Abstract

Several elementwise component transformations performed over primary color image components (RGB) before optical multichannel correlations are proposed to improve real-time multispectral pattern recognition. The first transformation is deduced from the theory of the optimal filter for object location and recognition extended to multispectral images. Several modifications of this transformation are studied. We investigate these transformations in terms of noise robustness and discrimination capability. Computer simulation with noisy input images for various kinds of correlation filter are presented to illustrate improvement of color pattern recognition by using the proposed transformations. Experimental results are also presented.

© 1997 Optical Society of America

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References

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  1. C. Warde, H. J. Caulfield, F. T. S. Yu, J. E. Ludman, “Real-time joint spectral–spatial matched filtering,” Opt. Commun. 49, 241–244 (1984).
    [CrossRef]
  2. F. T. S. Yu, White-Light Optical Signal Processing (Wiley, New York, 1985).
  3. E. Badiqué, Y. Komiya, N. Ohyama, J. Tsujiuchi, T. Honda, “Color image correlation,” Opt. Commun. 62, 181–186 (1987).
    [CrossRef]
  4. E. Badiqué, N. Ohyama, T. Honda, J. Tsujiuchi, “Color image correlation for spatial/spectral recognition and increasedselectivity,” Opt. Commun. 68, 91–96 (1988).
    [CrossRef]
  5. M. S. Millán, J. Campos, C. Ferreira, M. J. Yzuel, “Matched filter and phase-only filter performance in colour image recognition,” Opt. Commun. 73, 277–284 (1989).
    [CrossRef]
  6. C. Ferreira, M. S. Millán, M. J. Yzuel, J. Campos, “Experimental results in color pattern recognition by multichannel matchedfiltering,” Opt. Eng. 31, 2231–2238 (1992).
    [CrossRef]
  7. M. S. Millán, M. J. Yzuel, J. Campos, C. Ferreira, “Different strategies in optical recognition of polychromatic images,” Appl. Opt. 31, 2560–2567 (1992).
    [CrossRef] [PubMed]
  8. M. J. Yzuel, J. Campos, “Colour information in optical pattern recognition,” in Current Trends in Optics, J. C. Dainty, ed. (Academic, London, 1994), pp. 209–223.
  9. M. S. Millán, M. Corbalán, J. Romero, M. J. Yzuel, “Optical pattern recognition using color vision models,” Opt. Lett. 20, 1722–1724 (1995).
    [CrossRef]
  10. D. Mendlovic, P. Garcı́a-Martı́nez, J. Garcı́a, C. Ferreira, “Color encoding for polychromatic single-channel optical pattern recognition,” Appl. Opt. 34, 7538–7544 (1995).
    [CrossRef] [PubMed]
  11. K. Yamaba, Y. Miyake, “Color character recognition method based on human perception,” Opt. Eng. 32, 33–40 (1993).
    [CrossRef]
  12. S. K. Case, “Pattern recognition with wavelength-multiplexed filters,” Appl. Opt. 18, 1890–1894 (1979).
    [CrossRef] [PubMed]
  13. F. T. S. Yu, “Colour image recognition by spectral–spatial matched filtering,” Opt. Eng. 23, 690–695 (1984).
    [CrossRef]
  14. J. L. Horner, P. D. Gianino, “Phase-only matched filtering,” Appl. Opt. 23, 812–816 (1984).
    [CrossRef] [PubMed]
  15. L. P. Yaroslavsky, “The theory of optimal methods for localization of objects in pictures,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1993), Vol. XXXII, pp. 145–201.
  16. L. P. Yaroslavsky, “Is the phase-only filter and its modifications optimal in terms ofthe discrimination capability in pattern recognition?” Appl. Opt. 31, 1677–1679 (1992).
    [CrossRef] [PubMed]
  17. I. Moreno, V. Kober, V. Lashin, J. Campos, L. P. Yaroslavsky, M. J. Yzuel“Optical color pattern recognition with circular component decomposition,” Opt. Lett. 21, 498–500 (1996).
    [CrossRef] [PubMed]
  18. W. K. Pratt, Digital Image Processing (Wiley, New York, 1978).
  19. A. Vander-Lugt, “Signal detection by complex filters,” IEEE Trans. Inf. Theory IT-10, 139–145 (1965).
  20. G. W. Snedecor, W. G. Cochran, Statistical Methods (Iowa State University Press, Ames, Iowa, 1989).
  21. J. Campos, F. Turon, L. P. Yaroslavsky, M. J. Yzuel, “Some filters for reliable recognition and localization of objects byoptical correlators,” Int. J. Opt. Comput. 2, 341–365 (1991).
  22. S. Mazé, Ph. Réfrégier, “Noise robustness of optical correlation for amplitude or phase modulationof the input image,” Opt. Lett. 17, 426–428 (1992).
    [CrossRef]
  23. R. R. Kallman, D. H. Goldstein, “Phase-encoding input images for optical pattern recognition,” Opt. Eng. 33, 1806–1812 (1994).
    [CrossRef]
  24. K. Styczynski, J. Campos, M. J. Yzuel, C. Chalasinska-Macukow, “New arrangement for limited intensity pattern recognition with highdiffraction efficiency,” Opt. Commun. 118, 193–198 (1995).
    [CrossRef]

