Abstract

To implement real-time 3D reconstruction and displaying for polarization-modulated 3D imaging lidar system, an efficient subpixel registration based on maximum principal component analysis (MPCA) is proposed in this paper. With which only the maximum principal component is estimated to identify non-integer translations in spatial domain while other principal components affected by noise are ignored. Consequently, robustness and stability of the subpixel registration is implemented in presence of noise, while computational complexity is reduced and memory size is saved simultaneously, especially when the image size is large. Both simulated and real polarization-modulated images are used to verify the proposed method. Simulation results show that 0.01 pixels of the registration accuracy are implemented; meanwhile, experimental results show that the proposed method can effectively reconstruct the depth image in real world application.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]
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    [Crossref] [PubMed]
  3. Z. Chen, B. Liu, E. Liu, and Z. Peng, “Adaptive polarization-modulated method for high resolution 3D imaging,” IEEE Photonics Technol. Lett. 28(3), 295–298 (2016).
    [Crossref]
  4. B. Zitova and J. Flusser, “Image registration methods: a survey,” Image Vis. Comput. 21(11), 977–1000 (2003).
    [Crossref]
  5. J. Ma, H. Zhou, J. Zhao, Y. Gao, J. Jiang, and J. Tian, “Robust Feature Matching for Remote Sensing Image Registration via Locally Linear Transforming,” IEEE Trans. Geosci. Remote Sens. 53(12), 6469–6481 (2015).
    [Crossref]
  6. H. Foroosh, J. B. Zerubia, and M. Berthod, “Extension of Phase Correlation to Subpixel Registration,” IEEE Trans. Image Process. 11(3), 188–200 (2002).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  8. W. Tong, “Subpixel image registration with reduced bias,” Opt. Lett. 36(5), 763–765 (2011).
    [Crossref] [PubMed]
  9. R. C. Gonzalez, R. E. Woods, and S. L. Eddins, in Digital Image Processing Using MATLAB, 2nd ed. (McGraw-Hill, 2011).
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    [Crossref] [PubMed]
  11. W. S. Hoge, “A subspace identification extension to the phase correlation method,” IEEE Trans. Med. Imaging 22(2), 277–280 (2003).
    [Crossref] [PubMed]
  12. H. Foroosh and M. Balci, “Sub-pixel registration and estimation of local shifts directly in the Fourier domain,” in International Conf. Image Proces. (ICIP) (2004).
    [Crossref]
  13. H. S. Stone, M. T. Orchard, E. Chang, and S. A. Martucci, “A fast direct Fourier-based algorithm for subpixel registration of images,” IEEE Trans. Geosci. Remote Sens. 39(10), 2235–2243 (2001).
    [Crossref]
  14. P. Vandewalle, S. Susstrunk, and M. Vetterli, “A frequency domain approach to registration of aliased images with application to super-resolution,” EURASIP J. Appl. Signal Process. 71459, 1–14 (2006).
  15. X. Tong, Z. Ye, Y. Xu, S. Liu, L. Li, H. Xie, and T. Li, “A novel subpixel phase correlation method using singular value Decomposition and Unified Random Sample Consensus,” IEEE Trans. Geosci. Remote Sens. 53(8), 4143–4156 (2015).
    [Crossref]
  16. A. Yariv and P. Yeh, “Electro-optic Devices,” in Optical waves in crystals: Propagation and control of laser radiation, 1st ed. (Wiley, 2002).
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  18. C. M. Bishop, in Pattern Recognition and Machine Learning (Springer, 2006).
  19. S. Roweis, “EM algorithms for PCA and SPCA,” Adv. Neural Inf. Process. Syst. 1500, 626 (1998).
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    [Crossref]
  21. M. S. Robbins and B. J. Hadwen, “The noise performance of electron multiplying charge-coupled devices,” IEEE Trans. Electron Devices 50(5), 1227–1232 (2003).

