Abstract

A new two-dimensional phase unwrapping method, based on an iterative computational procedure, is proposed. The method, which is derived with the use of a global cost function to minimize the phase discontinuities in the unwrapped phase map, has shown to produce robust and reliable results on very noisy phase data. Preprocessing operations, such as noise cleaning or segmentation, will in many cases be superfluous but may be included.

© 1998 Optical Society of America

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References

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  1. J. M. Tribolet, “A new phase unwrapping algorithm,” IEEE Trans. Acoust. Speech Signal Process. ASSP-25, 170–177 (1977).
    [CrossRef]
  2. R. M. Goldstein, H. A. Zebker, C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23, 713–720 (1988).
    [CrossRef]
  3. J. M. Huntley, “Noise-immune phase unwrapping algorithm,” Appl. Opt. 28, 3268–3270 (1989).
    [CrossRef] [PubMed]
  4. H. T. Takajo, T. Takahashi, “Least-squares phase estimation from phase differences,” J. Opt. Soc. Am. A 5, 416–425 (1988).
    [CrossRef]
  5. D. C. Ghiglia, L. A. Romero, “Robust two-dimensional weighted and unweighted phase unwrapping using fast transforms and iterative methods,” J. Opt. Soc. Am. A 11, 107–117 (1994).
    [CrossRef]
  6. M. Hedley, D. Rosenfeld, “A new two-dimensional phase unwrapping algorithm for MRI images,” Magn. Reson. Med. 24, 177–181 (1992).
    [CrossRef] [PubMed]
  7. S. M. Song, S. Napel, N. J. Pelc, G. H. Glower, “Phase unwrapping of MR phase images using Poisson equation,” IEEE Trans. Image Process. 4, 667–676 (1995).
    [CrossRef]
  8. J. J. Gierloff, “Phase unwrapping by regions,” in Current Developments in Optical Engineering II, R. E. Fisher, W. J. Smith, eds., Proc. SPIE818, 2–9 (1987).
  9. P. G. Charette, I. W. Hunter, “Robust phase unwrapping method for phase images with high noise content,” Appl. Opt. 35, 3506–3513 (1996).
    [CrossRef] [PubMed]
  10. P. Stephenson, D. R. Burton, M. J. Lalor, “Data validation techniques in a tiled phase unwrapping algorithm,” Opt. Eng. 33, 3703–3708 (1994).
    [CrossRef]
  11. P. Ettl, K. Creath, “Comparison of phase-unwrapping algorithms by using gradient of first failure,” Appl. Opt. 35, 5108–5114 (1996).
    [CrossRef] [PubMed]
  12. Z. Liang, “A model-based method for phase unwrapping,” IEEE Trans. Med. Imaging 15, 893–897 (1996).
    [CrossRef] [PubMed]
  13. B. Friedlander, J. M. Francos, “Model based phase unwrapping of 2-D signals,” IEEE Trans. Signal Process. 44, 2999–3007 (1996).
    [CrossRef]
  14. J. S. Lim, “The discrete cosine transform,” in Two-Dimensional Signal and Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1990), pp. 148–157.
  15. J. S. Lim, “The window method,” in Two-Dimensional Signal and Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1990), pp. 202–213.
  16. O. Loffeld, C. Arndt, A. Hein, “Estimating the derivative of modulo-mapped phases,” presented at the European Space Agency Workshop on Applications of ERS SAR Interferometry, Zurich, September 30–October 2, 1996; also available at url: http://www.geo.unizh.ch/rsl/fringe96/papers/loffeld-et-al/ .
  17. A. K. Jain, “Morphological processing,” in Fundamentals of Digital Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1989), pp. 384–387.

1996 (4)

P. G. Charette, I. W. Hunter, “Robust phase unwrapping method for phase images with high noise content,” Appl. Opt. 35, 3506–3513 (1996).
[CrossRef] [PubMed]

P. Ettl, K. Creath, “Comparison of phase-unwrapping algorithms by using gradient of first failure,” Appl. Opt. 35, 5108–5114 (1996).
[CrossRef] [PubMed]

Z. Liang, “A model-based method for phase unwrapping,” IEEE Trans. Med. Imaging 15, 893–897 (1996).
[CrossRef] [PubMed]

B. Friedlander, J. M. Francos, “Model based phase unwrapping of 2-D signals,” IEEE Trans. Signal Process. 44, 2999–3007 (1996).
[CrossRef]

1995 (1)

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

1994 (2)

