Abstract

The implementation of non-zero-order joint-transform correlators (JTC’) is presented. The zero-order spectra (i.e., the autocorrelation power spectra) are removed from the joint-transform power spectrum by use of phase-shifting techniques by which the output diffraction and input spatial domain can more efficiently be utilized. Applications of the phase-shifting techniques to both conventional JTC’ and phase-transformed input JTC’ (PJTC’) are discussed. Compared with the conventional JTC, the PJTC has the advantages of higher light efficiency, a better signal-to-clutter ratio, and the simplicity to realize phase shifting. We anticipate that the proposed non-zero-order JTC’ should have a significant impact on the future development of more efficient JTC’.

© 1997 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. E. Rau, “Detection of differences in real distributions,” J. Opt. Soc. Am. 56, 1490–1494 (1966).
    [CrossRef]
  2. C. S. Weaver, J. W. Goodman, “Technique for optically convolving two functions,” Appl. Opt. 5, 1248–1249 (1966).
    [CrossRef] [PubMed]
  3. F. T. S. Yu, X. J. Lu, “A real-time programmable joint transform correlator,” Opt. Commun. 52, 10–16 (1984).
    [CrossRef]
  4. A. VanderLugt, “Signal detection by complex spatial filtering,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).
  5. F. T. S. Yu, S. Jutamulia, Optical Signal Processing, Computing and Neural Networks (Wiley-Interscience, New York, 1992).
  6. S. Jutamulia, G. M. Storti, D. A. Gregory, J. C. Kirsch, “Illumination-independent high-efficiency joint transform correlator,” Appl. Opt. 30, 4173–4175 (1991).
    [CrossRef] [PubMed]
  7. C. J. Kuo, “Joint transform correlator improved by means of the frequency-selective technique,” Opt. Eng. 33, 522–527 (1994).
    [CrossRef]
  8. F. T. S. Yu, S. Jutamulia, T. W. Lin, D. A. Gregory, “Adaptive real-time pattern recognition using a liquid crystal TV based joint transform correlator,” Appl. Opt. 26, 1370–1372 (1987).
    [CrossRef] [PubMed]
  9. A. Gregory, “Time multiplexed miniature optical correlator,” (U.S. Army Missile Command, Alabama, 1988).
  10. T. J. Grycewicz, “Applying time-modulation to the joint transform correlator,” Opt. Eng. 33, 1813–1820 (1994).
    [CrossRef]
  11. D. A. Gregory, J. C. Kirsch, E. C. Tam, “Full complex modulation using liquid-crystal televisions,” Appl. Opt. 31, 163–165 (1992).
    [CrossRef] [PubMed]
  12. G. Lu, “Study of phase-encoding techniques for joint transform correlator as applied to pattern recognition and classification,” Ph.D. dissertation (Pennsylvania State University, University Park, Pa., August1996), Chap. 4.
  13. J. H. Brunning, D. R. Herriott, J. E. Gallapfer, D. P. Rosenfeld, A. D. White, D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. 13, 2693–2703 (1974).
    [CrossRef]
  14. F. T. S. Yu, G. Lu, M. Lu, D. Zhao, “Application of position encoding to a complex joint transform correlator,” Appl. Opt. 34, 1386–1388 (1995).
    [CrossRef] [PubMed]
  15. F. T. S. Yu, T. Nagata, “Binary phase-only joint transform correlator,” Microwave Opt. Technol. Lett. 2, 15–17 (1989).
    [CrossRef]
  16. W. B. Hahn, D. L. Flannery, “Design elements of binary joint transform correlation and selected optimization techniques,” Opt. Eng. 31, 896–905 (1992).
    [CrossRef]
  17. C. J. Kuo, “Theoretical expression for the correlation signal of nonlinear joint transform correlators,” Appl. Opt. 31, 6264–6271 (1992).
    [CrossRef] [PubMed]
  18. F. T. S. Yu, F. Cheng, T. Nagata, D. A. Gregory, “Effects of fringe binarization of multiobject joint transform correlation,” Appl. Opt. 28, 2988–2990 (1989).
    [CrossRef] [PubMed]
  19. W. K. Pratt, Digital Image Processing (Wiley-Interscience, New York, 1991), Chap. 12, pp. 352–354.
  20. J. L. Horner, P. D. Gianino, “Phase-only matched filtering,” Appl. Opt. 23, 812–816 (1992).
    [CrossRef]
  21. R. Juday, S. E. Monroe, D. A. Gregory, “Optical correlation with phase encoding and phase filtering,” in Spatial Light Modulators and Applications II, U. Efron, ed., Proc. SPIE825, 149–156 (1987).
    [CrossRef]
  22. C. Hester, M. Temmen, “Phase phase implementation of optical correlator,” in Hybrid Image and Signal Processing II, D. P. Casesent, A. G. Tescher, eds., Proc. SPIE1297, 207–219 (1990).
  23. R. R. Kallman, D. H. Goldstein, “Phase-encoding input image for optical pattern recognition,” Opt. Eng. 33, 1806–1812 (1994).
    [CrossRef]
  24. G. Lu, Z. Zhang, F. T. S. Yu, “Phase encoded input joint transform correlator with improved pattern discriminability,” Opt. Lett. 20, 1307–1309 (1995).
    [CrossRef] [PubMed]
  25. G. Lu, F. T. S. Yu, “The performance of phase-transformed input joint transform correlator,” Appl. Opt. 35, 304–313 (1995).
    [CrossRef]
  26. Z. Zhang, G. Lu, F. T. S. Yu, “A simple method for measuring phase modulation in LCTVs,” Opt. Eng. 33, 3018–3022 (1994).
    [CrossRef]

