Abstract

We deal with the relation between two well-known topics in signal processing and computational vision: quadrature filters (QF’s) and Bayesian estimation with Markov random fields (MRF’s) as prior models. We present a new class of complex-valued MRF models such that the optimal estimators obtained with them correspond to the output of QF’s tuned at particular frequencies. It is shown that the machinery that has proven to be effective in classical (real-valued) MRF modeling may be generalized to the complex case in a straightforward way. To illustrate the power of this technique, we present complex MRF models that implement robust QF’s that exhibit good performance in situations in which ordinary linear, shift-invariant filters fail. These include robust filters that are relatively insensitive to edge effects and missing data and that can reliably estimate the local phase in singularity neighborhoods; we also present models for the specification of piecewise-smooth QF’s. Examples of applications to fringe pattern analysis, phase-based stereo reconstruction, and texture segmentation are presented as well.

© 1997 Optical Society of America

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References

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  1. S. Geman, D. Geman, “Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images,” IEEE Trans. Pattern Anal. Machine Intell. PAMI-6, 721–741 (1984).
    [CrossRef]
  2. J. Marroquin, S. Mitter, T. Poggio, “Probabilistic solution of ill-posed problems in computational vision,” J. Am. Stat. Assoc. 82, 76–89 (1987).
    [CrossRef]
  3. A. Blake, A. Zisserman, Visual Reconstruction (MIT Press, Cambridge, Mass., 1987).
  4. R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1978).
  5. D. Gabor, “Theory of communication,” J. Inst. Electr. Eng. 93, 429–457 (1946).
  6. D. J. Fleet, A. D. Jepson, M. R. M. Jenkin, “Phase-based disparity measurement,” Comput. Vision Graphics Image Process. 53, 198–210 (1991).
  7. T. Sanger, “Stereo disparity computation using Gabor filters,” Biol. Cybern. 59, 405–418 (1988).
    [CrossRef]
  8. 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]
  9. M. Turner, “Texture discrimination by Gabor functions,” Biol. Cybern. 55, 71–82 (1986).
    [PubMed]
  10. J. Beck, A. Sutter, R. Ivry, “Spatial frequency channels and perceptual grouping in texture segregation,” Comput. Vision Graphics and Image Process. 37, 299–325 (1987).
    [CrossRef]
  11. J. Bergen, E. Adelson, “Early vision and texture perception,” Nature (London) 333, 363–364 (1988).
    [CrossRef]
  12. I. Fogel, D. Sagi, “Gabor filters for texture discrimination,” Biol. Cybern. 61, 103–113 (1989).
    [CrossRef]
  13. 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]
  14. D. J. Heeger, “A model for the extraction of image flow,” J. Opt. Soc. Am. A 4, 1455–1471 (1987).
    [CrossRef] [PubMed]
  15. J. Besag, “Spatial interaction and the statistical analysis of lattice systems,” J. R. Stat. Soc. B 36, 192–326 (1974).
  16. G. H. Golub, C. F. Van Loan, Matrix Computations (Johns Hopkins U., Baltimore, 1990).
  17. J. L. Marroquin, M. Rivera, “Quadratic regularization functionals for phase unwrapping,” J. Opt. Soc. Am. A 12, 2393–2400 (1995).
    [CrossRef]
  18. J. L. Marroquin, “Deterministic interactive particle models for image processing and computer graphics,” Comput. Vision Graphics Image Process. 55, 408–417 (1993).
  19. K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1988), Vol. 26, pp. 349–393.
  20. D. Malacara, ed., Optical Shop Testing (Wiley, New York, 1992).
  21. A. N. Thikonov, V. Y. Arsenin, Solutions to Ill-posed Problems (Winston and Sons, Washington, D.C., 1977).
  22. M. Bertero, T. Poggio, V. Torre, “Ill-posed problems in early vision,” Proc. IEEE 76, 869–887 (1988).
    [CrossRef]
  23. W. E. L. Grimson, “A computational theory of visual surface interpolation,” Philos. Trans. R. Soc. London, Ser. B 298, 395–427 (1982).
    [CrossRef] [PubMed]
  24. D. Geman, G. Reynolds, “Constrained restoration and the recovery of discontinuities,” IEEE Trans. Pattern Anal. Machine Intell. 14, 367–383 (1992).
    [CrossRef]
  25. J. L. Marroquin, “Probabilistic solution of inverse problems,” (Massachusetts Institute of Technology, Cambridge, Mass., 1985).
  26. J. L. Marroquin, “The adaptive ICM algorithm,” , Centro de Investigación en Matematicas, Guonajuato, Mexico, 1996).
  27. T. Poggio, H. Voorhees, A. Yuille, “A regularized solution to edge detection,” (Massachusetts Institute of Technology, Cambridge, Mass., 1985).
  28. G. Wahba, Spline Models for Observational Data (Society for Industrial and Applied Mathematics, Philadelphia, 1990).