1996

1995

1994

R. R. Kallman, D. H. Goldstein, “Phase-encoding input images for optical pattern recognition,” Opt. Eng. 33, 1806–1812 (1994).
[CrossRef]

1993

K. Yamaba, Y. Miyake, “Color character recognition method based on human perception,” Opt. Eng. 32, 33–40 (1993).
[CrossRef]

1992

1991

J. Campos, F. Turon, L. P. Yaroslavsky, M. J. Yzuel, “Some filters for reliable recognition and localization of objects byoptical correlators,” Int. J. Opt. Comput. 2, 341–365 (1991).

1989

M. S. Millán, J. Campos, C. Ferreira, M. J. Yzuel, “Matched filter and phase-only filter performance in colour image recognition,” Opt. Commun. 73, 277–284 (1989).
[CrossRef]

1988

E. Badiqué, N. Ohyama, T. Honda, J. Tsujiuchi, “Color image correlation for spatial/spectral recognition and increasedselectivity,” Opt. Commun. 68, 91–96 (1988).
[CrossRef]

1987

E. Badiqué, Y. Komiya, N. Ohyama, J. Tsujiuchi, T. Honda, “Color image correlation,” Opt. Commun. 62, 181–186 (1987).
[CrossRef]

1984

C. Warde, H. J. Caulfield, F. T. S. Yu, J. E. Ludman, “Real-time joint spectral–spatial matched filtering,” Opt. Commun. 49, 241–244 (1984).
[CrossRef]

F. T. S. Yu, “Colour image recognition by spectral–spatial matched filtering,” Opt. Eng. 23, 690–695 (1984).
[CrossRef]

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

1979

1965

A. Vander-Lugt, “Signal detection by complex filters,” IEEE Trans. Inf. Theory IT-10, 139–145 (1965).

Badiqué, E.

E. Badiqué, N. Ohyama, T. Honda, J. Tsujiuchi, “Color image correlation for spatial/spectral recognition and increasedselectivity,” Opt. Commun. 68, 91–96 (1988).
[CrossRef]

E. Badiqué, Y. Komiya, N. Ohyama, J. Tsujiuchi, T. Honda, “Color image correlation,” Opt. Commun. 62, 181–186 (1987).
[CrossRef]

Campos, J.

I. Moreno, V. Kober, V. Lashin, J. Campos, L. P. Yaroslavsky, M. J. Yzuel“Optical color pattern recognition with circular component decomposition,” Opt. Lett. 21, 498–500 (1996).
[CrossRef] [PubMed]

K. Styczynski, J. Campos, M. J. Yzuel, C. Chalasinska-Macukow, “New arrangement for limited intensity pattern recognition with highdiffraction efficiency,” Opt. Commun. 118, 193–198 (1995).
[CrossRef]

M. S. Millán, M. J. Yzuel, J. Campos, C. Ferreira, “Different strategies in optical recognition of polychromatic images,” Appl. Opt. 31, 2560–2567 (1992).
[CrossRef] [PubMed]

C. Ferreira, M. S. Millán, M. J. Yzuel, J. Campos, “Experimental results in color pattern recognition by multichannel matchedfiltering,” Opt. Eng. 31, 2231–2238 (1992).
[CrossRef]

J. Campos, F. Turon, L. P. Yaroslavsky, M. J. Yzuel, “Some filters for reliable recognition and localization of objects byoptical correlators,” Int. J. Opt. Comput. 2, 341–365 (1991).

