Abstract

We investigate the relationship between local and global phase-unwrapping techniques. We demonstrate that both algorithms are based on line integration of an estimate of the phase gradient: the former evaluates the integral along a single path, and the latter averages integrals along many paths. This analysis is extended to the weighted case. We also give an interpretation of the errors caused by an incorrect phase-gradient estimate in terms of path-following integrals.

© 1997 Optical Society of America

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  1. R. H. Hudgin, “Wave-front reconstruction for compensated imaging,” J. Opt. Soc. Am. A 67, 370–375 (1977).
  2. H. A. Zebker, R. M. Goldstein, “Topographic mapping from interferometric synthetic aperture radar observations,” J. Geophys. Res. 91, 4993–4999 (1986).
    [CrossRef]
  3. Q. Lin, F. Vesecky, H. A. Zebker, “Comparison of elevation derived from INSAR data with DEM over largerelief terrain,” Int. J. Remote Sensing 15, 1775–1790 (1994).
    [CrossRef]
  4. R. M. Goldenstein, H. A. Zebker, C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23, 713–720 (1988).
    [CrossRef]
  5. G. Fornaro, G. Franceschetti, R. Lanari, D. Rossi, M. Tesauro, “Finite element method for interferometric SAR phase unwrapping,” presented at the EUROPTO '96 meeting, Taormina, Italy, September 23–26, 1996.
  6. H. Takajo, T. Takahashi, “Noniterative methods for obtaining the exact solution for the normalequation in least-squares phase estimation from phase difference,” J. Opt. Soc. Am. A 5, 1818–1827 (1988).
    [CrossRef]
  7. D. C. Ghiglia, L. A. Romero, “Direct phase estimation from phase difference using fast elliptic partialdifferential equation solvers,” Opt. Lett. 14, 1107–1109 (1989).
    [CrossRef] [PubMed]
  8. D. C. Ghiglia, L. A. Romero, “Robust two-dimensional weighted and unweighted phase unwrapping thatuses fast transform and iterative methods,” J. Opt. Soc. Am. A 11, 107–117 (1994).
    [CrossRef]
  9. M. D. Pritt, J. S. Shipman, “Least-squares two dimensional phase unwrapping using FFT's,” IEEE Trans. Geosci. Remote Sensing 32, 706–708 (1994).
    [CrossRef]
  10. G. Fornaro, G. Franceschetti, R. Lanari, “Interferometric SAR phase unwrapping using Green's formulation,” IEEE Trans. Geosci. Remote Sensing 34, 720–727 (1996).
    [CrossRef]
  11. U. Spagnolini, “2-D Phase unwrapping and instantaneous frequency estimation,” IEEE Trans. Geosci. Remote Sensing 33, 579–589 (1995).
    [CrossRef]
  12. G. Fornaro, G. Franceschetti, R. Lanari, E. Sansosti, “Robust phase unwrapping techniques: a comparison,” J. Opt. Soc. Am. A 13, 2355–2366 (1996).
    [CrossRef]
  13. C. Prati, M. Giani, N. Leuratti, “SAR Interferometry: a 2-D phase unwrapping technique based on phase and ab-solute values information,” presented at the International Geoscience and Remote Sensing Symposium (IGARSS) '90, Washington, D.C., May 20–24, 1990.
  14. W. Xu, I. Cumming, “A region growing algorithm for InSAR phase unwrapping,” presented at the International Geoscience and Remote Sensing Symposium (IGARSS) '96, Lincoln, Nebr., May 27–31, 1996.
  15. M. D. Pritt, “Phase unwrapping by means of multigrid techniques for interferometricSAR,” IEEE Trans. Geosci. Remote Sensing 34, 728–738 (1996).
    [CrossRef]
  16. D. J. Bone, “Fourier fringe analysis: the two-dimensional phase unwrapping problem,” Appl. Opt. 30, 3627–3632 (1991).
    [CrossRef] [PubMed]
  17. R. Cusak, J. M. Huntley, H. T. Goldrein, “Improved noise-immune phase-unwrapping algorithm,” Appl. Opt. 34, 781–789 (1995).
    [CrossRef]
  18. J. A. Quiroga, A. Gonzàlez-Cano, E. Bernabeu, “Stable-marriages algorithms for preprocessing phase maps with discontinuitysources,” Appl. Opt. 34, 5029–5038 (1995).
    [CrossRef] [PubMed]
  19. J. R. Buckland, J. M. Huntley, S. R. E. Turner, “Unwrapping noisy phase maps by use of a minimum-cost-matching algorithm,” Appl. Opt. 34, 5100–5108 (1995).
    [CrossRef] [PubMed]
  20. N. H. Ching, D. Rosenfeld, M. Braun, “Two-dimensional phase unwrapping algorithm using a minimum spanningtree algorithm,” IEEE Trans. Image Process. 1, 355–365 (1992).
    [CrossRef]

