Abstract

The problem of optically estimating an object’s position by using a charge-coupled device (CCD) array composed of square pixels Δx on a side is analyzed. The object’s image spot at the CCD is assumed to have a Gaussian intensity profile with a 1/e point at a radial distance of 2σs from the peak, and the CCD noise is modeled as Poisson-distributed, dark-current shot noise. A two-dimensional Cramér–Rao bound is developed and used to determine a lower limit for the mean-squared error of any unbiased position estimator, and the maximum-likelihood estimator is also derived. For the one-dimensional position-estimation problem the lower bound is shown to be minimum for a pixel-to-image size ratio Δx/σs of between 1 and 2 over a wide range of signal-to-noise ratios. Similarly for the two-dimensional problem, the optimum ratio is shown to lie between 1.5 and 2.5. As is customary in direct detection systems, it is also observed that the lower bound is a function of both the signal power and noise power separately and not just of their ratio. Finally, the maximum-likelihood estimator is shown to be independent of the signal and noise powers at high signal-to-noise ratios.

© 1986 Optical Society of America

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References

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  1. R. Stanton, J. Alexander, E. Dennison, T. Glavich, P. Salomon, R. Williamson, “ASTROS: a sub-arcsec CCD star tracker,” in State of the Art Imaging Arrays and their Applications, K. N. Prettyjohns, ed., Proc. Soc. Photo-Opt. Instrum. Eng.501, 256–282 (1984).
    [CrossRef]
  2. P. Salomon, T. Glavich, “Image signal processing in sub-pixel accuracy star trackers,” in Smart Sensors. II, B. S. Barbe, ed., Proc. Soc. Photo-Opt. Instrum. Eng.252, 64–74 (1980).
    [CrossRef]
  3. J. Cox, “Evaluation of peak location algorithms with subpixel accuracy for mosaic focal planes,” in Processing of Images and Data from Optical Sensors, W. H. Carter, ed., Proc. Soc. Photo-Opt. Instrum. Eng.292, 288–299 (1981).
    [CrossRef]
  4. E. Dennison, R. Stanton, “Ultra-precise star tracking using charged coupled devices (CCDs),” in Smart Sensors. II, B. S. Barbe, ed., Proc. Soc. Photo-Opt. Instrum. Eng.252, 54–63 (1980).
    [CrossRef]
  5. P. Salomon, W. Goss, “A microprocessor-controlled CCD star tracker,” presented at the AIAA 14th Aerospace Sciences Meeting, Washington, D.C., 1976.
  6. J. Kollodge, J. Sand, “Advanced star tracker design using the charge injection device,” presented at the Annual Rocky Mountain Guidance and Control Convention, Keystone, Colo., 1982.
  7. S. Grossman, R. Emmons, “Performance analysis and size optimization of focal planes for point-source tracking algorithm applications,” Opt. Eng. 23, 167–176 (1984).
    [CrossRef]
  8. D. Snyder, P. Fishman, “How to track a swarm of fireflies by observing their flashes,” IEEE Trans. Inf. Theory IT-10, 692–695 (1975).
    [CrossRef]
  9. C. Helstrom, “The detection and resolution of optical signals,” IEEE Trans. Inf. Theory IT-10, 275–287 (1964).
    [CrossRef]
  10. E. Farrell, C. Zimmerman, “Information limits of scanning optical systems,” in Optical and Electro-Optical Information Processing, J. T. Tippet, D. A. Berkowitz, L. C. Clapp, C. J. Koester, A. Vanderburgh, eds. (MIT Press, Cambridge, Mass., 1965), Chap. 35.
  11. T. McGarty, “The estimation of the center of gravity of a photon density profile in noise,” IEEE Trans. Aerosp. Electron. Syst. AES-5, 974–979 (1969).
    [CrossRef]
  12. M. Elbaum, P. Diament, “Estimation of image centroid, size and orientation with laser radar,” Appl. Opt. 16, 2433–2437 (1977).
    [CrossRef] [PubMed]
  13. M. Elbaum, M. Greenebaum, “Annular apertures for angular tracking,” Appl. Opt. 16, 2438–2440 (1977).
    [CrossRef] [PubMed]
  14. M. Elbaum, P. Diament, M. King, W. Edelson, “Maximum angular accuracy of pulsed laser radar in photocounting limit,” Appl. Opt. 16, 1982–1992 (1977).
    [CrossRef] [PubMed]
  15. D. Lubnau, “Maximum likelihood estimate of target angles for a conical scan tracking system in the presence of speckle,” Appl. Opt. 16, 184–186 (1977).
    [CrossRef] [PubMed]
  16. L. Kazovsky, “Beam position estimation by means of detector arrays,” Opt. Quantum Electron. 13, 201–208 (1981).
    [CrossRef]
  17. D. L. Snyder, Random Point Process (Wiley, New York, 1975).
  18. L. D. Vilesov, V. N. Veis, “Estimate of the angular position of a source of light when received by an array of photodetectors,” Izv. Vyssh. Uchebn. Zaved. Radioelektron. 26, 85–87 (1983).
  19. H. Van Trees, Detection, Estimation and Modulation Theory (Wiley, New York, 1968), Part I.
  20. C. W. Chen, “Optical Tracking Systems,” Ph.D. dissertation (Washington University, St. Louis, Missouri, 1982).
  21. C. W. Chen, D. L. Snyder, “A lower bound on the estimation performance for stochastic control systems,” IEEE Trans. Autom. Control AC-29, 577–559 (1984).