P. Stephenson, D. R. Burton, M. J. Lalor, “Data validation techniques in a tiled phase unwrapping algorithm,” Opt. Eng. 33, 3703–3708 (1994).
[CrossRef]

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

1992 (1)

M. Hedley, D. Rosenfeld, “A new two-dimensional phase unwrapping algorithm for MRI images,” Magn. Reson. Med. 24, 177–181 (1992).
[CrossRef] [PubMed]

1989 (1)

1988 (2)

H. T. Takajo, T. Takahashi, “Least-squares phase estimation from phase differences,” J. Opt. Soc. Am. A 5, 416–425 (1988).
[CrossRef]

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

1977 (1)

J. M. Tribolet, “A new phase unwrapping algorithm,” IEEE Trans. Acoust. Speech Signal Process. ASSP-25, 170–177 (1977).
[CrossRef]

Arndt, C.

O. Loffeld, C. Arndt, A. Hein, “Estimating the derivative of modulo-mapped phases,” presented at the European Space Agency Workshop on Applications of ERS SAR Interferometry, Zurich, September 30–October 2, 1996; also available at url: http://www.geo.unizh.ch/rsl/fringe96/papers/loffeld-et-al/ .

Burton, D. R.

P. Stephenson, D. R. Burton, M. J. Lalor, “Data validation techniques in a tiled phase unwrapping algorithm,” Opt. Eng. 33, 3703–3708 (1994).
[CrossRef]

Charette, P. G.

Creath, K.

Ettl, P.

Francos, J. M.

B. Friedlander, J. M. Francos, “Model based phase unwrapping of 2-D signals,” IEEE Trans. Signal Process. 44, 2999–3007 (1996).
[CrossRef]

Friedlander, B.

B. Friedlander, J. M. Francos, “Model based phase unwrapping of 2-D signals,” IEEE Trans. Signal Process. 44, 2999–3007 (1996).
[CrossRef]

Ghiglia, D. C.

Gierloff, J. J.

J. J. Gierloff, “Phase unwrapping by regions,” in Current Developments in Optical Engineering II, R. E. Fisher, W. J. Smith, eds., Proc. SPIE818, 2–9 (1987).

Glower, G. H.

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

Goldstein, R. M.

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

Hedley, M.

M. Hedley, D. Rosenfeld, “A new two-dimensional phase unwrapping algorithm for MRI images,” Magn. Reson. Med. 24, 177–181 (1992).
[CrossRef] [PubMed]

Hein, A.

O. Loffeld, C. Arndt, A. Hein, “Estimating the derivative of modulo-mapped phases,” presented at the European Space Agency Workshop on Applications of ERS SAR Interferometry, Zurich, September 30–October 2, 1996; also available at url: http://www.geo.unizh.ch/rsl/fringe96/papers/loffeld-et-al/ .

Hunter, I. W.

Huntley, J. M.

Jain, A. K.

A. K. Jain, “Morphological processing,” in Fundamentals of Digital Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1989), pp. 384–387.

Lalor, M. J.

P. Stephenson, D. R. Burton, M. J. Lalor, “Data validation techniques in a tiled phase unwrapping algorithm,” Opt. Eng. 33, 3703–3708 (1994).
[CrossRef]

Liang, Z.

Z. Liang, “A model-based method for phase unwrapping,” IEEE Trans. Med. Imaging 15, 893–897 (1996).
[CrossRef] [PubMed]

Lim, J. S.

J. S. Lim, “The discrete cosine transform,” in Two-Dimensional Signal and Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1990), pp. 148–157.

J. S. Lim, “The window method,” in Two-Dimensional Signal and Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1990), pp. 202–213.

Loffeld, O.

O. Loffeld, C. Arndt, A. Hein, “Estimating the derivative of modulo-mapped phases,” presented at the European Space Agency Workshop on Applications of ERS SAR Interferometry, Zurich, September 30–October 2, 1996; also available at url: http://www.geo.unizh.ch/rsl/fringe96/papers/loffeld-et-al/ .

Napel, S.

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

Pelc, N. J.

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

Romero, L. A.

Rosenfeld, D.

M. Hedley, D. Rosenfeld, “A new two-dimensional phase unwrapping algorithm for MRI images,” Magn. Reson. Med. 24, 177–181 (1992).
[CrossRef] [PubMed]

Song, S. M.

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

Stephenson, P.

P. Stephenson, D. R. Burton, M. J. Lalor, “Data validation techniques in a tiled phase unwrapping algorithm,” Opt. Eng. 33, 3703–3708 (1994).
[CrossRef]

Takahashi, T.