1995 (1)

1993 (1)

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

1992 (1)

D. Geman, G. Reynolds, “Constrained restoration and the recovery of discontinuities,” IEEE Trans. Pattern Anal. Machine Intell. 14, 367–383 (1992).
[CrossRef]

1991 (1)

D. J. Fleet, A. D. Jepson, M. R. M. Jenkin, “Phase-based disparity measurement,” Comput. Vision Graphics 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 (3)

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

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

M. Bertero, T. Poggio, V. Torre, “Ill-posed problems in early vision,” Proc. IEEE 76, 869–887 (1988).
[CrossRef]

1987 (3)

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

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

1986 (1)

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

1985 (1)

1984 (1)

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

1982 (2)

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]

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

1974 (1)

J. Besag, “Spatial interaction and the statistical analysis of lattice systems,” J. R. Stat. Soc. B 36, 192–326 (1974).

1946 (1)

D. Gabor, “Theory of communication,” J. Inst. Electr. Eng. 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.

Arsenin, V. Y.

A. N. Thikonov, V. Y. Arsenin, Solutions to Ill-posed Problems (Winston and Sons, Washington, D.C., 1977).

Beck, J.

J. Beck, A. Sutter, R. Ivry, “Spatial frequency channels and perceptual grouping in texture segregation,” Comput. Vision Graphics and 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.

Bertero, M.

M. Bertero, T. Poggio, V. Torre, “Ill-posed problems in early vision,” Proc. IEEE 76, 869–887 (1988).
[CrossRef]

Besag, J.

J. Besag, “Spatial interaction and the statistical analysis of lattice systems,” J. R. Stat. Soc. B 36, 192–326 (1974).

Blake, A.

A. Blake, A. Zisserman, Visual Reconstruction (MIT Press, Cambridge, Mass., 1987).

Bracewell, R. N.

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

Creath, K.

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

Fleet, D. J.

D. J. Fleet, A. D. Jepson, M. R. M. Jenkin, “Phase-based disparity measurement,” Comput. Vision Graphics 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. 93, 429–457 (1946).

Geman, D.

D. Geman, G. Reynolds, “Constrained restoration and the recovery of discontinuities,” IEEE Trans. Pattern Anal. Machine Intell. 14, 367–383 (1992).
[CrossRef]

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

Golub, G. H.

G. H. Golub, C. F. Van Loan, Matrix Computations (Johns Hopkins U., Baltimore, 1990).

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. Vision Graphics and 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. Vision Graphics Image Process. 53, 198–210 (1991).

Jepson, A. D.

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

Kobayashi, S.

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. Rivera, “Quadratic regularization functionals for phase unwrapping,” J. Opt. Soc. Am. A 12, 2393–2400 (1995).
[CrossRef]

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

J. L. Marroquin, “The adaptive ICM algorithm,” , Centro de Investigación en Matematicas, Guonajuato, Mexico, 1996).

J. L. Marroquin, “Probabilistic solution of inverse problems,” (Massachusetts Institute of Technology, Cambridge, Mass., 1985).

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]

Poggio, T.

M. Bertero, T. Poggio, V. Torre, “Ill-posed problems in early vision,” Proc. IEEE 76, 869–887 (1988).
[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]

T. Poggio, H. Voorhees, A. Yuille, “A regularized solution to edge detection,” (Massachusetts Institute of Technology, Cambridge, Mass., 1985).

Reynolds, G.

D. Geman, G. Reynolds, “Constrained restoration and the recovery of discontinuities,” IEEE Trans. Pattern Anal. Machine Intell. 14, 367–383 (1992).
[CrossRef]

Rivera, M.

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]

Sutter, A.

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

Takeda, M.

Thikonov, A. N.

A. N. Thikonov, V. Y. Arsenin, Solutions to Ill-posed Problems (Winston and Sons, Washington, D.C., 1977).

Torre, V.

M. Bertero, T. Poggio, V. Torre, “Ill-posed problems in early vision,” Proc. IEEE 76, 869–887 (1988).
[CrossRef]

Turner, M.

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

Van Loan, C. F.

