kemao, "Two-dimensional windowed Fourier transform for fringe pattern analysis: principles, applications and implementations," Opt. Lasers Eng. 45, 304-317 (2007).
[CrossRef]

Q. Kemao, "A simple phase unwrapping approach based on filtering by windowed Fourier transform: the phase near edges," Opt. Laser Technol. 39, 1364-1369 (2007).
[CrossRef]

C. Tang, W. Wang, H. Yan, and X. Gu, "Tangent least-square fitting filtering method for electrical speckle pattern interferometry phase fringe patterns," Appl. Opt. 46, 2907-2913 (2007).
[CrossRef] [PubMed]

2006

Q. Kemao and S. H. Soon, "A simple phase unwrapping approach based on filtering by windowed Fourier transform (II)," Key Eng. Mater. 326-328, 67-70 (2006).
[CrossRef]

2005

V. Rosso, F. Michel, V. Moreau, Y. Renotte, B. Tilkens, and Y. Lion, "Highlighting properties for filters for their application in temporal phase shifting interferometry," in Photonic Applications in Biosensing and Imaging, C. W. Chan, K. Yu, U. J. Krull, R. I. Hornsey, B. C. Wilson, and R. A. Weersink, eds., Proc. SPIE 5969, 59692M (2005).
[CrossRef]

K. Qian, S. H. Soon, and A. Asundi, "A simple phase unwrapping approach based on filtering by windowed Fourier transform," Opt. Laser Technol. 37, 458-462 (2005).
[CrossRef]

M. J. Huang and W. Sheu, "Histogram-data-orientated filter for inconsistency removal of interferometric phase maps," Opt. Eng. 44, 045602 (2005).
[CrossRef]

2004

F. Palacios, E. Goncalves, J. Ricardo, and J. L. Valin, "Adaptive filter to improve the performance of phase-unwrapping in digital holography," Opt. Commun. 238, 245-251 (2004).
[CrossRef]

2003

G. Chunsheng, Z. Zhaoda, and Z. Daiyin, "InSAR interferogram filtering based on PD operator," J. Nanjing University of Aeronautics and Astronautics 35, 72-76 (2003).

H. Y. Yun, C. K. Hong, and S. W. Chang, "Least-square phase estimation with multiple parameters in phase-shifting electronic speckle pattern interferometry," J. Opt. Soc. Am. A 20, 240-247 (2003).
[CrossRef]

2001

M. Servin, J. L. Marroquin, and F. J. Cuevas, "Fringe-follower regularized phase tracker for demodulation of closed-fringe interferograms," J. Opt. Soc. Am. A 18, 689-695 (2001).
[CrossRef]

F. Qian, X. Wang, X. Wang, and Y. Bu, "Adaptive filter for unwrapping noisy phase image in phase-stepping interferometry," Opt. Laser Technol. 33, 479-486 (2001).
[CrossRef]

1999

G. Bo, S. Dellepiane, and G. Beneventano, "A locally adaptive noise filtering approach for phase-unwrapping improvement," in SPIE Conference on SAR Image Analysis, Modeling, and Techniques, Florence, Italy, September 1999, Proc. SPIE 3869, 116-125 (1999).
[CrossRef]

H. A. Aebischer and S. Waldner, "A simple and effective method for filtering speckle-interferometric phase fringe patterns," Opt. Commun. 162, 205-210 (1999).
[CrossRef]

S. Mallat, A Wavelet Tour of Signal Processing, 2nd ed. (Academic, 1999).

M. Servin, F. J. Cuevas, D. Malacara, J. L. Marroguin, and R. Rodriguez-Vera, "Phase unwrapping through demodulation by use of the regularized phase-tracking technique," Appl. Opt. 38, 1934-1941 (1999).
[CrossRef]

1998

R. M. Goldstein and C. L. Werner, "Radar interferogram filtering for geophysical applications," Geophys. Res. Lett. 25, 4035-4038 (1998).
[CrossRef]

E. Trouve, J. Nicolas, and H. Maitre, "Improving phase unwrapping techniques by the use of local frequency estimates," IEEE Trans. Geosci. Remote Sens. 36, 1963-1972 (1998).
[CrossRef]

