Abstract

Phase unwrapping still plays an important role in many data-processing chains based on phase information. Here, we introduce a new phase unwrapping approach for noisy wrapped phase maps of continuous objects to improve the accuracy and computational time requirements of phase unwrapping using a rotational compensator (RC) method. The proposed algorithm is based on compensating the singularity of discontinuity sources. It uses direct compensation for adjoining singular point (SP) pairs and uses RC for other SP pairs. The performance of the proposed method is tested through both simulated and real wrapped phase data. The proposed algorithm is faster than the original algorithm with the RC and has proved efficiency compared to other phase unwrapping methods.

© 2011 Optical Society of America

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References

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    [CrossRef] [PubMed]
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    [CrossRef]
  3. 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]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  8. E. Cuche, P. Marquet, and C. Depeursinge, “Spatial filtering for zero-order and twin-image elimination in digital off-axis holography,” Appl. Opt. 39, 4070–4075 (2000).
    [CrossRef]
  9. J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, and D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. 13, 2693–2703 (1974).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  13. R. Cusack, J. M. Huntley, and H. T. Goldrein, “Improved noise-immune phase-unwrapping algorithm,” Appl. Opt. 34, 781–789 (1995).
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    [CrossRef]
  15. B. Gutmann and H. Weber, “Phase unwrapping with the branch-cut method: clustering of discontinuity sources and reverse simulated annealing,” Appl. Opt. 38, 5577–5593 (1999).
    [CrossRef]
  16. 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).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  29. Q. Kemao, 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]
  30. Q. Kemao, “Two-dimensional windowed Fourier transform for fringe pattern analysis: principles, applications and implementations,” Opt. Lasers Eng. 45, 304–317 (2007).
    [CrossRef]
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    [CrossRef] [PubMed]
  32. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes: The Art of Scientific Computing, 3rd edition (Cambridge University, 2007).

2010 (1)

2008 (1)

2007 (3)

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]

R. Yamaki and A. Hirose, “Singularity-spreading phase unwrapping,” IEEE Trans. Geosci. Remote Sens. 45, 3240–3251(2007).
[CrossRef]

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

2005 (1)

Q. Kemao, 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]

2002 (2)

C. L. Martinez and X. Fabergas, “Modeling and reduction of SAR interferometric phase noise in the wavelet domain,” IEEE Trans. Geosci. Remote Sens. 40, 2553–2566(2002).
[CrossRef]

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 (1)

C. L. Martinez, X. F. Canovas, and M. Chandra, “SAR interferometric phase noise reduction using wavelet transform,” Electron. Lett. 37, 649–651 (2001).
[CrossRef]

2000 (1)

1999 (2)

1998 (3)

M. Costantine, “A novel phase unwrapping method based on network programming,” IEEE Trans. Geosci. Remote Sens. 36, 813–821 (1998).
[CrossRef]

M. R. Goldstein and C. L. Werner, “Radar interferogram filtering for geophysical applications,” Geophys. Res. Lett. 25, 4035–4038 (1998).
[CrossRef]

J. S. Lee, K. P. Papathanassiou, T. L. Ainsworth, M. H. Grunes, and A. Reigber, “A new technique for noise filtering of SAR interferometric phase images,” IEEE Trans. Geosci. Remote Sens. 36, 1456–1465 (1998).
[CrossRef]

1995 (3)

1994 (1)

1993 (1)

K. E. Perry, Jr., and J. McKelvie, “A comparison of phase shifting and Fourier methods in the analysis of discontinuous fringe patterns,” Opt. Lasers Eng. 19, 269–284 (1993).
[CrossRef]

1989 (2)

1988 (3)

1985 (1)

1982 (1)

1979 (1)

1977 (2)

1974 (1)

Ainsworth, T. L.

J. S. Lee, K. P. Papathanassiou, T. L. Ainsworth, M. H. Grunes, and A. Reigber, “A new technique for noise filtering of SAR interferometric phase images,” IEEE Trans. Geosci. Remote Sens. 36, 1456–1465 (1998).
[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]

Asundi, A.

Q. Kemao, 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]

Brangaccio, D. J.

Breuckmann, B.

Bruning, J. H.

Buckland, J. R.

Burton, D. R.

Canovas, X. F.

C. L. Martinez, X. F. Canovas, and M. Chandra, “SAR interferometric phase noise reduction using wavelet transform,” Electron. Lett. 37, 649–651 (2001).
[CrossRef]

Chandra, M.

