Abstract

We present a hybrid three-dimensional (3D) unwrapping algorithm that combines the strengths of two other fast and robust existing techniques. In particular, a branch-cut surface algorithm and a path- following method have been integrated in a symbiotic way, still keeping execution times within a range that permits their use in real-time applications that need a relatively fast solution to the problem. First, branch-cut surfaces are calculated, disregarding partial residue loops that end at the boundary of the 3D phase volume. These partial loops are then used to define a quality for each image voxel. Finally, unwrapping proceeds along a path determined by a minimum spanning tree (MST). The MST is built according to the quality of the voxels and avoids crossing the branch-cut surfaces determined at the first step. The resulting technique shows a higher robustness than any of the two methods used in isolation. On the one hand, the 3D MST algorithm benefits from the branch-cut surfaces, which endows it with a higher robustness to noise and open-ended wraps. On the other hand, incorrectly placed surfaces due to open loops at the boundaries in the branch-cut surface approach disappear.

© 2009 Optical Society of America

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  2. M. Costantini, F. Malvarosa, F. Minati, L. Pietranera, and G. Milillo, “A three-dimensional phase unwrapping algorithm for processing of multitemporal sar interferometric measurements,” in Geoscience and Remote Sensing Symposium, 2002 (IEEE, 2002), Vol. 3, pp. 1741-1743.
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    [CrossRef] [PubMed]
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    [CrossRef]
  7. J. M. B. Dias and J. M. N. Leitão, “The ZπM algorithm: a method for interferometric image reconstruction in SAR/SAS,” IEEE Trans. Image Process. 11, 408-422 (2002).
    [CrossRef]
  8. L. Aiello, D. Riccio, P. Ferraro, S. Grilli, L. Sansone, G. Coppola, S. D. Nicola, and A. Finizio, “Green's formulation for robust phase unwrapping in digital holography,” Opt. Lasers Eng. 45, 750-755 (2007).
    [CrossRef]
  9. L. Ying, Z. Liang, D. Munson, R. Koetter, and B. Frey, “Unwrapping of MR phase images using a Markov random field model,” IEEE Trans. Med. Imaging 25, 128-136(2006).
    [CrossRef] [PubMed]
  10. J. Bioucas-Dias and G. Valadao, “Phase unwrapping via graph cuts,” IEEE Trans Image Process. 16, 698-709 (2007).
    [CrossRef]
  11. J. M. Huntley, “Three-dimensional noise-immune phase unwrapping algorithm,” Appl. Opt. 40, 3901-3908 (2001).
    [CrossRef]
  12. M. A. Herráez, J. G. Boticario, M. J. Lalor, and D. R. Burton, “Agglomerative clustering-based approach for two-dimensional phase unwrapping,” Appl. Opt. 44, 1129-1140 (2005).
    [CrossRef] [PubMed]
  13. A. Baldi, “Two-dimensional phase unwrapping by quad-tree decomposition,” Appl. Opt. 40, 1187-1194 (2001).
    [CrossRef]
  14. M. Jenkinson, “A fast, automated, n-dimensional phase unwrapping algorithm,” Magn. Reson. Med. 49, 193-197(2003).
    [CrossRef] [PubMed]
  15. M. A. Herráez, D. R. Burton, M. J. Lalor, and M. A. Gdeisat, “Robust, fast and effective two dimensional automatic phase unwrapping algorithm based on image decomposition,” Appl. Opt. 41, 7445-7455 (2002).
    [CrossRef] [PubMed]
  16. N. H. Ching, D. Rosenfeld, and M. Braun, “Two-dimensional phase unwrapping using a minimum spanning tree algorithm,” IEEE Trans. Image Process. 1, 355-365(1992).
    [CrossRef]
  17. L. An, Q.-S. Xiang, and S. Chavez, “A fast implementation of the minimum spanning tree method for phase unwrapping,” IEEE Trans. Med. Imaging 19, 805-808(2000).
    [CrossRef] [PubMed]
  18. M. A. Herráez, D. R. Burton, M. J. Lalor, and M. A. Gdeisat, “Fast two-dimensional phase-unwrapping algorithm based on sorting by reliability following a noncontinuous path,” Appl. Opt. 41, 7437-7444 (2002).
    [CrossRef] [PubMed]
  19. H. S. Abdul-Rahman, M. A. Gdeisat, D. R. Burton, M. J. Lalor, F. Lilley, and C. J. Moore, “Fast and robust three-dimensional best path phase unwrapping algorithm,” Appl. Opt. 46, 6623-6635 (2007).
    [CrossRef] [PubMed]
  20. R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23, 713-720 (1988).
    [CrossRef]
  21. H. W. Kuhn, “The Hungarian method for the assignment problem,” Naval Res. Log. Q. 2, 83-97 (1955).
    [CrossRef]
  22. “Extending the dynamic range of phase contrast magnetic resonance velocity imaging using advanced higher-dimensional phase unwrapping algorithms,” J. R. Soc. Interface 3, 415-427(2006).
    [CrossRef] [PubMed]
  23. M. F. Salfity, P. D. Ruiz, J. M. Huntley, M. J. Graves, R. Cusack, and D. A. 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]
  24. O. Marklund, J. M. Huntley, and R. Cusack, “Robust unwrapping algorithm for three-dimensional phase volumes of arbitrary shape containing knotted phase singularity loops,” Opt. Eng. 46, (2007).
    [CrossRef]
  25. A. Hooper and H. A. Zebker, “Phase unwrapping in three dimensions with application to InSAR time series,” J. Opt. Soc. Am. A 24, 2737-2747 (2007).
    [CrossRef]
  26. K. Brakke, “The surface evolver,” Exp. Math. 1, 141-165 (1992).
  27. M. T. Goodrich and R. Tamassia, “Kruskal's algorithm,” in Data Structures and Algorithms in Java, 4th ed. (Wiley, 2006), Chap. 13, Section 13.7, p. 632.
  28. S. Chavez, Q.-S. Xiang, and L. An, “Understanding phase maps in mri: a new cutline phase unwrapping method,” IEEE Trans. Med. Imaging 21, 966-977 (2002).
    [CrossRef] [PubMed]
  29. B. W. Silverman, Density Estimation for Statistics and Data Analysis (CRC, 1986).