1997

J. M. Huntley, "Random phase measurement errors in digital speckle pattern interferometry," Opt. Lasers Eng. 26, 131-150 (1997).
[CrossRef]

A. Capanni, L. Pezzati, D. Bertani, M. Cetica, and F. Francini, "Phase-shifting speckle interferometry: a noise reduction filter for phase unwrapping," Opt. Eng. 36, 2466-2472 (1997).
[CrossRef]

M. Servin, J. L. Marroguin, and F. J. Cucvas, "Demodulation of a single interferogram by use of a two-dimensional regularized phase-tracking technique," Appl. Opt. 36, 4540-4548 (1997).
[CrossRef] [PubMed]

1996

1986

J. Lee, "Speckle suppression and analysis for synthetic aperture radar images," Opt. Eng. 25, 636-643 (1986).

Aebischer, H. A.

H. A. Aebischer and S. Waldner, "A simple and effective method for filtering speckle-interferometric phase fringe patterns," Opt. Commun. 162, 205-210 (1999).
[CrossRef]

Asundi, A.

K. Qian, S. H. Soon, and A. Asundi, "A simple phase unwrapping approach based on filtering by windowed Fourier transform," Opt. Laser Technol. 37, 458-462 (2005).
[CrossRef]

Beneventano, G.

G. Bo, S. Dellepiane, and G. Beneventano, "A locally adaptive noise filtering approach for phase-unwrapping improvement," in SPIE Conference on SAR Image Analysis, Modeling, and Techniques, Florence, Italy, September 1999, Proc. SPIE 3869, 116-125 (1999).
[CrossRef]

Bertani, D.

A. Capanni, L. Pezzati, D. Bertani, M. Cetica, and F. Francini, "Phase-shifting speckle interferometry: a noise reduction filter for phase unwrapping," Opt. Eng. 36, 2466-2472 (1997).
[CrossRef]

Bo, G.

G. Bo, S. Dellepiane, and G. Beneventano, "A locally adaptive noise filtering approach for phase-unwrapping improvement," in SPIE Conference on SAR Image Analysis, Modeling, and Techniques, Florence, Italy, September 1999, Proc. SPIE 3869, 116-125 (1999).
[CrossRef]

Bu, Y.

F. Qian, X. Wang, X. Wang, and Y. Bu, "Adaptive filter for unwrapping noisy phase image in phase-stepping interferometry," Opt. Laser Technol. 33, 479-486 (2001).
[CrossRef]

Capanni, A.

A. Capanni, L. Pezzati, D. Bertani, M. Cetica, and F. Francini, "Phase-shifting speckle interferometry: a noise reduction filter for phase unwrapping," Opt. Eng. 36, 2466-2472 (1997).
[CrossRef]

Caramma, M.

Cetica, M.

A. Capanni, L. Pezzati, D. Bertani, M. Cetica, and F. Francini, "Phase-shifting speckle interferometry: a noise reduction filter for phase unwrapping," Opt. Eng. 36, 2466-2472 (1997).
[CrossRef]

Chang, S. W.

Chen, J.

S. Fu, H. Lin, J. Chen, and Q. Yu, "Influence of window size on the fringe orientation estimation," Opt. Commun. 272, 73-80 (2007).
[CrossRef]

Chunsheng, G.

G. Chunsheng, Z. Zhaoda, and Z. Daiyin, "InSAR interferogram filtering based on PD operator," J. Nanjing University of Aeronautics and Astronautics 35, 72-76 (2003).

Cucvas, F. J.

Cuevas, F. J.

Daiyin, Z.

G. Chunsheng, Z. Zhaoda, and Z. Daiyin, "InSAR interferogram filtering based on PD operator," J. Nanjing University of Aeronautics and Astronautics 35, 72-76 (2003).

Dellepiane, S.

G. Bo, S. Dellepiane, and G. Beneventano, "A locally adaptive noise filtering approach for phase-unwrapping improvement," in SPIE Conference on SAR Image Analysis, Modeling, and Techniques, Florence, Italy, September 1999, Proc. SPIE 3869, 116-125 (1999).
[CrossRef]

Francini, F.

A. Capanni, L. Pezzati, D. Bertani, M. Cetica, and F. Francini, "Phase-shifting speckle interferometry: a noise reduction filter for phase unwrapping," Opt. Eng. 36, 2466-2472 (1997).
[CrossRef]

Fu, S.

