Abstract

We propose a windowed Fourier-filtered and quality-guided phase-unwrapping algorithm that is an extension and improvement of our previous phase-unwrapping algorithm based on windowed Fourier transform [Opt. Laser Technol. 37, 458 (2005), Key Eng. Mater. 326–328, 67 (2006)]. First, the filtered amplitude is used as a real-valued quality map, rather than a binary mask, which makes the phase- unwrapping algorithm more tolerant to low-quality regions in a wrapped-phase map, and the process is more automatic. Second, the window size selection is considered, which enables the algorithm to be adapted to tackle different phase-unwrapping problems. A large window size is useful for removing noise, building long barriers along phase discontinuities, and identifying invalid regions, while a small window size is useful for preserving local features, such as small regions and high-quality narrow channels. Eight typical examples in Ghiglia and Pritt’s excellent book Two-Dimensional Phase Unwrapping: Theory, Algorithm and Software (Wiley, 1998) are used to evaluate the proposed algorithm. The proposed algorithm is able to unwrap all these examples successfully. The windowed Fourier ridges algorithm, another algorithm based on windowed Fourier transform, is also tested and found to be useful in building barriers along phase discontinuities.

© 2008 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithm and Software (Wiley, 1998).
  2. 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]
  3. K. Itoh, “Analysis of the phase unwrapping problem,” Appl. Opt. 21, 2470 (1982).
    [CrossRef] [PubMed]
  4. Q. Kemao, L. T. H. Nam, L. Feng, and S. H. Soon, “Comparative analysis on some filters for wrapped phase maps,” Appl. Opt. 46, 7412-7418 (2007).
    [CrossRef] [PubMed]
  5. 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]
  6. 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]
  7. 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]
  8. Q. Kemao, “Two-dimensional windowed Fourier transform for fringe pattern analysis: principles, applications and implementations,” Opt. Lasers Engi. 45, 304-317 (2007).
    [CrossRef]
  9. Q. Kemao, H. Wang, and W. Gao, “Windowed Fourier transform for fringe pattern analysis: theoretical analyses,” Appl. Opt. 47, 5408-5419 (2008).
  10. X. Su and W. Chen, “Reliability-guided phase unwrapping algorithm: a review,” Opt. Lasers Eng. 42, 245-261 (2004).
    [CrossRef]
  11. X. Su and L. Xue, “Phase unwrapping algorithm based on fringe frequency analysis in Fourier-transform profilometry,” Opt. Eng. 40, 637-643 (2001).
    [CrossRef]
  12. L. Xue and X. Su, “Phase unwrapping algorithm based on frequency analysis for measurement of a complex object by the phase-measuring-profilometry method,” Appl. Opt. 40, 1207-1215 (2001).
    [CrossRef]
  13. S. Li, W. Chen, and X. Su, “Reliability-guided phase unwrapping in wavelet-transofrm profilometry,” Appl. Opt. 47, 3369-3377 (2008).
    [CrossRef] [PubMed]
  14. Q. Kemao, “On window size selection in the windowed Fourier ridges algorithm,” Opt. Lasers Eng. 45, 1186-1192 (2007).
    [CrossRef]
  15. K. Qian, S. H. Soon, “Two-dimensional windowed Fourier frames for noise reduction in fringe pattern analysis,” Opt. Eng. 44, 075601 (2005).
    [CrossRef]
  16. Q. Kemao, “A simple phase unwrapping approach based on filtering by windowed Fourier transform: a note on the threshold selection,” Opt. Laser Technol. 40, 1091-1098 (2008).
    [CrossRef]
  17. 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]
  18. T. J. Flynn, “Two-dimensional phase unwrapping with minimum weighted discontinuity,” J. Opt. Soc. Am. A 14, 2692-2701 (1997).
    [CrossRef]

2008 (3)

Q. Kemao, H. Wang, and W. Gao, “Windowed Fourier transform for fringe pattern analysis: theoretical analyses,” Appl. Opt. 47, 5408-5419 (2008).

