Abstract

Phase unwrapping for a noisy image suffers from many singular points. Singularity-spreading methods are useful for the noisy image to regularize the singularity. However, the methods have a drawback of distorting phase distribution in a regular area that contains no singular points. When the singular points are confined in some local areas, the regular region is not distorted. This paper proposes a new phase unwrapping algorithm that uses a localized compensator obtained by clustering and by solving Poisson’s equation for the localized areas. The numerical results demonstrate that the proposed method can improve the accuracy compared with other singularity-spreading methods.

© 2012 Optical Society of America

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References

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  1. M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72, 156–160 (1982).
    [CrossRef]
  2. K. E. Perry 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]
  3. 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]
  4. 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]
  5. B. Breuckmann and W. Thieme, “Computer-aided analysis of holographic interferograms using the phase-shift method,” Appl. Opt. 24, 2145–2149 (1985).
    [CrossRef]
  6. J. Jiang, J. Cheng, Y. Zhou, and G. Chen, “Clustering-driven residue filter for profile measurement system,” J. Opt. Soc. Am. A 28, 214–221 (2011).
    [CrossRef]
  7. R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23, 713–720 (1988).
    [CrossRef]
  8. J. M. Huntley, “Noise-immune phase unwrapping algorithm,” Appl. Opt. 28, 3268–3270 (1989).
    [CrossRef]
  9. 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]
  10. R. Cusack, J. M. Huntley, and H. T. Goldrein, “Improved noise-immune phase-unwrapping algorithm,” Appl. Opt. 34, 781–789 (1995).
    [CrossRef]
  11. M. Costantine, “A novel phase unwrapping method based on network programming,” IEEE Trans. Geosci. Remote Sens. 36, 813–821 (1998).
    [CrossRef]
  12. 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]
  13. 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]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  22. 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]
  23. S. Heshmat, S. Tomioka, and S. Nishiyama, “A reliable phase unwrapping algorithm based on rotational and direct compensators,” Appl. Opt. 50, 6225–6233 (2011).
    [CrossRef]
  24. D. J. Bone, “Fourier fringe analysis: the two-dimensional phase unwrapping problem,” Appl. Opt. 30, 3627–3632 (1991).
    [CrossRef]
  25. P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1953), pp. 52–54.
  26. C. A. Brebia and S. Walker, Boundary Element Techniques in Engineering (Newnes-Butterworths, 1980).
  27. S. Tomioka and S. Nisiyama, “Analytical regularization of hypersingular integral for Helmholtz equation in boundary element method,” Eng. Anal. Bound. Elem. 34, 393–404 (2010).
    [CrossRef]
  28. M. Arai, T. Adachi, and H. Matsumoto, “Highly accurate analysis by boundary element method based on uniform gradient condition,” Trans. Jpn. Soc. Mech. Eng. A 61, 161–168 (1995), in Japanese.
    [CrossRef]
  29. M. Guiggiani, G. Krishinasamy, T. J. Rudolphi, and F. J. Rizz, “A general algorithm for the numerical solution of hypersingular boundary integral equations,” Trans. ASME 59, 604–614 (1992).
    [CrossRef]
  30. E. Anderson, Z. Bai, C. Bischof, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, S. Ostrouchov, and D. Sorensen, LAPACK Users’ Guide(Society for Industrial and Applied Mathematics, 1992).
  31. D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, 1998).

2011 (2)

2010 (2)

S. Tomioka and S. Nisiyama, “Analytical regularization of hypersingular integral for Helmholtz equation in boundary element method,” Eng. Anal. Bound. Elem. 34, 393–404 (2010).
[CrossRef]

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]

2007 (2)

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

2000 (1)

1999 (1)

1998 (1)

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

1995 (3)

1994 (1)

1993 (1)

K. E. Perry 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]

1992 (1)

M. Guiggiani, G. Krishinasamy, T. J. Rudolphi, and F. J. Rizz, “A general algorithm for the numerical solution of hypersingular boundary integral equations,” Trans. ASME 59, 604–614 (1992).
[CrossRef]

1991 (1)

1989 (2)

1988 (3)

1985 (1)

1982 (1)

1979 (1)

1977 (2)

1974 (1)

Adachi, T.

