Abstract

In this paper we propose a novel hybrid three-dimensional phase-unwrapping algorithm, which we refer to here as the three-dimensional best-path avoiding singularity loops (3DBPASL) algorithm. This algorithm combines the advantages and avoids the drawbacks of two well-known 3D phase-unwrapping algorithms, namely, the 3D phase-unwrapping noise-immune technique and the 3D phase-unwrapping best-path technique. The hybrid technique presented here is more robust than its predecessors since it not only follows a discrete unwrapping path depending on a 3D quality map, but it also avoids any singularity loops that may occur in the unwrapping path. Simulation and experimental results have shown that the proposed algorithm outperforms its parent techniques in terms of reliability and robustness.

© 2009 Optical Society of America

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  1. D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms and Software (Wiley, 1998).
  2. M. D. Pritt and J. S. Shipman, “Least-square two-dimensional phase unwrapping using FFTs,” IEEE Trans. Geosci. Remote Sens. 32, 706-708 (1994).
    [CrossRef]
  3. D. C. Ghiglia and L. A. Romero, “Minimum Lp-norm two-dimensional phase unwrapping,” J. Opt. Soc. Am. 13, 1999-2013 (1996).
    [CrossRef]
  4. R. Cusack, J. M. Huntley, and H. T. Goldrein, “Improved noise-immune phase-unwrapping algorithm,” Appl. Opt. 34, 781-789 (1995).
    [CrossRef] [PubMed]
  5. S. A. Karout, M. A. Gdeisat, D. R. Burton, and M. J. Lalor, “Two-dimensional phase unwrapping using a hybrid genetic algorithm,” Appl. Opt. 46, 730-743 (2007).
    [CrossRef] [PubMed]
  6. M. Arevalillo Herráez, D. R. Burton, M. J. Lalor, and D. B. Clegg, “Robust, simple, and fast algorithm for phase unwrapping,” Appl. Opt. 35, 5847-5852 (1996).
    [CrossRef] [PubMed]
  7. W. Xu and I. Cumming, “A region-growing algorithm for InSAR phase unwrapping,” IEEE Trans. Geosci. Remote Sens. 37, 124-134 (1999).
    [CrossRef]
  8. M. Arevalillo Herráez, D. R. Burton, M. J. Lalor, and M. A. Gdeisat, “Fast two-dimensional phase unwrapping algorithm based on sorting by reliability following a non-continuous path,” Appl. Opt. 41, 7437-7444 (2002).
    [CrossRef] [PubMed]
  9. K. Stetson, J. Wahid, and P. Gauthier, “Noise-immune phase unwrapping by use of calculated wrap regions,” Appl. Opt. 36, 4830-4838 (1997).
    [CrossRef] [PubMed]
  10. F. Lilley, M. J. Lalor, and D. R. Burton, “Robust fringe analysis system for human body shape measurement,” Opt. Eng. 39, 187-195 (2000).
    [CrossRef]
  11. M. Costanitini, F. Malvarosa, L. Minati, and G. Milillo, “A three dimensional phase unwrapping algorithm for processing of multitemproral SAR interferometric measurements,” IEEE Trans. Geosci. Remote Sens. 40, 1741-1743(2002).
  12. H. S. Abdul-Rahman, M. A. Gdeisat, D. R. Burton, M. J. Lalor, F. Lilley, and A. Abid, “Three-dimensional Fourier fringe analysis,” Opt. Las. Eng. 46, 446-455 (2008).
    [CrossRef]
  13. R. Cusack and N. Papadakis, “New robust three-dimensional phase unwrapping algorithm: application on magnetic field mapping and undistorting echo-planar images,” NeuroImage 16, 754-764 (2002).
    [CrossRef] [PubMed]
  14. X. Su, W. Chen, Q. Zhang, and Y. Chao, “Dynamic 3D-shape measurement method based on FTP,” Opt. Las. Eng. 36, 49-64 (2001).
    [CrossRef]
  15. J. M. Huntley, “Three-dimensional noise-immune phase unwrapping algorithm,” Appl. Opt. 40, 3901-3908 (2001).
    [CrossRef]
  16. M. Jenkinson, “Fast, automated, N-dimensional phase unwrapping algorithm,” Magn. Reson. Med. 49, 193-197(2003).
    [CrossRef] [PubMed]
  17. H. S. Abdul-Rahman, M. A. Gdeisat, D. R. Burton, and M. J. Lalor, “Fast three-dimensional phase unwrapping algorithm based on sorting by reliability following a non-continuous path,” Proc. SPIE 5856, 32-40 (2005).
    [CrossRef]
  18. H. S. Abdul-Rahman, M. A. Gdeisat, D. R. Burton, and M. J. Lalor, “Three-dimensional phase unwrapping algorithms: a comparison,” presented at the Photon06 Conference, Manchester, UK, 4-7 Sept. 2006.
  19. H. S. Abdul-Rahman, M. A. Gdeisat, D. R. Burton, M. J. Lalor, F. Lilley, and C. Moore, “Fast and robust three-dimensional best-path phase unwrapping algorithm,” Appl. Opt. 46, 6623-6635 (2007).
    [CrossRef] [PubMed]
  20. O. Marklund, J. Huntley, and R. Cusack, “Robust unwrapping algorithm for three-dimensional phase volumes of arbitrary shape containing knotted phase singularity loops,” Opt. Eng. 46, 085601 (2007).
    [CrossRef]
  21. M. Salfity, P. Ruiz, J. Huntley, M. Graves, R. Cusack, and D. Beauregard, “Branch cut surface placement for unwrapping of undersampled three-dimensional phase data: application to magnetic resonance imaging arterial flow mapping,” Appl. Opt. 45, 2711-2722 (2006).
    [CrossRef] [PubMed]
  22. H. S. Abdul-Rahman, “Three-dimensional Fourier fringe analysis and phase unwrapping,” PhD thesis (Liverpool John Moores University, 2007).
  23. URL:http://www.ljmu.ac.uk/GERI/90202.htm.