S. Fu, H. Lin, J. Chen, and Q. Yu, "Influence of window size on the fringe orientation estimation," Opt. Commun. 272, 73-80 (2007).
[CrossRef]

Ghiglia, D. C.

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithm, and Software (Wiley, 1998).

Goldstein, R. M.

R. M. Goldstein and C. L. Werner, "Radar interferogram filtering for geophysical applications," Geophys. Res. Lett. 25, 4035-4038 (1998).
[CrossRef]

Goncalves, E.

F. Palacios, E. Goncalves, J. Ricardo, and J. L. Valin, "Adaptive filter to improve the performance of phase-unwrapping in digital holography," Opt. Commun. 238, 245-251 (2004).
[CrossRef]

Gu, X.

Hong, C. K.

Huang, M. J.

M. J. Huang and W. Sheu, "Histogram-data-orientated filter for inconsistency removal of interferometric phase maps," Opt. Eng. 44, 045602 (2005).
[CrossRef]

Huntley, J. M.

J. M. Huntley, "Random phase measurement errors in digital speckle pattern interferometry," Opt. Lasers Eng. 26, 131-150 (1997).
[CrossRef]

Kemao, Q.

Q. Kemao, "Two-dimensional windowed Fourier transform for fringe pattern analysis: principles, applications and implementations," Opt. Lasers Eng. 45, 304-317 (2007).
[CrossRef]

Q. Kemao, "A simple phase unwrapping approach based on filtering by windowed Fourier transform: the phase near edges," Opt. Laser Technol. 39, 1364-1369 (2007).
[CrossRef]

Q. Kemao and S. H. Soon, "A simple phase unwrapping approach based on filtering by windowed Fourier transform (II)," Key Eng. Mater. 326-328, 67-70 (2006).
[CrossRef]

Kramer, R.

R. Kramer and O. Loffeld, "Presentation of an improved phase unwrapping algorithm based on Kalman filters combined with local slope estimation," in Proceedings of Fringe'96 ESA Workshop Applications of ERS SAR Interferometry (ESA, 1996).

Kujawinska, M.

M. Servin and M. Kujawinska, "Modern fringe pattern analysis in interferometry," Handbook of Optical Engineering, D. Malacara and B. J. Thompson, eds. (Dekker, 2001), Chap. 12, pp. 373-426.

Lee, J.

J. Lee, "Speckle suppression and analysis for synthetic aperture radar images," Opt. Eng. 25, 636-643 (1986).

Leonard, R. E.

C. M. Loeffler and R. E. Leonard, Jr., "Phase unwrapping via media filtering," in IEEE International Conference Acoustics, Speech, and Signal Process. (ICASSP) (IEEE, 1984), pp. 48.6.1-48.6.3.

Lin, H.

S. Fu, H. Lin, J. Chen, and Q. Yu, "Influence of window size on the fringe orientation estimation," Opt. Commun. 272, 73-80 (2007).
[CrossRef]

Lion, Y.

V. Rosso, F. Michel, V. Moreau, Y. Renotte, B. Tilkens, and Y. Lion, "Highlighting properties for filters for their application in temporal phase shifting interferometry," in Photonic Applications in Biosensing and Imaging, C. W. Chan, K. Yu, U. J. Krull, R. I. Hornsey, B. C. Wilson, and R. A. Weersink, eds., Proc. SPIE 5969, 59692M (2005).
[CrossRef]

Loeffler, C. M.

C. M. Loeffler and R. E. Leonard, Jr., "Phase unwrapping via media filtering," in IEEE International Conference Acoustics, Speech, and Signal Process. (ICASSP) (IEEE, 1984), pp. 48.6.1-48.6.3.

Loffeld, O.

R. Kramer and O. Loffeld, "Presentation of an improved phase unwrapping algorithm based on Kalman filters combined with local slope estimation," in Proceedings of Fringe'96 ESA Workshop Applications of ERS SAR Interferometry (ESA, 1996).

Maitre, H.