1984 (2)

S. Grossman, R. Emmons, “Performance analysis and size optimization of focal planes for point-source tracking algorithm applications,” Opt. Eng. 23, 167–176 (1984).
[CrossRef]

C. W. Chen, D. L. Snyder, “A lower bound on the estimation performance for stochastic control systems,” IEEE Trans. Autom. Control AC-29, 577–559 (1984).

1983 (1)

L. D. Vilesov, V. N. Veis, “Estimate of the angular position of a source of light when received by an array of photodetectors,” Izv. Vyssh. Uchebn. Zaved. Radioelektron. 26, 85–87 (1983).

1981 (1)

L. Kazovsky, “Beam position estimation by means of detector arrays,” Opt. Quantum Electron. 13, 201–208 (1981).
[CrossRef]

1977 (4)

1975 (1)

D. Snyder, P. Fishman, “How to track a swarm of fireflies by observing their flashes,” IEEE Trans. Inf. Theory IT-10, 692–695 (1975).
[CrossRef]

1969 (1)

T. McGarty, “The estimation of the center of gravity of a photon density profile in noise,” IEEE Trans. Aerosp. Electron. Syst. AES-5, 974–979 (1969).
[CrossRef]

1964 (1)

C. Helstrom, “The detection and resolution of optical signals,” IEEE Trans. Inf. Theory IT-10, 275–287 (1964).
[CrossRef]

Alexander, J.

R. Stanton, J. Alexander, E. Dennison, T. Glavich, P. Salomon, R. Williamson, “ASTROS: a sub-arcsec CCD star tracker,” in State of the Art Imaging Arrays and their Applications, K. N. Prettyjohns, ed., Proc. Soc. Photo-Opt. Instrum. Eng.501, 256–282 (1984).
[CrossRef]

Chen, C. W.

C. W. Chen, D. L. Snyder, “A lower bound on the estimation performance for stochastic control systems,” IEEE Trans. Autom. Control AC-29, 577–559 (1984).

C. W. Chen, “Optical Tracking Systems,” Ph.D. dissertation (Washington University, St. Louis, Missouri, 1982).

Cox, J.

J. Cox, “Evaluation of peak location algorithms with subpixel accuracy for mosaic focal planes,” in Processing of Images and Data from Optical Sensors, W. H. Carter, ed., Proc. Soc. Photo-Opt. Instrum. Eng.292, 288–299 (1981).
[CrossRef]

Dennison, E.

E. Dennison, R. Stanton, “Ultra-precise star tracking using charged coupled devices (CCDs),” in Smart Sensors. II, B. S. Barbe, ed., Proc. Soc. Photo-Opt. Instrum. Eng.252, 54–63 (1980).
[CrossRef]

R. Stanton, J. Alexander, E. Dennison, T. Glavich, P. Salomon, R. Williamson, “ASTROS: a sub-arcsec CCD star tracker,” in State of the Art Imaging Arrays and their Applications, K. N. Prettyjohns, ed., Proc. Soc. Photo-Opt. Instrum. Eng.501, 256–282 (1984).
[CrossRef]

Diament, P.

