Abstract

An image rotation and translation measurement technology based on a double phase-encoded joint transform correlator (DPEJTC) is proposed. The reference and the target images are Fourier transformed. Then the magnitude of the Fourier-transformed reference (MFR) and target (MFT) images are multiplied with a high-pass emphasis filter and transformed from Cartesian space into polar space. Rotation between the reference and the target image is obtained by measuring the emphasized MFR and MFT in polar coordinates by the DPEJTC. The target image is rotated by the rotation angle in the inverse orientation to get the rotation-correction target image. Finally, translation between the reference and the target image is obtained through measuring the reference and the rotation-correction target image by the DPEJTC. Results based on digital computation are given to verify our proposal. A possible optical setup is suggested.

© 2011 Optical Society of America

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References

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  1. K. Janschek, V. Tchernykh, S. Dyblenko, and B. Harnisch, “Compensation of the attitude instability effect on the imaging payload performance with optical correlators,” Acta Astronaut. 52, 965–974 (2003).
    [CrossRef]
  2. K. Janschek, V. Tchernykh, S. Dyblenko, G. Flandin, and B. Harnisch, “Compensation of focal plane image motion perturbations with optical correlator in feedback loop,” Proc. SPIE 5570, 280–288 (2004).
    [CrossRef]
  3. K. Janschek, V. Tchernykh, and S. Dyblenko, “Performance analysis of optomechatronic image stabilization for a compact space camera,” Control Eng. Pract. 15, 333–347 (2007).
    [CrossRef]
  4. M. Sambora and R. K. Martin, “Exploiting correlations in projection-based image registration,” Opt. Eng. 47, 077005(2008).
    [CrossRef]
  5. K. Takita, T. Aoki, Y. Sasaki, T. Higuchi, and K. Kobayashi, “High-accuracy subpixel image registration based on phase-only correlation,” IEICE Trans. Fundam. Electron. Commun. Comput. Sci. E86-A, 1925–1934 (2003).
    [CrossRef]
  6. S. L. Keeling and W. Ring, “Medical image registration and interpolation by optical flow with maximal rigidity,” J. Math. Imaging Vision 23, 47–65 (2005).
    [CrossRef]
  7. Y. Nie and K. K. Ma, “Adaptive rood pattern search for fast block-matching motion estimation,” IEEE Trans. Image Process. 11, 1442–1449 (2002).
    [CrossRef]
  8. K. Janschek, T. Boge, S. Dyblenko, and V. Tchernykh, “Image-based attitude determination using an optical correlator,” Proceedings of the 4th ESA International Conference (European Space Agency, 2000), pp. 487–492.
  9. V. Tchernykh, S. Dyblenko, K. Janschek, and B. Harnisch, “Optical correlator-based system for the real-time analysis of image motion in the focal plane of an Earth observation camera,” Proc. SPIE 4113, 23–31 (2000).
    [CrossRef]
  10. S. Dyblenko, K. Janschek, A. Kisselev, A. Sultanov, and V. Tchernykh, “Simulation of satellite landmark navigation on the base of optoelectronic image processing technique,” Proc. SPIE 5854, 178–186 (2005).
    [CrossRef]
  11. E. De Castro and C. Morandi, “Registration of translated and rotated images using finite Fourier transforms,” IEEE Trans. Pattern Anal. Machine Intell. PAMI-9, 700–703 (1987).
    [CrossRef]
  12. B. S. Reddy and B. N. Chatterji, “An FFT-based technique for translation, rotation, and scale-invariant image registration,” IEEE Trans. Image Process. 5, 1266–1271 (1996).
    [CrossRef] [PubMed]
  13. P. E. Zwicke and I. Kiss, “A new implementation of the Mellin transform and its application to radar classification of ships,” IEEE Trans. Pattern Anal. Machine Intell. PAMI-5, 191–199(1983).
    [CrossRef]
  14. Y. Sheng and J. Duvernoy, “Circular-Fourier-Radial-Mellin transform descriptors for pattern recognition,” J. Opt. Soc. Am. A 3, 885–888 (1986).
    [CrossRef] [PubMed]
  15. Y. Sheng and H. H. Arsenault, “Experiments on pattern recognition using invariant Fourier–Mellin descriptors,” J. Opt. Soc. Am. A 3, 771–776 (1986).
    [CrossRef] [PubMed]
  16. P. Réfrégier and B. Javidi, “Optical image encryption based on input plane and Fourier plane random encoding,” Opt. Lett. 20, 767–769 (1995).
    [CrossRef] [PubMed]
  17. M. S. Alam, “Fractional power fringe-adjusted joint transform correlation,” Opt. Eng. 34, 3208–3216 (1995).
    [CrossRef]
  18. T. M. Lehmann, C. Gonner, and K. Spitzer, “Survey: interpolation methods in medical image processing,” IEEE Trans. Med. Imaging 18, 1049–1075 (1999).
    [CrossRef]