Q. Kemao, “A simple phase unwrapping approach based on filtering by windowed Fourier transform: a note on the threshold selection,” Opt. Laser Technol. 40, 1091-1098 (2008).
[CrossRef]

S. Li, W. Chen, and X. Su, “Reliability-guided phase unwrapping in wavelet-transofrm profilometry,” Appl. Opt. 47, 3369-3377 (2008).
[CrossRef] [PubMed]

2007 (4)

Q. Kemao, L. T. H. Nam, L. Feng, and S. H. Soon, “Comparative analysis on some filters for wrapped phase maps,” Appl. Opt. 46, 7412-7418 (2007).
[CrossRef] [PubMed]

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, “On window size selection in the windowed Fourier ridges algorithm,” Opt. Lasers Eng. 45, 1186-1192 (2007).
[CrossRef]

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

2006 (1)

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 (2)

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]

K. Qian, S. H. Soon, “Two-dimensional windowed Fourier frames for noise reduction in fringe pattern analysis,” Opt. Eng. 44, 075601 (2005).
[CrossRef]

2004 (1)

X. Su and W. Chen, “Reliability-guided phase unwrapping algorithm: a review,” Opt. Lasers Eng. 42, 245-261 (2004).
[CrossRef]

2001 (2)

X. Su and L. Xue, “Phase unwrapping algorithm based on fringe frequency analysis in Fourier-transform profilometry,” Opt. Eng. 40, 637-643 (2001).
[CrossRef]

L. Xue and X. Su, “Phase unwrapping algorithm based on frequency analysis for measurement of a complex object by the phase-measuring-profilometry method,” Appl. Opt. 40, 1207-1215 (2001).
[CrossRef]

1999 (2)

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]

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]

1998 (1)

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

1997 (1)

1982 (1)

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]

Chen, W.

S. Li, W. Chen, and X. Su, “Reliability-guided phase unwrapping in wavelet-transofrm profilometry,” Appl. Opt. 47, 3369-3377 (2008).
[CrossRef] [PubMed]

X. Su and W. Chen, “Reliability-guided phase unwrapping algorithm: a review,” Opt. Lasers Eng. 42, 245-261 (2004).
[CrossRef]

Cuevas, F. J.

Feng, L.

Flynn, T. J.

Gao, W.

Q. Kemao, H. Wang, and W. Gao, “Windowed Fourier transform for fringe pattern analysis: theoretical analyses,” Appl. Opt. 47, 5408-5419 (2008).

Ghiglia, D. C.

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

Itoh, K.

Kemao, Q.

Q. Kemao, H. Wang, and W. Gao, “Windowed Fourier transform for fringe pattern analysis: theoretical analyses,” Appl. Opt. 47, 5408-5419 (2008).

Q. Kemao, “A simple phase unwrapping approach based on filtering by windowed Fourier transform: a note on the threshold selection,” Opt. Laser Technol. 40, 1091-1098 (2008).
[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, “Two-dimensional windowed Fourier transform for fringe pattern analysis: principles, applications and implementations,” Opt. Lasers Engi. 45, 304-317 (2007).
[CrossRef]

Q. Kemao, “On window size selection in the windowed Fourier ridges algorithm,” Opt. Lasers Eng. 45, 1186-1192 (2007).
[CrossRef]

Q. Kemao, L. T. H. Nam, L. Feng, and S. H. Soon, “Comparative analysis on some filters for wrapped phase maps,” Appl. Opt. 46, 7412-7418 (2007).
[CrossRef] [PubMed]

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]

Li, S.

Malacara, D.

Marroguin, J. L.

Nam, L. T. H.

Pritt, M. D.

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

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]

K. Qian, S. H. Soon, “Two-dimensional windowed Fourier frames for noise reduction in fringe pattern analysis,” Opt. Eng. 44, 075601 (2005).
[CrossRef]

Rodriguez-Vera, R.