2008

H. S. Abdul-Rahman, M. A. Gdeisat, D. R. Burton, M. J. Lalor, F. Lilley, and A. Abid, “Three-dimensional Fourier fringe analysis,” Opt. Las. Eng. 46, 446-455 (2008).
[CrossRef]

2007

2006

2005

H. S. Abdul-Rahman, M. A. Gdeisat, D. R. Burton, and M. J. Lalor, “Fast three-dimensional phase unwrapping algorithm based on sorting by reliability following a non-continuous path,” Proc. SPIE 5856, 32-40 (2005).
[CrossRef]

2003

M. Jenkinson, “Fast, automated, N-dimensional phase unwrapping algorithm,” Magn. Reson. Med. 49, 193-197(2003).
[CrossRef] [PubMed]

2002

M. Costanitini, F. Malvarosa, L. Minati, and G. Milillo, “A three dimensional phase unwrapping algorithm for processing of multitemproral SAR interferometric measurements,” IEEE Trans. Geosci. Remote Sens. 40, 1741-1743(2002).

R. Cusack and N. Papadakis, “New robust three-dimensional phase unwrapping algorithm: application on magnetic field mapping and undistorting echo-planar images,” NeuroImage 16, 754-764 (2002).
[CrossRef] [PubMed]

M. Arevalillo Herráez, D. R. Burton, M. J. Lalor, and M. A. Gdeisat, “Fast two-dimensional phase unwrapping algorithm based on sorting by reliability following a non-continuous path,” Appl. Opt. 41, 7437-7444 (2002).
[CrossRef] [PubMed]

2001

X. Su, W. Chen, Q. Zhang, and Y. Chao, “Dynamic 3D-shape measurement method based on FTP,” Opt. Las. Eng. 36, 49-64 (2001).
[CrossRef]

J. M. Huntley, “Three-dimensional noise-immune phase unwrapping algorithm,” Appl. Opt. 40, 3901-3908 (2001).
[CrossRef]

2000

F. Lilley, M. J. Lalor, and D. R. Burton, “Robust fringe analysis system for human body shape measurement,” Opt. Eng. 39, 187-195 (2000).
[CrossRef]

1999

W. Xu and I. Cumming, “A region-growing algorithm for InSAR phase unwrapping,” IEEE Trans. Geosci. Remote Sens. 37, 124-134 (1999).
[CrossRef]

1997

1996

1995

1994

M. D. Pritt and J. S. Shipman, “Least-square two-dimensional phase unwrapping using FFTs,” IEEE Trans. Geosci. Remote Sens. 32, 706-708 (1994).
[CrossRef]

Abdul-Rahman, H. S.

