Abstract

A novel autofocusing algorithm using the directional wavelet power spectrum is proposed for time delayed and integration charge coupled device (TDI CCD) space cameras, which overcomes the difficulty of focus measure for the real-time change of imaging scenes. Using the multiresolution and band-pass characteristics of wavelet transform to improve the power spectrum based on fast Fourier transform (FFT), the wavelet power spectrum is less sensitive to the variance of scenes. Moreover, the new focus measure can effectively eliminate the impact of image motion mismatching by the directional selection. We test the proposed method’s performance on synthetic images as well as a real ground experiment for one TDI CCD prototype camera, and compare it with the focus measure based on the existing FFT spectrum. The simulation results show that the new focus measure can effectively express the defocused states for the real remote sensing images. The error ratio is only 0.112, while the prevalent algorithm based on the FFT spectrum is as high as 0.4. Compared with the FFT-based method, the proposed algorithm performs at a high reliability in the real imaging experiments, where it reduces the instability from 0.600 to 0.161. Two experimental results demonstrate that the proposed algorithm has the characteristics of good monotonicity, high sensitivity, and accuracy. The new algorithm can satisfy the autofocusing requirements for TDI CCD space cameras.

© 2012 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. Z. H. Li and K. Y. Wu, “Autofocus system for space cameras,” Opt. Eng. 44, 053001 (2005).
    [CrossRef]
  2. J. F. Brenner, B. S. Dew, J. B. Horton, T. King, P. W. Neurath, and W. D. Selles, “An automated microscope for cytologic research a preliminary evaluation,” J. Histochem. Cytochem. 24, 100–111 (1976).
    [CrossRef]
  3. D. Vollath, “Automatic focusing by correlative methods,” J. Microsc. 147, 279–288 (1987).
    [CrossRef]
  4. M. Subbarao, T. Choi, and A. Nikzad, “A Focusing techniques,” Opt. Eng. 32, 2824–2836 (1993).
    [CrossRef]
  5. H. C. Zhang, C. Gould, B. Roberts, and M. McQuillan, “An objective image focus monitor for CD-SEM,” Proc. SPIE 6922, 692226 (2008).
    [CrossRef]
  6. C. M. Chen, C. M. Hong, and H. C. Chuang, “Efficient auto-focus algorithm utilizing discrete difference equation prediction model for digital still cameras,” IEEE Trans. Consumer Electron. 52, 1135–1143 (2006).
    [CrossRef]
  7. T. Aydin and Y. S. Akgul, “An occlusion insensitive adaptive focus measurement method,” Opt. Express 18, 14212–14224 (2010).
    [CrossRef]
  8. M. Kristan, J. Pers, M. Perse, and S. Kovaci, “A Bayes- spectral-entropy-based measure of camera focus using a discrete cosine transform,” Pattern Recog. Lett. 27, 1431–1439 (2006).
    [CrossRef]
  9. Y. N. Zhang, Y. Zhang, and C. Y. Wen, “A new focus measure method using moments,” Image Vision Comput. 18, 959–965 (2000).
    [CrossRef]
  10. Y. B. Tian, S. Kevin, and C. F. Wildsoet, “Performance of focus measures in the presence of nondefocus aberrations,” J. Opt. Soc. Am. A 24, B165–B173 (2007).
    [CrossRef]
  11. J. Kautsky, J. Flusser, B. Zitová, and Ŝ. Simberová, “A new wavelet-based measure of image focus,” Pattern Recog. Lett. 23, 1785–1794 (2002).
    [CrossRef]
  12. Y. B. Tian, “Autofocus using image phase congruency,” Opt. Express 19, 261–270 (2011).
    [CrossRef]
  13. NASA Glenn Research Center, “A compact microscope imaging system with intelligent controls,” https://technology.grc.nasa.gov/tech-detail-coded.php?cid=GR-0047 .
  14. Z. B. Zhao and J. H. Liu, “Power spectra based auto-focusing method for airborne optoelectronic platform,” Acta Optica Sinica 30, 3495–3500 (2010).
    [CrossRef]
  15. M. E. Rudnaya, R. M. M. Mattheij, J. M. L. Maubach, and H. G. ter Morsche, “Orientation identification of the power spectrum,” Opt. Eng. 50, 103201 (2011).
    [CrossRef]
  16. N. B. Nill and B. H. Bouzas, “Objective image quality measure derived from digital image power spectra,” Opt. Eng. 31, 813–825 (1992).
    [CrossRef]
  17. Z. G. Fan, S. Q. Chen, H. L. Hu, H. Chang, and Q. Fu, “Autofocus algorithm based on wavelet packet transform for infrared microscopy,” in Proceedings of IEEE Conference on Image and Signal Processing (IEEE, 2010), pp. 2510–2514.
  18. J. Pando and L. Z. Fang, “Discrete wavelet transform power spectrum estimator,” Phys. Rev. E 57, 3593–3601 (1998).
    [CrossRef]
  19. Y. Yitzhaky, I. Mor, A. Lantzman, and N. S. Kopeika, “Direct method for restoration of motion-blurred images,” J. Opt. Soc. Am. A 15, 1512–1519 (1998).
    [CrossRef]
  20. R. V. Shack, “The influence of image motion and shutter operation on the photographic transfer function,” Appl. Opt. 3, 1171–1181 (1964).
    [CrossRef]
  21. J. Q. Wang, P. Yu, C. X. Yan, J. Y. Ren, and B. He, “Space optical remote sensor image motion velocity vector computational modeling, error budget and synthesis,” Chin. Opt. Lett. 3, 414–417 (2005).