C. L. Martinez, X. F. Canovas, and M. Chandra, “SAR interferometric phase noise reduction using wavelet transform,” Electron. Lett. 37, 649–651 (2001).
[CrossRef]

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]

Costantine, M.

M. Costantine, “A novel phase unwrapping method based on network programming,” IEEE Trans. Geosci. Remote Sens. 36, 813–821 (1998).
[CrossRef]

Cuche, E.

Cusack, R.

Depeursinge, C.

Fabergas, X.

C. L. Martinez and X. Fabergas, “Modeling and reduction of SAR interferometric phase noise in the wavelet domain,” IEEE Trans. Geosci. Remote Sens. 40, 2553–2566(2002).
[CrossRef]

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes: The Art of Scientific Computing, 3rd edition (Cambridge University, 2007).

Fried, D. L.

Gallagher, J. E.

Gao, W.

Gdeisat, M. A.

Ghiglia, D. C.

Glover, G. H.

S. M. Song, S. Napel, N. J. Pelc, and G. H. Glover, “Phase unwrapping of MR phase images using Poisson equation,” IEEE Trans. Image Process. 4, 667–676 (1995).
[CrossRef]

Goldrein, H. T.

Goldstein, M. R.

M. R. Goldstein and C. L. Werner, “Radar interferogram filtering for geophysical applications,” Geophys. Res. Lett. 25, 4035–4038 (1998).
[CrossRef]

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]

Grunes, M. H.

J. S. Lee, K. P. Papathanassiou, T. L. Ainsworth, M. H. Grunes, and A. Reigber, “A new technique for noise filtering of SAR interferometric phase images,” IEEE Trans. Geosci. Remote Sens. 36, 1456–1465 (1998).
[CrossRef]

Gutmann, B.

Herriott, D. R.

Heshmat, S.

Hirose, A.

R. Yamaki and A. Hirose, “Singularity-spreading phase unwrapping,” IEEE Trans. Geosci. Remote Sens. 45, 3240–3251(2007).
[CrossRef]

Hudgin, R. H.

Hunt, B. R.

Huntley, J. M.

Ina, H.

Karout, S. A.

Kemao, Q.

Q. Kemao, W. Gao, and H. Wang, “Windowed Fourier-filtered and quality-guided phase-unwrapping algorithm,” Appl. Opt. 47, 5408–5428 (2008).
[CrossRef] [PubMed]

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

Q. Kemao, 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]

Kobayashi, S.

Lalor, M. J.

Lee, J. S.

J. S. Lee, K. P. Papathanassiou, T. L. Ainsworth, M. H. Grunes, and A. Reigber, “A new technique for noise filtering of SAR interferometric phase images,” IEEE Trans. Geosci. Remote Sens. 36, 1456–1465 (1998).
[CrossRef]

Lyuboshenko, I.

Maitre, H.

Marquet, P.

Martinez, C. L.

C. L. Martinez and X. Fabergas, “Modeling and reduction of SAR interferometric phase noise in the wavelet domain,” IEEE Trans. Geosci. Remote Sens. 40, 2553–2566(2002).
[CrossRef]

C. L. Martinez, X. F. Canovas, and M. Chandra, “SAR interferometric phase noise reduction using wavelet transform,” Electron. Lett. 37, 649–651 (2001).
[CrossRef]

McKelvie, J.

K. E. Perry, Jr., and J. McKelvie, “A comparison of phase shifting and Fourier methods in the analysis of discontinuous fringe patterns,” Opt. Lasers Eng. 19, 269–284 (1993).
[CrossRef]

Miyamoto, N.

Napel, S.

S. M. Song, S. Napel, N. J. Pelc, and G. H. Glover, “Phase unwrapping of MR phase images using Poisson equation,” IEEE Trans. Image Process. 4, 667–676 (1995).
[CrossRef]

Nishiyama, S.

Papathanassiou, K. P.

J. S. Lee, K. P. Papathanassiou, T. L. Ainsworth, M. H. Grunes, and A. Reigber, “A new technique for noise filtering of SAR interferometric phase images,” IEEE Trans. Geosci. Remote Sens. 36, 1456–1465 (1998).
[CrossRef]

Pelc, N. J.

S. M. Song, S. Napel, N. J. Pelc, and G. H. Glover, “Phase unwrapping of MR phase images using Poisson equation,” IEEE Trans. Image Process. 4, 667–676 (1995).
[CrossRef]

Perry, K. E.