H. S. Abdul-Rahman, M. A. Gdeisat, D. R. Burton, M. J. Lalor, F. Lilley, and A. Abid, “Three-dimensional Fourier fringe analysis,” Opt. Las. Eng. 46, 446-455 (2008).
[CrossRef]

H. S. Abdul-Rahman, M. A. Gdeisat, D. R. Burton, M. J. Lalor, F. Lilley, and C. Moore, “Fast and robust three-dimensional best-path phase unwrapping algorithm,” Appl. Opt. 46, 6623-6635 (2007).
[CrossRef] [PubMed]

H. S. Abdul-Rahman, M. A. Gdeisat, D. R. Burton, and M. J. Lalor, “Fast three-dimensional phase unwrapping algorithm based on sorting by reliability following a non-continuous path,” Proc. SPIE 5856, 32-40 (2005).
[CrossRef]

H. S. Abdul-Rahman, M. A. Gdeisat, D. R. Burton, and M. J. Lalor, “Three-dimensional phase unwrapping algorithms: a comparison,” presented at the Photon06 Conference, Manchester, UK, 4-7 Sept. 2006.

H. S. Abdul-Rahman, “Three-dimensional Fourier fringe analysis and phase unwrapping,” PhD thesis (Liverpool John Moores University, 2007).

Abid, A.

H. S. Abdul-Rahman, M. A. Gdeisat, D. R. Burton, M. J. Lalor, F. Lilley, and A. Abid, “Three-dimensional Fourier fringe analysis,” Opt. Las. Eng. 46, 446-455 (2008).
[CrossRef]

Beauregard, D.

Burton, D. R.

H. S. Abdul-Rahman, M. A. Gdeisat, D. R. Burton, M. J. Lalor, F. Lilley, and A. Abid, “Three-dimensional Fourier fringe analysis,” Opt. Las. Eng. 46, 446-455 (2008).
[CrossRef]

H. S. Abdul-Rahman, M. A. Gdeisat, D. R. Burton, M. J. Lalor, F. Lilley, and C. Moore, “Fast and robust three-dimensional best-path phase unwrapping algorithm,” Appl. Opt. 46, 6623-6635 (2007).
[CrossRef] [PubMed]

S. A. Karout, M. A. Gdeisat, D. R. Burton, and M. J. Lalor, “Two-dimensional phase unwrapping using a hybrid genetic algorithm,” Appl. Opt. 46, 730-743 (2007).
[CrossRef] [PubMed]

H. S. Abdul-Rahman, M. A. Gdeisat, D. R. Burton, and M. J. Lalor, “Fast three-dimensional phase unwrapping algorithm based on sorting by reliability following a non-continuous path,” Proc. SPIE 5856, 32-40 (2005).
[CrossRef]

M. Arevalillo Herráez, D. R. Burton, M. J. Lalor, and M. A. Gdeisat, “Fast two-dimensional phase unwrapping algorithm based on sorting by reliability following a non-continuous path,” Appl. Opt. 41, 7437-7444 (2002).
[CrossRef] [PubMed]

F. Lilley, M. J. Lalor, and D. R. Burton, “Robust fringe analysis system for human body shape measurement,” Opt. Eng. 39, 187-195 (2000).
[CrossRef]

M. Arevalillo Herráez, D. R. Burton, M. J. Lalor, and D. B. Clegg, “Robust, simple, and fast algorithm for phase unwrapping,” Appl. Opt. 35, 5847-5852 (1996).
[CrossRef] [PubMed]

H. S. Abdul-Rahman, M. A. Gdeisat, D. R. Burton, and M. J. Lalor, “Three-dimensional phase unwrapping algorithms: a comparison,” presented at the Photon06 Conference, Manchester, UK, 4-7 Sept. 2006.

Chao, Y.

X. Su, W. Chen, Q. Zhang, and Y. Chao, “Dynamic 3D-shape measurement method based on FTP,” Opt. Las. Eng. 36, 49-64 (2001).
[CrossRef]

Chen, W.

X. Su, W. Chen, Q. Zhang, and Y. Chao, “Dynamic 3D-shape measurement method based on FTP,” Opt. Las. Eng. 36, 49-64 (2001).
[CrossRef]

Clegg, D. B.

Costanitini, M.

M. Costanitini, F. Malvarosa, L. Minati, and G. Milillo, “A three dimensional phase unwrapping algorithm for processing of multitemproral SAR interferometric measurements,” IEEE Trans. Geosci. Remote Sens. 40, 1741-1743(2002).

Cumming, I.

W. Xu and I. Cumming, “A region-growing algorithm for InSAR phase unwrapping,” IEEE Trans. Geosci. Remote Sens. 37, 124-134 (1999).
[CrossRef]

Cusack, R.