1996 (3)

G. Fornaro, G. Franceschetti, R. Lanari, “Interferometric SAR phase unwrapping using Green's formulation,” IEEE Trans. Geosci. Remote Sensing 34, 720–727 (1996).
[CrossRef]

G. Fornaro, G. Franceschetti, R. Lanari, E. Sansosti, “Robust phase unwrapping techniques: a comparison,” J. Opt. Soc. Am. A 13, 2355–2366 (1996).
[CrossRef]

M. D. Pritt, “Phase unwrapping by means of multigrid techniques for interferometricSAR,” IEEE Trans. Geosci. Remote Sensing 34, 728–738 (1996).
[CrossRef]

1995 (4)

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

J. A. Quiroga, A. Gonzàlez-Cano, E. Bernabeu, “Stable-marriages algorithms for preprocessing phase maps with discontinuitysources,” Appl. Opt. 34, 5029–5038 (1995).
[CrossRef] [PubMed]

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

U. Spagnolini, “2-D Phase unwrapping and instantaneous frequency estimation,” IEEE Trans. Geosci. Remote Sensing 33, 579–589 (1995).
[CrossRef]

1994 (3)

D. C. Ghiglia, L. A. Romero, “Robust two-dimensional weighted and unweighted phase unwrapping thatuses fast transform and iterative methods,” J. Opt. Soc. Am. A 11, 107–117 (1994).
[CrossRef]

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

Q. Lin, F. Vesecky, H. A. Zebker, “Comparison of elevation derived from INSAR data with DEM over largerelief terrain,” Int. J. Remote Sensing 15, 1775–1790 (1994).
[CrossRef]

1992 (1)

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

1991 (1)

1989 (1)

1988 (2)

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

H. Takajo, T. Takahashi, “Noniterative methods for obtaining the exact solution for the normalequation in least-squares phase estimation from phase difference,” J. Opt. Soc. Am. A 5, 1818–1827 (1988).
[CrossRef]

1986 (1)

H. A. Zebker, R. M. Goldstein, “Topographic mapping from interferometric synthetic aperture radar observations,” J. Geophys. Res. 91, 4993–4999 (1986).
[CrossRef]

1977 (1)

R. H. Hudgin, “Wave-front reconstruction for compensated imaging,” J. Opt. Soc. Am. A 67, 370–375 (1977).

Bernabeu, E.

Bone, D. J.

Braun, M.

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

Buckland, J. R.

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

Ching, N. H.

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

Cumming, I.

W. Xu, I. Cumming, “A region growing algorithm for InSAR phase unwrapping,” presented at the International Geoscience and Remote Sensing Symposium (IGARSS) '96, Lincoln, Nebr., May 27–31, 1996.

Cusak, R.

Fornaro, G.

G. Fornaro, G. Franceschetti, R. Lanari, E. Sansosti, “Robust phase unwrapping techniques: a comparison,” J. Opt. Soc. Am. A 13, 2355–2366 (1996).
[CrossRef]

G. Fornaro, G. Franceschetti, R. Lanari, “Interferometric SAR phase unwrapping using Green's formulation,” IEEE Trans. Geosci. Remote Sensing 34, 720–727 (1996).
[CrossRef]

G. Fornaro, G. Franceschetti, R. Lanari, D. Rossi, M. Tesauro, “Finite element method for interferometric SAR phase unwrapping,” presented at the EUROPTO '96 meeting, Taormina, Italy, September 23–26, 1996.

Franceschetti, G.

G. Fornaro, G. Franceschetti, R. Lanari, “Interferometric SAR phase unwrapping using Green's formulation,” IEEE Trans. Geosci. Remote Sensing 34, 720–727 (1996).
[CrossRef]

G. Fornaro, G. Franceschetti, R. Lanari, E. Sansosti, “Robust phase unwrapping techniques: a comparison,” J. Opt. Soc. Am. A 13, 2355–2366 (1996).
[CrossRef]

G. Fornaro, G. Franceschetti, R. Lanari, D. Rossi, M. Tesauro, “Finite element method for interferometric SAR phase unwrapping,” presented at the EUROPTO '96 meeting, Taormina, Italy, September 23–26, 1996.

