Abstract

Relatively recent techniques that produce phase volumes have motivated the study of three-dimensional (3D) unwrapping algorithms that inherently incorporate the third dimension into the process. We propose a novel 3D unwrapping algorithm that can be considered to be a generalization of the minimum spanning tree (MST) approach. The technique combines characteristics of some of the most robust existing methods: it uses a quality map to guide the unwrapping process, a region growing mechanism to progressively unwrap the signal, and also cut surfaces to avoid error propagation. The approach has been evaluated in the context of noncontact measurement of dynamic objects, suggesting a better performance than MST-based approaches.

© 2010 Optical Society of America

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References

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  1. D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, 1998).
  2. D. C. Ghiglia and L. A. Romero, “Minimum LP-norm two-dimensional phase unwrapping,” J. Opt. Soc. Am. A 13, 1999-2013 (1996).
    [CrossRef]
  3. M. Pritt and J. Shipman, “Least-squares two-dimensional phase unwrapping using FFT's,” IEEE Trans. Geosci. Remote Sens. 32, 706-708 (1994).
    [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. 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]
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    [CrossRef] [PubMed]
  7. 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]
  8. 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]
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  11. R. Cusack and N. Papadakis, “New robust 3D phase unwrapping algorithms: application to magnetic field mapping and undistorting echoplanar images,” NeuroImage 16, 754-764 (2002).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]

2009 (2)

2007 (2)

2005 (1)

2003 (1)

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

2002 (4)

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]

R. Cusack and N. Papadakis, “New robust 3D 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]

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

2001 (1)

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 (2)

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 (1)

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] [PubMed]

Abdul-Rahman, H. S.

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]

Boticario, J. G.

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] [PubMed]

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] [PubMed]

Clegg, D. B.

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 IEEE International Geoscience and Remote Sensing Symposium, 2002 (IEEE, 2002), Vol. 3, pp. 1741-1743.

Cusack, R.

R. Cusack and N. Papadakis, “New robust 3D 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]

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.

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]

Lalor, M. J.

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 IEEE International Geoscience and Remote Sensing Symposium, 2002 (IEEE, 2002), Vol. 3, pp. 1741-1743.

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 IEEE International 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 IEEE International Geoscience and Remote Sensing Symposium, 2002 (IEEE, 2002), Vol. 3, pp. 1741-1743.

Moore, C. J.

Papadakis, N.

R. Cusack and N. Papadakis, “New robust 3D 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 IEEE International 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).

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] [PubMed]

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]

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]

Zebker, H. A.

Appl. Opt. (8)

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. (1)

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] [PubMed]

IEEE Trans. Med. Imaging (2)

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]

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]

J. Opt. Soc. Am. A (2)

Magn. Reson. Med. (1)

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

NeuroImage (1)

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

Opt. Lett. (1)

Other (2)

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

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 IEEE International Geoscience and Remote Sensing Symposium, 2002 (IEEE, 2002), Vol. 3, pp. 1741-1743.

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Figures (8)

Fig. 1
Fig. 1

First 3D phase map contains a noisy region located at its center. A transversal cut of the signal is shown for illustrative purposes.

Fig. 2
Fig. 2

Results of unwrapping the phase map in Fig. 1 using (a) a simple flood fill algorithm; (b) best path; (c) noise immune; and (d) the proposed approach.

Fig. 3
Fig. 3

3D plot of synthetic surface used in the second experiment.

Fig. 4
Fig. 4

Results of unwrapping the phase map generated by simulating fringe projection on the growing surface shown in Fig. 3 by using (a) a simple flood fill algorithm, (b) best path, (c) noise immune, and (d) the proposed approach.

Fig. 5
Fig. 5

Middle frame of simulated growing sphere.

Fig. 6
Fig. 6

Results of unwrapping the phase map in Fig. 5 using different number of frames, for the different algorithms being compared.

Fig. 7
Fig. 7

Frame 13th in the wrapped phase volume.

Fig. 8
Fig. 8

Results of unwrapping the phase map in Fig. 7 for different resolutions using the algorithms being compared.

Tables (2)

Tables Icon

Table 1 Execution Times (in Seconds) for Experiments

Tables Icon

Table 2 Memory Consumption (in Megabytes) for Experiments

Equations (15)

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2 π v . value u . value 2 π + 0.5 .
SD i , j , k = H 2 ( i , j , k ) + D 2 ( i , j , k ) + N 2 ( i , j , k ) + n = 1 10 D n 2 ( i , j , k ) ,
H ( i , j , k ) = ω ( φ ( i 1 , j , k ) φ ( i , j , k ) ) ω ( φ ( i , j , k ) φ ( i + 1 , j , k ) ) ,
V ( i , j , k ) = ω ( φ ( i , j 1 , k ) φ ( i , j , k ) ) ω ( φ ( i , j , k ) φ ( i , j + 1 , k ) ) ,
N ( i , j , k ) = ω ( φ ( i , j , k 1 ) φ ( i , j , k ) ) ω ( φ ( i , j , k ) φ ( i , j , k + 1 ) ) ,
D 1 ( i , j , k ) = ω ( φ ( i 1 , j 1 , k ) φ ( i , j , k ) ) ω ( φ ( i , j , k ) φ ( i + 1 , j + 1 , k ) ) ,
D 2 ( i , j , k ) = ω ( φ ( i + 1 , j 1 , k ) φ ( i , j , k ) ) ω ( φ ( i , j , k ) φ ( i 1 , j + 1 , k ) ) ,
D 3 ( i , j , k ) = ω ( φ ( i 1 , j 1 , k 1 ) φ ( i , j , k ) ) ω ( φ ( i , j , k ) φ ( i + 1 , j + 1 , k + 1 ) ) ,
D 4 ( i , j , k ) = ω ( φ ( i , j 1 , k 1 ) φ ( i , j , k ) ) ω ( φ ( i , j , k ) φ ( i , j + 1 , k + 1 ) ) ,
D 5 ( i , j , k ) = ω ( φ ( i + 1 , j 1 , k 1 ) φ ( i , j , k ) ) ω ( φ ( i , j , k ) φ ( i 1 , j + 1 , k + 1 ) ) ,
D 6 ( i , j , k ) = ω ( φ ( i 1 , j , k 1 ) φ ( i , j , k ) ) ω ( φ ( i , j , k ) φ ( i + 1 , j , k + 1 ) ) ,
D 7 ( i , j , k ) = ω ( φ ( i 1 , j + 1 , k 1 ) φ ( i , j , k ) ) ω ( φ ( i , j , k ) φ ( i + 1 , j 1 , k + 1 ) ) ,
D 8 ( i , j , k ) = ω ( φ ( i + 1 , j , k 1 ) φ ( i , j , k ) ) ω ( φ ( i , j , k ) φ ( i 1 , j , k + 1 ) ) ,
D 9 ( i , j , k ) = ω ( φ ( i , j + 1 , k 1 ) φ ( i , j , k ) ) ω ( φ ( i , j , k ) φ ( i , j 1 , k + 1 ) ) ,
D 10 ( i , j , k ) = ω ( φ ( i + 1 , j + 1 , k 1 ) φ ( i , j , k ) ) ω ( φ ( i , j , k ) φ ( i 1 , j 1 , k + 1 ) ) .

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