Abstract

The application of a regularization technique to filter synthesis in pattern recognition with synthetic discriminant function filters is presented. The proposed technique uses the stabilizing functional approach for two-dimensional ill-posed problems. Filter synthesis is thus formulated as the minimization of some relevant criteria with specified correlation values for some training input images and limited maximum value of a stabilizing functional. The choice of a particular stabilizing functional to be minimized is related to a priori knowledge regarding the pattern-recognition problem. The analogy between the regularization methods and optimal trade-off filters is also presented and is illustrated with numerical experiments.

© 1994 Optical Society of America

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  1. H. J. Caulfield, W. T. Maloney, “Improved discrimination in optical character recognition,” Appl. Opt. 8, 2354–2355 (1969).
    [Crossref] [PubMed]
  2. D. Casasent, D. Psaltis, “Position, rotation, and scale invariant optical correlation,” Appl. Opt. 15, 1795–1799 (1976).
    [Crossref] [PubMed]
  3. J. L. Horner, “Light utilization in optical correlators,” Appl. Opt. 21, 4511–4514 (1982).
    [Crossref] [PubMed]
  4. Y. N. Hsu, H. H. Arsenault, “Optical pattern recognition using circular harmonic expansion,” Appl. Opt. 21, 4016–4019 (1982).
    [Crossref] [PubMed]
  5. B. V. K. Vijaya Kumar, “Efficient approach for designing linear combination filters,” Appl. Opt. 22, 1445–1448 (1983).
    [Crossref]
  6. J. L. Horner, P. D. Gianino, “Phase-only matched filtering,” Appl. Opt. 23, 812–816 (1984).
    [Crossref] [PubMed]
  7. D. Casasent, “Unified synthetic discriminant function computation formulation,” Appl. Opt. 23, 1620–1627 (1984).
    [Crossref] [PubMed]
  8. F. M. Dickey, L. A. Romero, “Normalized correlation for pattern recognition,” Opt. Lett. 16, 1186–1188 (1991).
    [Crossref] [PubMed]
  9. F. M. Dickey, K. T. Stalker, “Binary phase-only filters; implications of bandwidth and uniqueness on performance,” J. Opt. Soc. Am. A 4, P69 (1987).
  10. F. M. Dickey, T. K. Stalker, J. J. Mason, “Bandwidth considerations for binary phase-only filters,” Appl. Opt. 27, 3811–3818 (1988).
    [Crossref] [PubMed]
  11. Z. Zouhir Bahri, B. V. K. Vijaya Kumar, “Fast algorithms for designing optical phase-only-filters (pof’s) and binary phase-only filters (bpof’s),” Appl. Opt. 29, 2992–2996 (1990).
    [Crossref]
  12. F. M. Dickey, B. V. K. Vijaya Kumar, L. A. Romero, J. M. Connelly, “Complex ternary matched filters yielding high signal-to-noise ratios,” Opt. Eng. 29, 994–1001 (1990).
    [Crossref]
  13. A. A. S. Awwal, M. A. Karim, S. R. Jahan, “Improved correlation discrimination using an amplitude modulated phase-only filter,” Appl. Opt. 29, 233–236 (1990).
    [Crossref] [PubMed]
  14. Y. N. Hsu, H. H. Arsenault, G. April, “Rotation-invariant digital pattern recognition using circular harmonic expansion,” Appl. Opt. 21, 4012–4015 (1982).
    [Crossref] [PubMed]
  15. B. V. K. Vijaya Kumar, “Minimum-variance synthetic discriminant functions,” J. Opt. Soc. Am. A3, 1579–1584 (1986).
  16. A. Mahalanobis, B. V. K. Vijaya Kumar, D. Casasent, “Minimum average correlation energy filters,” Appl. Opt. 26, 3633–3640 (1987).
    [Crossref] [PubMed]
  17. B. V. K. Vijaya Kumar, L. Hassebrook, “Performance measures for correlation filters,” Appl. Opt. 29, 2997–3006 (1990).
    [Crossref]