2011 (2)

M. E. Rudnaya, R. M. M. Mattheij, J. M. L. Maubach, and H. G. ter Morsche, “Orientation identification of the power spectrum,” Opt. Eng. 50, 103201 (2011).
[CrossRef]

Y. B. Tian, “Autofocus using image phase congruency,” Opt. Express 19, 261–270 (2011).
[CrossRef]

2010 (2)

T. Aydin and Y. S. Akgul, “An occlusion insensitive adaptive focus measurement method,” Opt. Express 18, 14212–14224 (2010).
[CrossRef]

Z. B. Zhao and J. H. Liu, “Power spectra based auto-focusing method for airborne optoelectronic platform,” Acta Optica Sinica 30, 3495–3500 (2010).
[CrossRef]

2008 (1)

H. C. Zhang, C. Gould, B. Roberts, and M. McQuillan, “An objective image focus monitor for CD-SEM,” Proc. SPIE 6922, 692226 (2008).
[CrossRef]

2007 (1)

2006 (2)

C. M. Chen, C. M. Hong, and H. C. Chuang, “Efficient auto-focus algorithm utilizing discrete difference equation prediction model for digital still cameras,” IEEE Trans. Consumer Electron. 52, 1135–1143 (2006).
[CrossRef]

M. Kristan, J. Pers, M. Perse, and S. Kovaci, “A Bayes- spectral-entropy-based measure of camera focus using a discrete cosine transform,” Pattern Recog. Lett. 27, 1431–1439 (2006).
[CrossRef]

2005 (2)

2002 (1)

J. Kautsky, J. Flusser, B. Zitová, and Ŝ. Simberová, “A new wavelet-based measure of image focus,” Pattern Recog. Lett. 23, 1785–1794 (2002).
[CrossRef]

2000 (1)

Y. N. Zhang, Y. Zhang, and C. Y. Wen, “A new focus measure method using moments,” Image Vision Comput. 18, 959–965 (2000).
[CrossRef]

1998 (2)

J. Pando and L. Z. Fang, “Discrete wavelet transform power spectrum estimator,” Phys. Rev. E 57, 3593–3601 (1998).
[CrossRef]