Ghiglia, D. C.

Giani, M.

C. Prati, M. Giani, N. Leuratti, “SAR Interferometry: a 2-D phase unwrapping technique based on phase and ab-solute values information,” presented at the International Geoscience and Remote Sensing Symposium (IGARSS) '90, Washington, D.C., May 20–24, 1990.

Goldenstein, R. M.

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

Goldrein, H. T.

Goldstein, R. M.

H. A. Zebker, R. M. Goldstein, “Topographic mapping from interferometric synthetic aperture radar observations,” J. Geophys. Res. 91, 4993–4999 (1986).
[CrossRef]

Gonzàlez-Cano, A.

Hudgin, R. H.

R. H. Hudgin, “Wave-front reconstruction for compensated imaging,” J. Opt. Soc. Am. A 67, 370–375 (1977).

Huntley, J. M.

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

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

Lanari, R.

G. Fornaro, G. Franceschetti, R. Lanari, E. Sansosti, “Robust phase unwrapping techniques: a comparison,” J. Opt. Soc. Am. A 13, 2355–2366 (1996).
[CrossRef]

G. Fornaro, G. Franceschetti, R. Lanari, “Interferometric SAR phase unwrapping using Green's formulation,” IEEE Trans. Geosci. Remote Sensing 34, 720–727 (1996).
[CrossRef]

G. Fornaro, G. Franceschetti, R. Lanari, D. Rossi, M. Tesauro, “Finite element method for interferometric SAR phase unwrapping,” presented at the EUROPTO '96 meeting, Taormina, Italy, September 23–26, 1996.

Leuratti, N.

C. Prati, M. Giani, N. Leuratti, “SAR Interferometry: a 2-D phase unwrapping technique based on phase and ab-solute values information,” presented at the International Geoscience and Remote Sensing Symposium (IGARSS) '90, Washington, D.C., May 20–24, 1990.

Lin, Q.

Q. Lin, F. Vesecky, H. A. Zebker, “Comparison of elevation derived from INSAR data with DEM over largerelief terrain,” Int. J. Remote Sensing 15, 1775–1790 (1994).
[CrossRef]

Prati, C.

C. Prati, M. Giani, N. Leuratti, “SAR Interferometry: a 2-D phase unwrapping technique based on phase and ab-solute values information,” presented at the International Geoscience and Remote Sensing Symposium (IGARSS) '90, Washington, D.C., May 20–24, 1990.

Pritt, M. D.

M. D. Pritt, “Phase unwrapping by means of multigrid techniques for interferometricSAR,” IEEE Trans. Geosci. Remote Sensing 34, 728–738 (1996).
[CrossRef]

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

Quiroga, J. A.

Romero, L. A.

Rosenfeld, D.

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

Rossi, D.

G. Fornaro, G. Franceschetti, R. Lanari, D. Rossi, M. Tesauro, “Finite element method for interferometric SAR phase unwrapping,” presented at the EUROPTO '96 meeting, Taormina, Italy, September 23–26, 1996.

Sansosti, E.

Shipman, J. S.

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

Spagnolini, U.

U. Spagnolini, “2-D Phase unwrapping and instantaneous frequency estimation,” IEEE Trans. Geosci. Remote Sensing 33, 579–589 (1995).
[CrossRef]

Takahashi, T.

Takajo, H.

Tesauro, M.

G. Fornaro, G. Franceschetti, R. Lanari, D. Rossi, M. Tesauro, “Finite element method for interferometric SAR phase unwrapping,” presented at the EUROPTO '96 meeting, Taormina, Italy, September 23–26, 1996.

Turner, S. R. E.

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

Vesecky, F.

Q. Lin, F. Vesecky, H. A. Zebker, “Comparison of elevation derived from INSAR data with DEM over largerelief terrain,” Int. J. Remote Sensing 15, 1775–1790 (1994).
[CrossRef]

Werner, C. L.

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

Xu, W.

W. Xu, I. Cumming, “A region growing algorithm for InSAR phase unwrapping,” presented at the International Geoscience and Remote Sensing Symposium (IGARSS) '96, Lincoln, Nebr., May 27–31, 1996.