2008 (1)

M. Sambora and R. K. Martin, “Exploiting correlations in projection-based image registration,” Opt. Eng. 47, 077005(2008).
[CrossRef]

2007 (1)

K. Janschek, V. Tchernykh, and S. Dyblenko, “Performance analysis of optomechatronic image stabilization for a compact space camera,” Control Eng. Pract. 15, 333–347 (2007).
[CrossRef]

2005 (2)

S. L. Keeling and W. Ring, “Medical image registration and interpolation by optical flow with maximal rigidity,” J. Math. Imaging Vision 23, 47–65 (2005).
[CrossRef]

S. Dyblenko, K. Janschek, A. Kisselev, A. Sultanov, and V. Tchernykh, “Simulation of satellite landmark navigation on the base of optoelectronic image processing technique,” Proc. SPIE 5854, 178–186 (2005).
[CrossRef]

2004 (1)

K. Janschek, V. Tchernykh, S. Dyblenko, G. Flandin, and B. Harnisch, “Compensation of focal plane image motion perturbations with optical correlator in feedback loop,” Proc. SPIE 5570, 280–288 (2004).
[CrossRef]

2003 (2)

K. Janschek, V. Tchernykh, S. Dyblenko, and B. Harnisch, “Compensation of the attitude instability effect on the imaging payload performance with optical correlators,” Acta Astronaut. 52, 965–974 (2003).
[CrossRef]

K. Takita, T. Aoki, Y. Sasaki, T. Higuchi, and K. Kobayashi, “High-accuracy subpixel image registration based on phase-only correlation,” IEICE Trans. Fundam. Electron. Commun. Comput. Sci. E86-A, 1925–1934 (2003).
[CrossRef]

2002 (1)

Y. Nie and K. K. Ma, “Adaptive rood pattern search for fast block-matching motion estimation,” IEEE Trans. Image Process. 11, 1442–1449 (2002).
[CrossRef]

2000 (1)

V. Tchernykh, S. Dyblenko, K. Janschek, and B. Harnisch, “Optical correlator-based system for the real-time analysis of image motion in the focal plane of an Earth observation camera,” Proc. SPIE 4113, 23–31 (2000).
[CrossRef]

1999 (1)

T. M. Lehmann, C. Gonner, and K. Spitzer, “Survey: interpolation methods in medical image processing,” IEEE Trans. Med. Imaging 18, 1049–1075 (1999).
[CrossRef]

1996 (1)

B. S. Reddy and B. N. Chatterji, “An FFT-based technique for translation, rotation, and scale-invariant image registration,” IEEE Trans. Image Process. 5, 1266–1271 (1996).
[CrossRef] [PubMed]

1995 (2)

1987 (1)

E. De Castro and C. Morandi, “Registration of translated and rotated images using finite Fourier transforms,” IEEE Trans. Pattern Anal. Machine Intell. PAMI-9, 700–703 (1987).
[CrossRef]

1986 (2)

1983 (1)

P. E. Zwicke and I. Kiss, “A new implementation of the Mellin transform and its application to radar classification of ships,” IEEE Trans. Pattern Anal. Machine Intell. PAMI-5, 191–199(1983).
[CrossRef]

Alam, M. S.