2016 (2)

Z. Chen, B. Liu, E. Liu, and Z. Peng, “Adaptive polarization-modulated method for high resolution 3D imaging,” IEEE Photonics Technol. Lett. 28(3), 295–298 (2016).
[Crossref]

Z. Chen, B. Liu, E. Liu, and Z. Peng, “Electro-optic modulation methods in range-gated active imaging,” Appl. Opt. 55(3), A184–A190 (2016).
[Crossref] [PubMed]

2015 (2)

J. Ma, H. Zhou, J. Zhao, Y. Gao, J. Jiang, and J. Tian, “Robust Feature Matching for Remote Sensing Image Registration via Locally Linear Transforming,” IEEE Trans. Geosci. Remote Sens. 53(12), 6469–6481 (2015).
[Crossref]

X. Tong, Z. Ye, Y. Xu, S. Liu, L. Li, H. Xie, and T. Li, “A novel subpixel phase correlation method using singular value Decomposition and Unified Random Sample Consensus,” IEEE Trans. Geosci. Remote Sens. 53(8), 4143–4156 (2015).
[Crossref]

2012 (1)

P. F. McManamon, “Review of ladar: a historic, yet emerging, sensor technology with rich phenomenology,” Opt. Eng. 51(6), 060901 (2012).
[Crossref]

2011 (1)

2008 (1)

2006 (1)

P. Vandewalle, S. Susstrunk, and M. Vetterli, “A frequency domain approach to registration of aliased images with application to super-resolution,” EURASIP J. Appl. Signal Process. 71459, 1–14 (2006).

2004 (1)

D. Dussault and P. Hoess, “Noise performance comparison of ICCD with CCD and EMCCD cameras,” Proc. SPIE 5563, 195–204 (2004).
[Crossref]

2003 (3)

M. S. Robbins and B. J. Hadwen, “The noise performance of electron multiplying charge-coupled devices,” IEEE Trans. Electron Devices 50(5), 1227–1232 (2003).

W. S. Hoge, “A subspace identification extension to the phase correlation method,” IEEE Trans. Med. Imaging 22(2), 277–280 (2003).
[Crossref] [PubMed]

B. Zitova and J. Flusser, “Image registration methods: a survey,” Image Vis. Comput. 21(11), 977–1000 (2003).
[Crossref]

2002 (1)

H. Foroosh, J. B. Zerubia, and M. Berthod, “Extension of Phase Correlation to Subpixel Registration,” IEEE Trans. Image Process. 11(3), 188–200 (2002).
[Crossref] [PubMed]

2001 (1)

H. S. Stone, M. T. Orchard, E. Chang, and S. A. Martucci, “A fast direct Fourier-based algorithm for subpixel registration of images,” IEEE Trans. Geosci. Remote Sens. 39(10), 2235–2243 (2001).
[Crossref]

1998 (2)

S. Roweis, “EM algorithms for PCA and SPCA,” Adv. Neural Inf. Process. Syst. 1500, 626 (1998).

P. Thévenaz, U. E. Ruttimann, and M. Unser, “A pyramid approach to subpixel registration based on intensity,” IEEE Trans. Image Process. 7(1), 27–41 (1998).
[Crossref] [PubMed]

Balci, M.

H. Foroosh and M. Balci, “Sub-pixel registration and estimation of local shifts directly in the Fourier domain,” in International Conf. Image Proces. (ICIP) (2004).
[Crossref]

Berthod, M.

H. Foroosh, J. B. Zerubia, and M. Berthod, “Extension of Phase Correlation to Subpixel Registration,” IEEE Trans. Image Process. 11(3), 188–200 (2002).
[Crossref] [PubMed]

Chang, E.

H. S. Stone, M. T. Orchard, E. Chang, and S. A. Martucci, “A fast direct Fourier-based algorithm for subpixel registration of images,” IEEE Trans. Geosci. Remote Sens. 39(10), 2235–2243 (2001).
[Crossref]

Chen, Z.

Z. Chen, B. Liu, E. Liu, and Z. Peng, “Adaptive polarization-modulated method for high resolution 3D imaging,” IEEE Photonics Technol. Lett. 28(3), 295–298 (2016).
[Crossref]

Z. Chen, B. Liu, E. Liu, and Z. Peng, “Electro-optic modulation methods in range-gated active imaging,” Appl. Opt. 55(3), A184–A190 (2016).
[Crossref] [PubMed]

Dussault, D.

D. Dussault and P. Hoess, “Noise performance comparison of ICCD with CCD and EMCCD cameras,” Proc. SPIE 5563, 195–204 (2004).
[Crossref]

Fienup, J. R.

Flusser, J.