Edelson, W.

Elbaum, M.

Emmons, R.

S. Grossman, R. Emmons, “Performance analysis and size optimization of focal planes for point-source tracking algorithm applications,” Opt. Eng. 23, 167–176 (1984).
[CrossRef]

Farrell, E.

E. Farrell, C. Zimmerman, “Information limits of scanning optical systems,” in Optical and Electro-Optical Information Processing, J. T. Tippet, D. A. Berkowitz, L. C. Clapp, C. J. Koester, A. Vanderburgh, eds. (MIT Press, Cambridge, Mass., 1965), Chap. 35.

Fishman, P.

D. Snyder, P. Fishman, “How to track a swarm of fireflies by observing their flashes,” IEEE Trans. Inf. Theory IT-10, 692–695 (1975).
[CrossRef]

Glavich, T.

P. Salomon, T. Glavich, “Image signal processing in sub-pixel accuracy star trackers,” in Smart Sensors. II, B. S. Barbe, ed., Proc. Soc. Photo-Opt. Instrum. Eng.252, 64–74 (1980).
[CrossRef]

R. Stanton, J. Alexander, E. Dennison, T. Glavich, P. Salomon, R. Williamson, “ASTROS: a sub-arcsec CCD star tracker,” in State of the Art Imaging Arrays and their Applications, K. N. Prettyjohns, ed., Proc. Soc. Photo-Opt. Instrum. Eng.501, 256–282 (1984).
[CrossRef]

Goss, W.

P. Salomon, W. Goss, “A microprocessor-controlled CCD star tracker,” presented at the AIAA 14th Aerospace Sciences Meeting, Washington, D.C., 1976.

Greenebaum, M.

Grossman, S.

S. Grossman, R. Emmons, “Performance analysis and size optimization of focal planes for point-source tracking algorithm applications,” Opt. Eng. 23, 167–176 (1984).
[CrossRef]

Helstrom, C.

C. Helstrom, “The detection and resolution of optical signals,” IEEE Trans. Inf. Theory IT-10, 275–287 (1964).
[CrossRef]

Kazovsky, L.

L. Kazovsky, “Beam position estimation by means of detector arrays,” Opt. Quantum Electron. 13, 201–208 (1981).
[CrossRef]

King, M.

Kollodge, J.

J. Kollodge, J. Sand, “Advanced star tracker design using the charge injection device,” presented at the Annual Rocky Mountain Guidance and Control Convention, Keystone, Colo., 1982.

Lubnau, D.

McGarty, T.

T. McGarty, “The estimation of the center of gravity of a photon density profile in noise,” IEEE Trans. Aerosp. Electron. Syst. AES-5, 974–979 (1969).
[CrossRef]

Salomon, P.

P. Salomon, W. Goss, “A microprocessor-controlled CCD star tracker,” presented at the AIAA 14th Aerospace Sciences Meeting, Washington, D.C., 1976.

R. Stanton, J. Alexander, E. Dennison, T. Glavich, P. Salomon, R. Williamson, “ASTROS: a sub-arcsec CCD star tracker,” in State of the Art Imaging Arrays and their Applications, K. N. Prettyjohns, ed., Proc. Soc. Photo-Opt. Instrum. Eng.501, 256–282 (1984).
[CrossRef]

P. Salomon, T. Glavich, “Image signal processing in sub-pixel accuracy star trackers,” in Smart Sensors. II, B. S. Barbe, ed., Proc. Soc. Photo-Opt. Instrum. Eng.252, 64–74 (1980).
[CrossRef]

Sand, J.

J. Kollodge, J. Sand, “Advanced star tracker design using the charge injection device,” presented at the Annual Rocky Mountain Guidance and Control Convention, Keystone, Colo., 1982.

Snyder, D.

D. Snyder, P. Fishman, “How to track a swarm of fireflies by observing their flashes,” IEEE Trans. Inf. Theory IT-10, 692–695 (1975).
[CrossRef]

Snyder, D. L.

