Abstract

A principled approach, based on Bayesian estimation theory and complex-valued Markov random-field prior models, is introduced for the design of a new class of adaptive quadrature filters. These filters are capable of adapting their tuning frequency to the local dominant spatial frequency of the input image while maintaining an arbitrarily narrow local frequency response; therefore they may be effectively used for the accurate recovery of the phase of broadband spatial-carrier fringe patterns, even when they are corrupted by a significant amount of noise. Also, by constraining the spatial variation of the adaptive frequency to be smooth, they permit the completely automatic recovery of local phase from single closed fringe pattern images, since the spurious discontinuities and sign reversals that one obtains from the classical Fourier-based methods are avoided in this case. Although the applications discussed here come from fringe pattern analysis in optics, these filters may also be useful in the solution of other problems, such as texture characterization and segmentation and the recovery of depth from stereoscopic pairs of images.

© 1997 Optical Society of America

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References

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  1. D. Gabor, “Theory of communication,” J. Inst. Electr. Eng. (London) 93, 429–457 (1946).
  2. M. Turner, “Texture discrimination by Gabor functions,” Biol. Cybern. 55, 71–82 (1986).
    [PubMed]
  3. J. Beck, A. Sutter, R. Ivry, “Spatial frequency channels and perceptual grouping in texture segregation,” Comput. Vis. Graph. Image Process. 37, 299–325 (1987).
    [CrossRef]
  4. J. Bergen, E. Adelson, “Early vision and texture perception,” Nature (London) 333, 363–364 (1988).
    [CrossRef]
  5. I. Fogel, D. Sagi, “Gabor filters for texture discrimination,” Biol. Cybern. 61, 103–113 (1989).
    [CrossRef]
  6. T. Sanger, “Stereo disparity computation using Gabor filters,” Biol. Cybern. 59, 405–418 (1988).
    [CrossRef]
  7. E. H. Adelson, J. R. Bergen, “Spatiotemporal energy models for the perception of motion,” J. Opt. Soc. Am. A 2, 284–299 (1985).
    [CrossRef] [PubMed]
  8. D. J. Heeger, “A model for the extraction of image flow,” J. Opt. Soc. Am. A 4, 1455–1471 (1987).
    [CrossRef] [PubMed]
  9. K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1988), Vol. 26, pp. 349–393.
  10. M. Takeda, H. Ina, S. Kobayashi, “Fourier transform methods of fringe pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72, 156–160 (1982).
    [CrossRef]
  11. Th. Kreis, “Digital holographic interference phase measurement using the Fourier transform method,” J. Opt. Soc. Am. A 3, 847–855 (1986).
    [CrossRef]
  12. D. J. Fleet, A. D. Jepson, M. R. M. Jenkin, “Phase-based disparity measurement,” Comput. Vis. Graph. Image Process. 53, 198–210 (1991).
  13. S. Geman, D. Geman, “Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images,” IEEE Trans. Pattern. Anal. Mach. Intell. PAMI-6, 721–741 (1984).
    [CrossRef]
  14. J. Marroquin, S. Mitter, T. Poggio, “Probabilistic solution of ill-posed problems in computational vision,” J. Am. Stat. Assoc. 82, 76–89 (1987).
    [CrossRef]
  15. J. L. Marroquin, M. Servin, J. E. Figueroa, “Robust quadrature filters,” J. Opt. Soc. Am. A 14, 779–791 (1997).
    [CrossRef]
  16. W. E. L. Grimson, “A computational theory of visual surface interpolation,” Philos. Trans. R. Soc. London, Ser. B 298, 395–427 (1982).
    [CrossRef] [PubMed]
  17. D. W. Robinson, G. T. Reid, eds., Interferogram Analysis: Digital Fringe Pattern Measurement Techniques (Institute of Physics, Bristol, UK, 1993).
  18. R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1978).
  19. A. Spik, D. W. Robinson, “Investigation of the cellular automata method for phase unwrapping and its implementation on an array processor,” Optics Lasers Eng. 14, 25–37 (1991).
  20. R. Jones, C. Wykes, Holographic and Speckle Interferometry (Cambridge U. Press, Cambridge, 1989).
  21. J. M. Ortega, W. C. Rheinholdt, Iterative Solution of Nonlinear Equations in Several Variables (Academic, New York, 1970).
  22. J. L. Marroquin, “Deterministic interactive particle models for image processing and computer graphics,” Comput. Vis. Graph. Image Process. 55, 408–417 (1993).
  23. S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
    [CrossRef] [PubMed]
  24. P. J. Burt, “The pyramid as a structure for efficient computation,” in Multiresolution Image Processing and Analysis, A. Rosenfeld, ed., Vol. 12 of Springer Series in Information Sciences (Springer, New York, 1984).
  25. D. Malacara, ed., Optical Shop Testing (Wiley, New York, 1992).
  26. C. S. Shanmugan, Digital and Analog Communication Systems (Wiley, New York, 1979).