1995 (3)

1994 (4)

Z. Zhang, G. Lu, F. T. S. Yu, “A simple method for measuring phase modulation in LCTVs,” Opt. Eng. 33, 3018–3022 (1994).
[CrossRef]

C. J. Kuo, “Joint transform correlator improved by means of the frequency-selective technique,” Opt. Eng. 33, 522–527 (1994).
[CrossRef]

T. J. Grycewicz, “Applying time-modulation to the joint transform correlator,” Opt. Eng. 33, 1813–1820 (1994).
[CrossRef]

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

1992 (4)

1991 (1)

1989 (2)

F. T. S. Yu, F. Cheng, T. Nagata, D. A. Gregory, “Effects of fringe binarization of multiobject joint transform correlation,” Appl. Opt. 28, 2988–2990 (1989).
[CrossRef] [PubMed]

F. T. S. Yu, T. Nagata, “Binary phase-only joint transform correlator,” Microwave Opt. Technol. Lett. 2, 15–17 (1989).
[CrossRef]

1987 (1)

1984 (1)

F. T. S. Yu, X. J. Lu, “A real-time programmable joint transform correlator,” Opt. Commun. 52, 10–16 (1984).
[CrossRef]

1974 (1)

J. H. Brunning, D. R. Herriott, J. E. Gallapfer, D. P. Rosenfeld, A. D. White, D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. 13, 2693–2703 (1974).
[CrossRef]

1966 (2)

1964 (1)

A. VanderLugt, “Signal detection by complex spatial filtering,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).

Brangaccio, D. J.

J. H. Brunning, D. R. Herriott, J. E. Gallapfer, D. P. Rosenfeld, A. D. White, D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. 13, 2693–2703 (1974).
[CrossRef]

Brunning, J. H.

J. H. Brunning, D. R. Herriott, J. E. Gallapfer, D. P. Rosenfeld, A. D. White, D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. 13, 2693–2703 (1974).
[CrossRef]

Cheng, F.

F. T. S. Yu, F. Cheng, T. Nagata, D. A. Gregory, “Effects of fringe binarization of multiobject joint transform correlation,” Appl. Opt. 28, 2988–2990 (1989).
[CrossRef] [PubMed]

Flannery, D. L.

W. B. Hahn, D. L. Flannery, “Design elements of binary joint transform correlation and selected optimization techniques,” Opt. Eng. 31, 896–905 (1992).
[CrossRef]

Gallapfer, J. E.

J. H. Brunning, D. R. Herriott, J. E. Gallapfer, D. P. Rosenfeld, A. D. White, D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. 13, 2693–2703 (1974).
[CrossRef]

Gianino, P. D.

Goldstein, D. H.

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

Goodman, J. W.

Gregory, A.

A. Gregory, “Time multiplexed miniature optical correlator,” (U.S. Army Missile Command, Alabama, 1988).

Gregory, D. A.