Takajo, H. T.

Tribolet, J. M.

J. M. Tribolet, “A new phase unwrapping algorithm,” IEEE Trans. Acoust. Speech Signal Process. ASSP-25, 170–177 (1977).
[CrossRef]

Werner, C. L.

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

Zebker, H. A.

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

Appl. Opt. (3)

IEEE Trans. Acoust. Speech Signal Process. (1)

J. M. Tribolet, “A new phase unwrapping algorithm,” IEEE Trans. Acoust. Speech Signal Process. ASSP-25, 170–177 (1977).
[CrossRef]

IEEE Trans. Image Process. (1)

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

IEEE Trans. Med. Imaging (1)

Z. Liang, “A model-based method for phase unwrapping,” IEEE Trans. Med. Imaging 15, 893–897 (1996).
[CrossRef] [PubMed]

IEEE Trans. Signal Process. (1)

B. Friedlander, J. M. Francos, “Model based phase unwrapping of 2-D signals,” IEEE Trans. Signal Process. 44, 2999–3007 (1996).
[CrossRef]

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

Magn. Reson. Med. (1)

M. Hedley, D. Rosenfeld, “A new two-dimensional phase unwrapping algorithm for MRI images,” Magn. Reson. Med. 24, 177–181 (1992).
[CrossRef] [PubMed]

Opt. Eng. (1)

P. Stephenson, D. R. Burton, M. J. Lalor, “Data validation techniques in a tiled phase unwrapping algorithm,” Opt. Eng. 33, 3703–3708 (1994).
[CrossRef]

Radio Sci. (1)

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

Other (5)

J. J. Gierloff, “Phase unwrapping by regions,” in Current Developments in Optical Engineering II, R. E. Fisher, W. J. Smith, eds., Proc. SPIE818, 2–9 (1987).

J. S. Lim, “The discrete cosine transform,” in Two-Dimensional Signal and Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1990), pp. 148–157.

J. S. Lim, “The window method,” in Two-Dimensional Signal and Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1990), pp. 202–213.

O. Loffeld, C. Arndt, A. Hein, “Estimating the derivative of modulo-mapped phases,” presented at the European Space Agency Workshop on Applications of ERS SAR Interferometry, Zurich, September 30–October 2, 1996; also available at url: http://www.geo.unizh.ch/rsl/fringe96/papers/loffeld-et-al/ .

A. K. Jain, “Morphological processing,” in Fundamentals of Digital Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1989), pp. 384–387.

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Figures (34)

Fig. 1
Fig. 1

(a) Original wrapped phase map ϕ(1), (b) detail (lower right quadrant) of the noisy wrapped phase map ϕ˜(1), (c) diagonal line ϕi,i(1) from the noise-free phase map ϕ(1), (d) corresponding line ϕ˜i,i(1) from the noisy phase map ϕ˜(1), (e) corresponding line v˜i,i(11) from the ULS estimate v˜(11), (f) corresponding line v˜i,i(12) from the estimate v˜(12) obtained with the proposed method.

Fig. 2
Fig. 2

(a) Incorrectly unwrapped phase surface with use of the ULS solution, (b) correctly unwrapped phase surface with use of the proposed method.

Fig. 3
Fig. 3

Signal-to-noise SNR(ΔKϕ˜) (in decibels) plotted versus km [neighborhood size |K|=(2km+1)2] for three noise distributions: (a) white Gaussian noise, (b) speckle noise, and (c) shot noise.

Fig. 4
Fig. 4

(a) Original wrapped phase map ϕ(2), (b) detail (lower left quadrant) of the noisy wrapped phase map ϕ˜(2), (c) unwrapped result φ˜(21) produced with K=κ, (d) unwrapped result φ˜(22) produced with the extended neighborhood K=Kmax, (e) diagonal line φ˜i,i(21) from the unwrapped phase map φ˜(21) (the dashed line represents the correct solution), (f) corresponding line φ˜i,i(22) from the unwrapped phase map φ˜(22) (the dashed line represents the correct solution).

Fig. 5
Fig. 5

(a) Original wrapped phase map ϕ(3), (b) detail (lower left quadrant) of the noisy wrapped phase map ϕ˜(3), (c) unwrapped result φ˜(31) produced with K=κ, (d) unwrapped result φ˜(32) produced with the extended neighborhood K=Kmax, (e) diagonal line φi,i(31) from the correct result φ(31) (obtained without noise), (f) corresponding line φ˜i,i(31) from the unwrapped phase map φ˜(31), (g) corresponding line φ˜i,i(32) from the unwrapped phase map φ˜(32). [Note the vertical scale in (e)–(g).]