M. Arai, T. Adachi, and H. Matsumoto, “Highly accurate analysis by boundary element method based on uniform gradient condition,” Trans. Jpn. Soc. Mech. Eng. A 61, 161–168 (1995), in Japanese.
[CrossRef]

Anderson, E.

E. Anderson, Z. Bai, C. Bischof, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, S. Ostrouchov, and D. Sorensen, LAPACK Users’ Guide(Society for Industrial and Applied Mathematics, 1992).

Arai, M.

M. Arai, T. Adachi, and H. Matsumoto, “Highly accurate analysis by boundary element method based on uniform gradient condition,” Trans. Jpn. Soc. Mech. Eng. A 61, 161–168 (1995), in Japanese.
[CrossRef]

Bai, Z.

E. Anderson, Z. Bai, C. Bischof, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, S. Ostrouchov, and D. Sorensen, LAPACK Users’ Guide(Society for Industrial and Applied Mathematics, 1992).

Bischof, C.

E. Anderson, Z. Bai, C. Bischof, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, S. Ostrouchov, and D. Sorensen, LAPACK Users’ Guide(Society for Industrial and Applied Mathematics, 1992).

Bone, D. J.

Brangaccio, D. J.

Brebia, C. A.

C. A. Brebia and S. Walker, Boundary Element Techniques in Engineering (Newnes-Butterworths, 1980).

Breuckmann, B.

Bruning, J. H.

Buckland, J. R.

Burton, D. R.

Chen, G.

Cheng, J.

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.

Demmel, J.

E. Anderson, Z. Bai, C. Bischof, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, S. Ostrouchov, and D. Sorensen, LAPACK Users’ Guide(Society for Industrial and Applied Mathematics, 1992).

Depeursinge, C.

Dongarra, J.

E. Anderson, Z. Bai, C. Bischof, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, S. Ostrouchov, and D. Sorensen, LAPACK Users’ Guide(Society for Industrial and Applied Mathematics, 1992).

Du Croz, J.

E. Anderson, Z. Bai, C. Bischof, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, S. Ostrouchov, and D. Sorensen, LAPACK Users’ Guide(Society for Industrial and Applied Mathematics, 1992).

Feshbach, H.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1953), pp. 52–54.

Fried, D. L.

Gallagher, J. E.

Gdeisat, M. A.

Ghiglia, D. C.

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]

Greenbaum, A.

E. Anderson, Z. Bai, C. Bischof, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, S. Ostrouchov, and D. Sorensen, LAPACK Users’ Guide(Society for Industrial and Applied Mathematics, 1992).

Guiggiani, M.

M. Guiggiani, G. Krishinasamy, T. J. Rudolphi, and F. J. Rizz, “A general algorithm for the numerical solution of hypersingular boundary integral equations,” Trans. ASME 59, 604–614 (1992).
[CrossRef]

Gutmann, B.

Hammarling, S.

E. Anderson, Z. Bai, C. Bischof, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, S. Ostrouchov, and D. Sorensen, LAPACK Users’ Guide(Society for Industrial and Applied Mathematics, 1992).

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.

Jiang, J.

Karout, S. A.

Kobayashi, S.

Krishinasamy, G.

M. Guiggiani, G. Krishinasamy, T. J. Rudolphi, and F. J. Rizz, “A general algorithm for the numerical solution of hypersingular boundary integral equations,” Trans. ASME 59, 604–614 (1992).
[CrossRef]

Lalor, M. J.

Marquet, P.

Matsumoto, H.

M. Arai, T. Adachi, and H. Matsumoto, “Highly accurate analysis by boundary element method based on uniform gradient condition,” Trans. Jpn. Soc. Mech. Eng. A 61, 161–168 (1995), in Japanese.
[CrossRef]

McKelvie, J.

K. E. Perry 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]

McKenney, A.

E. Anderson, Z. Bai, C. Bischof, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, S. Ostrouchov, and D. Sorensen, LAPACK Users’ Guide(Society for Industrial and Applied Mathematics, 1992).

Miyamoto, N.

Morse, P. M.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1953), pp. 52–54.

Nishiyama, S.

Nisiyama, S.

S. Tomioka and S. Nisiyama, “Analytical regularization of hypersingular integral for Helmholtz equation in boundary element method,” Eng. Anal. Bound. Elem. 34, 393–404 (2010).
[CrossRef]

Ostrouchov, S.