Grycewicz, T. J.

T. J. Grycewicz, “Applying time-modulation to the joint transform correlator,” Opt. Eng. 33, 1813–1820 (1994).
[CrossRef]

Hahn, W. B.

W. B. Hahn, D. L. Flannery, “Design elements of binary joint transform correlation and selected optimization techniques,” Opt. Eng. 31, 896–905 (1992).
[CrossRef]

Herriott, D. R.

J. H. Brunning, D. R. Herriott, J. E. Gallapfer, D. P. Rosenfeld, A. D. White, D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. 13, 2693–2703 (1974).
[CrossRef]

Hester, C.

C. Hester, M. Temmen, “Phase phase implementation of optical correlator,” in Hybrid Image and Signal Processing II, D. P. Casesent, A. G. Tescher, eds., Proc. SPIE1297, 207–219 (1990).

Horner, J. L.

Juday, R.

R. Juday, S. E. Monroe, D. A. Gregory, “Optical correlation with phase encoding and phase filtering,” in Spatial Light Modulators and Applications II, U. Efron, ed., Proc. SPIE825, 149–156 (1987).
[CrossRef]

Jutamulia, S.

Kallman, R. R.

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

Kirsch, J. C.

Kuo, C. J.

C. J. Kuo, “Joint transform correlator improved by means of the frequency-selective technique,” Opt. Eng. 33, 522–527 (1994).
[CrossRef]

C. J. Kuo, “Theoretical expression for the correlation signal of nonlinear joint transform correlators,” Appl. Opt. 31, 6264–6271 (1992).
[CrossRef] [PubMed]

Lin, T. W.

Lu, G.

F. T. S. Yu, G. Lu, M. Lu, D. Zhao, “Application of position encoding to a complex joint transform correlator,” Appl. Opt. 34, 1386–1388 (1995).
[CrossRef] [PubMed]

G. Lu, Z. Zhang, F. T. S. Yu, “Phase encoded input joint transform correlator with improved pattern discriminability,” Opt. Lett. 20, 1307–1309 (1995).
[CrossRef] [PubMed]

G. Lu, F. T. S. Yu, “The performance of phase-transformed input joint transform correlator,” Appl. Opt. 35, 304–313 (1995).
[CrossRef]

Z. Zhang, G. Lu, F. T. S. Yu, “A simple method for measuring phase modulation in LCTVs,” Opt. Eng. 33, 3018–3022 (1994).
[CrossRef]

G. Lu, “Study of phase-encoding techniques for joint transform correlator as applied to pattern recognition and classification,” Ph.D. dissertation (Pennsylvania State University, University Park, Pa., August1996), Chap. 4.

Lu, M.

F. T. S. Yu, G. Lu, M. Lu, D. Zhao, “Application of position encoding to a complex joint transform correlator,” Appl. Opt. 34, 1386–1388 (1995).
[CrossRef] [PubMed]

Lu, X. J.

F. T. S. Yu, X. J. Lu, “A real-time programmable joint transform correlator,” Opt. Commun. 52, 10–16 (1984).
[CrossRef]

Monroe, S. E.

R. Juday, S. E. Monroe, D. A. Gregory, “Optical correlation with phase encoding and phase filtering,” in Spatial Light Modulators and Applications II, U. Efron, ed., Proc. SPIE825, 149–156 (1987).
[CrossRef]

Nagata, T.

F. T. S. Yu, F. Cheng, T. Nagata, D. A. Gregory, “Effects of fringe binarization of multiobject joint transform correlation,” Appl. Opt. 28, 2988–2990 (1989).
[CrossRef] [PubMed]

F. T. S. Yu, T. Nagata, “Binary phase-only joint transform correlator,” Microwave Opt. Technol. Lett. 2, 15–17 (1989).
[CrossRef]

Pratt, W. K.

W. K. Pratt, Digital Image Processing (Wiley-Interscience, New York, 1991), Chap. 12, pp. 352–354.

Rau, J. E.

Rosenfeld, D. P.

J. H. Brunning, D. R. Herriott, J. E. Gallapfer, D. P. Rosenfeld, A. D. White, D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. 13, 2693–2703 (1974).
[CrossRef]

Storti, G. M.