Y. Yitzhaky, I. Mor, A. Lantzman, and N. S. Kopeika, “Direct method for restoration of motion-blurred images,” J. Opt. Soc. Am. A 15, 1512–1519 (1998).
[CrossRef]

1993 (1)

M. Subbarao, T. Choi, and A. Nikzad, “A Focusing techniques,” Opt. Eng. 32, 2824–2836 (1993).
[CrossRef]

1992 (1)

N. B. Nill and B. H. Bouzas, “Objective image quality measure derived from digital image power spectra,” Opt. Eng. 31, 813–825 (1992).
[CrossRef]

1987 (1)

D. Vollath, “Automatic focusing by correlative methods,” J. Microsc. 147, 279–288 (1987).
[CrossRef]

1976 (1)

J. F. Brenner, B. S. Dew, J. B. Horton, T. King, P. W. Neurath, and W. D. Selles, “An automated microscope for cytologic research a preliminary evaluation,” J. Histochem. Cytochem. 24, 100–111 (1976).
[CrossRef]

1964 (1)

Akgul, Y. S.

Aydin, T.

Bouzas, B. H.

N. B. Nill and B. H. Bouzas, “Objective image quality measure derived from digital image power spectra,” Opt. Eng. 31, 813–825 (1992).
[CrossRef]

Brenner, J. F.

J. F. Brenner, B. S. Dew, J. B. Horton, T. King, P. W. Neurath, and W. D. Selles, “An automated microscope for cytologic research a preliminary evaluation,” J. Histochem. Cytochem. 24, 100–111 (1976).
[CrossRef]

Chang, H.

Z. G. Fan, S. Q. Chen, H. L. Hu, H. Chang, and Q. Fu, “Autofocus algorithm based on wavelet packet transform for infrared microscopy,” in Proceedings of IEEE Conference on Image and Signal Processing (IEEE, 2010), pp. 2510–2514.

Chen, C. M.

C. M. Chen, C. M. Hong, and H. C. Chuang, “Efficient auto-focus algorithm utilizing discrete difference equation prediction model for digital still cameras,” IEEE Trans. Consumer Electron. 52, 1135–1143 (2006).
[CrossRef]

Chen, S. Q.

Z. G. Fan, S. Q. Chen, H. L. Hu, H. Chang, and Q. Fu, “Autofocus algorithm based on wavelet packet transform for infrared microscopy,” in Proceedings of IEEE Conference on Image and Signal Processing (IEEE, 2010), pp. 2510–2514.

Choi, T.

M. Subbarao, T. Choi, and A. Nikzad, “A Focusing techniques,” Opt. Eng. 32, 2824–2836 (1993).
[CrossRef]

Chuang, H. C.

C. M. Chen, C. M. Hong, and H. C. Chuang, “Efficient auto-focus algorithm utilizing discrete difference equation prediction model for digital still cameras,” IEEE Trans. Consumer Electron. 52, 1135–1143 (2006).
[CrossRef]

Dew, B. S.

J. F. Brenner, B. S. Dew, J. B. Horton, T. King, P. W. Neurath, and W. D. Selles, “An automated microscope for cytologic research a preliminary evaluation,” J. Histochem. Cytochem. 24, 100–111 (1976).
[CrossRef]

Fan, Z. G.

Z. G. Fan, S. Q. Chen, H. L. Hu, H. Chang, and Q. Fu, “Autofocus algorithm based on wavelet packet transform for infrared microscopy,” in Proceedings of IEEE Conference on Image and Signal Processing (IEEE, 2010), pp. 2510–2514.

Fang, L. Z.

J. Pando and L. Z. Fang, “Discrete wavelet transform power spectrum estimator,” Phys. Rev. E 57, 3593–3601 (1998).
[CrossRef]

Flusser, J.

J. Kautsky, J. Flusser, B. Zitová, and Ŝ. Simberová, “A new wavelet-based measure of image focus,” Pattern Recog. Lett. 23, 1785–1794 (2002).
[CrossRef]

Fu, Q.