E. Anderson, Z. Bai, C. Bischof, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, S. Ostrouchov, and D. Sorensen, LAPACK Users’ Guide(Society for Industrial and Applied Mathematics, 1992).

Perry, K. E.

K. E. Perry 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]

Pritt, M. D.

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

Rizz, F. J.

M. Guiggiani, G. Krishinasamy, T. J. Rudolphi, and F. J. Rizz, “A general algorithm for the numerical solution of hypersingular boundary integral equations,” Trans. ASME 59, 604–614 (1992).
[CrossRef]

Romero, L. A.

Rosenfeld, D. P.

Rudolphi, T. J.

M. Guiggiani, G. Krishinasamy, T. J. Rudolphi, and F. J. Rizz, “A general algorithm for the numerical solution of hypersingular boundary integral equations,” Trans. ASME 59, 604–614 (1992).
[CrossRef]

Sorensen, D.

E. Anderson, Z. Bai, C. Bischof, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, S. Ostrouchov, and D. Sorensen, LAPACK Users’ Guide(Society for Industrial and Applied Mathematics, 1992).

Takahashi, T.

Takajo, H.

Takeda, M.

Thieme, W.

Tomioka, S.

Turner, S. R. E.

Walker, S.

C. A. Brebia and S. Walker, Boundary Element Techniques in Engineering (Newnes-Butterworths, 1980).

Weber, H.

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]

White, A. D.

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]

Zhou, Y.

Appl. Opt. (11)

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]

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

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

D. J. Bone, “Fourier fringe analysis: the two-dimensional phase unwrapping problem,” Appl. Opt. 30, 3627–3632 (1991).
[CrossRef]

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]

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]

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]

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]

S. Heshmat, S. Tomioka, and S. Nishiyama, “A reliable phase unwrapping algorithm based on rotational and direct compensators,” Appl. Opt. 50, 6225–6233 (2011).
[CrossRef]

Eng. Anal. Bound. Elem. (1)

S. Tomioka and S. Nisiyama, “Analytical regularization of hypersingular integral for Helmholtz equation in boundary element method,” Eng. Anal. Bound. Elem. 34, 393–404 (2010).
[CrossRef]

IEEE Trans. Geosci. Remote Sens. (2)

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]

J. Opt. Soc. Am. (4)

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

Opt. Lasers Eng. (1)

K. E. Perry 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]

Trans. ASME (1)

M. Guiggiani, G. Krishinasamy, T. J. Rudolphi, and F. J. Rizz, “A general algorithm for the numerical solution of hypersingular boundary integral equations,” Trans. ASME 59, 604–614 (1992).
[CrossRef]

Trans. Jpn. Soc. Mech. Eng. A (1)

M. Arai, T. Adachi, and H. Matsumoto, “Highly accurate analysis by boundary element method based on uniform gradient condition,” Trans. Jpn. Soc. Mech. Eng. A 61, 161–168 (1995), in Japanese.
[CrossRef]

Other (4)

E. Anderson, Z. Bai, C. Bischof, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, S. Ostrouchov, and D. Sorensen, LAPACK Users’ Guide(Society for Industrial and Applied Mathematics, 1992).

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

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1953), pp. 52–54.

C. A. Brebia and S. Walker, Boundary Element Techniques in Engineering (Newnes-Butterworths, 1980).

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Figures (7)

Fig. 1.
Fig. 1.

Regularization of a quadrupole singularity: (a) original wrapped phase in the range (0.5,0.5] cycles; (b) differences between adjoining nodes shown with dashed arrows and singular points; (c) unwrapped phase by horizontal branch cuts; (d) unwrapped phase by vertical branch cuts; (e) unwrapped phase by rotational compensators; (f) unwrapped phase by localized compensators. In (c) and (d), the double lines indicate the branch cuts. From (c) to (f) the solid arrows indicate the compensators of which widths and lengths indicate magnitude.

Fig. 2.
Fig. 2.

Definition of local domain and flux density from singular points: (a) local domain including singular points, (b) flux distribution in the local domain. Outmost closed thick line is a boundary of the local domain, and the grids are segments of elementary loops. The dashed line in (a) represents a concave polygon that contains all singular points. The arrows in (b) are the flux density of which line width expresses the magnitude.

Fig. 3.
Fig. 3.