  18. H. L. Van Trees, Detection, Estimation and Modulation Theory (Wiley, New York, (1968).
  19. L. P. Yaroslavsky, “Is the phase-only filter and its modifications optimal in terms of the discrimination capability in pattern recognition?” Appl. Opt. 31, 1677–1679 (1992).
    [Crossref] [PubMed]
  20. B. V. K. Vijaya Kumar, F. M. Dickey, J. M. Delaurentis, “Correlation filters minimizing peak location errors,” J. Opt. Soc. Am. A 9, 678–682 (1992).
    [Crossref]
  21. Ph. Réfrégier, J. Figue, “Optimal trade-off filters for pattern recognition and comparison with Wiener approach,” Opt. Comput. Process. 1, 245–266 (1991).
  22. Ph. Réfrégier, “Filter design for optical pattern recognition: multicriteria optimization approach,” Opt. Lett. 15, 854–856 (1990).
    [Crossref] [PubMed]
  23. Ph. Réfrégier, “Optical pattern recognition: optimal trade-off circular harmonic filters,” Opt. Commun. 86, 113–118 (1991).
    [Crossref]
  24. Ph. Réfrégier, “Optimal trade-off filters for noise robustness, sharpness of the correlation peak, and Horner efficiency,” Opt. Lett. 16, 829–831 (1991).
    [Crossref] [PubMed]
  25. Ph. Réfrégier, “Optimal introduction of optical efficiency for pattern recognition filters,” in B. Javidi, ed., Optical Information Processing Systems and Architectures IV, Proc. Soc. Photo-Opt. Instrum. Eng.1772, 104–115 (1992).
    [Crossref]
  26. J. Figue, Ph. Réfrégier, “Influence of the noise model on correlation filters: peak sharpness and noise robustness,” Opt. Lett. 17,1476–1478 (1992).
    [Crossref] [PubMed]
  27. N. Boccara, Functional Analysis (Academic, Boston, Mass., (1990).
  28. R. O. Duda, P. E. Hart, Pattern Classification and Scene Analysis (Wiley, New York, (1973).
  29. W. K. Pratt, Digital Image Processing (Wiley, New York, (1978).
  30. J. V. Candy, Signal Processing, Electrical Engineering Series (McGraw-Hill, New York, (1988).
  31. A. Mahalanobis, D. P. Casasent, “Performances evaluation of minimum average correlation energy filters,” Appl. Opt. 30, 561–572 (1991).
    [Crossref] [PubMed]
  32. B. R. Frieden, “Restoring with maximum likelihood and maximum entropy,” J. Opt. Soc. Am. 62, 511–518 (1972).
    [Crossref] [PubMed]
  33. S. J. Wernecke, L. R. d’Addario, “Maximum entropy image reconstruction,” IEEE Trans. Comput. C-26351–364 (1977).
    [Crossref]
  34. M. Fleisher, U. Mahalab, J. Shamir, “Entropy optimized filter for pattern recognition,” Appl. Opt. 29, 2091–2098 (1990).
    [Crossref] [PubMed]
  35. A. N. Tikhonov, V. Y. Arsenin, Solutions of Ill-Posed Problems (Wiley, New York, (1977), pp. 857–896.
  36. N. B. Karayiannis, A. N. Venetsanopoulos, “Regularization theory in image restoration—the stabilizing functional approach,” IEEE Trans. Acoust. Speech Signal Process. 38, 1155–1179 (1990).
    [Crossref]
  37. G. Demoment, “Image reconstruction and restoration: overview of common estimation structures and problems,” IEEE Trans. Acoust. Speech Signal Process. 37, 2024–2036 (1989).
    [Crossref]

1992 (3)

1991 (5)

1990 (7)

1989 (1)

G. Demoment, “Image reconstruction and restoration: overview of common estimation structures and problems,” IEEE Trans. Acoust. Speech Signal Process. 37, 2024–2036 (1989).
[Crossref]

1988 (1)

1987 (2)

A. Mahalanobis, B. V. K. Vijaya Kumar, D. Casasent, “Minimum average correlation energy filters,” Appl. Opt. 26, 3633–3640 (1987).
[Crossref] [PubMed]

F. M. Dickey, K. T. Stalker, “Binary phase-only filters; implications of bandwidth and uniqueness on performance,” J. Opt. Soc. Am. A 4, P69 (1987).