B. Zitova and J. Flusser, “Image registration methods: a survey,” Image Vis. Comput. 21(11), 977–1000 (2003).
[Crossref]

Foroosh, H.

H. Foroosh, J. B. Zerubia, and M. Berthod, “Extension of Phase Correlation to Subpixel Registration,” IEEE Trans. Image Process. 11(3), 188–200 (2002).
[Crossref] [PubMed]

H. Foroosh and M. Balci, “Sub-pixel registration and estimation of local shifts directly in the Fourier domain,” in International Conf. Image Proces. (ICIP) (2004).
[Crossref]

Gao, Y.

J. Ma, H. Zhou, J. Zhao, Y. Gao, J. Jiang, and J. Tian, “Robust Feature Matching for Remote Sensing Image Registration via Locally Linear Transforming,” IEEE Trans. Geosci. Remote Sens. 53(12), 6469–6481 (2015).
[Crossref]

Guizar-Sicairos, M.

Hadwen, B. J.

M. S. Robbins and B. J. Hadwen, “The noise performance of electron multiplying charge-coupled devices,” IEEE Trans. Electron Devices 50(5), 1227–1232 (2003).

Hoess, P.

D. Dussault and P. Hoess, “Noise performance comparison of ICCD with CCD and EMCCD cameras,” Proc. SPIE 5563, 195–204 (2004).
[Crossref]

Hoge, W. S.

W. S. Hoge, “A subspace identification extension to the phase correlation method,” IEEE Trans. Med. Imaging 22(2), 277–280 (2003).
[Crossref] [PubMed]

Jiang, J.

J. Ma, H. Zhou, J. Zhao, Y. Gao, J. Jiang, and J. Tian, “Robust Feature Matching for Remote Sensing Image Registration via Locally Linear Transforming,” IEEE Trans. Geosci. Remote Sens. 53(12), 6469–6481 (2015).
[Crossref]

Li, L.

X. Tong, Z. Ye, Y. Xu, S. Liu, L. Li, H. Xie, and T. Li, “A novel subpixel phase correlation method using singular value Decomposition and Unified Random Sample Consensus,” IEEE Trans. Geosci. Remote Sens. 53(8), 4143–4156 (2015).
[Crossref]

Li, T.

X. Tong, Z. Ye, Y. Xu, S. Liu, L. Li, H. Xie, and T. Li, “A novel subpixel phase correlation method using singular value Decomposition and Unified Random Sample Consensus,” IEEE Trans. Geosci. Remote Sens. 53(8), 4143–4156 (2015).
[Crossref]

Liu, B.

Z. Chen, B. Liu, E. Liu, and Z. Peng, “Adaptive polarization-modulated method for high resolution 3D imaging,” IEEE Photonics Technol. Lett. 28(3), 295–298 (2016).
[Crossref]

Z. Chen, B. Liu, E. Liu, and Z. Peng, “Electro-optic modulation methods in range-gated active imaging,” Appl. Opt. 55(3), A184–A190 (2016).
[Crossref] [PubMed]

Liu, E.

Z. Chen, B. Liu, E. Liu, and Z. Peng, “Electro-optic modulation methods in range-gated active imaging,” Appl. Opt. 55(3), A184–A190 (2016).
[Crossref] [PubMed]

Z. Chen, B. Liu, E. Liu, and Z. Peng, “Adaptive polarization-modulated method for high resolution 3D imaging,” IEEE Photonics Technol. Lett. 28(3), 295–298 (2016).
[Crossref]

Liu, S.

X. Tong, Z. Ye, Y. Xu, S. Liu, L. Li, H. Xie, and T. Li, “A novel subpixel phase correlation method using singular value Decomposition and Unified Random Sample Consensus,” IEEE Trans. Geosci. Remote Sens. 53(8), 4143–4156 (2015).
[Crossref]

Ma, J.

J. Ma, H. Zhou, J. Zhao, Y. Gao, J. Jiang, and J. Tian, “Robust Feature Matching for Remote Sensing Image Registration via Locally Linear Transforming,” IEEE Trans. Geosci. Remote Sens. 53(12), 6469–6481 (2015).
[Crossref]

Martucci, S. A.

H. S. Stone, M. T. Orchard, E. Chang, and S. A. Martucci, “A fast direct Fourier-based algorithm for subpixel registration of images,” IEEE Trans. Geosci. Remote Sens. 39(10), 2235–2243 (2001).
[Crossref]

McManamon, P. F.