O. Marklund, J. Huntley, and R. Cusack, “Robust unwrapping algorithm for three-dimensional phase volumes of arbitrary shape containing knotted phase singularity loops,” Opt. Eng. 46, 085601 (2007).
[CrossRef]

M. Salfity, P. Ruiz, J. Huntley, M. Graves, R. Cusack, and D. Beauregard, “Branch cut surface placement for unwrapping of undersampled three-dimensional phase data: application to magnetic resonance imaging arterial flow mapping,” Appl. Opt. 45, 2711-2722 (2006).
[CrossRef] [PubMed]

R. Cusack and N. Papadakis, “New robust three-dimensional phase unwrapping algorithm: application on magnetic field mapping and undistorting echo-planar images,” NeuroImage 16, 754-764 (2002).
[CrossRef] [PubMed]

R. Cusack, J. M. Huntley, and H. T. Goldrein, “Improved noise-immune phase-unwrapping algorithm,” Appl. Opt. 34, 781-789 (1995).
[CrossRef] [PubMed]

Gauthier, P.

Gdeisat, M. A.

H. S. Abdul-Rahman, M. A. Gdeisat, D. R. Burton, M. J. Lalor, F. Lilley, and A. Abid, “Three-dimensional Fourier fringe analysis,” Opt. Las. Eng. 46, 446-455 (2008).
[CrossRef]

H. S. Abdul-Rahman, M. A. Gdeisat, D. R. Burton, M. J. Lalor, F. Lilley, and C. Moore, “Fast and robust three-dimensional best-path phase unwrapping algorithm,” Appl. Opt. 46, 6623-6635 (2007).
[CrossRef] [PubMed]

S. A. Karout, M. A. Gdeisat, D. R. Burton, and M. J. Lalor, “Two-dimensional phase unwrapping using a hybrid genetic algorithm,” Appl. Opt. 46, 730-743 (2007).
[CrossRef] [PubMed]

H. S. Abdul-Rahman, M. A. Gdeisat, D. R. Burton, and M. J. Lalor, “Fast three-dimensional phase unwrapping algorithm based on sorting by reliability following a non-continuous path,” Proc. SPIE 5856, 32-40 (2005).
[CrossRef]

M. Arevalillo Herráez, D. R. Burton, M. J. Lalor, and M. A. Gdeisat, “Fast two-dimensional phase unwrapping algorithm based on sorting by reliability following a non-continuous path,” Appl. Opt. 41, 7437-7444 (2002).
[CrossRef] [PubMed]

H. S. Abdul-Rahman, M. A. Gdeisat, D. R. Burton, and M. J. Lalor, “Three-dimensional phase unwrapping algorithms: a comparison,” presented at the Photon06 Conference, Manchester, UK, 4-7 Sept. 2006.

Ghiglia, D. C.

D. C. Ghiglia and L. A. Romero, “Minimum Lp-norm two-dimensional phase unwrapping,” J. Opt. Soc. Am. 13, 1999-2013 (1996).
[CrossRef]

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

Goldrein, H. T.

Graves, M.

Herráez, M. Arevalillo

Huntley, J.

O. Marklund, J. Huntley, and R. Cusack, “Robust unwrapping algorithm for three-dimensional phase volumes of arbitrary shape containing knotted phase singularity loops,” Opt. Eng. 46, 085601 (2007).
[CrossRef]

M. Salfity, P. Ruiz, J. Huntley, M. Graves, R. Cusack, and D. Beauregard, “Branch cut surface placement for unwrapping of undersampled three-dimensional phase data: application to magnetic resonance imaging arterial flow mapping,” Appl. Opt. 45, 2711-2722 (2006).
[CrossRef] [PubMed]

Huntley, J. M.

Jenkinson, M.

M. Jenkinson, “Fast, automated, N-dimensional phase unwrapping algorithm,” Magn. Reson. Med. 49, 193-197(2003).
[CrossRef] [PubMed]

Karout, S. A.

Lalor, M. J.

H. S. Abdul-Rahman, M. A. Gdeisat, D. R. Burton, M. J. Lalor, F. Lilley, and A. Abid, “Three-dimensional Fourier fringe analysis,” Opt. Las. Eng. 46, 446-455 (2008).
[CrossRef]

H. S. Abdul-Rahman, M. A. Gdeisat, D. R. Burton, M. J. Lalor, F. Lilley, and C. Moore, “Fast and robust three-dimensional best-path phase unwrapping algorithm,” Appl. Opt. 46, 6623-6635 (2007).
[CrossRef] [PubMed]