M. S. Alam, “Fractional power fringe-adjusted joint transform correlation,” Opt. Eng. 34, 3208–3216 (1995).
[CrossRef]

Aoki, T.

K. Takita, T. Aoki, Y. Sasaki, T. Higuchi, and K. Kobayashi, “High-accuracy subpixel image registration based on phase-only correlation,” IEICE Trans. Fundam. Electron. Commun. Comput. Sci. E86-A, 1925–1934 (2003).
[CrossRef]

Arsenault, H. H.

Boge, T.

K. Janschek, T. Boge, S. Dyblenko, and V. Tchernykh, “Image-based attitude determination using an optical correlator,” Proceedings of the 4th ESA International Conference (European Space Agency, 2000), pp. 487–492.

Chatterji, B. N.

B. S. Reddy and B. N. Chatterji, “An FFT-based technique for translation, rotation, and scale-invariant image registration,” IEEE Trans. Image Process. 5, 1266–1271 (1996).
[CrossRef] [PubMed]

De Castro, E.

E. De Castro and C. Morandi, “Registration of translated and rotated images using finite Fourier transforms,” IEEE Trans. Pattern Anal. Machine Intell. PAMI-9, 700–703 (1987).
[CrossRef]

Duvernoy, J.

Dyblenko, S.

K. Janschek, V. Tchernykh, and S. Dyblenko, “Performance analysis of optomechatronic image stabilization for a compact space camera,” Control Eng. Pract. 15, 333–347 (2007).
[CrossRef]

S. Dyblenko, K. Janschek, A. Kisselev, A. Sultanov, and V. Tchernykh, “Simulation of satellite landmark navigation on the base of optoelectronic image processing technique,” Proc. SPIE 5854, 178–186 (2005).
[CrossRef]

K. Janschek, V. Tchernykh, S. Dyblenko, G. Flandin, and B. Harnisch, “Compensation of focal plane image motion perturbations with optical correlator in feedback loop,” Proc. SPIE 5570, 280–288 (2004).
[CrossRef]

K. Janschek, V. Tchernykh, S. Dyblenko, and B. Harnisch, “Compensation of the attitude instability effect on the imaging payload performance with optical correlators,” Acta Astronaut. 52, 965–974 (2003).
[CrossRef]

V. Tchernykh, S. Dyblenko, K. Janschek, and B. Harnisch, “Optical correlator-based system for the real-time analysis of image motion in the focal plane of an Earth observation camera,” Proc. SPIE 4113, 23–31 (2000).
[CrossRef]

K. Janschek, T. Boge, S. Dyblenko, and V. Tchernykh, “Image-based attitude determination using an optical correlator,” Proceedings of the 4th ESA International Conference (European Space Agency, 2000), pp. 487–492.

Flandin, G.

K. Janschek, V. Tchernykh, S. Dyblenko, G. Flandin, and B. Harnisch, “Compensation of focal plane image motion perturbations with optical correlator in feedback loop,” Proc. SPIE 5570, 280–288 (2004).
[CrossRef]

Gonner, C.

T. M. Lehmann, C. Gonner, and K. Spitzer, “Survey: interpolation methods in medical image processing,” IEEE Trans. Med. Imaging 18, 1049–1075 (1999).
[CrossRef]

Harnisch, B.

K. Janschek, V. Tchernykh, S. Dyblenko, G. Flandin, and B. Harnisch, “Compensation of focal plane image motion perturbations with optical correlator in feedback loop,” Proc. SPIE 5570, 280–288 (2004).
[CrossRef]

K. Janschek, V. Tchernykh, S. Dyblenko, and B. Harnisch, “Compensation of the attitude instability effect on the imaging payload performance with optical correlators,” Acta Astronaut. 52, 965–974 (2003).
[CrossRef]

V. Tchernykh, S. Dyblenko, K. Janschek, and B. Harnisch, “Optical correlator-based system for the real-time analysis of image motion in the focal plane of an Earth observation camera,” Proc. SPIE 4113, 23–31 (2000).
[CrossRef]

Higuchi, T.