P. F. McManamon, “Review of ladar: a historic, yet emerging, sensor technology with rich phenomenology,” Opt. Eng. 51(6), 060901 (2012).
[Crossref]

Orchard, M. T.

H. S. Stone, M. T. Orchard, E. Chang, and S. A. Martucci, “A fast direct Fourier-based algorithm for subpixel registration of images,” IEEE Trans. Geosci. Remote Sens. 39(10), 2235–2243 (2001).
[Crossref]

Peng, Z.

Z. Chen, B. Liu, E. Liu, and Z. Peng, “Adaptive polarization-modulated method for high resolution 3D imaging,” IEEE Photonics Technol. Lett. 28(3), 295–298 (2016).
[Crossref]

Z. Chen, B. Liu, E. Liu, and Z. Peng, “Electro-optic modulation methods in range-gated active imaging,” Appl. Opt. 55(3), A184–A190 (2016).
[Crossref] [PubMed]

Robbins, M. S.

M. S. Robbins and B. J. Hadwen, “The noise performance of electron multiplying charge-coupled devices,” IEEE Trans. Electron Devices 50(5), 1227–1232 (2003).

Roweis, S.

S. Roweis, “EM algorithms for PCA and SPCA,” Adv. Neural Inf. Process. Syst. 1500, 626 (1998).

Ruttimann, U. E.

P. Thévenaz, U. E. Ruttimann, and M. Unser, “A pyramid approach to subpixel registration based on intensity,” IEEE Trans. Image Process. 7(1), 27–41 (1998).
[Crossref] [PubMed]

Stone, H. S.

H. S. Stone, M. T. Orchard, E. Chang, and S. A. Martucci, “A fast direct Fourier-based algorithm for subpixel registration of images,” IEEE Trans. Geosci. Remote Sens. 39(10), 2235–2243 (2001).
[Crossref]

Susstrunk, S.

P. Vandewalle, S. Susstrunk, and M. Vetterli, “A frequency domain approach to registration of aliased images with application to super-resolution,” EURASIP J. Appl. Signal Process. 71459, 1–14 (2006).

Thévenaz, P.

P. Thévenaz, U. E. Ruttimann, and M. Unser, “A pyramid approach to subpixel registration based on intensity,” IEEE Trans. Image Process. 7(1), 27–41 (1998).
[Crossref] [PubMed]

Thurman, S. T.

Tian, J.

J. Ma, H. Zhou, J. Zhao, Y. Gao, J. Jiang, and J. Tian, “Robust Feature Matching for Remote Sensing Image Registration via Locally Linear Transforming,” IEEE Trans. Geosci. Remote Sens. 53(12), 6469–6481 (2015).
[Crossref]

Tong, W.

Tong, X.

X. Tong, Z. Ye, Y. Xu, S. Liu, L. Li, H. Xie, and T. Li, “A novel subpixel phase correlation method using singular value Decomposition and Unified Random Sample Consensus,” IEEE Trans. Geosci. Remote Sens. 53(8), 4143–4156 (2015).
[Crossref]

Unser, M.

P. Thévenaz, U. E. Ruttimann, and M. Unser, “A pyramid approach to subpixel registration based on intensity,” IEEE Trans. Image Process. 7(1), 27–41 (1998).
[Crossref] [PubMed]

Vandewalle, P.

P. Vandewalle, S. Susstrunk, and M. Vetterli, “A frequency domain approach to registration of aliased images with application to super-resolution,” EURASIP J. Appl. Signal Process. 71459, 1–14 (2006).

Vetterli, M.

P. Vandewalle, S. Susstrunk, and M. Vetterli, “A frequency domain approach to registration of aliased images with application to super-resolution,” EURASIP J. Appl. Signal Process. 71459, 1–14 (2006).

Xie, H.

X. Tong, Z. Ye, Y. Xu, S. Liu, L. Li, H. Xie, and T. Li, “A novel subpixel phase correlation method using singular value Decomposition and Unified Random Sample Consensus,” IEEE Trans. Geosci. Remote Sens. 53(8), 4143–4156 (2015).
[Crossref]

Xu, Y.