E. Trouve, J. Nicolas, and H. Maitre, "Improving phase unwrapping techniques by the use of local frequency estimates," IEEE Trans. Geosci. Remote Sens. 36, 1963-1972 (1998).
[CrossRef]

E. Trouve, M. Caramma, and H. Maitre, "Fringe detection in noisy complex interferograms," Appl. Opt. 35, 3799-3806 (1996).
[CrossRef] [PubMed]

Malacara, D.

Mallat, S.

S. Mallat, A Wavelet Tour of Signal Processing, 2nd ed. (Academic, 1999).

Marroguin, J. L.

Marroquin, J. L.

Michel, F.

V. Rosso, F. Michel, V. Moreau, Y. Renotte, B. Tilkens, and Y. Lion, "Highlighting properties for filters for their application in temporal phase shifting interferometry," in Photonic Applications in Biosensing and Imaging, C. W. Chan, K. Yu, U. J. Krull, R. I. Hornsey, B. C. Wilson, and R. A. Weersink, eds., Proc. SPIE 5969, 59692M (2005).
[CrossRef]

Moreau, V.

V. Rosso, F. Michel, V. Moreau, Y. Renotte, B. Tilkens, and Y. Lion, "Highlighting properties for filters for their application in temporal phase shifting interferometry," in Photonic Applications in Biosensing and Imaging, C. W. Chan, K. Yu, U. J. Krull, R. I. Hornsey, B. C. Wilson, and R. A. Weersink, eds., Proc. SPIE 5969, 59692M (2005).
[CrossRef]

Nicolas, J.

E. Trouve, J. Nicolas, and H. Maitre, "Improving phase unwrapping techniques by the use of local frequency estimates," IEEE Trans. Geosci. Remote Sens. 36, 1963-1972 (1998).
[CrossRef]

Palacios, F.

F. Palacios, E. Goncalves, J. Ricardo, and J. L. Valin, "Adaptive filter to improve the performance of phase-unwrapping in digital holography," Opt. Commun. 238, 245-251 (2004).
[CrossRef]

Pezzati, L.

A. Capanni, L. Pezzati, D. Bertani, M. Cetica, and F. Francini, "Phase-shifting speckle interferometry: a noise reduction filter for phase unwrapping," Opt. Eng. 36, 2466-2472 (1997).
[CrossRef]

Pritt, M. D.

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithm, and Software (Wiley, 1998).

Qian, F.

F. Qian, X. Wang, X. Wang, and Y. Bu, "Adaptive filter for unwrapping noisy phase image in phase-stepping interferometry," Opt. Laser Technol. 33, 479-486 (2001).
[CrossRef]

Qian, K.

K. Qian, S. H. Soon, and A. Asundi, "A simple phase unwrapping approach based on filtering by windowed Fourier transform," Opt. Laser Technol. 37, 458-462 (2005).
[CrossRef]

Renotte, Y.

V. Rosso, F. Michel, V. Moreau, Y. Renotte, B. Tilkens, and Y. Lion, "Highlighting properties for filters for their application in temporal phase shifting interferometry," in Photonic Applications in Biosensing and Imaging, C. W. Chan, K. Yu, U. J. Krull, R. I. Hornsey, B. C. Wilson, and R. A. Weersink, eds., Proc. SPIE 5969, 59692M (2005).
[CrossRef]

Ricardo, J.

F. Palacios, E. Goncalves, J. Ricardo, and J. L. Valin, "Adaptive filter to improve the performance of phase-unwrapping in digital holography," Opt. Commun. 238, 245-251 (2004).
[CrossRef]

Rodriguez-Vera, R.

Rosso, V.

V. Rosso, F. Michel, V. Moreau, Y. Renotte, B. Tilkens, and Y. Lion, "Highlighting properties for filters for their application in temporal phase shifting interferometry," in Photonic Applications in Biosensing and Imaging, C. W. Chan, K. Yu, U. J. Krull, R. I. Hornsey, B. C. Wilson, and R. A. Weersink, eds., Proc. SPIE 5969, 59692M (2005).
[CrossRef]

Servin, M.

Sheu, W.

M. J. Huang and W. Sheu, "Histogram-data-orientated filter for inconsistency removal of interferometric phase maps," Opt. Eng. 44, 045602 (2005).
[CrossRef]

Soon, S. H.