Tam, E. C.

Temmen, M.

C. Hester, M. Temmen, “Phase phase implementation of optical correlator,” in Hybrid Image and Signal Processing II, D. P. Casesent, A. G. Tescher, eds., Proc. SPIE1297, 207–219 (1990).

VanderLugt, A.

A. VanderLugt, “Signal detection by complex spatial filtering,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).

Weaver, C. S.

White, A. D.

J. H. Brunning, D. R. Herriott, J. E. Gallapfer, D. P. Rosenfeld, A. D. White, D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. 13, 2693–2703 (1974).
[CrossRef]

Yu, F. T. S.

F. T. S. Yu, G. Lu, M. Lu, D. Zhao, “Application of position encoding to a complex joint transform correlator,” Appl. Opt. 34, 1386–1388 (1995).
[CrossRef] [PubMed]

G. Lu, F. T. S. Yu, “The performance of phase-transformed input joint transform correlator,” Appl. Opt. 35, 304–313 (1995).
[CrossRef]

G. Lu, Z. Zhang, F. T. S. Yu, “Phase encoded input joint transform correlator with improved pattern discriminability,” Opt. Lett. 20, 1307–1309 (1995).
[CrossRef] [PubMed]

Z. Zhang, G. Lu, F. T. S. Yu, “A simple method for measuring phase modulation in LCTVs,” Opt. Eng. 33, 3018–3022 (1994).
[CrossRef]

F. T. S. Yu, F. Cheng, T. Nagata, D. A. Gregory, “Effects of fringe binarization of multiobject joint transform correlation,” Appl. Opt. 28, 2988–2990 (1989).
[CrossRef] [PubMed]

F. T. S. Yu, T. Nagata, “Binary phase-only joint transform correlator,” Microwave Opt. Technol. Lett. 2, 15–17 (1989).
[CrossRef]

F. T. S. Yu, S. Jutamulia, T. W. Lin, D. A. Gregory, “Adaptive real-time pattern recognition using a liquid crystal TV based joint transform correlator,” Appl. Opt. 26, 1370–1372 (1987).
[CrossRef] [PubMed]

F. T. S. Yu, X. J. Lu, “A real-time programmable joint transform correlator,” Opt. Commun. 52, 10–16 (1984).
[CrossRef]

F. T. S. Yu, S. Jutamulia, Optical Signal Processing, Computing and Neural Networks (Wiley-Interscience, New York, 1992).

Zhang, Z.

G. Lu, Z. Zhang, F. T. S. Yu, “Phase encoded input joint transform correlator with improved pattern discriminability,” Opt. Lett. 20, 1307–1309 (1995).
[CrossRef] [PubMed]

Z. Zhang, G. Lu, F. T. S. Yu, “A simple method for measuring phase modulation in LCTVs,” Opt. Eng. 33, 3018–3022 (1994).
[CrossRef]

Zhao, D.

F. T. S. Yu, G. Lu, M. Lu, D. Zhao, “Application of position encoding to a complex joint transform correlator,” Appl. Opt. 34, 1386–1388 (1995).
[CrossRef] [PubMed]

Appl. Opt. (3)

J. H. Brunning, D. R. Herriott, J. E. Gallapfer, D. P. Rosenfeld, A. D. White, D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. 13, 2693–2703 (1974).
[CrossRef]

F. T. S. Yu, G. Lu, M. Lu, D. Zhao, “Application of position encoding to a complex joint transform correlator,” Appl. Opt. 34, 1386–1388 (1995).
[CrossRef] [PubMed]

F. T. S. Yu, F. Cheng, T. Nagata, D. A. Gregory, “Effects of fringe binarization of multiobject joint transform correlation,” Appl. Opt. 28, 2988–2990 (1989).
[CrossRef] [PubMed]

Appl. Opt. (7)

IEEE Trans. Inf. Theory (1)

A. VanderLugt, “Signal detection by complex spatial filtering,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).

J. Opt. Soc. Am. (1)

Microwave Opt. Technol. Lett. (1)

F. T. S. Yu, T. Nagata, “Binary phase-only joint transform correlator,” Microwave Opt. Technol. Lett. 2, 15–17 (1989).
[CrossRef]

Opt. Eng. (1)