1997 (1)

1993 (1)

J. L. Marroquin, “Deterministic interactive particle models for image processing and computer graphics,” Comput. Vis. Graph. Image Process. 55, 408–417 (1993).

1991 (2)

A. Spik, D. W. Robinson, “Investigation of the cellular automata method for phase unwrapping and its implementation on an array processor,” Optics Lasers Eng. 14, 25–37 (1991).

D. J. Fleet, A. D. Jepson, M. R. M. Jenkin, “Phase-based disparity measurement,” Comput. Vis. Graph. Image Process. 53, 198–210 (1991).

1989 (1)

I. Fogel, D. Sagi, “Gabor filters for texture discrimination,” Biol. Cybern. 61, 103–113 (1989).
[CrossRef]

1988 (2)

T. Sanger, “Stereo disparity computation using Gabor filters,” Biol. Cybern. 59, 405–418 (1988).
[CrossRef]

J. Bergen, E. Adelson, “Early vision and texture perception,” Nature (London) 333, 363–364 (1988).
[CrossRef]

1987 (3)

J. Beck, A. Sutter, R. Ivry, “Spatial frequency channels and perceptual grouping in texture segregation,” Comput. Vis. Graph. Image Process. 37, 299–325 (1987).
[CrossRef]

J. Marroquin, S. Mitter, T. Poggio, “Probabilistic solution of ill-posed problems in computational vision,” J. Am. Stat. Assoc. 82, 76–89 (1987).
[CrossRef]

D. J. Heeger, “A model for the extraction of image flow,” J. Opt. Soc. Am. A 4, 1455–1471 (1987).
[CrossRef] [PubMed]

1986 (2)

1985 (1)

1984 (1)

S. Geman, D. Geman, “Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images,” IEEE Trans. Pattern. Anal. Mach. Intell. PAMI-6, 721–741 (1984).
[CrossRef]

1983 (1)

S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef] [PubMed]

1982 (2)

W. E. L. Grimson, “A computational theory of visual surface interpolation,” Philos. Trans. R. Soc. London, Ser. B 298, 395–427 (1982).
[CrossRef] [PubMed]

M. Takeda, H. Ina, S. Kobayashi, “Fourier transform methods of fringe pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72, 156–160 (1982).
[CrossRef]

1946 (1)

D. Gabor, “Theory of communication,” J. Inst. Electr. Eng. (London) 93, 429–457 (1946).

Adelson, E.

J. Bergen, E. Adelson, “Early vision and texture perception,” Nature (London) 333, 363–364 (1988).
[CrossRef]

Adelson, E. H.

Beck, J.

J. Beck, A. Sutter, R. Ivry, “Spatial frequency channels and perceptual grouping in texture segregation,” Comput. Vis. Graph. Image Process. 37, 299–325 (1987).
[CrossRef]

Bergen, J.

J. Bergen, E. Adelson, “Early vision and texture perception,” Nature (London) 333, 363–364 (1988).
[CrossRef]

Bergen, J. R.

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1978).

Burt, P. J.

P. J. Burt, “The pyramid as a structure for efficient computation,” in Multiresolution Image Processing and Analysis, A. Rosenfeld, ed., Vol. 12 of Springer Series in Information Sciences (Springer, New York, 1984).

Creath, K.

K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1988), Vol. 26, pp. 349–393.

Figueroa, J. E.

Fleet, D. J.

D. J. Fleet, A. D. Jepson, M. R. M. Jenkin, “Phase-based disparity measurement,” Comput. Vis. Graph. Image Process. 53, 198–210 (1991).