M. S. Millán, J. Campos, C. Ferreira, M. J. Yzuel, “Matched filter and phase-only filter performance in colour image recognition,” Opt. Commun. 73, 277–284 (1989).
[CrossRef]

M. J. Yzuel, J. Campos, “Colour information in optical pattern recognition,” in Current Trends in Optics, J. C. Dainty, ed. (Academic, London, 1994), pp. 209–223.

Case, S. K.

Caulfield, H. J.

C. Warde, H. J. Caulfield, F. T. S. Yu, J. E. Ludman, “Real-time joint spectral–spatial matched filtering,” Opt. Commun. 49, 241–244 (1984).
[CrossRef]

Chalasinska-Macukow, C.

K. Styczynski, J. Campos, M. J. Yzuel, C. Chalasinska-Macukow, “New arrangement for limited intensity pattern recognition with highdiffraction efficiency,” Opt. Commun. 118, 193–198 (1995).
[CrossRef]

Cochran, W. G.

G. W. Snedecor, W. G. Cochran, Statistical Methods (Iowa State University Press, Ames, Iowa, 1989).

Corbalán, M.

Ferreira, C.

D. Mendlovic, P. Garcı́a-Martı́nez, J. Garcı́a, C. Ferreira, “Color encoding for polychromatic single-channel optical pattern recognition,” Appl. Opt. 34, 7538–7544 (1995).
[CrossRef] [PubMed]

M. S. Millán, M. J. Yzuel, J. Campos, C. Ferreira, “Different strategies in optical recognition of polychromatic images,” Appl. Opt. 31, 2560–2567 (1992).
[CrossRef] [PubMed]

C. Ferreira, M. S. Millán, M. J. Yzuel, J. Campos, “Experimental results in color pattern recognition by multichannel matchedfiltering,” Opt. Eng. 31, 2231–2238 (1992).
[CrossRef]

M. S. Millán, J. Campos, C. Ferreira, M. J. Yzuel, “Matched filter and phase-only filter performance in colour image recognition,” Opt. Commun. 73, 277–284 (1989).
[CrossRef]

Garci´a, J.

Garci´a-Marti´nez, P.

Gianino, P. D.

Goldstein, D. H.

R. R. Kallman, D. H. Goldstein, “Phase-encoding input images for optical pattern recognition,” Opt. Eng. 33, 1806–1812 (1994).
[CrossRef]

Honda, T.

E. Badiqué, N. Ohyama, T. Honda, J. Tsujiuchi, “Color image correlation for spatial/spectral recognition and increasedselectivity,” Opt. Commun. 68, 91–96 (1988).
[CrossRef]

E. Badiqué, Y. Komiya, N. Ohyama, J. Tsujiuchi, T. Honda, “Color image correlation,” Opt. Commun. 62, 181–186 (1987).
[CrossRef]

Horner, J. L.

Kallman, R. R.

R. R. Kallman, D. H. Goldstein, “Phase-encoding input images for optical pattern recognition,” Opt. Eng. 33, 1806–1812 (1994).
[CrossRef]

Kober, V.

Komiya, Y.

E. Badiqué, Y. Komiya, N. Ohyama, J. Tsujiuchi, T. Honda, “Color image correlation,” Opt. Commun. 62, 181–186 (1987).
[CrossRef]

Lashin, V.

Ludman, J. E.

C. Warde, H. J. Caulfield, F. T. S. Yu, J. E. Ludman, “Real-time joint spectral–spatial matched filtering,” Opt. Commun. 49, 241–244 (1984).
[CrossRef]

Mazé, S.

Mendlovic, D.

Millán, M. S.

M. S. Millán, M. Corbalán, J. Romero, M. J. Yzuel, “Optical pattern recognition using color vision models,” Opt. Lett. 20, 1722–1724 (1995).
[CrossRef]

C. Ferreira, M. S. Millán, M. J. Yzuel, J. Campos, “Experimental results in color pattern recognition by multichannel matchedfiltering,” Opt. Eng. 31, 2231–2238 (1992).
[CrossRef]

M. S. Millán, M. J. Yzuel, J. Campos, C. Ferreira, “Different strategies in optical recognition of polychromatic images,” Appl. Opt. 31, 2560–2567 (1992).
[CrossRef] [PubMed]

M. S. Millán, J. Campos, C. Ferreira, M. J. Yzuel, “Matched filter and phase-only filter performance in colour image recognition,” Opt. Commun. 73, 277–284 (1989).
[CrossRef]

Miyake, Y.