Z. G. Fan, S. Q. Chen, H. L. Hu, H. Chang, and Q. Fu, “Autofocus algorithm based on wavelet packet transform for infrared microscopy,” in Proceedings of IEEE Conference on Image and Signal Processing (IEEE, 2010), pp. 2510–2514.

Gould, C.

H. C. Zhang, C. Gould, B. Roberts, and M. McQuillan, “An objective image focus monitor for CD-SEM,” Proc. SPIE 6922, 692226 (2008).
[CrossRef]

He, B.

Hong, C. M.

C. M. Chen, C. M. Hong, and H. C. Chuang, “Efficient auto-focus algorithm utilizing discrete difference equation prediction model for digital still cameras,” IEEE Trans. Consumer Electron. 52, 1135–1143 (2006).
[CrossRef]

Horton, J. B.

J. F. Brenner, B. S. Dew, J. B. Horton, T. King, P. W. Neurath, and W. D. Selles, “An automated microscope for cytologic research a preliminary evaluation,” J. Histochem. Cytochem. 24, 100–111 (1976).
[CrossRef]

Hu, H. L.

Z. G. Fan, S. Q. Chen, H. L. Hu, H. Chang, and Q. Fu, “Autofocus algorithm based on wavelet packet transform for infrared microscopy,” in Proceedings of IEEE Conference on Image and Signal Processing (IEEE, 2010), pp. 2510–2514.

Kautsky, J.

J. Kautsky, J. Flusser, B. Zitová, and Ŝ. Simberová, “A new wavelet-based measure of image focus,” Pattern Recog. Lett. 23, 1785–1794 (2002).
[CrossRef]

Kevin, S.

King, T.

J. F. Brenner, B. S. Dew, J. B. Horton, T. King, P. W. Neurath, and W. D. Selles, “An automated microscope for cytologic research a preliminary evaluation,” J. Histochem. Cytochem. 24, 100–111 (1976).
[CrossRef]

Kopeika, N. S.

Kovaci, S.

M. Kristan, J. Pers, M. Perse, and S. Kovaci, “A Bayes- spectral-entropy-based measure of camera focus using a discrete cosine transform,” Pattern Recog. Lett. 27, 1431–1439 (2006).
[CrossRef]

Kristan, M.

M. Kristan, J. Pers, M. Perse, and S. Kovaci, “A Bayes- spectral-entropy-based measure of camera focus using a discrete cosine transform,” Pattern Recog. Lett. 27, 1431–1439 (2006).
[CrossRef]

Lantzman, A.

Li, Z. H.

Z. H. Li and K. Y. Wu, “Autofocus system for space cameras,” Opt. Eng. 44, 053001 (2005).
[CrossRef]

Liu, J. H.

Z. B. Zhao and J. H. Liu, “Power spectra based auto-focusing method for airborne optoelectronic platform,” Acta Optica Sinica 30, 3495–3500 (2010).
[CrossRef]

Mattheij, R. M. M.

M. E. Rudnaya, R. M. M. Mattheij, J. M. L. Maubach, and H. G. ter Morsche, “Orientation identification of the power spectrum,” Opt. Eng. 50, 103201 (2011).
[CrossRef]

Maubach, J. M. L.

M. E. Rudnaya, R. M. M. Mattheij, J. M. L. Maubach, and H. G. ter Morsche, “Orientation identification of the power spectrum,” Opt. Eng. 50, 103201 (2011).
[CrossRef]

McQuillan, M.

H. C. Zhang, C. Gould, B. Roberts, and M. McQuillan, “An objective image focus monitor for CD-SEM,” Proc. SPIE 6922, 692226 (2008).
[CrossRef]

Mor, I.

Neurath, P. W.