S. A. Karout, M. A. Gdeisat, D. R. Burton, and M. J. Lalor, “Two-dimensional phase unwrapping using a hybrid genetic algorithm,” Appl. Opt. 46, 730-743 (2007).
[CrossRef] [PubMed]

H. S. Abdul-Rahman, M. A. Gdeisat, D. R. Burton, and M. J. Lalor, “Fast three-dimensional phase unwrapping algorithm based on sorting by reliability following a non-continuous path,” Proc. SPIE 5856, 32-40 (2005).
[CrossRef]

M. Arevalillo Herráez, D. R. Burton, M. J. Lalor, and M. A. Gdeisat, “Fast two-dimensional phase unwrapping algorithm based on sorting by reliability following a non-continuous path,” Appl. Opt. 41, 7437-7444 (2002).
[CrossRef] [PubMed]

F. Lilley, M. J. Lalor, and D. R. Burton, “Robust fringe analysis system for human body shape measurement,” Opt. Eng. 39, 187-195 (2000).
[CrossRef]

M. Arevalillo Herráez, D. R. Burton, M. J. Lalor, and D. B. Clegg, “Robust, simple, and fast algorithm for phase unwrapping,” Appl. Opt. 35, 5847-5852 (1996).
[CrossRef] [PubMed]

H. S. Abdul-Rahman, M. A. Gdeisat, D. R. Burton, and M. J. Lalor, “Three-dimensional phase unwrapping algorithms: a comparison,” presented at the Photon06 Conference, Manchester, UK, 4-7 Sept. 2006.

Lilley, F.

H. S. Abdul-Rahman, M. A. Gdeisat, D. R. Burton, M. J. Lalor, F. Lilley, and A. Abid, “Three-dimensional Fourier fringe analysis,” Opt. Las. Eng. 46, 446-455 (2008).
[CrossRef]

H. S. Abdul-Rahman, M. A. Gdeisat, D. R. Burton, M. J. Lalor, F. Lilley, and C. Moore, “Fast and robust three-dimensional best-path phase unwrapping algorithm,” Appl. Opt. 46, 6623-6635 (2007).
[CrossRef] [PubMed]

F. Lilley, M. J. Lalor, and D. R. Burton, “Robust fringe analysis system for human body shape measurement,” Opt. Eng. 39, 187-195 (2000).
[CrossRef]

Malvarosa, F.

M. Costanitini, F. Malvarosa, L. Minati, and G. Milillo, “A three dimensional phase unwrapping algorithm for processing of multitemproral SAR interferometric measurements,” IEEE Trans. Geosci. Remote Sens. 40, 1741-1743(2002).

Marklund, O.

O. Marklund, J. Huntley, and R. Cusack, “Robust unwrapping algorithm for three-dimensional phase volumes of arbitrary shape containing knotted phase singularity loops,” Opt. Eng. 46, 085601 (2007).
[CrossRef]

Milillo, G.

M. Costanitini, F. Malvarosa, L. Minati, and G. Milillo, “A three dimensional phase unwrapping algorithm for processing of multitemproral SAR interferometric measurements,” IEEE Trans. Geosci. Remote Sens. 40, 1741-1743(2002).

Minati, L.

M. Costanitini, F. Malvarosa, L. Minati, and G. Milillo, “A three dimensional phase unwrapping algorithm for processing of multitemproral SAR interferometric measurements,” IEEE Trans. Geosci. Remote Sens. 40, 1741-1743(2002).

Moore, C.

Papadakis, N.

R. Cusack and N. Papadakis, “New robust three-dimensional phase unwrapping algorithm: application on magnetic field mapping and undistorting echo-planar images,” NeuroImage 16, 754-764 (2002).
[CrossRef] [PubMed]

Pritt, M. D.

M. D. Pritt and J. S. Shipman, “Least-square two-dimensional phase unwrapping using FFTs,” IEEE Trans. Geosci. Remote Sens. 32, 706-708 (1994).
[CrossRef]

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

Romero, L. A.

D. C. Ghiglia and L. A. Romero, “Minimum Lp-norm two-dimensional phase unwrapping,” J. Opt. Soc. Am. 13, 1999-2013 (1996).
[CrossRef]

Ruiz, P.

Salfity, M.

Shipman, J. S.

M. D. Pritt and J. S. Shipman, “Least-square two-dimensional phase unwrapping using FFTs,” IEEE Trans. Geosci. Remote Sens. 32, 706-708 (1994).
[CrossRef]

Stetson, K.

Su, X.

X. Su, W. Chen, Q. Zhang, and Y. Chao, “Dynamic 3D-shape measurement method based on FTP,” Opt. Las. Eng. 36, 49-64 (2001).
[CrossRef]

Wahid, J.