C. J. Kuo, “Joint transform correlator improved by means of the frequency-selective technique,” Opt. Eng. 33, 522–527 (1994).
[CrossRef]

Opt. Commun. (1)

F. T. S. Yu, X. J. Lu, “A real-time programmable joint transform correlator,” Opt. Commun. 52, 10–16 (1984).
[CrossRef]

Opt. Eng. (4)

W. B. Hahn, D. L. Flannery, “Design elements of binary joint transform correlation and selected optimization techniques,” Opt. Eng. 31, 896–905 (1992).
[CrossRef]

T. J. Grycewicz, “Applying time-modulation to the joint transform correlator,” Opt. Eng. 33, 1813–1820 (1994).
[CrossRef]

Z. Zhang, G. Lu, F. T. S. Yu, “A simple method for measuring phase modulation in LCTVs,” Opt. Eng. 33, 3018–3022 (1994).
[CrossRef]

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

Opt. Lett. (1)

Other (6)

R. Juday, S. E. Monroe, D. A. Gregory, “Optical correlation with phase encoding and phase filtering,” in Spatial Light Modulators and Applications II, U. Efron, ed., Proc. SPIE825, 149–156 (1987).
[CrossRef]

C. Hester, M. Temmen, “Phase phase implementation of optical correlator,” in Hybrid Image and Signal Processing II, D. P. Casesent, A. G. Tescher, eds., Proc. SPIE1297, 207–219 (1990).

W. K. Pratt, Digital Image Processing (Wiley-Interscience, New York, 1991), Chap. 12, pp. 352–354.

G. Lu, “Study of phase-encoding techniques for joint transform correlator as applied to pattern recognition and classification,” Ph.D. dissertation (Pennsylvania State University, University Park, Pa., August1996), Chap. 4.

F. T. S. Yu, S. Jutamulia, Optical Signal Processing, Computing and Neural Networks (Wiley-Interscience, New York, 1992).

A. Gregory, “Time multiplexed miniature optical correlator,” (U.S. Army Missile Command, Alabama, 1988).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (18)

Fig. 1
Fig. 1

Schematic diagrams of (a) the alignment of the input and reference functions, f(x, y) and r(x, y), respectively, in a conventional JTC and (b) the output correlation distribution from a conventional JTC.

Fig. 2
Fig. 2

Schematic diagrams of (a) the alignment of the input and reference functions under the assumption that the dc spectra in Eq. (1) are removed. (b) The corresponding output correlation distribution obtained from (a). (c) The output correlation distribution obtained from θ F (p, q) - θ R (p, q).

Fig. 3
Fig. 3

JTC architecture for the two-step phase-shifting technique.

Fig. 4
Fig. 4

JTC architecture for the three-step phase-shifting technique. AMSLM, amplitude-modulated SLM; PMSLM, phase-modulated SLM.

Fig. 5
Fig. 5

Input scene with a reference object for the NJTC simulations.

Fig. 6
Fig. 6

(a) JTPS corresponding to the input scene of Fig. 5. (b) ADJ(p, q) with c = 1, obtained with the two-step phase-shifting technique.

Fig. 7
Fig. 7

Output correlation distributions (a) obtained from Fig. 6(a) and (b) obtained from Fig. 6(b). The 16-pixel-wide center zone at each output is blocked and set to zero (also applied to other output correlation distributions).

Fig. 8
Fig. 8

CPI and the SCR plotted as functions of the normalization factor c.

Fig. 9
Fig. 9

(a) BDJ (p, q) of Fig. 5, as obtained from Eq. (11); (b) unipolar BDJ (p, q) and (c) bipolar BDJ (p, q) output correlations.

Fig. 10
Fig. 10

(a) Retrieved phase difference θ F (p, q) - θ R (p, q) + 2bp obtained by use of the three-step phase-shifting method. (b) Output correlation distribution obtained from (a). (c) Phase difference with the linear phase term 2bp removed. (d) Output correlation distribution obtained from (c).

Fig. 11
Fig. 11

Input for the simulation study of the NPJTC.

Fig. 12
Fig. 12

(a) JTPS of Fig. 11: (b) Bipolar BDJ (p, q) obtained from the two-step phase-shifting method. (c) Retrieved phase difference θ PFB (p, q) - θ PR (p, q) + 2bp by use of the three-step phase-shifting technique.