C. W. Chen, D. L. Snyder, “A lower bound on the estimation performance for stochastic control systems,” IEEE Trans. Autom. Control AC-29, 577–559 (1984).

D. L. Snyder, Random Point Process (Wiley, New York, 1975).

Stanton, R.

R. Stanton, J. Alexander, E. Dennison, T. Glavich, P. Salomon, R. Williamson, “ASTROS: a sub-arcsec CCD star tracker,” in State of the Art Imaging Arrays and their Applications, K. N. Prettyjohns, ed., Proc. Soc. Photo-Opt. Instrum. Eng.501, 256–282 (1984).
[CrossRef]

E. Dennison, R. Stanton, “Ultra-precise star tracking using charged coupled devices (CCDs),” in Smart Sensors. II, B. S. Barbe, ed., Proc. Soc. Photo-Opt. Instrum. Eng.252, 54–63 (1980).
[CrossRef]

Van Trees, H.

H. Van Trees, Detection, Estimation and Modulation Theory (Wiley, New York, 1968), Part I.

Veis, V. N.

L. D. Vilesov, V. N. Veis, “Estimate of the angular position of a source of light when received by an array of photodetectors,” Izv. Vyssh. Uchebn. Zaved. Radioelektron. 26, 85–87 (1983).

Vilesov, L. D.

L. D. Vilesov, V. N. Veis, “Estimate of the angular position of a source of light when received by an array of photodetectors,” Izv. Vyssh. Uchebn. Zaved. Radioelektron. 26, 85–87 (1983).

Williamson, R.

R. Stanton, J. Alexander, E. Dennison, T. Glavich, P. Salomon, R. Williamson, “ASTROS: a sub-arcsec CCD star tracker,” in State of the Art Imaging Arrays and their Applications, K. N. Prettyjohns, ed., Proc. Soc. Photo-Opt. Instrum. Eng.501, 256–282 (1984).
[CrossRef]

Zimmerman, C.

E. Farrell, C. Zimmerman, “Information limits of scanning optical systems,” in Optical and Electro-Optical Information Processing, J. T. Tippet, D. A. Berkowitz, L. C. Clapp, C. J. Koester, A. Vanderburgh, eds. (MIT Press, Cambridge, Mass., 1965), Chap. 35.

Appl. Opt. (4)

IEEE Trans. Aerosp. Electron. Syst. (1)

T. McGarty, “The estimation of the center of gravity of a photon density profile in noise,” IEEE Trans. Aerosp. Electron. Syst. AES-5, 974–979 (1969).
[CrossRef]

IEEE Trans. Autom. Control (1)

C. W. Chen, D. L. Snyder, “A lower bound on the estimation performance for stochastic control systems,” IEEE Trans. Autom. Control AC-29, 577–559 (1984).

IEEE Trans. Inf. Theory (2)

D. Snyder, P. Fishman, “How to track a swarm of fireflies by observing their flashes,” IEEE Trans. Inf. Theory IT-10, 692–695 (1975).
[CrossRef]

C. Helstrom, “The detection and resolution of optical signals,” IEEE Trans. Inf. Theory IT-10, 275–287 (1964).
[CrossRef]

Izv. Vyssh. Uchebn. Zaved. Radioelektron. (1)

L. D. Vilesov, V. N. Veis, “Estimate of the angular position of a source of light when received by an array of photodetectors,” Izv. Vyssh. Uchebn. Zaved. Radioelektron. 26, 85–87 (1983).

Opt. Eng. (1)

S. Grossman, R. Emmons, “Performance analysis and size optimization of focal planes for point-source tracking algorithm applications,” Opt. Eng. 23, 167–176 (1984).
[CrossRef]

Opt. Quantum Electron. (1)

L. Kazovsky, “Beam position estimation by means of detector arrays,” Opt. Quantum Electron. 13, 201–208 (1981).
[CrossRef]

Other (10)

D. L. Snyder, Random Point Process (Wiley, New York, 1975).

E. Farrell, C. Zimmerman, “Information limits of scanning optical systems,” in Optical and Electro-Optical Information Processing, J. T. Tippet, D. A. Berkowitz, L. C. Clapp, C. J. Koester, A. Vanderburgh, eds. (MIT Press, Cambridge, Mass., 1965), Chap. 35.