2007 (5)

L. Aiello, D. Riccio, P. Ferraro, S. Grilli, L. Sansone, G. Coppola, S. D. Nicola, and A. Finizio, “Green's formulation for robust phase unwrapping in digital holography,” Opt. Lasers Eng. 45, 750-755 (2007).
[CrossRef]

J. Bioucas-Dias and G. Valadao, “Phase unwrapping via graph cuts,” IEEE Trans Image Process. 16, 698-709 (2007).
[CrossRef]

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

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

A. Hooper and H. A. Zebker, “Phase unwrapping in three dimensions with application to InSAR time series,” J. Opt. Soc. Am. A 24, 2737-2747 (2007).
[CrossRef]

2006 (3)

“Extending the dynamic range of phase contrast magnetic resonance velocity imaging using advanced higher-dimensional phase unwrapping algorithms,” J. R. Soc. Interface 3, 415-427(2006).
[CrossRef] [PubMed]

M. F. Salfity, P. D. Ruiz, J. M. Huntley, M. J. Graves, R. Cusack, and D. A. 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]

L. Ying, Z. Liang, D. Munson, R. Koetter, and B. Frey, “Unwrapping of MR phase images using a Markov random field model,” IEEE Trans. Med. Imaging 25, 128-136(2006).
[CrossRef] [PubMed]

2005 (1)

2003 (1)

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

2002 (5)

M. A. Herráez, D. R. Burton, M. J. Lalor, and M. A. Gdeisat, “Robust, fast and effective two dimensional automatic phase unwrapping algorithm based on image decomposition,” Appl. Opt. 41, 7445-7455 (2002).
[CrossRef] [PubMed]

J. M. B. Dias and J. M. N. Leitão, “The ZπM algorithm: a method for interferometric image reconstruction in SAR/SAS,” IEEE Trans. Image Process. 11, 408-422 (2002).
[CrossRef]

R. Cusack and N. Papadakis, “New robust 3-d phase unwrapping algorithms: Application to magnetic field mapping and undistorting echoplanar images,” NeuroImage 16, 754-764(2002).
[CrossRef] [PubMed]

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

S. Chavez, Q.-S. Xiang, and L. An, “Understanding phase maps in mri: a new cutline phase unwrapping method,” IEEE Trans. Med. Imaging 21, 966-977 (2002).
[CrossRef] [PubMed]