Servin, M.

Soon, S. H.

Q. Kemao, L. T. H. Nam, L. Feng, and S. H. Soon, “Comparative analysis on some filters for wrapped phase maps,” Appl. Opt. 46, 7412-7418 (2007).
[CrossRef] [PubMed]

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]

K. Qian, S. H. Soon, “Two-dimensional windowed Fourier frames for noise reduction in fringe pattern analysis,” Opt. Eng. 44, 075601 (2005).
[CrossRef]

Su, X.

S. Li, W. Chen, and X. Su, “Reliability-guided phase unwrapping in wavelet-transofrm profilometry,” Appl. Opt. 47, 3369-3377 (2008).
[CrossRef] [PubMed]

X. Su and W. Chen, “Reliability-guided phase unwrapping algorithm: a review,” Opt. Lasers Eng. 42, 245-261 (2004).
[CrossRef]

L. Xue and X. Su, “Phase unwrapping algorithm based on frequency analysis for measurement of a complex object by the phase-measuring-profilometry method,” Appl. Opt. 40, 1207-1215 (2001).
[CrossRef]

X. Su and L. Xue, “Phase unwrapping algorithm based on fringe frequency analysis in Fourier-transform profilometry,” Opt. Eng. 40, 637-643 (2001).
[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, H.

Q. Kemao, H. Wang, and W. Gao, “Windowed Fourier transform for fringe pattern analysis: theoretical analyses,” Appl. Opt. 47, 5408-5419 (2008).

Xue, L.

X. Su and L. Xue, “Phase unwrapping algorithm based on fringe frequency analysis in Fourier-transform profilometry,” Opt. Eng. 40, 637-643 (2001).
[CrossRef]

L. Xue and X. Su, “Phase unwrapping algorithm based on frequency analysis for measurement of a complex object by the phase-measuring-profilometry method,” Appl. Opt. 40, 1207-1215 (2001).
[CrossRef]

Appl. Opt. (5)

J. Opt. Soc. Am. A (1)

Key Eng. Mater. (1)

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. (1)

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]

Opt. Eng. (2)

X. Su and L. Xue, “Phase unwrapping algorithm based on fringe frequency analysis in Fourier-transform profilometry,” Opt. Eng. 40, 637-643 (2001).
[CrossRef]

K. Qian, S. H. Soon, “Two-dimensional windowed Fourier frames for noise reduction in fringe pattern analysis,” Opt. Eng. 44, 075601 (2005).
[CrossRef]

Opt. Laser Technol. (3)

Q. Kemao, “A simple phase unwrapping approach based on filtering by windowed Fourier transform: a note on the threshold selection,” Opt. Laser Technol. 40, 1091-1098 (2008).
[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]

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]

Opt. Lasers Eng. (2)

X. Su and W. Chen, “Reliability-guided phase unwrapping algorithm: a review,” Opt. Lasers Eng. 42, 245-261 (2004).
[CrossRef]

Q. Kemao, “On window size selection in the windowed Fourier ridges algorithm,” Opt. Lasers Eng. 45, 1186-1192 (2007).
[CrossRef]

Opt. Lasers Engi. (1)

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

Other (2)

Q. Kemao, H. Wang, and W. Gao, “Windowed Fourier transform for fringe pattern analysis: theoretical analyses,” Appl. Opt. 47, 5408-5419 (2008).

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

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (10)

Fig. 1
Fig. 1

Unwrapping an experimental ESPI phase with a simulated invalid region: (a) a wrapped phase map ( 200 × 200 ), (b) the filtered wrapped phase map ( σ = 10 ), (c) the filtered amplitude ( σ = 10 ), (d) the unwrapped phase ( σ = 10 ).

