Abstract

What we believe to be a novel technique of branch-cut placement in the phase unwrapping is proposed. This approach is based on what we named residue vector, which is generated by a residue in a wrapped phase map and has an orientation that points out toward the balancing residue of opposite polarity. The residue vector can be used to guide the manner in which branch cuts are placed in phase unwrapping. Also, residue vector can be used for the determination of the weighting values used in different existing phase unwrapping methods such as minimum cost flow and least squares. The theoretical foundations of the residue-vector method are presented, and a branch-cut method using its information is developed and implemented. A general comparison is made between the residue-vector map and other existing quality maps.

© 2007 Optical Society of America

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References

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  1. J. M. Huntley, "Three-dimensional noise-immune phase unwrapping algorithm," Appl. Opt. 40, 3901-3908 (2001).
    [CrossRef]
  2. R. Cusack, J. M. Huntley, and H. T. Goldstein, "Improved noise-immune phase-unwrapping algorithm," Appl. Opt. 24, 781-789 (1995).
    [CrossRef]
  3. J. R. Buckland, J. M. Huntley, and J. M. Turner, "Unwrapping noisy phase maps by use of a minimum-cost-matching algorithm," Appl. Opt. 34, 5100-5108 (1995).
    [CrossRef] [PubMed]
  4. D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms and Software (Wiley, 1998).
  5. B. Gutmann, "Phase unwrapping with the branch-cut method: clustering of discontinuity sources and reverse simulated annealing," Appl. Opt. 38, 5577-5793 (1999).
    [CrossRef]
  6. R. Goldstein, H. Zebker, and C. Werner, "Satellite radar interferometry: two-dimensional phase unwrapping," Radio Sci. 23, 713-720 (1988).
    [CrossRef]
  7. C. Chen and H. Zebker, "Network approaches to the two-dimensional phase unwrapping: intractability and two new algorithms," Appl. Opt. 17, 401-414 (2000).
  8. J. Huntley, "Noise-immune phase unwrapping algorithm," Appl. Opt. 28, 3268-3270 (1989).
    [CrossRef] [PubMed]
  9. T. Flynn, "Consistent 2-D phase unwrapping guided by a quality map," in IEEE Proceedings of the 1996 International Geoscience and Remote Sensing Symposium, Lincoln (IEEE, 1996), pp. 2057-2059.
  10. M. Costantini, "A novel phase unwrapping method based on network programming," IEEE Trans. Geosci. Remote Sens. 36, 813-821 (1998).
    [CrossRef]
  11. T. Flynn, "Two-dimensional phase unwrapping with minimum weighted discontinuity," Appl. Opt. 14, 2691-2701 (1997).
  12. S. Chavez, Q. 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]
  13. M. Salfity, P. Ruiz, J. Huntley, M. Graves, R. Cusack, and D. Beauregard, "Branch-cut surface placement for unwrapping of under-sampled three-dimensional phase data: application to magnetic resonance imaging arterial flow mapping," Appl. Opt. 45, 2711-2721 (2006).
    [CrossRef] [PubMed]

2006

2002

S. Chavez, Q. 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

2000

C. Chen and H. Zebker, "Network approaches to the two-dimensional phase unwrapping: intractability and two new algorithms," Appl. Opt. 17, 401-414 (2000).

1999

1998

M. Costantini, "A novel phase unwrapping method based on network programming," IEEE Trans. Geosci. Remote Sens. 36, 813-821 (1998).
[CrossRef]

1997

1995

J. R. Buckland, J. M. Huntley, and J. M. Turner, "Unwrapping noisy phase maps by use of a minimum-cost-matching algorithm," Appl. Opt. 34, 5100-5108 (1995).
[CrossRef] [PubMed]

R. Cusack, J. M. Huntley, and H. T. Goldstein, "Improved noise-immune phase-unwrapping algorithm," Appl. Opt. 24, 781-789 (1995).
[CrossRef]

1989

1988

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

Appl. Opt.

IEEE Trans. Geosci. Remote Sens.

M. Costantini, "A novel phase unwrapping method based on network programming," IEEE Trans. Geosci. Remote Sens. 36, 813-821 (1998).
[CrossRef]

IEEE Trans. Med. Imaging

S. Chavez, Q. 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]

Radio Sci.

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

Other

T. Flynn, "Consistent 2-D phase unwrapping guided by a quality map," in IEEE Proceedings of the 1996 International Geoscience and Remote Sensing Symposium, Lincoln (IEEE, 1996), pp. 2057-2059.

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

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Figures (19)

Fig. 1
Fig. 1

(a) Visualizing residue calculation and (b) an interpixel network with a branch cut between dipole residues of opposite polarity.

Fig. 2
Fig. 2

Diagram illustrating that the proposed residue-vector map will be an aid to all existing phase unwrapping algorithms to create an optimum unwrapped solution.

Fig. 3
Fig. 3

(a) Simulated spiral 257 × 257 wrapped phase map computer simulated object from Ghiglia and Pritt [4], (b) its corresponding residue map, a wrapped phase gradient in the (c) x direction, (d) y direction sense, (e) original 3D surface of the spiral, and (f) magnified version of the residue vector of a single residue.

Fig. 4
Fig. 4

Illustration of how to distinguish between positive and negative residues for the residue-vector direction.