G. H. Golub, C. F. Van Loan, Matrix Computations (Johns Hopkins U., Baltimore, 1990).

Voorhees, H.

T. Poggio, H. Voorhees, A. Yuille, “A regularized solution to edge detection,” (Massachusetts Institute of Technology, Cambridge, Mass., 1985).

Wahba, G.

G. Wahba, Spline Models for Observational Data (Society for Industrial and Applied Mathematics, Philadelphia, 1990).

Yuille, A.

T. Poggio, H. Voorhees, A. Yuille, “A regularized solution to edge detection,” (Massachusetts Institute of Technology, Cambridge, Mass., 1985).

Zisserman, A.

A. Blake, A. Zisserman, Visual Reconstruction (MIT Press, Cambridge, Mass., 1987).

Biol. Cybern. (3)

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]

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

Comput. Vision Graphics and Image Process. (1)

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

Comput. Vision Graphics Image Process. (2)

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

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

IEEE Trans. Pattern Anal. Machine Intell. (2)

D. Geman, G. Reynolds, “Constrained restoration and the recovery of discontinuities,” IEEE Trans. Pattern Anal. Machine Intell. 14, 367–383 (1992).
[CrossRef]

S. Geman, D. Geman, “Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images,” IEEE Trans. Pattern Anal. Machine 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. (1)

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

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

J. R. Stat. Soc. B (1)

J. Besag, “Spatial interaction and the statistical analysis of lattice systems,” J. R. Stat. Soc. B 36, 192–326 (1974).

Nature (London) (1)

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

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]

Proc. IEEE (1)

M. Bertero, T. Poggio, V. Torre, “Ill-posed problems in early vision,” Proc. IEEE 76, 869–887 (1988).
[CrossRef]

Other (10)

J. L. Marroquin, “Probabilistic solution of inverse problems,” (Massachusetts Institute of Technology, Cambridge, Mass., 1985).

J. L. Marroquin, “The adaptive ICM algorithm,” , Centro de Investigación en Matematicas, Guonajuato, Mexico, 1996).

T. Poggio, H. Voorhees, A. Yuille, “A regularized solution to edge detection,” (Massachusetts Institute of Technology, Cambridge, Mass., 1985).

G. Wahba, Spline Models for Observational Data (Society for Industrial and Applied Mathematics, Philadelphia, 1990).

A. Blake, A. Zisserman, Visual Reconstruction (MIT Press, Cambridge, Mass., 1987).

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

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

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

A. N. Thikonov, V. Y. Arsenin, Solutions to Ill-posed Problems (Winston and Sons, Washington, D.C., 1977).

G. H. Golub, C. F. Van Loan, Matrix Computations (Johns Hopkins U., Baltimore, 1990).

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Figures (12)

Fig. 1
Fig. 1

(a) Relation between the width σ of a Gabor filter and the equivalent regularization parameter λ for first-order (dashed curve) and second-order (solid curve) models. (b) Shapes of the corresponding frequency responses (for σ=1): Gabor filter (dashed curve), second-order model (solid curve), and first-order model (dotted–dashed curve).

Fig. 2
Fig. 2

Space-varying kernel generated by a one-dimensional, second-order model on a lattice of size 64. (a) Kernel corresponding to pixel 32 (i.e., central row of the inverse matrix Q2-1; see the text). (b) Kernel corresponding to pixel 2 (second row of Q2-1). (c) Solid curve: kernel corresponding to pixel 32 when observations at pixels 32 and 33 are missing; dashed curve: kernel of (a) presented for comparison purposes. (d) Magnitude of the Fourier transforms (local frequency responses) of the kernels of (a) (dashed curve) and (b) (dotted–dashed curve). The solid curve corresponds to the frequency response on an infinite lattice, given by Eq. (13).

Fig. 3
Fig. 3

Magnitude of the frequency response of a directional filter (coded as a gray level) obtained with a second-order potential. The parameters are λ1=100, λ2=0.01, u0=v0=0.5π, and θ=π/4.

Fig. 4
Fig. 4

Magnitude of the frequency response of a second-order RQF [Eq. (13); dashed curve] and of a second-order RQF corrected for low tuning frequency [Eq. (18); solid curve]. The parameters are λ=20 and ω0=0.2π.

Fig. 5
Fig. 5

Synthetic interferogram of a wave front with a pure spherical aberration.

Fig. 6
Fig. 6

Plots (a) and (b) show the central line of the ideal wave front that modulates the interferogram and the corresponding profile obtained from Fig. 5. Plots (c) and (d) show the absolute phase error obtained with an ordinary Gabor filter and with a second-order RQF, respectively.