Xu, W.

W. Xu and I. Cumming, “A region-growing algorithm for InSAR phase unwrapping,” IEEE Trans. Geosci. Remote Sens. 37, 124-134 (1999).
[CrossRef]

Zhang, Q.

X. Su, W. Chen, Q. Zhang, and Y. Chao, “Dynamic 3D-shape measurement method based on FTP,” Opt. Las. Eng. 36, 49-64 (2001).
[CrossRef]

Appl. Opt.

R. Cusack, J. M. Huntley, and H. T. Goldrein, “Improved noise-immune phase-unwrapping algorithm,” Appl. Opt. 34, 781-789 (1995).
[CrossRef] [PubMed]

S. A. Karout, M. A. Gdeisat, D. R. Burton, and M. J. Lalor, “Two-dimensional phase unwrapping using a hybrid genetic algorithm,” Appl. Opt. 46, 730-743 (2007).
[CrossRef] [PubMed]

M. Arevalillo Herráez, D. R. Burton, M. J. Lalor, and D. B. Clegg, “Robust, simple, and fast algorithm for phase unwrapping,” Appl. Opt. 35, 5847-5852 (1996).
[CrossRef] [PubMed]

M. Arevalillo Herráez, D. R. Burton, M. J. Lalor, and M. A. Gdeisat, “Fast two-dimensional phase unwrapping algorithm based on sorting by reliability following a non-continuous path,” Appl. Opt. 41, 7437-7444 (2002).
[CrossRef] [PubMed]

K. Stetson, J. Wahid, and P. Gauthier, “Noise-immune phase unwrapping by use of calculated wrap regions,” Appl. Opt. 36, 4830-4838 (1997).
[CrossRef] [PubMed]

J. M. Huntley, “Three-dimensional noise-immune phase unwrapping algorithm,” Appl. Opt. 40, 3901-3908 (2001).
[CrossRef]

H. S. Abdul-Rahman, M. A. Gdeisat, D. R. Burton, M. J. Lalor, F. Lilley, and C. Moore, “Fast and robust three-dimensional best-path phase unwrapping algorithm,” Appl. Opt. 46, 6623-6635 (2007).
[CrossRef] [PubMed]

M. Salfity, P. Ruiz, J. Huntley, M. Graves, R. Cusack, and D. Beauregard, “Branch cut surface placement for unwrapping of undersampled three-dimensional phase data: application to magnetic resonance imaging arterial flow mapping,” Appl. Opt. 45, 2711-2722 (2006).
[CrossRef] [PubMed]

IEEE Trans. Geosci. Remote Sens.

M. Costanitini, F. Malvarosa, L. Minati, and G. Milillo, “A three dimensional phase unwrapping algorithm for processing of multitemproral SAR interferometric measurements,” IEEE Trans. Geosci. Remote Sens. 40, 1741-1743(2002).

M. D. Pritt and J. S. Shipman, “Least-square two-dimensional phase unwrapping using FFTs,” IEEE Trans. Geosci. Remote Sens. 32, 706-708 (1994).
[CrossRef]

W. Xu and I. Cumming, “A region-growing algorithm for InSAR phase unwrapping,” IEEE Trans. Geosci. Remote Sens. 37, 124-134 (1999).
[CrossRef]

J. Opt. Soc. Am.

D. C. Ghiglia and L. A. Romero, “Minimum Lp-norm two-dimensional phase unwrapping,” J. Opt. Soc. Am. 13, 1999-2013 (1996).
[CrossRef]

Magn. Reson. Med.

M. Jenkinson, “Fast, automated, N-dimensional phase unwrapping algorithm,” Magn. Reson. Med. 49, 193-197(2003).
[CrossRef] [PubMed]

NeuroImage

R. Cusack and N. Papadakis, “New robust three-dimensional phase unwrapping algorithm: application on magnetic field mapping and undistorting echo-planar images,” NeuroImage 16, 754-764 (2002).
[CrossRef] [PubMed]

Opt. Eng.

F. Lilley, M. J. Lalor, and D. R. Burton, “Robust fringe analysis system for human body shape measurement,” Opt. Eng. 39, 187-195 (2000).
[CrossRef]

O. Marklund, J. Huntley, and R. Cusack, “Robust unwrapping algorithm for three-dimensional phase volumes of arbitrary shape containing knotted phase singularity loops,” Opt. Eng. 46, 085601 (2007).
[CrossRef]

Opt. Las. Eng.