1986 (1)

B. V. K. Vijaya Kumar, “Minimum-variance synthetic discriminant functions,” J. Opt. Soc. Am. A3, 1579–1584 (1986).

1984 (2)

1983 (1)

1982 (3)

1977 (1)

S. J. Wernecke, L. R. d’Addario, “Maximum entropy image reconstruction,” IEEE Trans. Comput. C-26351–364 (1977).
[Crossref]

1976 (1)

1972 (1)

1969 (1)

April, G.

Arsenault, H. H.

Arsenin, V. Y.

A. N. Tikhonov, V. Y. Arsenin, Solutions of Ill-Posed Problems (Wiley, New York, (1977), pp. 857–896.

Awwal, A. A. S.

Boccara, N.

N. Boccara, Functional Analysis (Academic, Boston, Mass., (1990).

Candy, J. V.

J. V. Candy, Signal Processing, Electrical Engineering Series (McGraw-Hill, New York, (1988).

Casasent, D.

Casasent, D. P.

Caulfield, H. J.

Connelly, J. M.

F. M. Dickey, B. V. K. Vijaya Kumar, L. A. Romero, J. M. Connelly, “Complex ternary matched filters yielding high signal-to-noise ratios,” Opt. Eng. 29, 994–1001 (1990).
[Crossref]

d’Addario, L. R.

S. J. Wernecke, L. R. d’Addario, “Maximum entropy image reconstruction,” IEEE Trans. Comput. C-26351–364 (1977).
[Crossref]

Delaurentis, J. M.

Demoment, G.

G. Demoment, “Image reconstruction and restoration: overview of common estimation structures and problems,” IEEE Trans. Acoust. Speech Signal Process. 37, 2024–2036 (1989).
[Crossref]

Dickey, F. M.

B. V. K. Vijaya Kumar, F. M. Dickey, J. M. Delaurentis, “Correlation filters minimizing peak location errors,” J. Opt. Soc. Am. A 9, 678–682 (1992).
[Crossref]

F. M. Dickey, L. A. Romero, “Normalized correlation for pattern recognition,” Opt. Lett. 16, 1186–1188 (1991).
[Crossref] [PubMed]

F. M. Dickey, B. V. K. Vijaya Kumar, L. A. Romero, J. M. Connelly, “Complex ternary matched filters yielding high signal-to-noise ratios,” Opt. Eng. 29, 994–1001 (1990).
[Crossref]

F. M. Dickey, T. K. Stalker, J. J. Mason, “Bandwidth considerations for binary phase-only filters,” Appl. Opt. 27, 3811–3818 (1988).
[Crossref] [PubMed]

F. M. Dickey, K. T. Stalker, “Binary phase-only filters; implications of bandwidth and uniqueness on performance,” J. Opt. Soc. Am. A 4, P69 (1987).

Duda, R. O.

R. O. Duda, P. E. Hart, Pattern Classification and Scene Analysis (Wiley, New York, (1973).

Figue, J.

J. Figue, Ph. Réfrégier, “Influence of the noise model on correlation filters: peak sharpness and noise robustness,” Opt. Lett. 17,1476–1478 (1992).
[Crossref] [PubMed]

Ph. Réfrégier, J. Figue, “Optimal trade-off filters for pattern recognition and comparison with Wiener approach,” Opt. Comput. Process. 1, 245–266 (1991).

Fleisher, M.

Frieden, B. R.

Gianino, P. D.

Hart, P. E.

R. O. Duda, P. E. Hart, Pattern Classification and Scene Analysis (Wiley, New York, (1973).

Hassebrook, L.

Horner, J. L.

Hsu, Y. N.