Fig. 2
Fig. 2

Unwrapping an experimental IFSAR phase: (a) a wrapped-phase map ( 512 × 512 ) from [1] with permission of John Wiley and Sons, Inc., (b) the filtered wrapped-phase map with ( σ = 2 ), (c) the filtered amplitude ( σ = 2 ), (d) the unwrapped phase ( σ = 2 ), (e) the filtered wrapped-phase map ( σ = 10 ), (f) the filtered amplitude ( σ = 10 ), (g) the unwrapped phase ( σ = 10 ), (h) the unwrapped phase with invalid regions identified ( σ = 10 ).

Fig. 3
Fig. 3

Unwrapping a MRI phase of a head: (a) a wrapped-phase map ( 256 × 256 ) from [1] with permission of John Wiley and Sons, Inc., (b) the filtered wrapped-phase map ( σ = 2 ), (c) the filtered amplitude ( σ = 2 ), (d) the unwrapped phase ( σ = 2 ).

Fig. 4
Fig. 4

Unwrapping a MRI phase of a knee: (a) a wrapped-phase map ( 256 × 256 ) from [1] with permission of John Wiley and Sons, Inc., (b) the filtered wrapped-phase map ( σ = 10 ), (c) the filtered amplitude ( σ = 10 ), (d) the unwrapped phase with invalid regions identified ( σ = 10 ).

Fig. 5
Fig. 5

Unwrapping a simulated IFSAR phase of Long’s Peak: (a) a wrapped-phase map ( 152 × 458 ) from [1] with permission of John Wiley and Sons, Inc., (b) the filtered wrapped-phase map ( σ = 2 ), (c) the filtered amplitude ( σ = 2 ), (d) the unwrapped phase ( σ = 2 ).

Fig. 6
Fig. 6

Unwrapping a simulated IFSAR phase of Isolation Peak (a) a wrapped-phase map ( 157 × 458 ) from [1] with permission of John Wiley and Sons, Inc., (b) the filtered wrapped-phase map ( σ = 2 ), (c) the filtered amplitude ( σ = 2 ), (d) the unwrapped phase ( σ = 2 ), (e) the known correlation map generated from simulation from [1] with permission of John Wiley and Sons, Inc.

Fig. 7
Fig. 7

Unwrapping a simulated phase of sheared planes: (a) a wrapped-phase map ( 257 × 257 ) from [1] with permission of John Wiley and Sons, Inc., (b) the filtered wrapped-phase map ( σ = 10 ), (c) the filtered amplitude ( σ = 10 ), (d) the unwrapped phase ( σ = 10 ).

Fig. 8
Fig. 8

Unwrapping a simulated noisy phase of sheared planes: (a) a wrapped-phase map ( 257 × 257 ) from [1] with permission of John Wiley and Sons, Inc., (b) the filtered wrapped-phase map ( σ = 10 ), (c) the filtered amplitude ( σ = 10 ), (d) the unwrapped phase ( σ = 10 ).

Fig. 9
Fig. 9

Unwrapping a simulated phase with a spiral discontinuity: (a) a wrapped-phase map ( 257 × 257 ) from [1] with permission of John Wiley and Sons, Inc., (b) the filtered wrapped-phase map ( σ = 10 ), (c) the filtered amplitude ( σ = 10 ), (d) the unwrapped phase ( σ = 10 ).

Fig. 10
Fig. 10

Windowed Fourier ridges as quality maps: (a) MRI head ( σ = 2 ), (b) sheared planes ( σ = 10 ), (c) noisy shear planes ( σ = 10 ), (d) spiral shear ( σ = 10 ).

Tables (1)

Tables Icon

Table 1 Selection of Preferred Window Size

Equations (3)

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

f ( x , y ) = b ( x , y ) exp [ j φ w ( x , y ) ] ,
f ¯ ( x , y ) = IWFT { WFT [ f ( x , y ) ] ¯ } ,
f ¯ ( x , y ) = b ¯ ( x , y ) exp [ j φ w ¯ ( x , y ) ] ,

Metrics