K. Yamaba, Y. Miyake, “Color character recognition method based on human perception,” Opt. Eng. 32, 33–40 (1993).
[CrossRef]

Moreno, I.

Ohyama, N.

E. Badiqué, N. Ohyama, T. Honda, J. Tsujiuchi, “Color image correlation for spatial/spectral recognition and increasedselectivity,” Opt. Commun. 68, 91–96 (1988).
[CrossRef]

E. Badiqué, Y. Komiya, N. Ohyama, J. Tsujiuchi, T. Honda, “Color image correlation,” Opt. Commun. 62, 181–186 (1987).
[CrossRef]

Pratt, W. K.

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

Réfrégier, Ph.

Romero, J.

Snedecor, G. W.

G. W. Snedecor, W. G. Cochran, Statistical Methods (Iowa State University Press, Ames, Iowa, 1989).

Styczynski, K.

K. Styczynski, J. Campos, M. J. Yzuel, C. Chalasinska-Macukow, “New arrangement for limited intensity pattern recognition with highdiffraction efficiency,” Opt. Commun. 118, 193–198 (1995).
[CrossRef]

Tsujiuchi, J.

E. Badiqué, N. Ohyama, T. Honda, J. Tsujiuchi, “Color image correlation for spatial/spectral recognition and increasedselectivity,” Opt. Commun. 68, 91–96 (1988).
[CrossRef]

E. Badiqué, Y. Komiya, N. Ohyama, J. Tsujiuchi, T. Honda, “Color image correlation,” Opt. Commun. 62, 181–186 (1987).
[CrossRef]

Turon, F.

J. Campos, F. Turon, L. P. Yaroslavsky, M. J. Yzuel, “Some filters for reliable recognition and localization of objects byoptical correlators,” Int. J. Opt. Comput. 2, 341–365 (1991).

Vander-Lugt, A.

A. Vander-Lugt, “Signal detection by complex filters,” IEEE Trans. Inf. Theory IT-10, 139–145 (1965).

Warde, C.

C. Warde, H. J. Caulfield, F. T. S. Yu, J. E. Ludman, “Real-time joint spectral–spatial matched filtering,” Opt. Commun. 49, 241–244 (1984).
[CrossRef]

Yamaba, K.

K. Yamaba, Y. Miyake, “Color character recognition method based on human perception,” Opt. Eng. 32, 33–40 (1993).
[CrossRef]

Yaroslavsky, L. P.

I. Moreno, V. Kober, V. Lashin, J. Campos, L. P. Yaroslavsky, M. J. Yzuel“Optical color pattern recognition with circular component decomposition,” Opt. Lett. 21, 498–500 (1996).
[CrossRef] [PubMed]

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

J. Campos, F. Turon, L. P. Yaroslavsky, M. J. Yzuel, “Some filters for reliable recognition and localization of objects byoptical correlators,” Int. J. Opt. Comput. 2, 341–365 (1991).

L. P. Yaroslavsky, “The theory of optimal methods for localization of objects in pictures,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1993), Vol. XXXII, pp. 145–201.

Yu, F. T. S.

F. T. S. Yu, “Colour image recognition by spectral–spatial matched filtering,” Opt. Eng. 23, 690–695 (1984).
[CrossRef]

C. Warde, H. J. Caulfield, F. T. S. Yu, J. E. Ludman, “Real-time joint spectral–spatial matched filtering,” Opt. Commun. 49, 241–244 (1984).
[CrossRef]

F. T. S. Yu, White-Light Optical Signal Processing (Wiley, New York, 1985).

Yzuel, M. J.

I. Moreno, V. Kober, V. Lashin, J. Campos, L. P. Yaroslavsky, M. J. Yzuel“Optical color pattern recognition with circular component decomposition,” Opt. Lett. 21, 498–500 (1996).
[CrossRef] [PubMed]

M. S. Millán, M. Corbalán, J. Romero, M. J. Yzuel, “Optical pattern recognition using color vision models,” Opt. Lett. 20, 1722–1724 (1995).
[CrossRef]

K. Styczynski, J. Campos, M. J. Yzuel, C. Chalasinska-Macukow, “New arrangement for limited intensity pattern recognition with highdiffraction efficiency,” Opt. Commun. 118, 193–198 (1995).
[CrossRef]

M. S. Millán, M. J. Yzuel, J. Campos, C. Ferreira, “Different strategies in optical recognition of polychromatic images,” Appl. Opt. 31, 2560–2567 (1992).
[CrossRef] [PubMed]

C. Ferreira, M. S. Millán, M. J. Yzuel, J. Campos, “Experimental results in color pattern recognition by multichannel matchedfiltering,” Opt. Eng. 31, 2231–2238 (1992).
[CrossRef]

J. Campos, F. Turon, L. P. Yaroslavsky, M. J. Yzuel, “Some filters for reliable recognition and localization of objects byoptical correlators,” Int. J. Opt. Comput. 2, 341–365 (1991).