2001 (2)

2000 (1)

L. An, Q.-S. Xiang, and S. Chavez, “A fast implementation of the minimum spanning tree method for phase unwrapping,” IEEE Trans. Med. Imaging 19, 805-808(2000).
[CrossRef] [PubMed]

1996 (1)

1995 (1)

1994 (1)

M. Pritt and J. Shipman, “Least-squares two-dimensional phase unwrapping using FFT's,” IEEE Trans. Geosci. Remote Sens. , 32, 706-708 (1994).
[CrossRef]

1992 (2)

N. H. Ching, D. Rosenfeld, and M. Braun, “Two-dimensional phase unwrapping using a minimum spanning tree algorithm,” IEEE Trans. Image Process. 1, 355-365(1992).
[CrossRef]

K. Brakke, “The surface evolver,” Exp. Math. 1, 141-165 (1992).

1988 (1)

R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23, 713-720 (1988).
[CrossRef]

1955 (1)

H. W. Kuhn, “The Hungarian method for the assignment problem,” Naval Res. Log. Q. 2, 83-97 (1955).
[CrossRef]

Abdul-Rahman, H. S.

Aiello, L.

L. Aiello, D. Riccio, P. Ferraro, S. Grilli, L. Sansone, G. Coppola, S. D. Nicola, and A. Finizio, “Green's formulation for robust phase unwrapping in digital holography,” Opt. Lasers Eng. 45, 750-755 (2007).
[CrossRef]

An, L.

S. Chavez, Q.-S. Xiang, and L. An, “Understanding phase maps in mri: a new cutline phase unwrapping method,” IEEE Trans. Med. Imaging 21, 966-977 (2002).
[CrossRef] [PubMed]

L. An, Q.-S. Xiang, and S. Chavez, “A fast implementation of the minimum spanning tree method for phase unwrapping,” IEEE Trans. Med. Imaging 19, 805-808(2000).
[CrossRef] [PubMed]

Baldi, A.

Beauregard, D. A.

Bioucas-Dias, J.

J. Bioucas-Dias and G. Valadao, “Phase unwrapping via graph cuts,” IEEE Trans Image Process. 16, 698-709 (2007).
[CrossRef]

Boticario, J. G.

Brakke, K.

K. Brakke, “The surface evolver,” Exp. Math. 1, 141-165 (1992).

Braun, M.

N. H. Ching, D. Rosenfeld, and M. Braun, “Two-dimensional phase unwrapping using a minimum spanning tree algorithm,” IEEE Trans. Image Process. 1, 355-365(1992).
[CrossRef]

Burton, D. R.

Chavez, S.

S. Chavez, Q.-S. Xiang, and L. An, “Understanding phase maps in mri: a new cutline phase unwrapping method,” IEEE Trans. Med. Imaging 21, 966-977 (2002).
[CrossRef] [PubMed]

L. An, Q.-S. Xiang, and S. Chavez, “A fast implementation of the minimum spanning tree method for phase unwrapping,” IEEE Trans. Med. Imaging 19, 805-808(2000).
[CrossRef] [PubMed]

Ching, N. H.

N. H. Ching, D. Rosenfeld, and M. Braun, “Two-dimensional phase unwrapping using a minimum spanning tree algorithm,” IEEE Trans. Image Process. 1, 355-365(1992).
[CrossRef]

Coppola, G.

L. Aiello, D. Riccio, P. Ferraro, S. Grilli, L. Sansone, G. Coppola, S. D. Nicola, and A. Finizio, “Green's formulation for robust phase unwrapping in digital holography,” Opt. Lasers Eng. 45, 750-755 (2007).
[CrossRef]

Costantini, M.

M. Costantini, F. Malvarosa, F. Minati, L. Pietranera, and G. Milillo, “A three-dimensional phase unwrapping algorithm for processing of multitemporal sar interferometric measurements,” in Geoscience and Remote Sensing Symposium, 2002 (IEEE, 2002), Vol. 3, pp. 1741-1743.

Cusack, R.

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

M. F. Salfity, P. D. Ruiz, J. M. Huntley, M. J. Graves, R. Cusack, and D. A. 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 3-d phase unwrapping algorithms: Application to magnetic field mapping and undistorting echoplanar 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]

Dias, J. M. B.