Jahan, S. R.

Karayiannis, N. B.

N. B. Karayiannis, A. N. Venetsanopoulos, “Regularization theory in image restoration—the stabilizing functional approach,” IEEE Trans. Acoust. Speech Signal Process. 38, 1155–1179 (1990).
[Crossref]

Karim, M. A.

Mahalab, U.

Mahalanobis, A.

Maloney, W. T.

Mason, J. J.

Pratt, W. K.

W. K. Pratt, Digital Image Processing (Wiley, New York, (1978).

Psaltis, D.

Réfrégier, Ph.

Romero, L. A.

F. M. Dickey, L. A. Romero, “Normalized correlation for pattern recognition,” Opt. Lett. 16, 1186–1188 (1991).
[Crossref] [PubMed]

F. M. Dickey, B. V. K. Vijaya Kumar, L. A. Romero, J. M. Connelly, “Complex ternary matched filters yielding high signal-to-noise ratios,” Opt. Eng. 29, 994–1001 (1990).
[Crossref]

Shamir, J.

Stalker, K. T.

F. M. Dickey, K. T. Stalker, “Binary phase-only filters; implications of bandwidth and uniqueness on performance,” J. Opt. Soc. Am. A 4, P69 (1987).

Stalker, T. K.

Tikhonov, A. N.

A. N. Tikhonov, V. Y. Arsenin, Solutions of Ill-Posed Problems (Wiley, New York, (1977), pp. 857–896.

Van Trees, H. L.

H. L. Van Trees, Detection, Estimation and Modulation Theory (Wiley, New York, (1968).

Venetsanopoulos, A. N.

N. B. Karayiannis, A. N. Venetsanopoulos, “Regularization theory in image restoration—the stabilizing functional approach,” IEEE Trans. Acoust. Speech Signal Process. 38, 1155–1179 (1990).
[Crossref]

Vijaya Kumar, B. V. K.

Wernecke, S. J.

S. J. Wernecke, L. R. d’Addario, “Maximum entropy image reconstruction,” IEEE Trans. Comput. C-26351–364 (1977).
[Crossref]

Yaroslavsky, L. P.

Zouhir Bahri, Z.

Appl. Opt. (16)

H. J. Caulfield, W. T. Maloney, “Improved discrimination in optical character recognition,” Appl. Opt. 8, 2354–2355 (1969).
[Crossref] [PubMed]

D. Casasent, D. Psaltis, “Position, rotation, and scale invariant optical correlation,” Appl. Opt. 15, 1795–1799 (1976).
[Crossref] [PubMed]

J. L. Horner, “Light utilization in optical correlators,” Appl. Opt. 21, 4511–4514 (1982).
[Crossref] [PubMed]

Y. N. Hsu, H. H. Arsenault, “Optical pattern recognition using circular harmonic expansion,” Appl. Opt. 21, 4016–4019 (1982).
[Crossref] [PubMed]

B. V. K. Vijaya Kumar, “Efficient approach for designing linear combination filters,” Appl. Opt. 22, 1445–1448 (1983).
[Crossref]

J. L. Horner, P. D. Gianino, “Phase-only matched filtering,” Appl. Opt. 23, 812–816 (1984).
[Crossref] [PubMed]

D. Casasent, “Unified synthetic discriminant function computation formulation,” Appl. Opt. 23, 1620–1627 (1984).
[Crossref] [PubMed]

F. M. Dickey, T. K. Stalker, J. J. Mason, “Bandwidth considerations for binary phase-only filters,” Appl. Opt. 27, 3811–3818 (1988).
[Crossref] [PubMed]

Z. Zouhir Bahri, B. V. K. Vijaya Kumar, “Fast algorithms for designing optical phase-only-filters (pof’s) and binary phase-only filters (bpof’s),” Appl. Opt. 29, 2992–2996 (1990).
[Crossref]

A. A. S. Awwal, M. A. Karim, S. R. Jahan, “Improved correlation discrimination using an amplitude modulated phase-only filter,” Appl. Opt. 29, 233–236 (1990).
[Crossref] [PubMed]