Fogel, I.

I. Fogel, D. Sagi, “Gabor filters for texture discrimination,” Biol. Cybern. 61, 103–113 (1989).
[CrossRef]

Gabor, D.

D. Gabor, “Theory of communication,” J. Inst. Electr. Eng. (London) 93, 429–457 (1946).

Gelatt, C. D.

S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef] [PubMed]

Geman, D.

S. Geman, D. Geman, “Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images,” IEEE Trans. Pattern. Anal. Mach. Intell. PAMI-6, 721–741 (1984).
[CrossRef]

Geman, S.

S. Geman, D. Geman, “Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images,” IEEE Trans. Pattern. Anal. Mach. Intell. PAMI-6, 721–741 (1984).
[CrossRef]

Grimson, W. E. L.

W. E. L. Grimson, “A computational theory of visual surface interpolation,” Philos. Trans. R. Soc. London, Ser. B 298, 395–427 (1982).
[CrossRef] [PubMed]

Heeger, D. J.

Ina, H.

Ivry, R.

J. Beck, A. Sutter, R. Ivry, “Spatial frequency channels and perceptual grouping in texture segregation,” Comput. Vis. Graph. Image Process. 37, 299–325 (1987).
[CrossRef]

Jenkin, M. R. M.

D. J. Fleet, A. D. Jepson, M. R. M. Jenkin, “Phase-based disparity measurement,” Comput. Vis. Graph. Image Process. 53, 198–210 (1991).

Jepson, A. D.

D. J. Fleet, A. D. Jepson, M. R. M. Jenkin, “Phase-based disparity measurement,” Comput. Vis. Graph. Image Process. 53, 198–210 (1991).

Jones, R.

R. Jones, C. Wykes, Holographic and Speckle Interferometry (Cambridge U. Press, Cambridge, 1989).

Kirkpatrick, S.

S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef] [PubMed]

Kobayashi, S.

Kreis, Th.

Marroquin, J.

J. Marroquin, S. Mitter, T. Poggio, “Probabilistic solution of ill-posed problems in computational vision,” J. Am. Stat. Assoc. 82, 76–89 (1987).
[CrossRef]

Marroquin, J. L.

J. L. Marroquin, M. Servin, J. E. Figueroa, “Robust quadrature filters,” J. Opt. Soc. Am. A 14, 779–791 (1997).
[CrossRef]

J. L. Marroquin, “Deterministic interactive particle models for image processing and computer graphics,” Comput. Vis. Graph. Image Process. 55, 408–417 (1993).

Mitter, S.

J. Marroquin, S. Mitter, T. Poggio, “Probabilistic solution of ill-posed problems in computational vision,” J. Am. Stat. Assoc. 82, 76–89 (1987).
[CrossRef]

Ortega, J. M.

J. M. Ortega, W. C. Rheinholdt, Iterative Solution of Nonlinear Equations in Several Variables (Academic, New York, 1970).

Poggio, T.

J. Marroquin, S. Mitter, T. Poggio, “Probabilistic solution of ill-posed problems in computational vision,” J. Am. Stat. Assoc. 82, 76–89 (1987).
[CrossRef]

Rheinholdt, W. C.

J. M. Ortega, W. C. Rheinholdt, Iterative Solution of Nonlinear Equations in Several Variables (Academic, New York, 1970).

Robinson, D. W.

A. Spik, D. W. Robinson, “Investigation of the cellular automata method for phase unwrapping and its implementation on an array processor,” Optics Lasers Eng. 14, 25–37 (1991).

Sagi, D.

I. Fogel, D. Sagi, “Gabor filters for texture discrimination,” Biol. Cybern. 61, 103–113 (1989).
[CrossRef]

Sanger, T.

T. Sanger, “Stereo disparity computation using Gabor filters,” Biol. Cybern. 59, 405–418 (1988).
[CrossRef]

Servin, M.

Shanmugan, C. S.

C. S. Shanmugan, Digital and Analog Communication Systems (Wiley, New York, 1979).

Spik, A.

A. Spik, D. W. Robinson, “Investigation of the cellular automata method for phase unwrapping and its implementation on an array processor,” Optics Lasers Eng. 14, 25–37 (1991).