X. Tong, Z. Ye, Y. Xu, S. Liu, L. Li, H. Xie, and T. Li, “A novel subpixel phase correlation method using singular value Decomposition and Unified Random Sample Consensus,” IEEE Trans. Geosci. Remote Sens. 53(8), 4143–4156 (2015).
[Crossref]

Ye, Z.

X. Tong, Z. Ye, Y. Xu, S. Liu, L. Li, H. Xie, and T. Li, “A novel subpixel phase correlation method using singular value Decomposition and Unified Random Sample Consensus,” IEEE Trans. Geosci. Remote Sens. 53(8), 4143–4156 (2015).
[Crossref]

Zerubia, J. B.

H. Foroosh, J. B. Zerubia, and M. Berthod, “Extension of Phase Correlation to Subpixel Registration,” IEEE Trans. Image Process. 11(3), 188–200 (2002).
[Crossref] [PubMed]

Zhao, J.

J. Ma, H. Zhou, J. Zhao, Y. Gao, J. Jiang, and J. Tian, “Robust Feature Matching for Remote Sensing Image Registration via Locally Linear Transforming,” IEEE Trans. Geosci. Remote Sens. 53(12), 6469–6481 (2015).
[Crossref]

Zhou, H.

J. Ma, H. Zhou, J. Zhao, Y. Gao, J. Jiang, and J. Tian, “Robust Feature Matching for Remote Sensing Image Registration via Locally Linear Transforming,” IEEE Trans. Geosci. Remote Sens. 53(12), 6469–6481 (2015).
[Crossref]

Zitova, B.

B. Zitova and J. Flusser, “Image registration methods: a survey,” Image Vis. Comput. 21(11), 977–1000 (2003).
[Crossref]

Adv. Neural Inf. Process. Syst. (1)

S. Roweis, “EM algorithms for PCA and SPCA,” Adv. Neural Inf. Process. Syst. 1500, 626 (1998).

Appl. Opt. (1)

EURASIP J. Appl. Signal Process. (1)

P. Vandewalle, S. Susstrunk, and M. Vetterli, “A frequency domain approach to registration of aliased images with application to super-resolution,” EURASIP J. Appl. Signal Process. 71459, 1–14 (2006).

IEEE Photonics Technol. Lett. (1)

Z. Chen, B. Liu, E. Liu, and Z. Peng, “Adaptive polarization-modulated method for high resolution 3D imaging,” IEEE Photonics Technol. Lett. 28(3), 295–298 (2016).
[Crossref]

IEEE Trans. Electron Devices (1)

M. S. Robbins and B. J. Hadwen, “The noise performance of electron multiplying charge-coupled devices,” IEEE Trans. Electron Devices 50(5), 1227–1232 (2003).

IEEE Trans. Geosci. Remote Sens. (3)

X. Tong, Z. Ye, Y. Xu, S. Liu, L. Li, H. Xie, and T. Li, “A novel subpixel phase correlation method using singular value Decomposition and Unified Random Sample Consensus,” IEEE Trans. Geosci. Remote Sens. 53(8), 4143–4156 (2015).
[Crossref]

J. Ma, H. Zhou, J. Zhao, Y. Gao, J. Jiang, and J. Tian, “Robust Feature Matching for Remote Sensing Image Registration via Locally Linear Transforming,” IEEE Trans. Geosci. Remote Sens. 53(12), 6469–6481 (2015).
[Crossref]

H. S. Stone, M. T. Orchard, E. Chang, and S. A. Martucci, “A fast direct Fourier-based algorithm for subpixel registration of images,” IEEE Trans. Geosci. Remote Sens. 39(10), 2235–2243 (2001).
[Crossref]

IEEE Trans. Image Process. (2)

H. Foroosh, J. B. Zerubia, and M. Berthod, “Extension of Phase Correlation to Subpixel Registration,” IEEE Trans. Image Process. 11(3), 188–200 (2002).
[Crossref] [PubMed]

P. Thévenaz, U. E. Ruttimann, and M. Unser, “A pyramid approach to subpixel registration based on intensity,” IEEE Trans. Image Process. 7(1), 27–41 (1998).
[Crossref] [PubMed]

IEEE Trans. Med. Imaging (1)

W. S. Hoge, “A subspace identification extension to the phase correlation method,” IEEE Trans. Med. Imaging 22(2), 277–280 (2003).
[Crossref] [PubMed]