Zebker, H. A.

Q. Lin, F. Vesecky, H. A. Zebker, “Comparison of elevation derived from INSAR data with DEM over largerelief terrain,” Int. J. Remote Sensing 15, 1775–1790 (1994).
[CrossRef]

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

H. A. Zebker, R. M. Goldstein, “Topographic mapping from interferometric synthetic aperture radar observations,” J. Geophys. Res. 91, 4993–4999 (1986).
[CrossRef]

Appl. Opt. (1)

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

Appl. Opt. (3)

IEEE Trans. Image Process. (1)

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

IEEE Trans. Geosci. Remote Sensing (1)

M. D. Pritt, “Phase unwrapping by means of multigrid techniques for interferometricSAR,” IEEE Trans. Geosci. Remote Sensing 34, 728–738 (1996).
[CrossRef]

IEEE Trans. Geosci. Remote Sensing (1)

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

IEEE Trans. Geosci. Remote Sensing (2)

G. Fornaro, G. Franceschetti, R. Lanari, “Interferometric SAR phase unwrapping using Green's formulation,” IEEE Trans. Geosci. Remote Sensing 34, 720–727 (1996).
[CrossRef]

U. Spagnolini, “2-D Phase unwrapping and instantaneous frequency estimation,” IEEE Trans. Geosci. Remote Sensing 33, 579–589 (1995).
[CrossRef]

Int. J. Remote Sensing (1)

Q. Lin, F. Vesecky, H. A. Zebker, “Comparison of elevation derived from INSAR data with DEM over largerelief terrain,” Int. J. Remote Sensing 15, 1775–1790 (1994).
[CrossRef]

J. Geophys. Res. (1)

H. A. Zebker, R. M. Goldstein, “Topographic mapping from interferometric synthetic aperture radar observations,” J. Geophys. Res. 91, 4993–4999 (1986).
[CrossRef]

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

Opt. Lett. (1)

Radio Sci. (1)

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

Other (3)

G. Fornaro, G. Franceschetti, R. Lanari, D. Rossi, M. Tesauro, “Finite element method for interferometric SAR phase unwrapping,” presented at the EUROPTO '96 meeting, Taormina, Italy, September 23–26, 1996.

C. Prati, M. Giani, N. Leuratti, “SAR Interferometry: a 2-D phase unwrapping technique based on phase and ab-solute values information,” presented at the International Geoscience and Remote Sensing Symposium (IGARSS) '90, Washington, D.C., May 20–24, 1990.

W. Xu, I. Cumming, “A region growing algorithm for InSAR phase unwrapping,” presented at the International Geoscience and Remote Sensing Symposium (IGARSS) '96, Lincoln, Nebr., May 27–31, 1996.

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Figures (7)

Fig. 1
Fig. 1

Relevance of local integration.

Fig. 2
Fig. 2

Relevance of the use of Green's first identity in polar coordinates.

Fig. 3
Fig. 3

Multiconnected domain.

Fig. 4
Fig. 4

Relevance of error propagation resulting from a line of phase discontinuity.

Fig. 5
Fig. 5

Mirrored domain.

Fig. 6
Fig. 6

Relevance of a rectangular domain with different locations, (A) and (B), of a corrupted region.

Fig. 7
Fig. 7

Error functions obtained in presence (solid curves) and in the absence (dashed–dotted curves) of boundary information for the corrupted region. (a) Case (A) of Fig. 6, (b) case (B) of Fig. 6. The phase jump is 10π rad.

Equations (29)