K. Takita, T. Aoki, Y. Sasaki, T. Higuchi, and K. Kobayashi, “High-accuracy subpixel image registration based on phase-only correlation,” IEICE Trans. Fundam. Electron. Commun. Comput. Sci. E86-A, 1925–1934 (2003).
[CrossRef]

Janschek, K.

K. Janschek, V. Tchernykh, and S. Dyblenko, “Performance analysis of optomechatronic image stabilization for a compact space camera,” Control Eng. Pract. 15, 333–347 (2007).
[CrossRef]

S. Dyblenko, K. Janschek, A. Kisselev, A. Sultanov, and V. Tchernykh, “Simulation of satellite landmark navigation on the base of optoelectronic image processing technique,” Proc. SPIE 5854, 178–186 (2005).
[CrossRef]

K. Janschek, V. Tchernykh, S. Dyblenko, G. Flandin, and B. Harnisch, “Compensation of focal plane image motion perturbations with optical correlator in feedback loop,” Proc. SPIE 5570, 280–288 (2004).
[CrossRef]

K. Janschek, V. Tchernykh, S. Dyblenko, and B. Harnisch, “Compensation of the attitude instability effect on the imaging payload performance with optical correlators,” Acta Astronaut. 52, 965–974 (2003).
[CrossRef]

V. Tchernykh, S. Dyblenko, K. Janschek, and B. Harnisch, “Optical correlator-based system for the real-time analysis of image motion in the focal plane of an Earth observation camera,” Proc. SPIE 4113, 23–31 (2000).
[CrossRef]

K. Janschek, T. Boge, S. Dyblenko, and V. Tchernykh, “Image-based attitude determination using an optical correlator,” Proceedings of the 4th ESA International Conference (European Space Agency, 2000), pp. 487–492.

Javidi, B.

Keeling, S. L.

S. L. Keeling and W. Ring, “Medical image registration and interpolation by optical flow with maximal rigidity,” J. Math. Imaging Vision 23, 47–65 (2005).
[CrossRef]

Kiss, I.

P. E. Zwicke and I. Kiss, “A new implementation of the Mellin transform and its application to radar classification of ships,” IEEE Trans. Pattern Anal. Machine Intell. PAMI-5, 191–199(1983).
[CrossRef]

Kisselev, A.

S. Dyblenko, K. Janschek, A. Kisselev, A. Sultanov, and V. Tchernykh, “Simulation of satellite landmark navigation on the base of optoelectronic image processing technique,” Proc. SPIE 5854, 178–186 (2005).
[CrossRef]

Kobayashi, K.

K. Takita, T. Aoki, Y. Sasaki, T. Higuchi, and K. Kobayashi, “High-accuracy subpixel image registration based on phase-only correlation,” IEICE Trans. Fundam. Electron. Commun. Comput. Sci. E86-A, 1925–1934 (2003).
[CrossRef]

Lehmann, T. M.

T. M. Lehmann, C. Gonner, and K. Spitzer, “Survey: interpolation methods in medical image processing,” IEEE Trans. Med. Imaging 18, 1049–1075 (1999).
[CrossRef]

Ma, K. K.

Y. Nie and K. K. Ma, “Adaptive rood pattern search for fast block-matching motion estimation,” IEEE Trans. Image Process. 11, 1442–1449 (2002).
[CrossRef]

Martin, R. K.

M. Sambora and R. K. Martin, “Exploiting correlations in projection-based image registration,” Opt. Eng. 47, 077005(2008).
[CrossRef]

Morandi, C.

E. De Castro and C. Morandi, “Registration of translated and rotated images using finite Fourier transforms,” IEEE Trans. Pattern Anal. Machine Intell. PAMI-9, 700–703 (1987).
[CrossRef]

Nie, Y.