Fig. 7
Fig. 7

(a) A row of a noisy fringe pattern modulated by a triangular phase object (see the text). (b) Output of a RQF tuned at the fundamental frequency of the pattern (real part: solid curve; imaginary part: dashed curve) without magnitude equalization (for ν=0). (c) Corresponding recovered phase. (d) and (e): Same as (b) and (c) but for ν=100.

Fig. 8
Fig. 8

(a) One row of the left image of a synthetic stereo pair with the location of the binary features described in the text. (b) Output of a linear RQF (real part: solid curve; imaginary part: dashed curve). (c) Output of a RQF with magnitude equalization. (d) Reconstructed phase difference (disparity) with magnitude equalization (solid curve) and without it (dashed curve) the true disparity appears as a dotted curve.

Fig. 9
Fig. 9

Cliques of the line lattice associated with nonzero potentials (× indicates a line site, and ○ indicates a pixel site). (a) C2 cliques, (b) C4 cliques.

Fig. 10
Fig. 10

(a) One-dimensional signal with different textures in the left and right halves. (b) Output of a linear RQF tuned at ω1 (magnitude: solid curve; real part: dashed curve; imaginary part: dotted–dashed curve), with λ=64. (c) Same as (b), but with λ=16. (d) Same as (b), but for a truncated quadratic RQF with λ=900.

Fig. 11
Fig. 11

(a) Two-dimensional texture mosaic: The magnitude of the frequency component (u0=1, v0=1) is different in the rectangle at the center of the pattern. (b) Magnitude (coded as a gray level) of the output of a linear RQF, tuned at (1, 1), with λ=100. (c) Boundaries found by the adaptive ICM algorithm with functional (25) [parameters: λ=2500, μ=γ=0, δ=0.1, and θ(0)=0.1]. (d) Same as (b), but for the corresponding nonlinear RQF.

Fig. 12
Fig. 12

(a) Synthetic fringe pattern with magnitude and phase discontinuities. (b) Discontinuities found by the adaptive ICM algorithm with functional (25) are superimposed as white pixels [parameters: λ=1000, μ=γ=0, δ=0.1, and θ(0)=0.05]. (c) Recovered phase. (d) Recovered phase with a linear RQF and λ=100.

Equations (61)