Fig. 6
Fig. 6

(a) Phase map ϕ˜(4), (b) partially unwrapped phase map φ˜(41), (c) diagonal line ϕ˜i,i(4), (d) diagonal line φ˜i,i(41).

Fig. 7
Fig. 7

(a) Diagonal lines v˜i,i(42) (solid line) and [v˜(42)]i,i (dashed line), (b) diagonal lines (Γv˜(42))i,i (solid line) and [Γv˜(42)]i,i (dashed line).

Fig. 8
Fig. 8

(a) Diagonal line v˜i,i(43), (b) corresponding line [Γv˜(43)]i,i.

Fig. 9
Fig. 9

(a) Detail (lower left quadrant) of the noisy wrapped phase map ϕ˜(5), (b) diagonal line (φ˜i,i(51)) from the ULS solution φ˜(51), (c) φ˜1(52) produced after one iteration with K=κ, (d) diagonal line (φ˜28(52))i,i, (e) φ˜1(53) produced after one iteration with |K|=11×11, (f) diagonal line (φ˜8(53))i,i.

Fig. 10
Fig. 10

(a) Diagonal line (v˜1,0(1))i,i, (b) corresponding line vi,i(1), (c) corresponding line (η1,0)i,i.

Fig. 11
Fig. 11

(a) Interval from the diagonal line (η1,1)i,i, (b) corresponding interval (η1,2)i,i, (c) corresponding interval (η1,4)i,i.

Fig. 12
Fig. 12

(a) Interval from the diagonal line ζi,i, (b) corresponding interval (ζ+2π[η1,4])i,i.

Fig. 13
Fig. 13

(a) Wrapped phase map ϕ˜(6), (b) real-valued estimate v˜1,30(61). (The phase map was used with kind permission from Henrik Saldner, Division of Experimental Mechanics, Luleå University of Technology.)

Fig. 14
Fig. 14

(a) Row (ϕ˜175,j(6)), (b) row (v˜1,0(61))175,j from the initial estimate v˜1,0(61), (c) corresponding row (v˜1,10(61))175,j from the result v˜1,10(61) obtained after ten iterations with the use of Eq. (5.1), (d) corresponding row (v˜1.30(61))175,j.

Fig. 15
Fig. 15

(a) Wrapped test map ϕ˜(7), (b) detail (lower left quadrant) of ϕ˜(7), (c) corresponding detail of the partially unwrapped phase map φ˜(71) (remaining discontinuities are marked with white lines), (d) corresponding detail of φ˜(711).

Fig. 16
Fig. 16

(a) Noisy wrapped phase map ϕ˜(8), (b) unwrapped phase map φ˜(81) (the GOFF is marked with a black line).

Fig. 17
Fig. 17

(a) Correct unwrapped phase surface, (b) unwrapped phase surface φ˜(81).

Fig. 18
Fig. 18

Result of the 50 trials described in the text (the mean value is marked with a dashed line).

Fig. 19
Fig. 19

(a) Result for σ=0.6275 with the use of |K|=5×5, (b) result for σ=0.5490 with the use of |K|=5×5. (In both cases the mean value is marked with a dashed line.)

Fig. 20
Fig. 20

(a) Original wrapped test map ϕ˜(9), (b) diagonal from the upper left to the lower right corner of ϕ˜(9).

Fig. 21
Fig. 21

(a) Inconsistency map corresponding to ϕ˜(9), (b) processed inconsistency map.

Fig. 22
Fig. 22

Uˆφ˜n(91)/XY (Frobenius norm) plotted versus n.

Fig. 23
Fig. 23

(a) Unwrapped test map φ˜(91), (b) diagonal from the upper left to the lower right corner of φ˜(91).

Fig. 24
Fig. 24

(a) Details of (a) (cos ϕ)256,j (the noise-free case), (b) (cos ϕ˜)256,j(1) (Gaussian white noise), (c) (cos ϕ˜)256,j(2) (speckle noise), (d) (cos ϕ˜)256,j(3)) (shot noise).

Fig. 25
Fig. 25

(a) Detail (lower left quadrant) of the noisy wrapped phase map ϕ˜(10) (Gaussian white noise), (b) unwrapped phase map φ˜(101).

Fig. 26
Fig. 26

(a) Detail (lower left quadrant) of the noisy wrapped phase map ϕ˜(11) (speckle noise), (b) unwrapped phase map φ˜(111).