J. M. B. Dias and J. M. N. Leitão, “The ZπM algorithm: a method for interferometric image reconstruction in SAR/SAS,” IEEE Trans. Image Process. 11, 408-422 (2002).
[CrossRef]

Ferraro, P.

L. Aiello, D. Riccio, P. Ferraro, S. Grilli, L. Sansone, G. Coppola, S. D. Nicola, and A. Finizio, “Green's formulation for robust phase unwrapping in digital holography,” Opt. Lasers Eng. 45, 750-755 (2007).
[CrossRef]

Finizio, A.

L. Aiello, D. Riccio, P. Ferraro, S. Grilli, L. Sansone, G. Coppola, S. D. Nicola, and A. Finizio, “Green's formulation for robust phase unwrapping in digital holography,” Opt. Lasers Eng. 45, 750-755 (2007).
[CrossRef]

Frey, B.

L. Ying, Z. Liang, D. Munson, R. Koetter, and B. Frey, “Unwrapping of MR phase images using a Markov random field model,” IEEE Trans. Med. Imaging 25, 128-136(2006).
[CrossRef] [PubMed]

Gdeisat, M. A.

Ghiglia, D. C.

D. C. Ghiglia and L. A. Romero, “Minimum lp-norm two-dimensional phase unwrapping,” J. Opt. Soc. Am. A 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.

Goldstein, R. M.

R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23, 713-720 (1988).
[CrossRef]

Goodrich, M. T.

M. T. Goodrich and R. Tamassia, “Kruskal's algorithm,” in Data Structures and Algorithms in Java, 4th ed. (Wiley, 2006), Chap. 13, Section 13.7, p. 632.

Graves, M. J.

Grilli, S.

L. Aiello, D. Riccio, P. Ferraro, S. Grilli, L. Sansone, G. Coppola, S. D. Nicola, and A. Finizio, “Green's formulation for robust phase unwrapping in digital holography,” Opt. Lasers Eng. 45, 750-755 (2007).
[CrossRef]

Herráez, M. A.

Hooper, A.

Huntley, J. M.

Jenkinson, M.

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

Koetter, R.

L. Ying, Z. Liang, D. Munson, R. Koetter, and B. Frey, “Unwrapping of MR phase images using a Markov random field model,” IEEE Trans. Med. Imaging 25, 128-136(2006).
[CrossRef] [PubMed]

Kuhn, H. W.

H. W. Kuhn, “The Hungarian method for the assignment problem,” Naval Res. Log. Q. 2, 83-97 (1955).
[CrossRef]

Lalor, M. J.

Leitão, J. M. N.

J. M. B. Dias and J. M. N. Leitão, “The ZπM algorithm: a method for interferometric image reconstruction in SAR/SAS,” IEEE Trans. Image Process. 11, 408-422 (2002).
[CrossRef]

Liang, Z.

L. Ying, Z. Liang, D. Munson, R. Koetter, and B. Frey, “Unwrapping of MR phase images using a Markov random field model,” IEEE Trans. Med. Imaging 25, 128-136(2006).
[CrossRef] [PubMed]

Lilley, F.

Malvarosa, F.

M. Costantini, F. Malvarosa, F. Minati, L. Pietranera, and G. Milillo, “A three-dimensional phase unwrapping algorithm for processing of multitemporal sar interferometric measurements,” in Geoscience and Remote Sensing Symposium, 2002 (IEEE, 2002), Vol. 3, pp. 1741-1743.

Marklund, O.

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

Milillo, G.

M. Costantini, F. Malvarosa, F. Minati, L. Pietranera, and G. Milillo, “A three-dimensional phase unwrapping algorithm for processing of multitemporal sar interferometric measurements,” in Geoscience and Remote Sensing Symposium, 2002 (IEEE, 2002), Vol. 3, pp. 1741-1743.

Minati, F.

M. Costantini, F. Malvarosa, F. Minati, L. Pietranera, and G. Milillo, “A three-dimensional phase unwrapping algorithm for processing of multitemporal sar interferometric measurements,” in Geoscience and Remote Sensing Symposium, 2002 (IEEE, 2002), Vol. 3, pp. 1741-1743.

Moore, C. J.

Munson, D.

L. Ying, Z. Liang, D. Munson, R. Koetter, and B. Frey, “Unwrapping of MR phase images using a Markov random field model,” IEEE Trans. Med. Imaging 25, 128-136(2006).
[CrossRef] [PubMed]

Nicola, S. D.