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

g(x)=Af(x)+n(x),xS,
Pg|f(f)=Pn(Af-g)=1Kexp-xS [(Af)(x)-g(x)]2/2s2,
Pf(f)=1Zexp-C VC(f),
Pf|g(f)=1Zexp-12s2xS[(Af)(x)-g(x)]2-C VC(f)
U(f)=xS Φf,g(x)+λ C VC(f),
Vab(f)=[f(a)-f(b)]2;
Vabc(f)=[-f(a)+2 f(b)-f(c)]2,
Vpqrs(f)=2[-f(p)+f(q)-f(r)+f(s)]2,
f(x)+λ[-f(x-1)+2 f(x)-f(x+1)]=g(x).
F(ω){1+λ[-exp(-iω)+2-exp(iω)]}=G(ω),
H(ω)=11+2λ(1-cos ω),
H(ω)=πσexp-(ω-ω0)24σ2,
h(x)=[exp(-σ2x2)][cos(ω0x)+i sin(ω0x)].
H(ω)=21+2λ[1-cos(ω-ω0)],
f(x)-2g(x)+λ[-f(x-1)exp(iω0)+2 f(x)-f(x+1)exp(-iω0)]=0.
U(f)=12xS|f(x)-2g(x)|2+λ2x=2N|f(x)-f(x-1)exp(iω0)|2,
|f(x)exp(-iω0x)-f(x-1)exp(-iω0x)exp(iω0)|2=|f(x)-f(x-1)exp(iω0)|2.
H(ω)=21+λ{6+2 cos[2(ω-ω0)]-8 cos(ω-ω0)}.
f(x)-2g(x)+λ[6f(x)+f(x+2)exp(-2iω0)+f(x-2)exp(2iω0)-4[f(x+1)exp(-iω0)+f(x-1)exp(iω0)]=0,
U(f)=12xS|f(x)-2g(x)|2+λ2x=2N-1|-f(x-1)×exp(iω0)+2 f(x)-f(x+1)exp(-iω0)|2,
-f(x-1)exp(iω0)+2 f(x)-f(x+1)exp(-iω0)=0,
f(1)-2g(1)+λ[f(1)-f(2)exp(-iω0)]=0,
f(N)-2g(N)+λ[f(N)-f(N-1)exp(iω0)]=0.
V(f)=λ1|(c+s)f(x, y)-cf(x-1, y)exp(iu0)-sf(x, y-1)exp(iv0)|2+λ2|(c-s)f(x, y)+sf(x-1, y)exp(iu0)-cf(x, y-1)exp(iv0)|2,
V(f)=λ1|c2Fxx+2scFxy+s2Fyy|2+λ2|s2Fxx-2scFxy+c2Fyy|2,
Fxx=f(x-1, y)exp(iu0)-2 f(x, y)+f(x+1, y)exp(i0),
Fyy=f(x, y-1)exp(iv0)-2 f(x, y)+f(x, y+1)exp(iv0),
Fxy=f(x, y)-f(x-1, y)exp(iu0)-f(x, y-1)exp(iv0)+f(x-1, y-1)exp[i(u0+v0)]
U(f)=12[x, y]f(x)-f(y)-2[g(x)-g(y)]2+λ2C VC(f),
H(ω)=2(1-cos ω)(1-cos ω)+λ[1-cos(ω-ω0)]
H(ω)=2(1-cos ω)(1-cos ω)+λ{3+cos[2(ω-ω0)]-4 cos(ω-ω0)}
ϕ(x)=arctan[ψ(x)/φ(x)].
Φ(f)=k ck[Dαkf(x)]2,
Ψ(f)=k ck|Dαk[f(x)exp(-iΩ0·x)]|2,
Ψ(f)=|D11[f(x)exp(-iΩ0·x)]|2+2|D12[f(x)exp(-iΩ0·x)]|2+|D22[f(x)exp(-iΩ0·x)]|2.
U(f)=x1x2 |f(x)-2g(x)|2 dx+λ x1x2 |D1[f(x)exp(-iΩ0x)]|2 dx.
(1+λΩ02)f+2iλΩ0f-λf=2g(x).
H(Ω)=21+λ(Ω-Ω0)2
H(Ω)=11+λΩ2,
U(f)=x1x2 |f(x)-g(x)|2 dx+λ x1x2 |f(x)|2 dx.
U(f)=j=1n |f(xj)-2gj|2+λ  Ψ(f).
k=1n [(1-αk)φ(zk)+αkφ(zk+1)-2g(xk)]2+k=1n [(1-αk)ψ(zk)+αkψ(zk+1)]2,
Ve(f(x), f(y))=[|f(x)|-|f(y)|]2,
U(f)=12xS |f(x)-2g(x)|2+λ2C VC(f)+ν [x,y] [|f(x)|-|f(y)|]2,
g(x)=sgn[cos(0.28π[x+ϕ(x)])]+ξ,
U(f, l)=xS[f(x)-g(x)]2+λ [x,y][f(x)-f(y)]2×(1-lxy)+θ [x,y] lxy+Cl VCl(l),
VC2(l)=γifΣrsC2 lrs>10otherwise,
VC4(l)=-μifΣrsC4 lrs=2δifΣrsC4 lrs=30otherwise,
|f(x)-f(x-1)exp(iω0)|2
f(x-1)=B exp[iω0(x-1)],f(x)=A exp(iω0x),
f(x-1)=B exp[iω0(x-1)],
f(x)=B exp[i(ω0x+a)],
U(f, l)=xS|f(x)-2g(x)|2+λ [x,y] |f(x)-f(y)×exp[iω0(x-y)]|2(1-lxy)+θ [x,y] lxy+Cl VCl(l),
g(x)=cos(ω0x)+cos(ω2x)+0.3 cos(ω3x)cos(ω1x)+cos(ω2x)+0.3 cos(ω3x)forx<25forx25,
U(f)=j=1n |f(xj)-2gj|2+λ D k ck|Dαk[f(x)exp(-iΩ0·x)]|2,
|Dαk[f(x)exp(-iΩ0·x)]|2=[Dαkϕˆ(x)]2+[Dαkψˆ(x)]2
|f(xj)-2gj|2=[ϕˆ(x)-ηj]2+[ψˆ(x)-ξj]2,
U1(ϕˆ)=j=1n [ϕˆ(xj)-ηj]2+λ D k ck[Dαkϕˆ(x)]2,
U2(ψˆ)=j=1n [ψˆ(xj)-ξj]2+λ D k ck[Dαkψˆ(x)]2,
j=1n ajx-xj2 ln(x-xj)+b·x+a0,
f*(x)=[ϕˆ*(x)+iψˆ*(x)]exp(iΩ0·x).

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