M. S. Millán, J. Campos, C. Ferreira, M. J. Yzuel, “Matched filter and phase-only filter performance in colour image recognition,” Opt. Commun. 73, 277–284 (1989).
[CrossRef]

M. J. Yzuel, J. Campos, “Colour information in optical pattern recognition,” in Current Trends in Optics, J. C. Dainty, ed. (Academic, London, 1994), pp. 209–223.

Appl. Opt.

IEEE Trans. Inf. Theory

A. Vander-Lugt, “Signal detection by complex filters,” IEEE Trans. Inf. Theory IT-10, 139–145 (1965).

Int. J. Opt. Comput.

J. Campos, F. Turon, L. P. Yaroslavsky, M. J. Yzuel, “Some filters for reliable recognition and localization of objects byoptical correlators,” Int. J. Opt. Comput. 2, 341–365 (1991).

Opt. Commun.

C. Warde, H. J. Caulfield, F. T. S. Yu, J. E. Ludman, “Real-time joint spectral–spatial matched filtering,” Opt. Commun. 49, 241–244 (1984).
[CrossRef]

E. Badiqué, Y. Komiya, N. Ohyama, J. Tsujiuchi, T. Honda, “Color image correlation,” Opt. Commun. 62, 181–186 (1987).
[CrossRef]

E. Badiqué, N. Ohyama, T. Honda, J. Tsujiuchi, “Color image correlation for spatial/spectral recognition and increasedselectivity,” Opt. Commun. 68, 91–96 (1988).
[CrossRef]

M. S. Millán, J. Campos, C. Ferreira, M. J. Yzuel, “Matched filter and phase-only filter performance in colour image recognition,” Opt. Commun. 73, 277–284 (1989).
[CrossRef]

K. Styczynski, J. Campos, M. J. Yzuel, C. Chalasinska-Macukow, “New arrangement for limited intensity pattern recognition with highdiffraction efficiency,” Opt. Commun. 118, 193–198 (1995).
[CrossRef]

Opt. Eng.

C. Ferreira, M. S. Millán, M. J. Yzuel, J. Campos, “Experimental results in color pattern recognition by multichannel matchedfiltering,” Opt. Eng. 31, 2231–2238 (1992).
[CrossRef]

K. Yamaba, Y. Miyake, “Color character recognition method based on human perception,” Opt. Eng. 32, 33–40 (1993).
[CrossRef]

F. T. S. Yu, “Colour image recognition by spectral–spatial matched filtering,” Opt. Eng. 23, 690–695 (1984).
[CrossRef]

R. R. Kallman, D. H. Goldstein, “Phase-encoding input images for optical pattern recognition,” Opt. Eng. 33, 1806–1812 (1994).
[CrossRef]

Opt. Lett.

Other

F. T. S. Yu, White-Light Optical Signal Processing (Wiley, New York, 1985).

G. W. Snedecor, W. G. Cochran, Statistical Methods (Iowa State University Press, Ames, Iowa, 1989).

L. P. Yaroslavsky, “The theory of optimal methods for localization of objects in pictures,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1993), Vol. XXXII, pp. 145–201.

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

M. J. Yzuel, J. Campos, “Colour information in optical pattern recognition,” in Current Trends in Optics, J. C. Dainty, ed. (Academic, London, 1994), pp. 209–223.

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Figures (8)

Fig. 1
Fig. 1

Optoelectronic systems for color pattern recognition with (a) addition of intensities, (b) addition of amplitudes. (R, G, B) are the primary-color signals, (T0, T1, T2) is the elementwise transformed signal, Hi is the transfer function of a linear system in the ith transformed channel, (i=0, 1, 2), Ci is the correlation output in the ith transformed channel (i=0, 1, 2), CU is the sum of the amplitudes of the correlation outputs, and OUT is the unified correlation plane.