Q. Kemao and S. H. Soon, "A simple phase unwrapping approach based on filtering by windowed Fourier transform (II)," Key Eng. Mater. 326-328, 67-70 (2006).
[CrossRef]

K. Qian, S. H. Soon, and A. Asundi, "A simple phase unwrapping approach based on filtering by windowed Fourier transform," Opt. Laser Technol. 37, 458-462 (2005).
[CrossRef]

Tang, C.

Tilkens, B.

V. Rosso, F. Michel, V. Moreau, Y. Renotte, B. Tilkens, and Y. Lion, "Highlighting properties for filters for their application in temporal phase shifting interferometry," in Photonic Applications in Biosensing and Imaging, C. W. Chan, K. Yu, U. J. Krull, R. I. Hornsey, B. C. Wilson, and R. A. Weersink, eds., Proc. SPIE 5969, 59692M (2005).
[CrossRef]

Trouve, E.

E. Trouve, J. Nicolas, and H. Maitre, "Improving phase unwrapping techniques by the use of local frequency estimates," IEEE Trans. Geosci. Remote Sens. 36, 1963-1972 (1998).
[CrossRef]

E. Trouve, M. Caramma, and H. Maitre, "Fringe detection in noisy complex interferograms," Appl. Opt. 35, 3799-3806 (1996).
[CrossRef] [PubMed]

Valin, J. L.

F. Palacios, E. Goncalves, J. Ricardo, and J. L. Valin, "Adaptive filter to improve the performance of phase-unwrapping in digital holography," Opt. Commun. 238, 245-251 (2004).
[CrossRef]

Waldner, S.

H. A. Aebischer and S. Waldner, "A simple and effective method for filtering speckle-interferometric phase fringe patterns," Opt. Commun. 162, 205-210 (1999).
[CrossRef]

Wang, W.

Wang, X.

F. Qian, X. Wang, X. Wang, and Y. Bu, "Adaptive filter for unwrapping noisy phase image in phase-stepping interferometry," Opt. Laser Technol. 33, 479-486 (2001).
[CrossRef]

F. Qian, X. Wang, X. Wang, and Y. Bu, "Adaptive filter for unwrapping noisy phase image in phase-stepping interferometry," Opt. Laser Technol. 33, 479-486 (2001).
[CrossRef]

Werner, C. L.

R. M. Goldstein and C. L. Werner, "Radar interferogram filtering for geophysical applications," Geophys. Res. Lett. 25, 4035-4038 (1998).
[CrossRef]

Yan, H.

Yu, Q.

S. Fu, H. Lin, J. Chen, and Q. Yu, "Influence of window size on the fringe orientation estimation," Opt. Commun. 272, 73-80 (2007).
[CrossRef]

Yun, H. Y.

Zhaoda, Z.

G. Chunsheng, Z. Zhaoda, and Z. Daiyin, "InSAR interferogram filtering based on PD operator," J. Nanjing University of Aeronautics and Astronautics 35, 72-76 (2003).

Appl. Opt.

Geophys. Res. Lett.

R. M. Goldstein and C. L. Werner, "Radar interferogram filtering for geophysical applications," Geophys. Res. Lett. 25, 4035-4038 (1998).
[CrossRef]

IEEE Trans. Geosci. Remote Sens.

E. Trouve, J. Nicolas, and H. Maitre, "Improving phase unwrapping techniques by the use of local frequency estimates," IEEE Trans. Geosci. Remote Sens. 36, 1963-1972 (1998).
[CrossRef]

J. Nanjing University of Aeronautics and Astronautics

G. Chunsheng, Z. Zhaoda, and Z. Daiyin, "InSAR interferogram filtering based on PD operator," J. Nanjing University of Aeronautics and Astronautics 35, 72-76 (2003).

J. Opt. Soc. Am. A

Key Eng. Mater.

Q. Kemao and S. H. Soon, "A simple phase unwrapping approach based on filtering by windowed Fourier transform (II)," Key Eng. Mater. 326-328, 67-70 (2006).
[CrossRef]

Opt. Commun.