J. F. Brenner, B. S. Dew, J. B. Horton, T. King, P. W. Neurath, and W. D. Selles, “An automated microscope for cytologic research a preliminary evaluation,” J. Histochem. Cytochem. 24, 100–111 (1976).
[CrossRef]

Nikzad, A.

M. Subbarao, T. Choi, and A. Nikzad, “A Focusing techniques,” Opt. Eng. 32, 2824–2836 (1993).
[CrossRef]

Nill, N. B.

N. B. Nill and B. H. Bouzas, “Objective image quality measure derived from digital image power spectra,” Opt. Eng. 31, 813–825 (1992).
[CrossRef]

Pando, J.

J. Pando and L. Z. Fang, “Discrete wavelet transform power spectrum estimator,” Phys. Rev. E 57, 3593–3601 (1998).
[CrossRef]

Pers, J.

M. Kristan, J. Pers, M. Perse, and S. Kovaci, “A Bayes- spectral-entropy-based measure of camera focus using a discrete cosine transform,” Pattern Recog. Lett. 27, 1431–1439 (2006).
[CrossRef]

Perse, M.

M. Kristan, J. Pers, M. Perse, and S. Kovaci, “A Bayes- spectral-entropy-based measure of camera focus using a discrete cosine transform,” Pattern Recog. Lett. 27, 1431–1439 (2006).
[CrossRef]

Ren, J. Y.

Roberts, B.

H. C. Zhang, C. Gould, B. Roberts, and M. McQuillan, “An objective image focus monitor for CD-SEM,” Proc. SPIE 6922, 692226 (2008).
[CrossRef]

Rudnaya, M. E.

M. E. Rudnaya, R. M. M. Mattheij, J. M. L. Maubach, and H. G. ter Morsche, “Orientation identification of the power spectrum,” Opt. Eng. 50, 103201 (2011).
[CrossRef]

Selles, W. D.

J. F. Brenner, B. S. Dew, J. B. Horton, T. King, P. W. Neurath, and W. D. Selles, “An automated microscope for cytologic research a preliminary evaluation,” J. Histochem. Cytochem. 24, 100–111 (1976).
[CrossRef]

Shack, R. V.

Simberová, S.

J. Kautsky, J. Flusser, B. Zitová, and Ŝ. Simberová, “A new wavelet-based measure of image focus,” Pattern Recog. Lett. 23, 1785–1794 (2002).
[CrossRef]

Subbarao, M.

M. Subbarao, T. Choi, and A. Nikzad, “A Focusing techniques,” Opt. Eng. 32, 2824–2836 (1993).
[CrossRef]

ter Morsche, H. G.

M. E. Rudnaya, R. M. M. Mattheij, J. M. L. Maubach, and H. G. ter Morsche, “Orientation identification of the power spectrum,” Opt. Eng. 50, 103201 (2011).
[CrossRef]

Tian, Y. B.

Vollath, D.

D. Vollath, “Automatic focusing by correlative methods,” J. Microsc. 147, 279–288 (1987).
[CrossRef]

Wang, J. Q.

Wen, C. Y.

Y. N. Zhang, Y. Zhang, and C. Y. Wen, “A new focus measure method using moments,” Image Vision Comput. 18, 959–965 (2000).
[CrossRef]

Wildsoet, C. F.

Wu, K. Y.

Z. H. Li and K. Y. Wu, “Autofocus system for space cameras,” Opt. Eng. 44, 053001 (2005).
[CrossRef]

Yan, C. X.

Yitzhaky, Y.

Yu, P.

Zhang, H. C.

H. C. Zhang, C. Gould, B. Roberts, and M. McQuillan, “An objective image focus monitor for CD-SEM,” Proc. SPIE 6922, 692226 (2008).
[CrossRef]

Zhang, Y.

Y. N. Zhang, Y. Zhang, and C. Y. Wen, “A new focus measure method using moments,” Image Vision Comput. 18, 959–965 (2000).
[CrossRef]

Zhang, Y. N.