Y. Nie and K. K. Ma, “Adaptive rood pattern search for fast block-matching motion estimation,” IEEE Trans. Image Process. 11, 1442–1449 (2002).
[CrossRef]

Reddy, B. S.

B. S. Reddy and B. N. Chatterji, “An FFT-based technique for translation, rotation, and scale-invariant image registration,” IEEE Trans. Image Process. 5, 1266–1271 (1996).
[CrossRef] [PubMed]

Réfrégier, P.

Ring, W.

S. L. Keeling and W. Ring, “Medical image registration and interpolation by optical flow with maximal rigidity,” J. Math. Imaging Vision 23, 47–65 (2005).
[CrossRef]

Sambora, M.

M. Sambora and R. K. Martin, “Exploiting correlations in projection-based image registration,” Opt. Eng. 47, 077005(2008).
[CrossRef]

Sasaki, Y.

K. Takita, T. Aoki, Y. Sasaki, T. Higuchi, and K. Kobayashi, “High-accuracy subpixel image registration based on phase-only correlation,” IEICE Trans. Fundam. Electron. Commun. Comput. Sci. E86-A, 1925–1934 (2003).
[CrossRef]

Sheng, Y.

Spitzer, K.

T. M. Lehmann, C. Gonner, and K. Spitzer, “Survey: interpolation methods in medical image processing,” IEEE Trans. Med. Imaging 18, 1049–1075 (1999).
[CrossRef]

Sultanov, A.

S. Dyblenko, K. Janschek, A. Kisselev, A. Sultanov, and V. Tchernykh, “Simulation of satellite landmark navigation on the base of optoelectronic image processing technique,” Proc. SPIE 5854, 178–186 (2005).
[CrossRef]

Takita, K.

K. Takita, T. Aoki, Y. Sasaki, T. Higuchi, and K. Kobayashi, “High-accuracy subpixel image registration based on phase-only correlation,” IEICE Trans. Fundam. Electron. Commun. Comput. Sci. E86-A, 1925–1934 (2003).
[CrossRef]

Tchernykh, V.

K. Janschek, V. Tchernykh, and S. Dyblenko, “Performance analysis of optomechatronic image stabilization for a compact space camera,” Control Eng. Pract. 15, 333–347 (2007).
[CrossRef]

S. Dyblenko, K. Janschek, A. Kisselev, A. Sultanov, and V. Tchernykh, “Simulation of satellite landmark navigation on the base of optoelectronic image processing technique,” Proc. SPIE 5854, 178–186 (2005).
[CrossRef]

K. Janschek, V. Tchernykh, S. Dyblenko, G. Flandin, and B. Harnisch, “Compensation of focal plane image motion perturbations with optical correlator in feedback loop,” Proc. SPIE 5570, 280–288 (2004).
[CrossRef]

K. Janschek, V. Tchernykh, S. Dyblenko, and B. Harnisch, “Compensation of the attitude instability effect on the imaging payload performance with optical correlators,” Acta Astronaut. 52, 965–974 (2003).
[CrossRef]

V. Tchernykh, S. Dyblenko, K. Janschek, and B. Harnisch, “Optical correlator-based system for the real-time analysis of image motion in the focal plane of an Earth observation camera,” Proc. SPIE 4113, 23–31 (2000).
[CrossRef]

K. Janschek, T. Boge, S. Dyblenko, and V. Tchernykh, “Image-based attitude determination using an optical correlator,” Proceedings of the 4th ESA International Conference (European Space Agency, 2000), pp. 487–492.

Zwicke, P. E.