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ϕ ( r ) = ϕ 0 + P 0 P ( r ) d c s ( r ) c ˆ , ϕ 0 = ϕ ( P 0 ) ,
s - ϕ 2 = min .
ϕ ( r ) = - S d S s ( r ) g ( r - r ) + C d c ϕ ( r C )   g ( r - r C ) n , r S ,
2 g ( r ) = δ ( r ) .
g ( r ) = 1 2 π   ln ( 1 / r ) ,
g ( r ) = - 1 2 π   r ˆ r .
ϕ ( 0 ) = C d c ϕ ( r M ,   θ )   r ˆ n ˆ 2 π r M - 0 2 π d θ 0 r M r d r s ( r ,   θ )   r ˆ 2 π r
= 1 2 π   0 2 π d θ ϕ ( r M ,   θ ) - 1 2 π 0 2 π d θ 0 r M d r s ( r ,   θ ) r ˆ = 1 2 π   0 2 π d θ ϕ ( r M ,   θ ) + r M 0 d r s ( r ,   θ ) r ˆ ,
r M C ,
ϕ ( 0 ) = 1 2 π   0 2 π d θ [ ϕ ( r M ) + ϕ ( 0 ) - ϕ ( r M ) ] .
ϕ ˜ ( x ,   y ) = - S d S s ˜ ( x ,   y ) g ˜ ( x - x ,   y - y ) ,
2 g ˜ ( x ,   y ) = p , q = - + δ ( x - pa ,   y - qb ) .
g ˜ ( x ,   y ) = p , q = - + g ( x - pa ,   y - qb ) = rep a , b [ g ( x ,   y ) ] ,
ϕ ˜ ( x ,   y ) = - - a / 2 a / 2 d x - b / 2 b / 2 d y s ˜ ( x ,   y ) p , q = - + g ( x - x - pa ,   y - y - qb ) .
ϕ ˜ ( x ,   y ) = - p , q = - + - a / 2 + pa a / 2 + pa d x - b / 2 + qb b / 2 + qb d y [ s ˜ ( x - pa , y - qb ) g ( x - x ,   y - y ) ] = - p , q = - + - a / 2 + pa a / 2 + pa d x - b / 2 + qb b / 2 + qb d y [ s ˜ ( x ,   y ) g ( x - x ,   y - y ) ] = - - + d x - + d y [ s ˜ ( x ,   y ) g ( x - x ,   y - y ) ] ,
s ˜ ( x + pa ,   y + qb ) = s ˜ ( x ,   y ) .
ϕ ˜ ( 0 ) = - 0 2 π d θ 0 + r d r s ˜ ( r ,   θ )   r ˆ 2 π r = 1 2 π   0 2 π d θ + 0 d r s ˜ ( r ,   θ ) r ˆ .
x = - a / 2 , y = t , t [ - b / 2 ,   b / 2 [ ,
x = t , t [ - a / 2 ,   a / 2 [ , y = - b / 2 ,
w ϕ - w s 2 = min ,
ϕ ( r ) = - S - W d S s ( r ) g ( r - r ) + C d c ϕ ( r C )   g ( r - r C ) n + C 1 d c ϕ ( r C )   g ( r - r C ) n , r S ,
ϕ ( 0 ) = ϕ 2 π - Δ α ( 0 ) + 1 2 π   Δ α d θ r M r C + d r s ( r ,   θ ) r ˆ + r C - 0 d r s ( r ,   θ ) r ˆ - 1 2 π   Δ α d θ ϕ ( r C + ,   θ ) + 1 2 π   Δ α d θ ϕ ( r C - ,   θ ) + 1 2 π   Δ α d θ ϕ ( r M ,   θ ) .
ϕ ( 0 ) = ϕ 2 π - Δ α ( 0 ) + 1 2 π   Δ α d θ ϕ ( r M ,   θ ) + r M r C + d r s ( r ,   θ ) r ˆ + [ ϕ ( r C - ,   θ ) - ϕ ( r C + ,   θ ) ] + r C - 0 d r s ( r ,   θ ) r ˆ .
ϕ = s + 2 π N δ ( y - y 0 ) rect x - x 0 2 L y ˆ .
( x ,   y ) = ϕ ( x ,   y ) - ϕ ˆ ( x ,   y ) .
( x ,   y ) = ϕ ( x ,   y ) - P 0 P ( x ,   y ) d c s c ˆ = 2 π N P 0 P ( x ,   y ) d c δ ( y - y 0 ) × rect x - x 0 2 L y ˆ c ˆ ,
( x ,   y ) = N Δ α ( x ,   y ) .
ϕ ˜ = s ˜ + rep 2 a , 2 b 2 π N δ ( y - y 0 ) rect x - x 0 2 L + δ ( y - y 0 ) rect a - x - x 0 2 L + δ ( b - y - y 0 ) rect x - x 0 2 L + δ ( b - y - y 0 ) rect a - x - x 0 2 L y ˆ .
˜ ( x ,   y ) = rep 2 a , 2 b [ N Δ α ( x ,   y ) ] + rep 2 a , 2 b [ N Δ α ( a - x ,   y ) ] + rep 2 a , 2 b [ N Δ α ( x ,   b - y ) ] + rep 2 a , 2 b [ N Δ α ( a - x ,   b - y ) ] .

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