H. A. Aebischer and S. Waldner, "A simple and effective method for filtering speckle-interferometric phase fringe patterns," Opt. Commun. 162, 205-210 (1999).
[CrossRef]

F. Palacios, E. Goncalves, J. Ricardo, and J. L. Valin, "Adaptive filter to improve the performance of phase-unwrapping in digital holography," Opt. Commun. 238, 245-251 (2004).
[CrossRef]

S. Fu, H. Lin, J. Chen, and Q. Yu, "Influence of window size on the fringe orientation estimation," Opt. Commun. 272, 73-80 (2007).
[CrossRef]

Opt. Eng.

J. Lee, "Speckle suppression and analysis for synthetic aperture radar images," Opt. Eng. 25, 636-643 (1986).

A. Capanni, L. Pezzati, D. Bertani, M. Cetica, and F. Francini, "Phase-shifting speckle interferometry: a noise reduction filter for phase unwrapping," Opt. Eng. 36, 2466-2472 (1997).
[CrossRef]

M. J. Huang and W. Sheu, "Histogram-data-orientated filter for inconsistency removal of interferometric phase maps," Opt. Eng. 44, 045602 (2005).
[CrossRef]

Opt. Laser Technol.

F. Qian, X. Wang, X. Wang, and Y. Bu, "Adaptive filter for unwrapping noisy phase image in phase-stepping interferometry," Opt. Laser Technol. 33, 479-486 (2001).
[CrossRef]

Q. Kemao, "A simple phase unwrapping approach based on filtering by windowed Fourier transform: the phase near edges," Opt. Laser Technol. 39, 1364-1369 (2007).
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V. Rosso, F. Michel, V. Moreau, Y. Renotte, B. Tilkens, and Y. Lion, "Highlighting properties for filters for their application in temporal phase shifting interferometry," in Photonic Applications in Biosensing and Imaging, C. W. Chan, K. Yu, U. J. Krull, R. I. Hornsey, B. C. Wilson, and R. A. Weersink, eds., Proc. SPIE 5969, 59692M (2005).
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Figures (1)

Fig. 1
Fig. 1

Filtering by the SCAF, (a) simulated noisy wrapped phase map, (b) phase filtered by the SCAF with M = N = 2 and 10 repetitions, (c) phase compensated by a horizontal local frequency of 0.5 and then filtered, (d) phase filtered by the WFR.

Tables (1)

Tables Icon

Table 1 Parameter Settings for Numerical Comparison of Different Filters

Equations (112)