P. E. Zwicke and I. Kiss, “A new implementation of the Mellin transform and its application to radar classification of ships,” IEEE Trans. Pattern Anal. Machine Intell. PAMI-5, 191–199(1983).
[CrossRef]

Acta Astronaut. (1)

K. Janschek, V. Tchernykh, S. Dyblenko, and B. Harnisch, “Compensation of the attitude instability effect on the imaging payload performance with optical correlators,” Acta Astronaut. 52, 965–974 (2003).
[CrossRef]

Control Eng. Pract. (1)

K. Janschek, V. Tchernykh, and S. Dyblenko, “Performance analysis of optomechatronic image stabilization for a compact space camera,” Control Eng. Pract. 15, 333–347 (2007).
[CrossRef]

IEEE Trans. Image Process. (2)

Y. Nie and K. K. Ma, “Adaptive rood pattern search for fast block-matching motion estimation,” IEEE Trans. Image Process. 11, 1442–1449 (2002).
[CrossRef]

B. S. Reddy and B. N. Chatterji, “An FFT-based technique for translation, rotation, and scale-invariant image registration,” IEEE Trans. Image Process. 5, 1266–1271 (1996).
[CrossRef] [PubMed]

IEEE Trans. Med. Imaging (1)

T. M. Lehmann, C. Gonner, and K. Spitzer, “Survey: interpolation methods in medical image processing,” IEEE Trans. Med. Imaging 18, 1049–1075 (1999).
[CrossRef]

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

P. E. Zwicke and I. Kiss, “A new implementation of the Mellin transform and its application to radar classification of ships,” IEEE Trans. Pattern Anal. Machine Intell. PAMI-5, 191–199(1983).
[CrossRef]

E. De Castro and C. Morandi, “Registration of translated and rotated images using finite Fourier transforms,” IEEE Trans. Pattern Anal. Machine Intell. PAMI-9, 700–703 (1987).
[CrossRef]

IEICE Trans. Fundam. Electron. Commun. Comput. Sci. (1)

K. Takita, T. Aoki, Y. Sasaki, T. Higuchi, and K. Kobayashi, “High-accuracy subpixel image registration based on phase-only correlation,” IEICE Trans. Fundam. Electron. Commun. Comput. Sci. E86-A, 1925–1934 (2003).
[CrossRef]

J. Math. Imaging Vision (1)

S. L. Keeling and W. Ring, “Medical image registration and interpolation by optical flow with maximal rigidity,” J. Math. Imaging Vision 23, 47–65 (2005).
[CrossRef]

J. Opt. Soc. Am. A (2)

Opt. Eng. (2)

M. S. Alam, “Fractional power fringe-adjusted joint transform correlation,” Opt. Eng. 34, 3208–3216 (1995).
[CrossRef]

M. Sambora and R. K. Martin, “Exploiting correlations in projection-based image registration,” Opt. Eng. 47, 077005(2008).
[CrossRef]

Opt. Lett. (1)

Proc. SPIE (3)

V. Tchernykh, S. Dyblenko, K. Janschek, and B. Harnisch, “Optical correlator-based system for the real-time analysis of image motion in the focal plane of an Earth observation camera,” Proc. SPIE 4113, 23–31 (2000).
[CrossRef]

S. Dyblenko, K. Janschek, A. Kisselev, A. Sultanov, and V. Tchernykh, “Simulation of satellite landmark navigation on the base of optoelectronic image processing technique,” Proc. SPIE 5854, 178–186 (2005).
[CrossRef]

K. Janschek, V. Tchernykh, S. Dyblenko, G. Flandin, and B. Harnisch, “Compensation of focal plane image motion perturbations with optical correlator in feedback loop,” Proc. SPIE 5570, 280–288 (2004).
[CrossRef]

Other (1)

K. Janschek, T. Boge, S. Dyblenko, and V. Tchernykh, “Image-based attitude determination using an optical correlator,” Proceedings of the 4th ESA International Conference (European Space Agency, 2000), pp. 487–492.

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Figures (9)

Fig. 1
Fig. 1

(a) Reference image, (b) target image.

Fig. 2
Fig. 2

(a) MFR multiplied by the high-pass filter, (b) MFT multiplied by the high-pass filter, (c) Enlarged part of (a), (d) Enlarged part of (b).

Fig. 3
Fig. 3

(a) Fig. 2a transformed to polar space; (b) Fig. 2b transformed to polar space.