Fig. 2
Fig. 2

Performance of elementwise transformations in terms of (a) expected value and (b) standard deviation of the transformed signal versus standard deviation of the input noise. Here and in subsequent figures, St.D. stands for standard deviation.

Fig. 3
Fig. 3

Performance of elementwise transformations in terms of (a) SNR and (b) NMSE versus SNR of the centered signal.

Fig. 4
Fig. 4

Test RGB color scene components: (a) Red, (b) Green, (c) Blue. The target is the object B4.

Fig. 5
Fig. 5

Autocorrelation peak as a function of the standard deviation of input noise obtained with (a) RGB, (b) CNT, (c) CNT–N, (d) CW, (e) CW–N, (f) QSVD, and (g) QSVD–N transformations.

Fig. 6
Fig. 6

Scheme of the optical convergent correlator with two SLM's: HWP, half-wave plate; POL, polarizer; L, lens.

Fig. 7
Fig. 7

Correlation intensity distributions obtained with a QSVD–N preprocessed color scene and with a POF. (a) Absolute-valued input signal, computer simulation; (b) absolute-valued input signal, optical experiment; (c) phase-encoded input signal, computer simulation; (d) phase-encoded input signal, optical experiment.

Fig. 8
Fig. 8

Discrimination capability as a function of the standard deviation of input noise obtained with the CMF, the POF, the OC, and (a) RGB, (b) CNT, (c) CNT–N, (d) CW, (e) CW–N, (f) QSVD, and (g) QSVD–N transformations.

Tables (4)

Tables Icon

Table 1 Correlation Peaks and DC Obtained with the CMF for the Cases of Addition of Amplitudes and Addition of Intensities

Tables Icon

Table 2 Correlation Peaks and DC Obtained with the POF for the Cases of Addition of Amplitudes and Addition of Intensities

Tables Icon

Table 3 Correlation Peaks and DC Obtained with the OC for the Cases of Addition of Amplitudes and Addition of Intensities

Tables Icon

Table 4 Recognition Results with a Threshold for DC of 0.5 and SNR1 a

Equations (59)