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φ W ( x , y )
C ( x , y ) = cos [ φ W ( x , y ) ] ,
S ( x , y ) = sin [ φ W ( x , y ) ] .
φ W ( x , y ) = arctan [ S ( x , y ) , C ( x , y ) ] ,
arctan ( , )
e ( x , y ) = C ( x , y ) + j S ( x , y ) = exp [ j φ W ( x , y ) ] ,
j = 1
φ W ( x , y ) = arctan { Im [ e ( x , y ) ] , Re [ e ( x , y ) ] } angle [ e ( x , y ) ] ,
Re ( )
Im ( )
U ( x , y ) = ( x + u , y + v ) ( N x , y L ) { ( C ( x + u , y + v ) cos [ φ ( x , y ) + ω x ( x , y ) u + ω y ( x , y ) v ] ) 2 + ( S ( x + u , y + v ) sin [ φ ( x , y ) + ω x ( x , y ) u + ω y ( x , y ) v ] ) 2 + λ [ φ ( x + u , y + v ) φ ( x , y ) ω x ( x , y ) u ω y ( x , y ) v ] 2 m ( x + u , y + v ) } ,
N x , y
( x , y )
m ( x , y )
( x , y )
ω x ( x , y )
ω y ( x , y )
( x , y )
N x , y
[ φ ( x , y ) , ω x ( x , y ) , ω y ( x , y ) ] = arg min ϕ , ξ , η v = N N u = M M { [ C ( x + u , y + v ) cos ( ϕ + ξ u + η v ) ] 2 +  [ S ( x + u , y + v ) sin ( ϕ + ξ u + η v ) ] 2 } ,
α = arg   min   β U ( β )
U ( β )
N x , y L
N x , y
[ φ ( x , y ) , ω x ( x , y ) , ω y ( x , y ) ] = arg min ϕ , ξ , η v = N N u = M M | [ C ( x + u , y + v ) cos ( ϕ + ξ u + η v ) ] + j [ S ( x + u , y + v ) sin ( ϕ + ξ u + η v ) ] | 2 = arg min ϕ , ξ , η v = N N u = M M | [ C ( x + u , y + v ) + j S ( x + u , y + v ) ] [ cos ( ϕ + ξ u + η v ) + j sin ( ϕ + ξ u + η v ) ] | 2 = arg min ϕ , ξ , η v = N N u = M M | e ( x + u , y + v ) exp ( j ϕ + j ξ u + j η v ) | 2 ,
| |
e ( x , y )
[ φ ( x , y ) , ω x ( x , y ) , ω y ( x , y ) ]
[ φ ( x , y ) , ω x ( x , y ) , ω y ( x , y ) ]
= arg min ϕ , ξ , η v = N N u = M M [ 2 2 C ( x + u , y + v ) × cos ( ϕ + ξ u + η v ) 2 S ( x + u , y + v ) × sin ( ϕ + ξ u + η v ) ]
= arg min ϕ , ξ , η v = N N u = M M [ C ( x + u , y + v ) × cos ( ϕ + ξ u + η v ) S ( x + u , y + v ) × sin ( ϕ + ξ u + η v ) ] ,
[ φ ( x , y ) , ω x ( x , y ) , ω y ( x , y ) ] = arg min ϕ , ξ , η v = N N u = M M [ 2 e ( x + u , y + v ) ×  exp ( j ϕ j ξ u j η v ) e * ( x + u , y + v ) ×  exp ( j ϕ + j ξ u + j η v ) ] = arg min ϕ , ξ , η v = N N u = M M { e ( x + u , y + v ) ×  exp ( j ϕ j ξ u j η v ) [ e ( x + u , y + v ) ×  exp ( j ϕ j ξ u j η v ) ] * } = arg max ϕ , ξ , η Re [ v = N N u = M M e ( x + u , y + v ) ×  exp ( j ϕ j ξ u j η v ) ] = arg max ϕ , ξ , η Re [ exp ( j ϕ ) v = N N u = M M e ( x + u , y + v ) ×  exp ( j ξ u j η v ) ] ,
v = N N u = M M e ( x + u , y + v ) exp ( j ξ u j η v ) = r ( x , y ) exp [ j φ 0 ( x , y ) ] ,
r ( x , y )
φ 0 ( x , y )
v = N N u = M M e ( x + u , y + v ) exp ( j ξ u j η v )
[ φ ( x , y ) , ω x ( x , y ) , ω y ( x , y ) ] = arg max ϕ , ξ , η Re { exp ( j ϕ ) r ( x , y ) exp [ j φ 0 ( x , y ) ] } = arg max ϕ , ξ , η Re { r ( x , y ) exp [ j φ 0 ( x , y ) j ϕ ] } = arg max ϕ , ξ , η r ( x , y ) cos [ φ 0 ( x , y ) ϕ ] .
[ φ ( x , y ) , ω x ( x , y ) , ω y ( x , y ) ]
[ ω x ( x , y ) , ω y ( x , y ) ] = arg max ξ , η r ( x , y ) = arg max ξ , η | v = N N u = M M e ( x + u , y + v ) ×  exp ( j ξ u j η v ) | ,