Y. N. Zhang, Y. Zhang, and C. Y. Wen, “A new focus measure method using moments,” Image Vision Comput. 18, 959–965 (2000).
[CrossRef]

Zhao, Z. B.

Z. B. Zhao and J. H. Liu, “Power spectra based auto-focusing method for airborne optoelectronic platform,” Acta Optica Sinica 30, 3495–3500 (2010).
[CrossRef]

Zitová, B.

J. Kautsky, J. Flusser, B. Zitová, and Ŝ. Simberová, “A new wavelet-based measure of image focus,” Pattern Recog. Lett. 23, 1785–1794 (2002).
[CrossRef]

Acta Optica Sinica (1)

Z. B. Zhao and J. H. Liu, “Power spectra based auto-focusing method for airborne optoelectronic platform,” Acta Optica Sinica 30, 3495–3500 (2010).
[CrossRef]

Appl. Opt. (1)

Chin. Opt. Lett. (1)

IEEE Trans. Consumer Electron. (1)

C. M. Chen, C. M. Hong, and H. C. Chuang, “Efficient auto-focus algorithm utilizing discrete difference equation prediction model for digital still cameras,” IEEE Trans. Consumer Electron. 52, 1135–1143 (2006).
[CrossRef]

Image Vision Comput. (1)

Y. N. Zhang, Y. Zhang, and C. Y. Wen, “A new focus measure method using moments,” Image Vision Comput. 18, 959–965 (2000).
[CrossRef]

J. Histochem. Cytochem. (1)

J. F. Brenner, B. S. Dew, J. B. Horton, T. King, P. W. Neurath, and W. D. Selles, “An automated microscope for cytologic research a preliminary evaluation,” J. Histochem. Cytochem. 24, 100–111 (1976).
[CrossRef]

J. Microsc. (1)

D. Vollath, “Automatic focusing by correlative methods,” J. Microsc. 147, 279–288 (1987).
[CrossRef]

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

Opt. Eng. (4)

Z. H. Li and K. Y. Wu, “Autofocus system for space cameras,” Opt. Eng. 44, 053001 (2005).
[CrossRef]

M. E. Rudnaya, R. M. M. Mattheij, J. M. L. Maubach, and H. G. ter Morsche, “Orientation identification of the power spectrum,” Opt. Eng. 50, 103201 (2011).
[CrossRef]

N. B. Nill and B. H. Bouzas, “Objective image quality measure derived from digital image power spectra,” Opt. Eng. 31, 813–825 (1992).
[CrossRef]

M. Subbarao, T. Choi, and A. Nikzad, “A Focusing techniques,” Opt. Eng. 32, 2824–2836 (1993).
[CrossRef]

Opt. Express (2)

Pattern Recog. Lett. (2)

J. Kautsky, J. Flusser, B. Zitová, and Ŝ. Simberová, “A new wavelet-based measure of image focus,” Pattern Recog. Lett. 23, 1785–1794 (2002).
[CrossRef]

M. Kristan, J. Pers, M. Perse, and S. Kovaci, “A Bayes- spectral-entropy-based measure of camera focus using a discrete cosine transform,” Pattern Recog. Lett. 27, 1431–1439 (2006).
[CrossRef]

Phys. Rev. E (1)

J. Pando and L. Z. Fang, “Discrete wavelet transform power spectrum estimator,” Phys. Rev. E 57, 3593–3601 (1998).
[CrossRef]

Proc. SPIE (1)

H. C. Zhang, C. Gould, B. Roberts, and M. McQuillan, “An objective image focus monitor for CD-SEM,” Proc. SPIE 6922, 692226 (2008).
[CrossRef]

Other (2)

NASA Glenn Research Center, “A compact microscope imaging system with intelligent controls,” https://technology.grc.nasa.gov/tech-detail-coded.php?cid=GR-0047 .