Example of cluster merging. The rectangular frame shows the image domain. The circles are SPs, and the squares are VSPs. The open ones and filled ones indicate positive and negative charges, respectively. The closed curves with thin lines show the original clusters after several reputations. The new clusters shown as the closed curves with thick lines are formed by cluster merging. The closed curves with dashed lines are intermediate clusters. The arrows represent the typical linkages from host SPs to their partners.

Fig. 4.
Fig. 4.

Schematics of cluster splitting: (a) an example of cluster merging progress and (b) cluster splitting. The open circles and filled circles are SPs (or VSPs) with positive charges and those with negative ones, respectively. The lines between SPs are their linkages. The closed curves with solid and dashed lines represent the zero-charged clusters and the charged clusters, respectively. The clusters in (a) are those after every stage, and the outmost cluster is the final result of the cluster merging process. The clusters split into two shown in (b) are formed by the cluster splitting algorithm. In (b), dashed lines and solid lines indicate the splittable linkages and the unsplittable linkages, respectively.

Fig. 5.
Fig. 5.

Phase unwrapping simulation: (a) original wrapped phase map, (b) unwrapped phase map by the LS-DCT method, (c) connections of the clustered tree, (d) unwrapped phase map by the method using an LC.

Fig. 6.
Fig. 6.

Unwrapping an experimental IFSAR phase: (a) a wrapped phase map (512×512) from [31] (copyright 1998 by John Wiley and Sons, Inc. Reproduced with permission of John Wiley and Sons, Inc.), (b) the clustered trees, (c) the unwrapped phase by the method using the RC, (d) the unwrapped phase by the method using an LC. The images (c′) and (d′) are the unwrapped phase with contour lines where the step is 2π radian.

Fig. 7.
Fig. 7.

Estimation of phase shift due to candle flame: (a) fringe pattern obtained by a Mach–Zehnder interferometer with a candle flame; (b) wrapped phase obtained from the fringe pattern by a Fourier transformed filter and by background phase shifting; (c)–(f) unwrapped phase distributions with different methods; in each unwrapped result, the reference point is positioned at the bottom left corner at which the unwrapped phases by all methods are same. The candle has a core at the bottom center in each figure.

Tables (1)

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Table 1. Accuracy Comparison among Algorithms by Planar Function Fitting

Equations (37)

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

ϕw=W{ϕ}ϕInt[ϕ2π]2π.
g·s^1ΔlW{Δϕw}=1Δl(ΔϕInt[Δϕ2π]2π)=ϕ·s^Int[Δϕ2π]2πΔl,
Cg·s^dl=2πkmk.
g=ϕ+×A,
·A=0.
c×A,
C(g+c)·s^dl=0,
Cc·s^dl=2πkmk.
ϕ(r)=ϕ(r0)+r0rg·s^dl+r0rc·s^dl.
kmk=0.
c·s^=(×A)·(z^×n^)=n^·Az,
n^·Az=0.
Cc·s^dl=S2AzdS.
mk=Smkδ(rrk)dS.
2Az=2πkmkδ(rrk).
r0rc·s^dl=r0rs^×z^·Azdl.
Q(V(sp))=Q(sp),
Ci={si,p,vi,p},{V(si,p)}={vi,p}+{v¯i,p},si,pCi,vi,pCi,v¯i,pCi,
QipQ(si,p)+pQ(vi,p).
Li,jminp,q,Q=±1{Dist(si,p+Q,sj,qQ)},
d¯iQiminp{Dist(si,p+Qi,v¯i,pQi)},
di+Qimaxp{Dist(si,pQi,vi,p+Qi)}.
LCP=mini{minj{Li,j},d¯iQi}forQi0,QjQi,ij,
di+Qi>LCP,
d¯iQi<LCP,
Dist(vi,p+,vi,q)<Dist(si,p,vi,p+)+Dist(si,q+,vi,q),
cAz·n^dl=2πkSmkδ(rrk)dS,
ek,j=(Az)k,j(Az)k,j.
j=14((Az)k,jek,j)·n^k,j=2πmk.
ek,j=ek,j=ei,
n^k,j=dk,id^i,
De=r,
(D)k,i=dk,i,(e)i=d^i·ei,(r)k=2πmkidk,id^i·(Az)i.
minimizei|(e)i|2.
(Az)k,j=(Az)k,jek,j.
c=mcm,
I(r)=I0+I12(ei(ϕ(r)+kbg·r)+ei(ϕ(r)+kbg·r)),

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