φ ( x , y ) = φ 0 ( x , y ) = angle { v = N N u = M M e ( x + u , y + v ) ×  exp [ j ω x ( x , y ) u j ω y ( x , y ) v ] } .
v = N N u = M M e ( x + u , y + v ) exp ( j ξ u j η v )
v = N N u = M M e ( x + u , y + v ) exp ( j ξ u j η v ) = v = u = e ( x + u , y + v ) g ( u , v ) exp ( j ξ u j η v ) = t = s = e ( s , t ) g ( s x , t y ) ×  exp [ j ξ ( s x ) j η ( t y ) ] = v = u = e ( u , v ) g ( u x , v y ) ×  exp [ j ξ ( u x ) j η ( v y ) ] = exp ( j ξ x + j η y ) v = u = e ( u , v ) g ( u x , v y ) ×  exp ( j ξ u j η v ) = exp ( j ξ x + j η y ) S e ( x , y ;  ξ , η ) .
g ( u , v )
g ( u , v ) = { 1 , M u M , N v N 0 , otherwise ,
s = x + u
t = y + v
s u
t v
exp ( j ξ x + j η y )
S e ( x , y ;  ξ , η )
e ( x , y )
S e ( x , y ;  ξ , η ) = v = u = e ( u , v ) g ( u x , v y ) ×  exp ( j ξ u j η v ) .
[ ω x ( x , y ) , ω y ( x , y ) ] = arg max ξ , η | S e ( x , y ;    ξ , η ) | ,
φ ( x , y ) = angle { S e [ x , y ;    ω x ( x , y ) , ω y ( x , y ) ] } + ω x ( x , y ) x + ω y ( x , y ) y ,
[ ω x ( x , y ) , ω y ( x , y ) ] = arg max ξ , η | S e ( u , v ;  ξ , η ) | = arg max ξ , η | v = N N u = M M e ( x + u , y + v ) exp ( j ξ u j η v ) | = arg max ξ , η | v = N N u = M M e ( x + u , y + v ) exp ( j ξ u j η v ) | 2 = arg max ξ , η n = 2 N 2 N m = 2 M 2 M | v | N | v + n | N | u | M | u + m | M e ( x + u , y + v ) ×  e * ( x + u m , y + v n ) exp ( j ξ m j η n ) arg max ξ , η n = D N D N m = D M D M v = N N u = M M e ( x + u , y + v ) ×  e * ( x + u m , y + v n ) exp ( j ξ m j η n ) = arg max ξ , η | n = D N D N m = D M D M γ ( x , y ;  m , n ) exp ( j ξ m j η n ) |
= arg max ξ , η | Γ ( x , y ;  ξ , η ) | .
( ξ , η )
| f ( ξ , η ) |
| f ( ξ , η ) | 2
D m m D M
D N n D N
M u M
N n N
γ ( x , y ;  m , n ) = v = N N u = M M e ( x + u , y + v ) × e * ( x + u m , y + v n ) ,
Γ ( x , y ;  ξ , η )
γ ( x , y ;  m , n )
e ( k ) ( x , y ) = v = N N u = M M e ( k 1 ) ( x + u , y + v ) | e ( k 1 ) ( x + u , y + v ) | ,
e ( k 1 ) ( x , y )
( k 1 ) th
e ( 0 ) ( x , y ) = e ( x , y )
1 M
N 3
φ ( x , y ) = 0.005 [ ( x 127 ) 2 + ( y 127 ) 2 ] .
M = N = 2
e ( k ) ( x , y ;  ξ , η ) = v = N N u = M M exp ( j ξ u η v ) × e ( k 1 ) ( x + u , y + v ) | e ( k 1 ) ( x + u , y + v ) | ,
exp ( j ξ u η v )
( ξ , η )
( 0 , 0 )
( ξ , η ) = ( 0.5 , 0 )
( ξ , η )
e ( 1 ) ( x , y ;  ξ , η ) = v = N N u = M M e ( 0 ) ( x + u , y + v ) ×  exp ( j ξ u j η v ) = v = N N u = M M e ( x + u , y + v ) ×  exp ( j ξ u j η v ) .
( ξ , η )
( x , y )
[ ξ opt ( x , y ) , η opt ( x , y ) ] = arg max ξ , η f [ v = N N u = M M e ( x + u , y + v ) × exp ( j ξ u η v ) ] ,
f ( )
f ( ) = | |
ω x
ω y
5 × 10 5 rad / pixel
5 × 10 5 rad / pixel
0.013   rad
0 .05   rad
ω x
ω y
0.009 rad / pixel
0.009 rad / pixel
0.004   rad
ξ [ 3.1 , 3.1 ]
η [ 3.1 , 3.1 ]
ϕ [ 3.1 , 3.1 ]
ξ [ ξ 0 0.3 , ξ 0 + 0.3 ]
η [ η 0 0.3 , η 0 + 0.3 ]
ϕ = [ ϕ 0 1.5 , ϕ 0 + 1.5 ]
ξ 0
η 0 , and   ϕ 0
ξ [ 1.5 , 1.5 ]
η [ 1.5 , 1.5 ]
ξ [ 1.5 , 1.5 ]
η [ 1.5 , 1.5 ]
D M = D N = 3
M = N = 2
0.5

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