Sutter, A.

J. Beck, A. Sutter, R. Ivry, “Spatial frequency channels and perceptual grouping in texture segregation,” Comput. Vis. Graph. Image Process. 37, 299–325 (1987).
[CrossRef]

Takeda, M.

Turner, M.

M. Turner, “Texture discrimination by Gabor functions,” Biol. Cybern. 55, 71–82 (1986).
[PubMed]

Vecchi, M. P.

S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef] [PubMed]

Wykes, C.

R. Jones, C. Wykes, Holographic and Speckle Interferometry (Cambridge U. Press, Cambridge, 1989).

Biol. Cybern. (3)

I. Fogel, D. Sagi, “Gabor filters for texture discrimination,” Biol. Cybern. 61, 103–113 (1989).
[CrossRef]

T. Sanger, “Stereo disparity computation using Gabor filters,” Biol. Cybern. 59, 405–418 (1988).
[CrossRef]

M. Turner, “Texture discrimination by Gabor functions,” Biol. Cybern. 55, 71–82 (1986).
[PubMed]

Comput. Vis. Graph. Image Process. (3)

J. Beck, A. Sutter, R. Ivry, “Spatial frequency channels and perceptual grouping in texture segregation,” Comput. Vis. Graph. Image Process. 37, 299–325 (1987).
[CrossRef]

J. L. Marroquin, “Deterministic interactive particle models for image processing and computer graphics,” Comput. Vis. Graph. Image Process. 55, 408–417 (1993).

D. J. Fleet, A. D. Jepson, M. R. M. Jenkin, “Phase-based disparity measurement,” Comput. Vis. Graph. Image Process. 53, 198–210 (1991).

IEEE Trans. Pattern. Anal. Mach. Intell. (1)

S. Geman, D. Geman, “Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images,” IEEE Trans. Pattern. Anal. Mach. Intell. PAMI-6, 721–741 (1984).
[CrossRef]

J. Am. Stat. Assoc. (1)

J. Marroquin, S. Mitter, T. Poggio, “Probabilistic solution of ill-posed problems in computational vision,” J. Am. Stat. Assoc. 82, 76–89 (1987).
[CrossRef]

J. Inst. Electr. Eng. (London) (1)

D. Gabor, “Theory of communication,” J. Inst. Electr. Eng. (London) 93, 429–457 (1946).

J. Opt. Soc. Am. (1)

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

Nature (London) (1)

J. Bergen, E. Adelson, “Early vision and texture perception,” Nature (London) 333, 363–364 (1988).
[CrossRef]

Optics Lasers Eng. (1)

A. Spik, D. W. Robinson, “Investigation of the cellular automata method for phase unwrapping and its implementation on an array processor,” Optics Lasers Eng. 14, 25–37 (1991).

Philos. Trans. R. Soc. London, Ser. B (1)

W. E. L. Grimson, “A computational theory of visual surface interpolation,” Philos. Trans. R. Soc. London, Ser. B 298, 395–427 (1982).
[CrossRef] [PubMed]

Science (1)

S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef] [PubMed]

Other (8)

P. J. Burt, “The pyramid as a structure for efficient computation,” in Multiresolution Image Processing and Analysis, A. Rosenfeld, ed., Vol. 12 of Springer Series in Information Sciences (Springer, New York, 1984).

D. Malacara, ed., Optical Shop Testing (Wiley, New York, 1992).

C. S. Shanmugan, Digital and Analog Communication Systems (Wiley, New York, 1979).

K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1988), Vol. 26, pp. 349–393.

D. W. Robinson, G. T. Reid, eds., Interferogram Analysis: Digital Fringe Pattern Measurement Techniques (Institute of Physics, Bristol, UK, 1993).

R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1978).

R. Jones, C. Wykes, Holographic and Speckle Interferometry (Cambridge U. Press, Cambridge, 1989).

J. M. Ortega, W. C. Rheinholdt, Iterative Solution of Nonlinear Equations in Several Variables (Academic, New York, 1970).