Image Vis. Comput. (1)

B. Zitova and J. Flusser, “Image registration methods: a survey,” Image Vis. Comput. 21(11), 977–1000 (2003).
[Crossref]

Opt. Eng. (1)

P. F. McManamon, “Review of ladar: a historic, yet emerging, sensor technology with rich phenomenology,” Opt. Eng. 51(6), 060901 (2012).
[Crossref]

Opt. Lett. (2)

Proc. SPIE (1)

D. Dussault and P. Hoess, “Noise performance comparison of ICCD with CCD and EMCCD cameras,” Proc. SPIE 5563, 195–204 (2004).
[Crossref]

Other (5)

A. Yariv and P. Yeh, “Electro-optic Devices,” in Optical waves in crystals: Propagation and control of laser radiation, 1st ed. (Wiley, 2002).

M. Born and E. Wolf, “Optics of crystals,” in Principles of Optics, 7th (expanded) ed. (Cambridge U., 1999).

C. M. Bishop, in Pattern Recognition and Machine Learning (Springer, 2006).

R. C. Gonzalez, R. E. Woods, and S. L. Eddins, in Digital Image Processing Using MATLAB, 2nd ed. (McGraw-Hill, 2011).

H. Foroosh and M. Balci, “Sub-pixel registration and estimation of local shifts directly in the Fourier domain,” in International Conf. Image Proces. (ICIP) (2004).
[Crossref]

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Figures (6)

Fig. 1
Fig. 1 Schematic diagram of polarization-modulated 3D imaging. Here, the electro-optic modulators manipulate polarization state of returned-light to perform range-gated imaging; meanwhile, dual EMCCD cameras acquire polarization-modulated images for 3D reconstruction in channel X and Y, respectively.
Fig. 2
Fig. 2 The polarization-modulated images and demodulated images in the simulation system. (a) distorted image (cos2-modulated image); (b) reference image (sin2-modulated image); (c) polarization-demodulated image with feature-based registration; (d) polarization-demodulated image with area-based registration.
Fig. 3
Fig. 3 Results of 3D reconstruction with high noise level. (a), (b) and (c) are the depth images represented by RPCM, RSVD and RMPCA, which are reconstructed by performing the PCM, SVD and MPCA registration methods, respectively. Besides, (d), (e) and (f) are the range error images derived by performing (RPCM-RTRUE), (RSVD-RTRUE) and (RMPCA-RTRUE) operations.
Fig. 4
Fig. 4 The experimental system for polarization-modulated 3D imaging.
Fig. 5
Fig. 5 The polarization-modulated images and demodulated images in the experimental system. (a) cos2-modulated image; (b) sin2-modulated image; (c) polarization-demodulated image with feature-based registration and (d) polarization-demodulated image with area-based registration.
Fig. 6
Fig. 6 The towers’ 3D structure derived from the polarization-modulated 3D imaging system. (a) The gray image obtained from conventional CCD camera during day-time; (b) the depth image, corresponding to the designated area in gray image, is reconstructed from the two polarization-modulated images.

Tables (2)

Tables Icon

Table 1 Performance evaluation among the PCM, SVD and MPCA registration methods

Tables Icon

Table 2 System parameters for polarization-modulated 3D imaging.

Equations (10)

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

θ= π V π V 1 ( t ),( 0 V 1 ( t ) V π )
θ= π L ( R R Base ),( R Base R R Base +L )
{ I X = I REC cos 2 ( π 2 · R R Base L ) I Y = I REC sin 2 ( π 2 · R R Base L ) ,( R Base R R Base +L )
R= R Base + 2L π arctan( I Y I X )
I REC = I X + I Y
Q( u,v )= F( u,v )G ( u,v ) * | F( u,v )G ( u,v ) * |
X D×N = Q ( u , v )| HD 2 u H+D 2 1, WN 2 v W+N 2 1
X D×N = P D×K Z K×N + μ X I D×N +ε
{ Z K×N = R K×K 1 P D×K T ( X D×N μ X I D×N ) P D×K =( X D×N μ X I D×N ) Z K×N T ( Z K×N Z K×N T ) 1 X D×N = P D×K Z K×N + μ X I D×N
R K×K = P D×K T P D×K + σ 2 I K×K

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