X. Su, W. Chen, Q. Zhang, and Y. Chao, “Dynamic 3D-shape measurement method based on FTP,” Opt. Las. Eng. 36, 49-64 (2001).
[CrossRef]

H. S. Abdul-Rahman, M. A. Gdeisat, D. R. Burton, M. J. Lalor, F. Lilley, and A. Abid, “Three-dimensional Fourier fringe analysis,” Opt. Las. Eng. 46, 446-455 (2008).
[CrossRef]

Proc. SPIE

H. S. Abdul-Rahman, M. A. Gdeisat, D. R. Burton, and M. J. Lalor, “Fast three-dimensional phase unwrapping algorithm based on sorting by reliability following a non-continuous path,” Proc. SPIE 5856, 32-40 (2005).
[CrossRef]

Other

H. S. Abdul-Rahman, M. A. Gdeisat, D. R. Burton, and M. J. Lalor, “Three-dimensional phase unwrapping algorithms: a comparison,” presented at the Photon06 Conference, Manchester, UK, 4-7 Sept. 2006.

H. S. Abdul-Rahman, “Three-dimensional Fourier fringe analysis and phase unwrapping,” PhD thesis (Liverpool John Moores University, 2007).

URL:http://www.ljmu.ac.uk/GERI/90202.htm.

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

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Figures (18)

Fig. 1
Fig. 1

Closed singularity loops and partial singularity loops in the phase volume.

Fig. 2
Fig. 2

Example of loops ambiguity in a 3 × 3 × 3 wrapped phase volume.

Fig. 3
Fig. 3

Singularity loop ambiguity resulting from a C-shaped loop.

Fig. 4
Fig. 4

Definition of zero-weighted edges.

Fig. 5
Fig. 5

Definition of closed singularity loops and partial singularity loops in the phase volume.

Fig. 6
Fig. 6

Example of a partial singularity loop that needs to be closed.

Fig. 7
Fig. 7

Demonstration of closing partial loops.

Fig. 8
Fig. 8

(a) U-shaped segment and (b) L-shaped segment.

Fig. 9
Fig. 9

Flow chart for processing a closed loop.

Fig. 10
Fig. 10

Demonstration of processing a closed loop.

Fig. 11
Fig. 11

Example 1 of replacing U-shaped segments.

Fig. 12
Fig. 12

Example 2 to illustrate the process of replacing U-shaped segments.

Fig. 13
Fig. 13

(a) Example 3 to illustrate the process of replacing U-shaped segments and (b) top view of the figure in (a).

Fig. 14
Fig. 14

(a) Example 4 of replacing U-shaped segments and (b) side view of the figure in (a).

Fig. 15
Fig. 15

L-shaped segment and its replacement.

Fig. 16
Fig. 16

Identifying L-shaped segment replacement using U-shaped segments rules. (a) Set of U-shaped segment and its equivalent, (b) L-turn segment and its equivalent, (c) new U-shaped segment and its equivalent.

Fig. 17
Fig. 17

Results for frames 46, 55, and 84 of a simulated object shown in the first, second, and third columns, respectively. Row (a) shows the wrapped-phase maps. A selection of different unwrapped phase maps are then shown that have been produced by the following respective unwrapping algorithms: row (b) Cusack’s algorithm, row (c) PRELUDE algorithm, row (d) Huntley’s algorithm, row (e) the best-path algorithm, and row (f) the 3DBPASL algorithm.

Fig. 18
Fig. 18

Results for frames 0, 15, and 24 of the RANDO phantom’s face real experimental example, with added noise, shown in the first, second, and third columns, respectively. Row (a) shows the wrapped-phase maps. A selection of different unwrapped phase maps are then shown that have been produced by the following respective unwrapping algorithms: row (b) Cusack’s algorithm, row (c) PRELUDE algorithm, row (d) Huntley’s algorithm, row (e) the best-path algorithm, and row (f) the 3DBPASL algorithm.