Y. N. Hsu, H. H. Arsenault, G. April, “Rotation-invariant digital pattern recognition using circular harmonic expansion,” Appl. Opt. 21, 4012–4015 (1982).
[Crossref] [PubMed]

L. P. Yaroslavsky, “Is the phase-only filter and its modifications optimal in terms of the discrimination capability in pattern recognition?” Appl. Opt. 31, 1677–1679 (1992).
[Crossref] [PubMed]

A. Mahalanobis, B. V. K. Vijaya Kumar, D. Casasent, “Minimum average correlation energy filters,” Appl. Opt. 26, 3633–3640 (1987).
[Crossref] [PubMed]

B. V. K. Vijaya Kumar, L. Hassebrook, “Performance measures for correlation filters,” Appl. Opt. 29, 2997–3006 (1990).
[Crossref]

A. Mahalanobis, D. P. Casasent, “Performances evaluation of minimum average correlation energy filters,” Appl. Opt. 30, 561–572 (1991).
[Crossref] [PubMed]

M. Fleisher, U. Mahalab, J. Shamir, “Entropy optimized filter for pattern recognition,” Appl. Opt. 29, 2091–2098 (1990).
[Crossref] [PubMed]

IEEE Trans. Acoust. Speech Signal Process. (2)

N. B. Karayiannis, A. N. Venetsanopoulos, “Regularization theory in image restoration—the stabilizing functional approach,” IEEE Trans. Acoust. Speech Signal Process. 38, 1155–1179 (1990).
[Crossref]

G. Demoment, “Image reconstruction and restoration: overview of common estimation structures and problems,” IEEE Trans. Acoust. Speech Signal Process. 37, 2024–2036 (1989).
[Crossref]

IEEE Trans. Comput. (1)

S. J. Wernecke, L. R. d’Addario, “Maximum entropy image reconstruction,” IEEE Trans. Comput. C-26351–364 (1977).
[Crossref]

J. Opt. Soc. Am. (1)

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

F. M. Dickey, K. T. Stalker, “Binary phase-only filters; implications of bandwidth and uniqueness on performance,” J. Opt. Soc. Am. A 4, P69 (1987).

B. V. K. Vijaya Kumar, F. M. Dickey, J. M. Delaurentis, “Correlation filters minimizing peak location errors,” J. Opt. Soc. Am. A 9, 678–682 (1992).
[Crossref]

B. V. K. Vijaya Kumar, “Minimum-variance synthetic discriminant functions,” J. Opt. Soc. Am. A3, 1579–1584 (1986).

Opt. Commun. (1)

Ph. Réfrégier, “Optical pattern recognition: optimal trade-off circular harmonic filters,” Opt. Commun. 86, 113–118 (1991).
[Crossref]

Opt. Comput. Process. (1)

Ph. Réfrégier, J. Figue, “Optimal trade-off filters for pattern recognition and comparison with Wiener approach,” Opt. Comput. Process. 1, 245–266 (1991).

Opt. Eng. (1)

F. M. Dickey, B. V. K. Vijaya Kumar, L. A. Romero, J. M. Connelly, “Complex ternary matched filters yielding high signal-to-noise ratios,” Opt. Eng. 29, 994–1001 (1990).
[Crossref]

Opt. Lett. (4)

Other (7)

N. Boccara, Functional Analysis (Academic, Boston, Mass., (1990).

R. O. Duda, P. E. Hart, Pattern Classification and Scene Analysis (Wiley, New York, (1973).

W. K. Pratt, Digital Image Processing (Wiley, New York, (1978).

J. V. Candy, Signal Processing, Electrical Engineering Series (McGraw-Hill, New York, (1988).

Ph. Réfrégier, “Optimal introduction of optical efficiency for pattern recognition filters,” in B. Javidi, ed., Optical Information Processing Systems and Architectures IV, Proc. Soc. Photo-Opt. Instrum. Eng.1772, 104–115 (1992).
[Crossref]