Fig. 27
Fig. 27

(a) Detail (lower left quadrant) of the noisy wrapped phase map ϕ˜(12) (shot noise), (b) unwrapped phase map φ˜(121).

Fig. 28
Fig. 28

(a) Wrapped test map ϕ˜(13), (b) detail (lower left quadrant) of ϕ˜(13).

Fig. 29
Fig. 29

Real-valued estimate v˜1,10(131).

Fig. 30
Fig. 30

Wrapped test map ϕ˜(14), (b) detail (lower left quadrant) of ϕ˜(14).

Fig. 31
Fig. 31

Real-valued estimate v˜1,40(121).

Fig. 32
Fig. 32

Unwrapped phase map φ˜(121).

Fig. 33
Fig. 33

Flow chart for Eq. (4.6).

Fig. 34
Fig. 34

Flow chart for Eq. (5.5).

Tables (4)

Tables Icon

Table 1 R(Fv˜) Evaluated from the Phase Map ϕ˜(4) with the Use of Different Neighborhood Sizes |K|=(2km+1)(2lm+1)

Tables Icon

Table 2 Minimum, Mean, and Maximum Number of Iterations n Required for Various Neighborhood Sizes |K|

Tables Icon

Table 3 Required Number of Iterations m and Dynamic Range of the Final Results for Various Neighborhood Sizes |K|

Tables Icon

Table 4 R(Fṽ) Evaluated from the Phase Map ϕ˜(8) with the Use of Different Neighborhood Sizes |K|=(2km+1)(2lm+1)

Equations (32)

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φi,j=ϕi,j+2πvi,j,
=(i, j)I(k, l)Kci,j,k,lwk,l{ϑi,j,k,l+(v˜i,j-v˜i-k,j-1)}2,
ϑi,j,k,l=signϕi,j-ϕi-k,j-lTϕ.
ci,j,k,l=1,(i-k, j-l)I0,otherwise.
sign(x)=1,x>00,x=0-1,x<0
v˜i,j=(k, l)Kci,j,k,lwk,l{ϑi,j,k,l+(v˜i,j-v˜i-k,j-l)}=0
(ΔKϕ)i,j=(k, l)Kci,j,k,lwk,lϕi,j-ϕi-k,j-lTϕ,
-(ΔKϕ)i,j=(k, l)Kci,j,k,lwk,l(v˜i,j-v˜i-k,j-l).
hk,l=-wk,l,(k, l)(0, 0)(k, l)K\(0, 0)wk,l,(k, l)=(0, 0),
-Φp,q=Hp,qNp,q,
Hp,q=k=0kmaxl=0lmax{4-2(δk+δl)}wk,l(1-Cp,q,k,l),
Cp,q,k,l=cosπpkXcosπqlY.
Np,q=Gp,qΦp,q,
wk,l=1,(k, l)κ0,otherwise,
ϕ˜=D(W(φ+ζ)),
(ΔKϕ˜)i,j=(k, l)Kϕ˜i,j-ϕ˜i-k,j-lTϕ=(k, l)K(vi,j-vi-k,j-l)+(k, l)Kξi,j,k,l,
SNR(x˜)=20 log10(x/x˜-x),
wk=0.54-0.46 cos(π(k+kmax)/kmax),
φ˜=(E+2πU)ϕ˜,
Uϕ˜=IDCT(G·DCT(-ΔKϕ˜))=v˜,
Uˆϕ˜=[Γv˜]
(Γv˜)i,j=v˜i,j+μ(v˜),
μ(v˜)=-min(Fv˜),R(Fv˜)>TΓ0,otherwise,
φ˜n+1=(E+2πUˆ)φ˜n
Tϕ=2π,n=13π,n>1
v˜m+1=-U1(v˜m)
v˜n,m=-U1(-U1((Uπ(φ˜n-1)))),
v˜n,0=vn+ηn,0,
[(v˜n,m)i,j,k0,l0]=[(vn)i,j,k0,l0+(ηn,mL)i,j,k0,l0+(ηn,mH)i,j,k0,l0]
(Γv˜n)i,j=(v˜n,m)i,j+μ(v˜n,m)
ιi,j=ϑi,j,k,l+ϑi,j+1,k,l+ϑi-1,j+1,k,l+ϑi-1,j,k,l,
(Vφ˜)i,j=1|K|-1(k, l)K{φ˜i-k,j-l-(Mφ˜)i,j}2,

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