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Figures (11)

Fig. 1
Fig. 1

(a) Physical model for the one-dimensional membrane regularization function extended to the complex case: The estimated field is modeled as a system of particles (white circles) allowed to move in copies of the complex plane; the particles are connected to the observations (black circles) and to neighboring particles by linear springs (bold lines). (b) Model for the AQF regularization function obtained from model (a) by rotating the copies of the complex plane by an angle equal to the local tuning frequency (see the text).

Fig. 2
Fig. 2

Three- and four-site cliques for the thin-plate two-dimensional model (see the text). 

Fig. 3
Fig. 3

(a) Phase-modulated signal with two quasi-monochromatic components (see the text), (b) power spectra of components 1 (solid curve) and 2 (dashed curve), (c) first component extracted by the AQF, (d) phase recovered from the first component (solid curve) and true modulating phase (dashed curve).

Fig. 4
Fig. 4

(a) Noisy fringe pattern obtained by modulating a vertical grating with the quadratic phase shown (wrapped) in (b) and with additive uniform observation noise; (c) real part of the output of a narrow-band QF applied to (a); (d) reconstructed wrapped phase; (e) and (f) same as (c) and (d), respectively, for a broadband QF.

Fig. 5
Fig. 5

(a) Real part of the output of the AQF described in the text applied to the pattern of Fig. 1(a), (b) reconstructed wrapped phase [compare with Fig. 1(b)].

Fig. 6
Fig. 6

(a) Fringe pattern generated by the wrapped phase shown in (b), (c) phase recovered by the HTF (Fourier method), (d) phase recovered by the AQF with λ=5 and μ=50.

Fig. 7
Fig. 7

(a) First component of the ω field that corresponds to the solution found by the HTF method for the data of Fig. 4(a); dark pixels indicate negative values, and light pixels indicate positive ones. (b) Same as (a) when the energy function is optimized with respect to ω while maintaining f=fHTF fixed. (c) Same as (a), but for the AQF method.

Fig. 8
Fig. 8

(a) 64×64 noisy fringe pattern generated by the wrapped phase shown in (b), which is formed by the sum of five Gaussian functions with centers at (32, 32), (19, 19), (45, 45), (19, 45), and (45, 19), with weights equal to 20, 20, -20, -20, and -20, respectively, plus additive noise uniformly distributed in the interval [-1, 1]; (c) phase recovered by the HTF (Fourier method); (d) phase recovered by the AQF with λ=5 and μ=50.

Fig. 9
Fig. 9

(a) Interferogram of an early stage of the manufacturing process of a flat mirror obtained with a Wyko interferometer, (b) phase recovered by the AQF method, (c) phase recovered by the HTF method.  

Fig. 10
Fig. 10

(a) 256×256 image of an ESPI fringe pattern, (b) real part of the field recovered by the AQF method (see the text), (c) phase recovered by the AQF, (d) phase recovered by the HTF method after filtering the fringe pattern with a Gaussian filter that reduces the bandwidth by a factor of 2 (σ=256/6 in the frequency domain).

Fig. 11
Fig. 11

Phase configuration that corresponds to a local minimum of Eq. (9) for the data of Fig. 4.

Equations (58)