Fig. 5
Fig. 5

Residue vector of a pair of two opposite polarity dipole residues caused by a high level of phase noise; this image is created from the interferometric synthetic aperture radar wrapped phase map in Ghiglia and Pritt [4]; (a) dx gradient phase map, (b) emphasis of how the residue vector in the dx gradient phase map, (c) residue distribution map of the same dx gradient phase map, and (d) emphasis of the pair of dipoles in the residue distribution.

Fig. 6
Fig. 6

(a) Original 3D simulated object of a quarter-pyramid with a square hole in the middle, (b) its wrapped phase map, (c) its downsampled wrapped phase, (d) illustrative emphasis of the behavior of the residue vector in the (e) dx and (f) dy gradient phase maps.

Fig. 7
Fig. 7

Residue vector between dipole residues taken from Fig. 3(d) shows a constant vector charge shared between the residues of the dipole (a) and (b) present the residue-vector charge varying with the nature of discontinuity whether descending or ascending.

Fig. 8
Fig. 8

(a) Overlapped wrapped phase gradient of dx and dy calculated from the wrapped phase map in Fig. 6(b), (b) schematic showing a constant vector charge shared between the residues of the dipole, (c) schematic showing how the zero vector is created, (d) residue-vector orientation of the four top corner residues in (a), and (e) schematic of the residue-vector orientation of the four top corner residues.

Fig. 9
Fig. 9

(a) Original 3D simulated object, (b) its wrapped phase map, a section of the dy wrapped phase gradient map of (b) marked by a box is used to show the (c) residue vector behavior at low phase noise and the (d) residue-vector behavior at very high phase noise.

Fig. 10
Fig. 10

(Color online) (a) Monopole residue vector extending from the monopole residue to the border (white curve), (b) similar image but with increased contrast to make the residue vector more visible, (c) original fairy's image illustrating the position of the fairy's hand with the monopole residue, and (d) its corresponding wrapped phase map.

Fig. 11
Fig. 11

(a) Magnitude of the wrapped phase gradient of the wrapped phase map in Figs. 9(b) and 9(b) combined wrapped phase gradient map of Fig. 9(b) (middle section of the image) using the maximum of both the dx and dy wrapped phase gradient (for illustration only).

Fig. 12
Fig. 12

Combined wrapped phase gradient map of the middle section of the wrapped phase map in Fig. 9(b) using the maximum of both the dx and dy wrapped phase gradient for illustration only scaled down by (a) 3 × 3 pixels and (b) 5 × 5 pixels.

Fig. 13
Fig. 13

Branch-cut placement methods used to connect two residues; (a) original dy gradient map taken from Fig. 3(d), (b) incorrect branch-cut placement using straight-line cuts, and (c) correct branch-cut placement obeying the residue-vector rule.

Fig. 14
Fig. 14

(Color online) Monopole residue correct branch-cut placement implemented on the dy wrapped phase gradient map of the wrapped phase map in Fig. 10(d).

Fig. 15
Fig. 15

Zero-weighted mask of the wrapped phase map in Fig. 3(a) using (a) phase variance quality map [4] (arrows point at some of the nonmasked zero vector) and (b) residue-vector map showing the position of the extracted residue-vector pixels in the gradient phase map.

Fig. 16
Fig. 16

(a) Branch cuts produced by Flynn's algorithm [4, 11] with zero weights provided by the mask of the minimum phase variance quality map (b) branch cuts produced by Flynn's algorithm with zero weights provided by the mask of the residue-vector map and their corresponding unwrapped phase map in (c) and (d), respectively.

Fig. 17
Fig. 17

Implementation of the branch-cut placement of the mask-cut method [4, 9] using the wrapped map of Fig. 10(d) shows an (a) incorrect placement of the branch cut, (b) phase distortion in the unwrapped phase map, and (c) 3D surface of the unwrapped phase with phase distortion.

Fig. 18
Fig. 18

(a) Branch cuts produced by Flynn's algorithm [4, 11] with zero weights provided by the mask of the maximum phase gradient quality map and (b) its corresponding wrapped phase.

Fig. 19
Fig. 19

(a) Branch cuts produced by Flynn's algorithm [4, 11] with zero weights provided by the mask of the residue-vector map and (b) its corresponding wrapped phase.

Equations (35)

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π
+ π
2 π
2 π
2 π
i M ^ Φ ( p i ) = 0 ,
Φ ( p i )
p i { P }
^ Φ ( p i )
2 π
2 × 2
n = 0
± 1
i M ^ Φ ( p i ) = 2 π n .
| ^ Φ | > π
+ π / π
± 2 π
3 × 3
5 × 5
b N b c p N b p ^ Φ ( b , p ) = 0 ,
^ Φ ( b , p )
2 × 2
N b c
N b p
E = i N j M | k x ( i , j ) | + i N j M | k y ( i , j ) | ,
k x ( i , j ) = Int ( Ψ ( i , j ) Ψ ( i 1 , j ) ^ Φ ( i , j ) 2 π )
k y ( i , j ) = Int ( Ψ ( i , j ) Ψ ( i , j 1 ) ^ Φ ( i , j ) 2 π )
Ψ ( i , j )
I n t ( )
min   E If   and   only   if { max   N r f min   N b r .
N r f
N b r
257 × 257
3 × 3
5 × 5

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