H. L. Van Trees, Detection, Estimation and Modulation Theory (Wiley, New York, (1968).

A. N. Tikhonov, V. Y. Arsenin, Solutions of Ill-Posed Problems (Wiley, New York, (1977), pp. 857–896.

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Figures (9)

Fig. 1
Fig. 1

Schematic representation of the effect of the stabilizing functional on the modulus of the filter as a function of the frequencies k. | B ̂ k | represents a typical modulus of the denominator of the filter (see text for more details). From top to bottom, four cases are represented, a, No stabilizing functional: the corresponding filter diverges for the high frequencies; b, the minimum-norm stabilizing functional Ĥk = α; c, the minimum-variation stabilizing functional Ĥk = [1 − cos(2πk/N)]2; d, the minimum-elastic-energy stabilizing functional Ĥk = k4.

Fig. 2
Fig. 2

Image of the airplane used for SDF synthesis (four images of the four in-plane rotation angles; 0, 10, 20, and 30 deg are used in the training set).

Fig. 3
Fig. 3

Input image used for testing the different SDF filters.

Fig. 4
Fig. 4

Correlation peaks obtained with an OT SDF filter optimized for a white-noise model when there is no noise in the input image.

Fig. 5
Fig. 5

Locations of the correlation peaks and of the corresponding airplanes in the input image.

Fig. 6
Fig. 6

Correlation peaks obtained with an OT SDF filter optimized for a white-noise model with the input image of Fig. 3.

Fig. 7
Fig. 7

Correlation peaks obtained with a SDF filter optimized with a 1/k4 noise model with the input image of Fig. 3.

Fig. 8
Fig. 8

Correlation peaks obtained with a SDF filter optimized with a 1/k4 noise model with use of the minimum-norm stabilizing function.

Fig. 9
Fig. 9

Correlation peaks obtained with a SDF filter optimized with a 1/k4 noise model with use of the minimum-elastic-energy stabilizing function.

Equations (58)