L. Aiello, D. Riccio, P. Ferraro, S. Grilli, L. Sansone, G. Coppola, S. D. Nicola, and A. Finizio, “Green's formulation for robust phase unwrapping in digital holography,” Opt. Lasers Eng. 45, 750-755 (2007).
[CrossRef]

Papadakis, N.

R. Cusack and N. Papadakis, “New robust 3-d phase unwrapping algorithms: Application to magnetic field mapping and undistorting echoplanar images,” NeuroImage 16, 754-764(2002).
[CrossRef] [PubMed]

Pietranera, L.

M. Costantini, F. Malvarosa, F. Minati, L. Pietranera, and G. Milillo, “A three-dimensional phase unwrapping algorithm for processing of multitemporal sar interferometric measurements,” in Geoscience and Remote Sensing Symposium, 2002 (IEEE, 2002), Vol. 3, pp. 1741-1743.

Pritt, M.

M. Pritt and J. Shipman, “Least-squares two-dimensional phase unwrapping using FFT's,” IEEE Trans. Geosci. Remote Sens. , 32, 706-708 (1994).
[CrossRef]

Pritt, M. D.

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

Riccio, D.

L. Aiello, D. Riccio, P. Ferraro, S. Grilli, L. Sansone, G. Coppola, S. D. Nicola, and A. Finizio, “Green's formulation for robust phase unwrapping in digital holography,” Opt. Lasers Eng. 45, 750-755 (2007).
[CrossRef]

Romero, L. A.

Rosenfeld, D.

N. H. Ching, D. Rosenfeld, and M. Braun, “Two-dimensional phase unwrapping using a minimum spanning tree algorithm,” IEEE Trans. Image Process. 1, 355-365(1992).
[CrossRef]

Ruiz, P. D.

Salfity, M. F.

Sansone, L.

L. Aiello, D. Riccio, P. Ferraro, S. Grilli, L. Sansone, G. Coppola, S. D. Nicola, and A. Finizio, “Green's formulation for robust phase unwrapping in digital holography,” Opt. Lasers Eng. 45, 750-755 (2007).
[CrossRef]

Shipman, J.

M. Pritt and J. Shipman, “Least-squares two-dimensional phase unwrapping using FFT's,” IEEE Trans. Geosci. Remote Sens. , 32, 706-708 (1994).
[CrossRef]

Silverman, B. W.

B. W. Silverman, Density Estimation for Statistics and Data Analysis (CRC, 1986).

Tamassia, R.

M. T. Goodrich and R. Tamassia, “Kruskal's algorithm,” in Data Structures and Algorithms in Java, 4th ed. (Wiley, 2006), Chap. 13, Section 13.7, p. 632.

Valadao, G.

J. Bioucas-Dias and G. Valadao, “Phase unwrapping via graph cuts,” IEEE Trans Image Process. 16, 698-709 (2007).
[CrossRef]

Werner, C. L.

R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23, 713-720 (1988).
[CrossRef]

Xiang, Q.-S.

S. Chavez, Q.-S. Xiang, and L. An, “Understanding phase maps in mri: a new cutline phase unwrapping method,” IEEE Trans. Med. Imaging 21, 966-977 (2002).
[CrossRef] [PubMed]

L. An, Q.-S. Xiang, and S. Chavez, “A fast implementation of the minimum spanning tree method for phase unwrapping,” IEEE Trans. Med. Imaging 19, 805-808(2000).
[CrossRef] [PubMed]

Ying, L.

L. Ying, Z. Liang, D. Munson, R. Koetter, and B. Frey, “Unwrapping of MR phase images using a Markov random field model,” IEEE Trans. Med. Imaging 25, 128-136(2006).
[CrossRef] [PubMed]

Zebker, H. A.