R. Stanton, J. Alexander, E. Dennison, T. Glavich, P. Salomon, R. Williamson, “ASTROS: a sub-arcsec CCD star tracker,” in State of the Art Imaging Arrays and their Applications, K. N. Prettyjohns, ed., Proc. Soc. Photo-Opt. Instrum. Eng.501, 256–282 (1984).
[CrossRef]

P. Salomon, T. Glavich, “Image signal processing in sub-pixel accuracy star trackers,” in Smart Sensors. II, B. S. Barbe, ed., Proc. Soc. Photo-Opt. Instrum. Eng.252, 64–74 (1980).
[CrossRef]

J. Cox, “Evaluation of peak location algorithms with subpixel accuracy for mosaic focal planes,” in Processing of Images and Data from Optical Sensors, W. H. Carter, ed., Proc. Soc. Photo-Opt. Instrum. Eng.292, 288–299 (1981).
[CrossRef]

E. Dennison, R. Stanton, “Ultra-precise star tracking using charged coupled devices (CCDs),” in Smart Sensors. II, B. S. Barbe, ed., Proc. Soc. Photo-Opt. Instrum. Eng.252, 54–63 (1980).
[CrossRef]

P. Salomon, W. Goss, “A microprocessor-controlled CCD star tracker,” presented at the AIAA 14th Aerospace Sciences Meeting, Washington, D.C., 1976.

J. Kollodge, J. Sand, “Advanced star tracker design using the charge injection device,” presented at the Annual Rocky Mountain Guidance and Control Convention, Keystone, Colo., 1982.

H. Van Trees, Detection, Estimation and Modulation Theory (Wiley, New York, 1968), Part I.

C. W. Chen, “Optical Tracking Systems,” Ph.D. dissertation (Washington University, St. Louis, Missouri, 1982).

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Figures (2)

Fig. 1
Fig. 1

Lower bound on performance of 1-D CCD optical position estimator.

Fig. 2
Fig. 2

Lower bound on performance of 2-D CCD optical position estimator.

Equations (60)