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

c h , x = h x .
h x l = d l , l = 1 , , P
X h = d ,
X = [ x 1 , x 2 , , x P ]
X ̂ h ̂ = d ,
σ N 2 = | c 0 ( x + n ) h | 2 = h S h ,
σ N 2 = ĥ Ŝ ĥ ,
ACPE = k = 1 N ĥ k * D ̂ k ĥ k = ĥ D ̂ ĥ ,
D ̂ k = l = 1 P | x ̂ k l | 2 .
E ( h ) = h B h ,
E ( ĥ ) = h ̂ B ĥ .
ĥ = B ̂ 1 X ̂ ( X ̂ B ̂ 1 X ̂ ) 1 d ,
B ̂ = ( 1 μ ) Ŝ + μ D ̂ .
b l = n = 1 P [ ( X ̂ B ̂ 1 X ̂ ) 1 ] l , n d n ;
ĥ k = l = 1 P b l x ̂ k l B k .
ĥ k = β ( x ̂ k / B ̂ k ) ,
B ̂ k = ( 1 μ ) Ŝ k + μ D ̂ k .
δ h ̂ k β [ B ̂ k μ | x ̂ k | 2 ( B ̂ k ) 2 δ x ̂ k μ x ̂ k 2 ( B ̂ k ) 2 δ x ̂ k * ] .
δ c = β k x ̂ k B ̂ k δ x ̂ k .
ĥ opt = argmin E ( ĥ ) ĥ D ̂ ,
ĥ reg = argmin [ E ( ĥ ) + α Ω ( ĥ ) ] , ĥ D ̂ .
Ω ( ĥ ) ɛ ,
ĥ 1 = argmin [ E ( ĥ ) + α Ω ( ĥ ) ] , ĥ D ̂
ĥ 2 = argmin [ E ( ĥ ) + ( α + d α ) Ω ( ĥ ) ] , ĥ D ̂ .
E ( ĥ 1 ) + α Ω ( ĥ 1 ) E ( ĥ 2 ) + α Ω ( ĥ 2 ) , E ( ĥ 1 ) + ( α + d α ) Ω ( ĥ 1 ) E ( ĥ 2 ) + ( α + d α ) Ω ( ĥ 2 ) ;
d α Ω ( ĥ 1 ) d α Ω ( h ̂ 2 ) ,
δ c = h ( x + δ x ) h x = h δ x .
δ c h δ x ,
Ω ( h ) = h 2 ,
ĥ reg = argmin ( ĥ B ̂ ĥ + α ĥ ĥ ) , ĥ D ̂ .
ĥ reg = ( B ̂ + α I d ) 1 X ̂ [ X ̂ ( B ̂ + α I d ) 1 X ̂ ] 1 d ,
ĥ k = β x ̂ k B ̂ k + α ,
δ ĥ k β B ̂ k + α μ | x ̂ k | 2 [ B ̂ k + α ] 2 δ x ̂ k μ x ̂ k 2 [ B ̂ k + α ] 2 δ x ̂ k * .
δ x = g hp x ,
δ c = h g hp x .
δ c h g hp x ,
Ω ( h ) = h g hp 2 .
ĥ reg = argmin ( ĥ B ̂ ĥ + α ĥ Ĝ ĥ ) , ĥ D ̂ ,
ĥ reg = ( B ̂ + α Ĝ ̂ ) 1 X ̂ [ X ̂ ( B ̂ + α Ĝ ) 1 X ̂ ] 1 d .
ĥ = β x ̂ k B ̂ k + α Ĝ k ,
δ h ̂ k B ̂ k + α Ĝ k μ | x ̂ k | 2 ( B ̂ k + α Ĝ k ) 2 δ x ̂ k . μ x ̂ k 2 ( B ̂ k + α Ĝ k ) 2 δ x ̂ k * .
| 2 h ( u , υ ) u 2 + 2 h ( u , υ ) υ 2 | 2 d u d υ .
ĥ ( k u , k υ ) = h ( u , υ ) exp [ i 2 π ( k u u + k υ υ ) ] d u d υ ,
16 π 4 | ( k u 2 + k υ 2 ) ĥ ( k u , k υ ) | 2 d u d υ .
Ω ( ĥ ) = k u k υ ( k u 2 + k υ 2 ) 2 | ĥ ( k u , k υ ) | 2 .
ĥ reg = ( B ̂ + α L ̂ ) 1 X ̂ [ X ̂ ( B ̂ + α L ̂ ) 1 X ̂ ] 1 d ,
L ̂ ( k u , k υ ) = ( k u 2 + k υ 2 ) 2 .
Ŝ k k = { S γ / k γ if k > 0 0 otherwise .
ĥ k = β x ̂ k μ | x ̂ k | 2 + ( 1 μ ) S γ / k γ ,
ĥ k reg = x ̂ k μ | x ̂ k | 2 + ( 1 μ ) S γ / k γ + α Ĥ k ,
Ĝ k = 4 [ 1 cos ( 2 π k / N ) ] 2 ,
ĥ reg = ( B ̂ + α Ĥ ) 1 X ̂ [ X ̂ ( B ̂ + α Ĥ ) 1 X ̂ ] 1 d
E l = δ c l 2 = k | ĥ k | 2 | x ̂ k | 2 Ĝ k ,
E = l E l = k | ĥ k | 2 D ̂ k Ĝ k ,
h ̂ OT = ( B ̂ + α D ̂ Ĝ ) 1 X ̂ [ X ̂ ( B ̂ + α D ̂ Ĝ ) 1 X ̂ ] 1 d .
E elast = l k u k υ ( k u 2 + k υ 2 ) 2 | ĥ ( k u , k υ ) | 2 | x ̂ l ( k u , k υ ) | 2 ,
E elast = ĥ D ̂ L ̂ ĥ .
ĥ OT = ( B ̂ + α D ̂ L ̂ ) 1 X ̂ [ X ̂ ( B ̂ + α D ̂ L ̂ ) 1 X ̂ ] 1 d .

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