Fig. 13
Fig. 13

Output correlation distributions (a) obtained from Fig. 12(b) and (b) obtained from Fig. 12(c).

Fig. 14
Fig. 14

(a) Input for the simulation study of the NPJTC with a binarized reference. (b) Output correlation distribution obtained from the bipolar BDJ (p, q) of (a) obtained by use of the two-step phase-shifting technique.

Fig. 15
Fig. 15

Inputs for the NJTC experiments by use of the two-step phase-shifting method with (a) the original reference object and (b) the π-phase-shifted reference object.

Fig. 16
Fig. 16

JTPS’s of (a) Fig. 15(a) and (b) Fig. 15(b) and output correlations obtained (c) from (a) and (d) from (b). Notice that (c) and (d) are essentially identical. (e) ADJ (p, q) obtained from (a) and (b). (f) Unipolar BDJ (p, q) obtained from (e). Output correlations obtained (g) from (e) and (h) from (f).

Fig. 17
Fig. 17

Inputs for the NPJTC experiments by use of the two-step phase-shifting method with (a) the original reference object and (b) the π-phase-shifted reference object.

Fig. 18
Fig. 18

JTPS’s of (a) Fig. 17(a) and (b) Fig. 17(b). Output correlations obtained (c) from Fig. 17(a) and (d) from Fig. 17(b). (e) Bipolar BDJ (p, q) obtained from (a) and (b). (f) Output correlation distribution obtained from (e).

Equations (42)