K. E. Perry, Jr., and J. McKelvie, “A comparison of phase shifting and Fourier methods in the analysis of discontinuous fringe patterns,” Opt. Lasers Eng. 19, 269–284 (1993).
[CrossRef]

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes: The Art of Scientific Computing, 3rd edition (Cambridge University, 2007).

Reigber, A.

J. S. Lee, K. P. Papathanassiou, T. L. Ainsworth, M. H. Grunes, and A. Reigber, “A new technique for noise filtering of SAR interferometric phase images,” IEEE Trans. Geosci. Remote Sens. 36, 1456–1465 (1998).
[CrossRef]

Romero, L. A.

Rosenfeld, D. P.

Song, S. M.

S. M. Song, S. Napel, N. J. Pelc, and G. H. Glover, “Phase unwrapping of MR phase images using Poisson equation,” IEEE Trans. Image Process. 4, 667–676 (1995).
[CrossRef]

Soon, S. H.

Q. Kemao, 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]

Takahashi, T.

Takajo, H.

Takeda, M.

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes: The Art of Scientific Computing, 3rd edition (Cambridge University, 2007).

Thieme, W.

Tomioka, S.

Turner, S. R. E.

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes: The Art of Scientific Computing, 3rd edition (Cambridge University, 2007).

Wang, H.

Weber, H.

Werner, C. L.

M. R. Goldstein and C. L. Werner, “Radar interferogram filtering for geophysical applications,” Geophys. Res. Lett. 25, 4035–4038 (1998).
[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]

White, A. D.

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]

Yamaki, R.

R. Yamaki and A. Hirose, “Singularity-spreading phase unwrapping,” IEEE Trans. Geosci. Remote Sens. 45, 3240–3251(2007).
[CrossRef]

Zebker, H. A.

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. (10)

J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, and D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. 13, 2693–2703 (1974).
[CrossRef] [PubMed]

B. Breuckmann and W. Thieme, “Computer-aided analysis of holographic interferograms using the phase-shift method,” Appl. Opt. 24, 2145–2149 (1985).
[CrossRef] [PubMed]

J. M. Huntley, “Noise-immune phase unwrapping algorithm,” Appl. Opt. 28, 3268–3270 (1989).
[CrossRef] [PubMed]

B. Gutmann and H. Weber, “Phase unwrapping with the branch-cut method: clustering of discontinuity sources and reverse simulated annealing,” Appl. Opt. 38, 5577–5593 (1999).
[CrossRef]

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

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

E. Cuche, P. Marquet, and C. Depeursinge, “Spatial filtering for zero-order and twin-image elimination in digital off-axis holography,” Appl. Opt. 39, 4070–4075 (2000).
[CrossRef]

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]

Q. Kemao, W. Gao, and H. Wang, “Windowed Fourier-filtered and quality-guided phase-unwrapping algorithm,” Appl. Opt. 47, 5408–5428 (2008).
[CrossRef] [PubMed]

S. Tomioka, S. Heshmat, N. Miyamoto, and S. Nishiyama, “Phase unwrapping for noisy phase maps using rotational compensator with virtual singular points,” Appl. Opt. 49, 4735–4745 (2010).
[CrossRef] [PubMed]

Electron. Lett. (1)

C. L. Martinez, X. F. Canovas, and M. Chandra, “SAR interferometric phase noise reduction using wavelet transform,” Electron. Lett. 37, 649–651 (2001).
[CrossRef]

Geophys. Res. Lett. (1)

M. R. Goldstein and C. L. Werner, “Radar interferogram filtering for geophysical applications,” Geophys. Res. Lett. 25, 4035–4038 (1998).
[CrossRef]

IEEE Trans. Geosci. Remote Sens. (4)

J. S. Lee, K. P. Papathanassiou, T. L. Ainsworth, M. H. Grunes, and A. Reigber, “A new technique for noise filtering of SAR interferometric phase images,” IEEE Trans. Geosci. Remote Sens. 36, 1456–1465 (1998).
[CrossRef]

M. Costantine, “A novel phase unwrapping method based on network programming,” IEEE Trans. Geosci. Remote Sens. 36, 813–821 (1998).
[CrossRef]

R. Yamaki and A. Hirose, “Singularity-spreading phase unwrapping,” IEEE Trans. Geosci. Remote Sens. 45, 3240–3251(2007).
[CrossRef]