Fig. 4
Fig. 4

(a) Phase-encoded Fig. 3a, (b) Fig. 4a overlaid with Fig. 3b, (c) 2D correlation output of rotation measurement, (d) 3D correlation output.

Fig. 5
Fig. 5

(a) Rotation-correction target image, (b) Phase-encoded reference image, (c) Fig. 5a overlaid with Fig. 5b, (d) 3D output of translation measurement.

Fig. 6
Fig. 6

Errors of rotation.

Fig. 7
Fig. 7

Errors of translation on x axis.

Fig. 8
Fig. 8

Errors of translation on y axis.

Fig. 9
Fig. 9

Possible optical setup.

Tables (1)

Tables Icon

Table 1 Results by DPEJTC Compared with Phase-Correlation (PC)

Equations (21)

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

Θ ( u , v ) = exp [ j Φ ( u , v ) ] ,
ϕ ( x , y ) = i f f t 2 [ Θ ( u , v ) ] .
ϒ ( x , y ) = r ( x , y ) ϕ ( x , y ) ,
f ( x , y ) = ϒ ( x , y ) + t ( x + x i , y + y i ) .
F ( u , v ) = f f t 2 [ ϒ ( x , y ) + t ( x + x i , y + y i ) ] = f f t 2 [ r ( x , y ) ϕ ( x , y ) ] + f f t 2 [ t ( x + x i , y + y i ) ] = R ( u , v ) Θ ( u , v ) + T ( u , v ) exp ( i u x i + i v y i ) .
JPS ( u , v ) = | F ( u , v ) | 2 = | R ( u , v ) | 2 + | T ( u , v ) | 2 + R ( u , v ) Θ ( u , v ) T ( u , v ) * exp ( i u x i i v y i ) + R ( u , v ) * Θ ( u , v ) * T ( u , v ) exp ( i u x i + i v y i ) ,
PJPS = JPS ( u , v ) Θ ( u , v ) = | R ( u , v ) | 2 Θ ( u , v ) + | T ( u , v ) | 2 Θ ( u , v ) + R ( u , v ) Θ 2 ( u , v ) T ( u , v ) * exp ( i u x i i v y i ) + R ( u , v ) * T ( u , v ) exp ( i u x i + i v y i ) .
H fpfaf ( u , v ) = B ( u , v ) [ A ( u , v ) + | R ( u , v ) | m ] 1 ,
H ( u , v ) = | R ( u , v ) | 1 .
HPJPS ( u , v ) = R ( u , v ) φ R ( u , v ) Θ ( u , v ) + 1 | R ( u , v ) | T ( u , v ) 2 Θ ( u , v ) + φ R ( u , v ) Θ 2 ( u , v ) T ( u , v ) * exp ( i u x i i v y i ) + φ R ( u , v ) * T ( u , v ) exp ( i u x i + i v y i ) ] ,
φ R ( u , v ) = exp ( i u φ x + i v φ y ) .
C ( u , v ) = T ( u , v ) exp [ i u ( x i + φ x ) + i v ( y i + φ y ) ] .
c ( x , y ) = t ( x ( x i + φ x ) , y ( y i + φ y ) ) .
{ x ideal = φ x y ideal = φ y .
{ x peak = ( x i + φ x ) y peak = ( y i + φ y ) .
{ x i = x peak x ideal y i = y peak y ideal .
t = t ( x cos θ 0 + y sin θ 0 + x i , x sin θ 0 + y cos θ 0 + y i ) .
{ R = R ( u , v ) T = T ( u cos θ 0 + v sin θ 0 , u sin θ 0 + v cos θ 0 ) exp ( i u x i + i v y i ) .
H ( u , v ) = ( 1 X ( u , v ) ) 2 ,
{ M 1 = M 1 ( u , v ) H ( u , v ) M 2 = M 2 ( u cos θ 0 + v sin θ 0 , u sin θ 0 + v cos θ 0 ) H ( u , v ) .
θ = Δ θ × 360 / 256 ,

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