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

g(x)=b(x)cos[ω0x+α(x)]=b(x)2 (exp{i[ω0x+α(x)]}+exp{-i[ω0x+α(x)]}),
f(x)=φ(x)+iψ(x)b(x)2 exp{i[ω0x+α(x)]},
M(x)=[φ2(x)+ψ2(x)]1/2
ϕ(x)=arctan[ψ(x)/φ(x)].
g(x)=b(x)cos[ω(x)x+αˆ(x)],
Pf,ω|g(f, ω)=1K Pg|f,ω(f, ω)Pf,ω(f, ω),
Pg|f,ω(f)=1K exp-xS Φ(f(x), g(x)),
Pf,ω(f, ω)=1Z exp[-Uf,ω(f, ω)].
Uf,ω(f, ω)=C VC(f, ω).
U(f, ω)=xS Φ(f(x), g(x))+C VC(f, ω).
U(f )=xS |f(x)-g(x)|2+λ [x,y] |f(x)-f(y)|2,
U(f )=xS |f(x)-g(x)|2+λ [x,y] |f(x)-f(y)exp[iω0(x-y)]|2.
Vxy(f)=|f(x)-f(y)exp[iω0(x-y)]|2=f(x)exp-i2 ω0(x-y)-f(y)expi2 ω0(x-y)2
U(f)=xS |f(x)-2g(x)|2+λ [x,y] f(x)exp-i2 ω0(x-y)-f(y)expi2 ω0(x-y)2.
Vxy(f )=f(x)exp-i2 ω(x)(x-y)-f(y)×expi2 ω(y)(x-y)2,
Vabc(ω)=|-ω(a)+2ω(b)-ω(c)|2,
Vpqrs(ω)=2|-ω(p)+ω(q)-ω(r)+ω(s)|2
U(f, ω)=xS |f(x)-2g(x)|2+λ [x,y] f(x)exp-i2 ω(x)(x-y)-f(y)expi2 ω(y)(x-y)2+μ[a,b,c] Vabc(ω)+[p,q,r,s] Vpqrs(ω),
U(f, ω)=[x,y] |f(x)-f(y)-2[g(x)-g(y)]|2+λ [x,y] f(x)exp-i2 ω(x)(x-y)-f(y)expi2 ω(y)(x-y)2+μ[a,b,c] Vabc(ω)+[p,q,r,s] Vpqrs(ω).
f(x)=M(x)exp[ϕ(x)],
ϕ(x)=ω(x)x+α(x),
g(x)=cos[(a1+bx)x]+cos[(a2+bx)x],
g(x)=a(x)+b(x)cos[ϕ(x)]+n(x),
ϕ(x)=ω0x+α(x),
H(ω)=1forω00otherwise.
g(x)=b(x)cos[ϕ(x)]=b(x)2 {exp[iϕ(x)]+exp[-iϕ(x)]}
ω^c(t)=ωc(t)ifωc1(t)>0-ωc(t)otherwise,
ϕ(x)/x1ω1(x)=0.
ϕ(x, y)=30exp-(x-16)2+(y-16)2409.6+exp-(x-48)2+(y-48)2409.6,
θ(x)=(f(x), ω(x))=(φ(x), ψ(x), ω1(x), ω2(x))
2θt2=-θU(θ)-2k θt,
θ(t+1)=Aθ(t)+Bθ(t-1)+CθU(θ(t)),
A=2kh+1,B=kh-1kh+1,C=-h2kh+1,
g(x):=2[g(x)-minyS g(y)]maxyS g(y)-minyS g(y)-1,
N(x)={yS:|x-y|=1},
Nh(x)={yS:y1=x1and|y2-x2|=1},
Nv(x)={yS:y2=x2and|y1-x1|=1},
Nhv(x)={(x1, x2),(x1+1, x2),(x1, x2+1),
(x1+1, x2+1)}S;
axy=φ(x)+φ(y)Gxy+ψ(y)Hxy,
bxy=ψ(x)-φ(y)Hxy+ψ(y)Gxy,
Gxy=-sxysyx-cxycyx,
Hxy=-cxysyx+cyxsxy,
cxy=cosω(x)(x-y)2,
sxy=sinω(x)(x-y)2.
Uω(x)=Uω1(x), Uω2(x)T.
Uφ(x)=φ(x)-2g(x)+λ yN(x) axy,
Uψ(x)=ψ(x)+λ yN(x) bxy,
Uω(x)=λ yN(x)[axyψ(x)-bxyφ(x)](x-y)+μd(x),
dk(x)=yNh(x) Vh(ωk, y)-2Vh(ωk,x)+yNv(x) Vv(ωk, y)-2Vv(ωk, x)+yNhv(x) Vhv(ωk, y)
Vh(ωk, y)
=ωk(y1, y2-1)-2ωk(y1, y2)+ωk(y1, y2+1)ifNh(y)S0otherwise,
Vv(ωk, y)
=ωk(y1-1, y2)-2ωk(y1, y2)+ωk(y1+1, y2)ifNv(y)S0otherwise,
Vhv(ωk, y)
=ωk(y1, y2)-ωk(y1+1, y2)-ωk(y1, y2+1)+ωk(y1+1, y2+1)ifNhv(y)S0otherwise.
Uφ(x)=yN(x) {φ(x)-φ(y)-2[g(x)-g(y)]}+λ yN(x) axy,
Uψ(x)=yN(x) [ψ(x)-ψ(y)]+λ yN(x) bxy.

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