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

{tk(x, y)}=T[{sk(x, y)}].
Hopl(fc, fx, fy)=O*(fc, fx, fy)AV(|S(fc, fx, fy)|2),
H(fc, fx, fy)=O*(fc, fx, fy)|S(fc, fx, fy)|2,
H(fc, fx, fy)=H1(fc, fx, fy)H2(fc, fx, fy),
H1(fc, fx, fy)=1|S(fc, fx, fy)|,
H2(fc, fx, fy)=O*(fc, fx, fy)|S(fc, fx, fy)|.
Hs(fc, fx, fy)=1|S(fc, fx, fy)|,
Ho(fc, fx, fy)=1|O(fc, fx, fy)|,
dk(x, y)=13 m=02 S¯m(x, y)exp(i2πkm/3),
S¯m(x, y)=Sm(x, y)/|Sm(x, y)|,
Sm(x, y)=k=02 sk(x, y)exp(-i2πmk/3).
d0(x, y)=1+2 s0(x, y)-s1(x, y)+s2(x, y)2k=02sk2(x, y)-s0(x, y)s1(x, y)-s0(x, y)s2(x, y)-s1(x, y)s2(x, y)1/2,
d1(x, y)=1+2 s1(x, y)-s0(x, y)+s2(x, y)2k=02sk2(x, y)-s0(x, y)s1(x, y)-s0(x, y)s2(x, y)-s1(x, y)s2(x, y)1/2,
d2(x, y)=1+2 s2(x, y)-s0(x, y)+s1(x, y)2k=02sk2(x, y)-s0(x, y)s1(x, y)-s0(x, y)s2(x, y)-s1(x, y)s2(x, y)1/2,
k=02[dk(x, y)]2=1,
d0(x, y)d1(x, y)+d0(x, y)d2(x, y)
+d1(x, y)d2(x, y)=0.
s0(x, y)-s1(x, y)+s2(x, y)2,
s1(x, y)-s0(x, y)+s2(x, y)2,
s2(x, y)-s0(x, y)+s1(x, y)2.
z0(x, y)=s0(x, y)-s1(x, y)+s2(x, y)2,
z1(x, y)=s1(x, y)-s0(x, y)+s2(x, y)2,
z2(x, y)=s2(x, y)-s0(x, y)+s1(x, y)2.
q0(x, y)=s0(x, y)-s1(x, y)+s2(x, y)2k=02sk2(x, y)-s0(x, y)s1(x, y)-s0(x, y)s2(x, y)-s1(x, y)s2(x, y)1/2,
q1(x, y)=s1(x, y)-s0(x, y)+s2(x, y)2k=02sk2(x, y)-s0(x, y)s1(x, y)-s0(x, y)s2(x, y)-s1(x, y)s2(x, y)1/2,
q2(x, y)=s2(x, y)-s0(x, y)+s1(x, y)2k=02sk2(x, y)-s0(x, y)s1(x, y)-s0(x, y)s2(x, y)-s1(x, y)s2(x, y)1/2.
k=02[ek(x, y)]2=CEk=02[zk(x, y)]2,
e0(x, y)e1(x, y)+e0(x, y)e2(x, y)
+e1(x, y)e2(x, y)=0,
e0(x, y)=[s0(x, y)-s1(x, y)]+[s0(x, y)-s2(x, y)]+k=02sk2(x, y)-s0(x, y)s1(x, y)-s0(x, y)s2(x, y)-s1(x, y)s2(x, y)1/2,
e1(x, y)=[s1(x, y)-s0(x, y)]+[s1(x, y)-s2(x, y)]+k=02sk2(x, y)-s0(x, y)s1(x, y)-s0(x, y)s2(x, y)-s1(x, y)s2(x, y)1/2,
e2(x, y)=[s2(x, y)-s0(x, y)]+[s2(x, y)-s1(x, y)]+k=02sk2(x, y)-s0(x, y)s1(x, y)-s0(x, y)s2(x, y)-s1(x, y)s2(x, y)1/2.
λ(x, y)=sTs=k=02[sk(x, y)]2.
s=ψv=n=02vn(x, y)ψn(x, y),
vT={λ(x, y),0,0}.
s^k(x, y)=sk(x, y)+nk(x, y).
{t^k(x, y)}=T[{s^k(x,y)}].
SNRT[t^k(x, y)]={E[t^k(x, y)]}2var[t^k(x, y)],
NMSET[t^k(x, y)]=E[tk(x, y)-t^k(x, y)]2[tk(x, y)]2.
c^k(x, y)=ck(x, y)+bk(x, y),
ck(x, y)=sk(ξ, η)hk(x-ξ, y-η)dξdη
bk(x, y)=nk(ξ, η)hk(x-ξ, y-η)dξdη
Bk(x, y)=Rk(x, y) * hk(x, y) * hk(-x, -y).
E[OUT(x, y)]=Ek|c^k(x, y)|2=k|ck(x, y)|2+kBk(0, 0).
E[OUTZ(x, y)]=Ek|z^k(x, y) * hk(x, y)|2=k|zk(x, y) * hk(x, y)|2+kBzk(0, 0),
Bzk(x, y)=19 [4Rk(x, y)+R(k-1)mod 3(x, y)+R(k+1)mod 3(x, y)] * hk(x, y) * hk(-x, -y).
t^k(x, y)=tdk(x, y)+tnk(x, y),
E[OUTT(x, y)]=Ek|t^k(x, y) * hk(x, y)|2=k|tdk(x, y) * hk(x, y)|2+kBtk(x, y),
Btk(x, y)=E[tnk(ξ, η)tnk(α, β)]×hk(x-ξ,y-η)×hk(x-α, y-β)dξdηdαdβ
E[OUT(x, y)]=Ekc^k(x, y)2
=kck(x, y)2+kBk(0, 0).
E[OUTZ(x, y)]=Ekz^k(x, y) * hk(x, y)2
=kzk(x, y) * hk(x, y)2+kBzk(0, 0).
E[OUTT(x, y)]=Ekt^k(x, y) * hk(x, y)2=ktdk(x, y) * hk(x, y)2+Bt(x, y),
Bt(x, y)=kmE[tkn(ξ, η)tmn(α, β)]×hk(x-ξ, y-η)×hm(x-α, y-β)dξdηdαdβ
Rk(x, y)=σn2ρk(|x|+|y|),
[OUTT(x, y)¯-tu(N)σ¯/N,OUTT(x, y)¯+tn(N)σ¯/N],
DC=1-|CB|2|CO|2,
HkOC(f1, f2)=Ok*(f1, f2)|Sk(f1, f2)|2+|Ok(f1, f2)|2,

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