Equations (44)

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r x = [ ψ i , j , k ψ i , j + 1 , k 2 π ] + [ ψ i , j + 1 , k ψ i , j + 1 , k + 1 2 π ] + [ ψ i , j + 1 , k + 1 ψ i , j , k + 1 2 π ] + [ ψ i , j , k + 1 ψ i , j , k 2 π ] ,
r y = [ ψ i , j , k ψ i , j , k + 1 2 π ] + [ ψ i , j , k + 1 ψ i + 1 , j , k + 1 2 π ] + [ ψ i + 1 , j , k + 1 ψ i + 1 , j , k 2 π ] + [ ψ i + 1 , j , k ψ i , j , k 2 π ] ,
r z = [ ψ i , j , k ψ i + 1 , j , k 2 π ] + [ ψ i + 1 , j , k ψ i + 1 , j + 1 , k 2 π ] + [ ψ i + 1 , j + 1 , k ψ i , j + 1 , k 2 π ] + [ ψ i , j + 1 , k ψ i , j , k 2 π ] ,
r c 1 { type = t c 1 sign = s c 1 index = ( i c 1 , j c 1 , k c 1 ) } r c { type = t c sign = s c index = ( i c , j c , k c ) } r c + 1 { type = t c + 1 sign = s c + 1 index = ( i c + 1 , j c + 1 , k c + 1 ) } .
{ r c 1 . type = r c + 1 . type r c . type r c 1 . sign = r c + 1 . sign r c 1 . index r c + 1 . index = 1 } .
r c { type = t c sign = s c index = ( i c , j c , k c ) } r c + 1 { type = t c + 1 sign = s c + 1 index = ( i c + 1 , j c + 1 , k c + 1 ) } .
{ r c . type r c + 1 . type }
r c 1 { type = x sign = + v e index = ( i , j , k ) } r c { type = z sign = v e index = ( i , j , k ) } r c + 1 { type = x sign = v e index = ( i , j , k 1 ) } .
r n { type = z sign = v e index = ( i 1 , j , k ) } .
z w e { type = y index = ( i , j , k ) } .
r n . type = r c . type ,
r n . sign = r c . sign ,
z w e . type ( r c . type or r c ± 1 . type ) .
r c 1 { type = z sign = v e index = ( i , j , k + 1 ) } r c { type = x sign = v e index = ( i , j , k ) } r c + 1 { type = z sign = + v e index = ( i 1 , j , k + 1 ) } .
r n { type = x sign = v e index = ( i , j , k + 1 ) } ,
z w e { type = y index = ( i , j , k + 1 ) } .
r c . index r n . index = 1.
r n . index = { i ± λ x j ± λ y k ± λ z } ,
λ x = { 1 , if   r c + 1 . type = x 0 , otherwise ,
λ y = { 1 , if   r c + 1 . type = y 0 , otherwise ,
λ z = { 1 , if   r c + 1 . type = z 0 , otherwise .
r c 1 { type = x sign = v e index = ( i + 1 , j 1 , k ) } r c { type = y sign = + v e index = ( i , j , k ) } r c + 1 { type = x sign = + v e index = ( i + 1 , j , k ) } .
r n { type = y sign = + v e index = ( i + 1 , j , k ) } .
z w e { type = z index = ( i + 1 , j , k ) } .
r n . index = { i + λ x × ( r c + 1 . sign ) i + λ y × ( r c + 1 . sign) i + λ z × ( r c + 1 . sign) } ,
r c 1 { type = z sign = v e index = ( i , j 1 , k + 1 ) } r c { type = y sign = + v e index = ( i , j , k ) } r c + 1 { type = z sign = + v e index = ( i , j , k + 1 ) } .
r n { type = y sign = + v e index = ( i , j , k + 1 ) } .
z w e { type = x index = ( i , j , k + 1 ) } .
r c 1 r c r c + 1 ,
r n . type = r c . type ,
r n . sign = r c . sign ,
r n . index = { i + λ x × ( r c + 1 . sign ) j + λ y × ( r c + 1 . sign) k + λ z × ( r c + 1 . sign) } ,
λ x = { 1 , if r c + 1 . type = x 0 , otherwise ,
λ y = { 1 , if r c + 1 . type = y 0 , otherwise ,
λ z = { 1 , if r c + 1 . type = z 0 , otherwise .
z w e . type ( r c . type or r c + 1 . type ) ,
z w e . index = { r c 1 . index if   r c . sign = v e r c + 1 . index if   r c . sign = + v e .
r c r c + 1 ,
r ¯ n 2 . type = r n 2 . type ,
r ¯ n 2 . sign = r n 2 . sign ,
r ¯ n 2 . index = r n 2 . index ;
z ( i , j , t ) = 10 × [ σ 1 ( t ) · sin [ x ( i , j ) ] x ( i , j ) + σ 2 ( t ) · sin [ y ( i , j ) ] y ( i , j ) ] ,
σ 1 ( t ) = 1.50 ( 0.01 × ( t + 1 ) ) ,
σ 2 ( t ) = 0.49 + ( 0.01 × ( t + 1 ) ) ,

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