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

J p ,   q = F p ,   q 2 + R p ,   q 2 + F p ,   q R * p ,   q exp 2 jbp + F * p ,   q R p ,   q exp - 2 jbp ,
b 3 l f + l r 2 ,
J 0 p ,   q = F p ,   q 2 + R p ,   q 2 + F p ,   q R * p ,   q exp 2 jbp + F * p ,   q R p ,   q exp - 2 jbp .
J π p ,   q = F p ,   q 2 + R p ,   q 2 - F p ,   q R * p ,   q exp 2 jbp - F * p ,   q R p ,   q exp - 2 jbp .
DJ p ,   q = J 0 p ,   q - J π p ,   q = 2 F p ,   q R * p ,   q exp 2 jbp + 2 F * p ,   q R p ,   q exp - 2 jbp = 4 F p ,   q R p ,   q cos 2 bp + θ F p ,   q - θ R p ,   q ,
j 0 x ,   y = G / 2 + f x + b ,   y + r x - b ,   y 0 , j π x ,   y = G / 2 + f x + b ,   y - r x - b ,   y 0 .
DJ p ,   q = J 0 p ,   q - J π p ,   q = 4 F p ,   q R p ,   q cos 2 bp + θ F p ,   q - θ R p ,   q + 2 GA δ p ,   q R p ,   q cos bp - θ R p ,   q ,
δ p ,   q 1 p = 0 ,   q = 0 0 others .
ADJ p ,   q = cDJ p ,   q + G / 2 0 ,   for all p ,   q ,
C α ,   β cf α + 2 b ,   β r α ,   β + cf - α + 2 b , - β r - α , - β ) + δ α ,   β G / 2 .
BDJ p ,   q = 1 DJ p ,   q > 0 0   or - 1 DJ p ,   q 0 .
J 0 p ,   q = F p ,   q 2 + R p ,   q 2 + 2 F p ,   q R p ,   q cos 2 bp + θ F p ,   q - θ R p ,   q .
J 2 π / 3 p ,   q = F p ,   q 2 + R p ,   q 2 + 2 F p ,   q R p ,   q cos 2 bp + θ F p ,   q - θ R p ,   q + 2 π / 3 .
J 4 π / 3 p ,   q = F p ,   q 2 + R p ,   q 2 + 2 F p ,   q R p ,   q cos 2 bp + θ F p ,   q - θ R p ,   q + 4 π / 3 .
θ F p ,   q - θ R p ,   q = tan - 1 3   J 4 π / 3 p ,   q - J 2 π / 3 p ,   q 2 J 0 p ,   q - J 4 π / 3 p ,   q - J 2 π / 3 p ,   q - 2 bp .
PJ p ,   q = exp j θ F p ,   q - j θ R p ,   q ,
f x ,   y = r x ,   y + n x ,   y ,
exp j θ F p ,   q = R p ,   q F p ,   q exp j θ R p ,   q + N p ,   q F p ,   q exp j θ N p ,   q ,
PJ p ,   q = R p ,   q F p ,   q + N p ,   q F p ,   q × exp j θ N p ,   q - θ R p ,   q ,
ADJ p ,   q = O cDJ p ,   q < - 128 cDJ p ,   q + 128 - 128 cDJ p ,   q 127 255 cDJ p ,   q > 127 .
pf x ,   y = exp jT f x ,   y ,
T f x ,   y = f x ,   y - G min G max - G min π ,
w f x ,   y = 1 x ,   y f x ,   y 0 x ,   y f x ,   y     an object ,
w r x ,   y = 1 x ,   y r x ,   y 0 x ,   y r x ,   y     the reference ,
w b x ,   y = 1 - w f x + b ,   y - w r x - b ,   y .
j x ,   y = pf x + b ,   y + pr x - b ,   y + pb x ,   y w b x ,   y ,
pf x ,   y = exp jT f x ,   y w f x ,   y , pr x ,   y = exp j ϕ r x ,   y w r x ,   y , pb x ,   y = exp j π x + y .
j 0 x ,   y = pf x + b ,   y + pr x - b ,   y + pb x ,   y w b x ,   y
J 0 p ,   q = PF p ,   q 2 + PR p ,   q 2 + PB p ,   q 2 + 2 PF p ,   q PR p ,   q cos 2 bp + θ PF p ,   q - θ PR p ,   q + 2 PR p ,   q W b p - π , × q - π cos bp + θ b p ,   q - θ PR p ,   q + 2 PF p ,   q W b p - π , q - π × cos bp + θ PF p ,   q - θ b p ,   q ,
j π x ,   y = pf x + b ,   y - pr x - b ,   y + pb x ,   y w b x ,   y
J π p ,   q = PF p ,   q 2 + PR p ,   q 2 + PB p ,   q 2 - 2 PF p ,   q PR p ,   q cos 2 bp + θ PF p ,   q - θ PR p ,   q - 2 PR p ,   q W b p - π ,   q - π cos bp + θ b p ,   q - θ PR p ,   q + 2 PF p ,   q W b p - π ,   q - π cos bp + θ PF p ,   q - θ b p ,   q .
DJ p ,   q = J 0 p ,   q - J π p ,   q = 4 PF p ,   q PR p ,   q cos 2 pb + θ PF p ,   q - θ PR p ,   q + 4 PR p ,   q W b p - π , × q - π cos bp + θ b p ,   q - θ PR p ,   q .
DJ p ,   q 4 PF p ,   q PR p ,   q × cos 2 pb + θ PF p ,   q - θ PR p ,   q ,
pfb x + b ,   y = pf x + b ,   y + pb x ,   y w b x ,   y ,
j 0 x ,   y = pfb x + b ,   y + pr x - b ,   y
J 0 p ,   q = PFB p ,   q 2 + PR p ,   q 2 + 2 PFB p ,   q PR p ,   q cos 2 bp + θ PFB p ,   q - θ PR p ,   q ,
j 2 π / 3 x ,   y = pfb x + b ,   y + pr x - b ,   y exp j 2 π / 3
J 2 π / 3 p ,   q = PFB p ,   q 2 + PR p ,   q 2 + 2 PFB p ,   q PR p ,   q cos 2 bp + θ PFB p ,   q - θ PR p ,   q + 2 π / 3 .
j 4 π / 3 x ,   y = pfb x + b ,   y + pr x - b ,   y exp j 4 π / 3
J 4 π / 3 p ,   q = PFB p ,   q 2 + PR p ,   q 2 + 2 PFB p ,   q PR p ,   q cos 2 bp + θ PFB p ,   q - θ PR p ,   q + 4 π / 3 .
θ PFB p ,   q - θ PR p ,   q = tan - 1 3   J 4 π / 3 p ,   q - J 2 π / 3 p ,   q 2 J 0 p ,   q - J 4 π / 3 p ,   q - J 2 π / 3 p ,   q - 2 bp .
PJ p ,   q = exp j θ PFB p ,   q - j θ PR p ,   q

Metrics