Z. G. Fan, S. Q. Chen, H. L. Hu, H. Chang, and Q. Fu, “Autofocus algorithm based on wavelet packet transform for infrared microscopy,” in Proceedings of IEEE Conference on Image and Signal Processing (IEEE, 2010), pp. 2510–2514.

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 (8)

Fig. 1.
Fig. 1.

2D DWT for three images with different defocusing amount: (a) relative focusing code is 0; (b) relative focusing code is 30; and (c) relative focusing code is 60.

Fig. 2.
Fig. 2.

Image motion mismatching image and its 2D DWT.

Fig. 3.
Fig. 3.

Schematic diagram of 2D power spectrum to 1D conversion. The 1D power spectrum is generated by averaging the power contained within cycles of radial distance. The radial distance ρ=0.707cycles/pixel corresponds to the largest inscribed circle of the 2D power spectrum.

Fig. 4.
Fig. 4.

Image taken by some TDI CCD aeronautic camera.

Fig. 5.
Fig. 5.

Normalized power spectra for the four subpictures in Fig. 3: (a) FFT spectrum; (b) the proposed DWT spectrum.

Fig. 6.
Fig. 6.

Power spectra of focusing and defocusing images. (a) FFT power spectra; (b) DWT spectra. The bold lines in upside correspond to the four focusing subpictures, and the thin lines in downside correspond to their defocusing images. The sharp transitions at 0.707cycles/pixel are caused by the 2D power spectra to 1D conversion.

Fig. 7.
Fig. 7.

Experiment samples.

Fig. 8.
Fig. 8.

Comparison of two methods’ definition evaluation results for images with different scenes and defocusing amounts.

Tables (2)

Tables Icon

Table 1. Error Ratio for Two Methods

Tables Icon

Table 2. STD of Two Focus Measures

Equations (20)

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

ε(x,y)=I(x,y)I0I0,
ψ1(x,y)=φ(x)ϕ(y),ψ2(x,y)=ϕ(x)φ(y),ψ3(x,y)=ϕ(x)ϕ(y).
ψi,m;j,nl(x,y)={φi,m(x)φj,n(y),l=0ψi,ml(x)ψj,nl(y),l=1,2,3i,j0;i,j,m,nare all integer,
ψi,ml(x)=(2iM)1/2ψl(2iMxm),ψi,nl(y)=(2iN)1/2ψl(2iNyn).
ε(x,y)=i,j=0m,n=l=03wi,jl(m,n)ψi,m;j,nl(x,y).
wi,jl(m,n)=x=y=ε(x,y)ψi,m;j,nl(x,y),
1MNx=1My=1N|ε(x,y)|2=i,j=01MNm=02i1n=02j1l=03|wi,jl(m,n)|2.
Pi,j1MNm=02i1n=02j1l=03|wi,jl(m,n)|2.
MTFmatch=sin(π2NvcvNΔvPvP)π2NvcvNΔvPvP,
Pi,j1MNm=02i1n=02j1|wi,j1(m,n)|2.
F(u,v)=1MNx=1My=1Nε(x,y)e2πiyvNe2πixuM,
1MNx=1My=1N|ε(x,y)|2=u=v=|F(u,v)|2.
P(u,v)=|F(u,v)|2.
P(ρ)=log(1nρμ2MNθ=ππ|F(ρ,θ)|2),
I(x,y)=f(x,y)h(x,y),
h(x,y)={1/πR2,x2+y2R20,x2+y2>R2.
H(u,v)=exp[12ρ2(u,v)a2].
P(ρ)=1nρμ2MNθ=ππ|T(F(f(x,y)h(x,y)))|2=1nρμ2MNθ=ππ|T(F(f(x,y))×F(h(x,y)))|2=1nρμ2MNθ=ππT(|F(u,v)|2)×T(|H(u,v)|2).
QPSS(1)=ρ=0.050.5Pi,j.
QPSS(2)=ρ=0.050.5ρPi,j.

Metrics