C. L. Martinez and X. Fabergas, “Modeling and reduction of SAR interferometric phase noise in the wavelet domain,” IEEE Trans. Geosci. Remote Sens. 40, 2553–2566(2002).
[CrossRef]

IEEE Trans. Image Process. (1)

S. M. Song, S. Napel, N. J. Pelc, and G. H. Glover, “Phase unwrapping of MR phase images using Poisson equation,” IEEE Trans. Image Process. 4, 667–676 (1995).
[CrossRef]

IEEE Trans. Med. Imaging (1)

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. (4)

J. Opt. Soc. Am. A (4)

Opt. Laser Technol. (1)

Q. Kemao, 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)

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

K. E. Perry, Jr., and J. McKelvie, “A comparison of phase shifting and Fourier methods in the analysis of discontinuous fringe patterns,” Opt. Lasers Eng. 19, 269–284 (1993).
[CrossRef]

Opt. Lett. (1)

Radio Sci. (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]

Other (1)

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes: The Art of Scientific Computing, 3rd edition (Cambridge University, 2007).

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Figures (5)

Fig. 1
Fig. 1

Existence of the branch cuts between the adjoining SPs and the concept of direct compensation. Open and filled squares represent positive and negative SPs, respectively. The thick dashed line denotes the branch cut that connects two SPs of opposite signs. Compensator position is denoted by thick arrows. The thin arrows show the direction and distribution of compensators for the segments of each SP, where S and S denote the residues of the SPs.

Fig. 2
Fig. 2

Complex cases for the position patterns of SP pairs and the DCs for the adjoining SPs pairs. Open and filled squares represent positive and negative SPs, respectively. (a) SPs are distributed in discrete values. (b) SPs are distributed by using the USP technique. In (b), the thick dashed line denotes the branch cut that connects two SPs of opposite signs, and the DC positions are denoted by thick arrows.

Fig. 3
Fig. 3

Comparison of the unwrapped phase results for simulated phase data: (a) original phase data, (b) wrapped data, (c) positions of all SP pairs, (d) positions of the pairs of nonadjoining SPs, (e) positions of the pairs of adjoining SPs, (f) unwrapped result by LS-DCT, (g) unwrapped result by RC, and (h) unwrapped result by RC + DC (proposed). In (a), (b), and (f)–(h), the phase increases with the increases of brightness. In (f)–(h), contour lines of the phase with the interval of one cycle are also shown.

Fig. 4
Fig. 4

Required computational time of each algorithm for various image sizes. The horizontal axis N denotes one-dimensional area size in pixels. RC shows the required time cost for RC method, RC + DC shows the required execution time for the proposed method, and LS-DCT shows the required time cost for LS-DCT method. The computational time is measured with a PC including an Intel Core 2 DUO central processing unit (CPU) with a 2.13 GHz clock in the single CPU operation mode.

Fig. 5
Fig. 5

Unwrapped phase result of experimental data for candle flame: (a) wrapped data, (b) SPs distribution map (positive and negative SPs are represented by white and black dots, respectively), (c) unwrapped result of the RC algorithm, and (d) unwrapped result of RC + DC (proposed algorithm).

Tables (2)

Tables Icon

Table 1 Comparison of the Accuracy for the Simulation Data Shown in Fig. 3

Tables Icon

Table 2 Comparison of the Execution Time Cost between RC Algorithm and the Proposed Algorithm a

Equations (11)

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

^ Ψ i = Δ Ψ i Int [ Δ Ψ i 2 π ] 2 π , Δ Ψ i = Ψ i + 1 Ψ i ,
Φ M = Φ 0 + i = 0 M 1 ^ Ψ i .
i = 0 3 ^ Ψ i = 2 π S ,
i = 0 N 1 ( ^ Ψ i + C i ) = 0 .
C j i R = S j ( θ i + 1 , j θ i , j ) ,
C i R = j = 1 N s C j i R .
Φ M = Φ 0 + i = 0 M 1 ( ^ Ψ i + C i ) ,
| C j i R | 1 R .
C j i D = { T j i π S j when the segment number  i is a member of the loop of the   j -th SP, which belongs to the adjoining pair ; 0 otherwise ;
m j i = δ j adj + ( i ) + δ j adj ( i ) .
C i = j = 1 N s m ¯ j i · C j i R + j = 1 N s m j i · C j i D ,

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