A. Hooper and H. A. Zebker, “Phase unwrapping in three dimensions with application to InSAR time series,” J. Opt. Soc. Am. A 24, 2737-2747 (2007).
[CrossRef]

R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23, 713-720 (1988).
[CrossRef]

Appl. Opt. (8)

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

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

M. A. Herráez, J. G. Boticario, M. J. Lalor, and D. R. Burton, “Agglomerative clustering-based approach for two-dimensional phase unwrapping,” Appl. Opt. 44, 1129-1140 (2005).
[CrossRef] [PubMed]

A. Baldi, “Two-dimensional phase unwrapping by quad-tree decomposition,” Appl. Opt. 40, 1187-1194 (2001).
[CrossRef]

M. A. Herráez, D. R. Burton, M. J. Lalor, and M. A. Gdeisat, “Robust, fast and effective two dimensional automatic phase unwrapping algorithm based on image decomposition,” Appl. Opt. 41, 7445-7455 (2002).
[CrossRef] [PubMed]

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

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

M. F. Salfity, P. D. Ruiz, J. M. Huntley, M. J. Graves, R. Cusack, and D. A. 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]

Exp. Math. (1)

K. Brakke, “The surface evolver,” Exp. Math. 1, 141-165 (1992).

IEEE Trans Image Process. (1)

J. Bioucas-Dias and G. Valadao, “Phase unwrapping via graph cuts,” IEEE Trans Image Process. 16, 698-709 (2007).
[CrossRef]

IEEE Trans. Geosci. Remote Sens. (1)

M. Pritt and J. Shipman, “Least-squares two-dimensional phase unwrapping using FFT's,” IEEE Trans. Geosci. Remote Sens. , 32, 706-708 (1994).
[CrossRef]

IEEE Trans. Image Process. (2)

J. M. B. Dias and J. M. N. Leitão, “The ZπM algorithm: a method for interferometric image reconstruction in SAR/SAS,” IEEE Trans. Image Process. 11, 408-422 (2002).
[CrossRef]

N. H. Ching, D. Rosenfeld, and M. Braun, “Two-dimensional phase unwrapping using a minimum spanning tree algorithm,” IEEE Trans. Image Process. 1, 355-365(1992).
[CrossRef]

IEEE Trans. Med. Imaging (3)

L. An, Q.-S. Xiang, and S. Chavez, “A fast implementation of the minimum spanning tree method for phase unwrapping,” IEEE Trans. Med. Imaging 19, 805-808(2000).
[CrossRef] [PubMed]

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Figures (8)

Fig. 1
Fig. 1

Construction of a phase volume with two loops that break at boundaries. (a) A 2D phase map with two residues of different sign, (b) a 3D phase map build by stacking a pile of these 2D phase maps, and (c) incomplete loops in the 3D volume.

Fig. 2
Fig. 2

Resulting branch-cut surface when (a) each loop in Fig. 1c is closed on itself and when (b) ends are allowed to connect to other incomplete loops.

Fig. 3
Fig. 3

Example of a problem with pairing partial loops. One of two residues caused by a true surface discontinuity exits and reenters the volume. (a) Illustration in 3D. (b) A 2D view of the loops from the top of the volume.

Fig. 4
Fig. 4

Visual example of computing reliability values according to the nearest neighbor rule (distance from a voxel to its nearest residue). For clarity, the process is illustrated in two dimensions, on a mesh with only the two residues shown in (a).

Fig. 5
Fig. 5

Results of unwrapping a synthetic phase volume composed of 64 frames with a cube of random noise at the center: (a) middle frame, (b)–(f) unwrapping results with (b) flood-fill algorithm, (c) MST, (d) NN, (e) Huntley, and (f) proposed.

Fig. 6
Fig. 6

Results of unwrapping a synthetic phase volume that represents a growing sphere: (a) middle frame, (b)–(f) unwrapping results with (b) flood-fill algorithm, (c) MST, (d) NN, (e) Huntley, and (f) proposed.

Fig. 7
Fig. 7

Results of unwrapping a synthetic phase volume that represents a moving complex surface. Two representative frames are shown, with different noise intensities.

Fig. 8
Fig. 8

Results of unwrapping a real phase volume that resulted from the analysis of real fringe patterns projected onto a mannequin.

Tables (3)

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Table 1 Algorithm 1. MSTunwrapping. MST-Based Technique for Phase Unwrapping

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Table 2 Algorithm 2. calculateEdgeReliability. Algorithm Used to Define the Reliability of the Edges

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Table 3 Algorithm 3. Outline of the Proposed Technique

Equations (4)

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2 π v . phase - u . phase 2 π + 0.5
d NN ( v ) = 1 / N V ( v - NN ( v ) ) ,
reliability ( v ) = v - NN ( v ) ,
reliability ( e ) = reliability ( v 1 ) + reliability ( v 2 ) .

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