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[ ˆ x ( c ) x ] p ( c | x , y ) d c = 0 ,
x [ ˆ x ( c ) x ] p ( c | x , y ) d c = 0 ,
[ ˆ x ( c ) x ] p ( c | x , y ) x d c p ( c | x , y ) d c = 0 ,
[ ˆ x ( c ) x ] ln p ( c | x , y ) x p ( c | x , y ) d c = p ( c | x , y ) d c ,
[ ˆ x ( c ) x ] ln p ( c | x , y ) x p ( c | x , y ) d c = 1.
y [ ˆ x ( c ) x ] p ( c | x , y ) d c = 0 ,
[ ˆ x ( c ) x ] p ( c | x , y ) y d c = 0 , [ ˆ x ( c ) x ) ] ln p ( c | x , y ) y p ( c | x , y ) d c = 0.
Z Δ ¯ [ ˆ x ( c ) x ln p ( c | x , y ) x ] ,
W Δ ¯ [ ˆ x ( c ) x ln p ( c | x , y ) x ln p ( c | x , y ) y ] ,
a 1 Δ ¯ [ ˆ x ( c ) x ] ,
a 2 Δ ¯ ln p ( c | x , y ) x ,
a 3 Δ ¯ ln p ( c | x , y ) y .
E [ a 1 a 2 ] = 1 ,
E [ a 1 a 3 ] = 0 ,
det E [ ZZ T ] = | E [ a 1 2 ] 1 1 E [ a 2 2 ] | 0 ,
det E [ W W T ] = | E [ a 1 2 ] 1 0 1 E [ a 2 2 ] E [ a 2 a 3 ] 0 E [ a 3 a 2 ] E [ a 3 2 ] | > 0 ,
E [ a 1 2 ] E [ a 2 2 ] 1 , E [ ( ˆ x ( c ) x ) 2 ] 1 E [ ( ln p ( c | x , y ) x ) 2 ] .
E [ a 1 2 ] | E [ a 2 2 ] E [ a 2 a 3 ] E [ a 3 a 2 ] E [ a 3 2 ] | E [ a 3 2 ] 0
E [ ( ˆ x ( c ) x ) 2 ] E { [ ln p ( c | x , y ) y ] 2 } E { [ ln p ( c | x , y ) x ] 2 } E { [ ln p ( c | x , y ) y ] 2 } E { [ ln p ( c | x , y ) x ln p ( c | x , y ) y ] 2 } .
S ( x , y , x , y ) = ( 2 π σ s 2 ) 1 exp [ ( x x ) 2 2 σ s 2 ] exp [ ( y y ) 2 2 σ s 2 ] ,
S ( x , x ) = ( 2 π σ s 2 ) 1 / 2 exp [ ( x x ) 2 2 σ s 2 ] ,
S ( y , y ) = ( 2 π σ s 2 ) 1 / 2 exp [ ( y y ) 2 2 σ s 2 ] .
S ( x , y , x , y ) = S ( x , x ) S ( y , y ) .
g i j ( x , y ) = λ s g i ( x ) g j ( y ) ,
g i ( x ) Δ ¯ x i Δ x 2 x i + Δ x 2 S ( x , x ) d x ,
g i ( y ) Δ ¯ y j Δ x 2 y j + Δ x 2 S ( y , y ) d y
p ( c i | x ) = exp [ λ s g i ( x ) λ N ] [ λ s g i ( x ) + λ N ] c i c i ! ,
ln p ( c | x ) = ln i p ( c i | x ) = i [ λ s g i ( x ) + λ N ] j ln c i ! + i c i ln [ λ s g i ( x ) + λ N ] .
i g i ( x ) = 1 ,
Γ Δ ¯ ln p ( c | x ) x = i c i λ s g i ( x ) λ s g i ( x ) + λ N ,
g i ( x ) Δ ¯ x g i ( x ) .
E [ Γ 2 ] = i [ λ s g i ( x ) λ s g i ( x ) + λ N ] 2 E [ c i 2 ] + i j j i λ s g i ( x ) λ s g i ( x ) + λ N λ s g j ( x ) λ s g j ( x ) + λ N E [ c i ] E [ c j ] .
E [ c i ] = λ s g i ( x ) + λ N ,
E [ c i 2 ] = [ λ s g i ( x ) + λ N ] [ 1 + λ s g i ( x ) + λ N ] .
E [ Γ 2 ] = [ i λ s g i ( x ) ] 2 + i [ λ s g i ( x ) ] 2 λ s g i ( x ) + λ N .
i g i ( x ) = 0 ,
E [ ( ˆ x ( c ) x ) 2 ] 1 E [ Γ 2 ] = 1 i [ λ s g i ( x ) ] 2 λ s g i ( x ) + λ N .
Δ x 2 x Δ x 2
1 i [ λ s g i ( x ) ] 2 λ s g i ( x ) + λ N = 1 i [ λ s g i ( x + n Δ x ) ] 2 λ s g i ( x + n Δ x ) + λ N
lim L 1 L L / 2 L / 2 1 i [ λ s g i ( x ) 2 ] λ s g i ( x ) + λ N d x = 1 Δ x Δ x 2 Δ x 2 1 i [ λ s g i ( x ) 2 ] λ s g i ( x ) + λ N d x .
normalized rms error 1 - D = [ 1 Δ x Δ x / 2 Δ x / 2 E [ ( ˆ x ( c ) x ) 2 ] d x σ s ] 1 / 2 .
p ( c i j | x , y ) = exp [ λ s g i ( x ) g j ( y ) λ N ] [ λ s g i ( x ) g i ( y ) + λ N ] c i j c i j ! .
ln p ( c | x , y ) = ln i j p ( c i j | x , y ) = i j [ λ s g i ( x ) g j ( y ) + λ N ] i j ln c i j ! + i j c i j ln [ λ s g i ( x ) g j ( y ) + λ N ] .
i j g i ( x ) g j ( y ) = 1 ,
Q x Δ ¯ ln p ( c | x , y ) x = i j c i j λ s g i ( x ) g j ( y ) λ s g i ( x ) g j ( y ) + λ N ,
Q y Δ ¯ ln p ( c | x , y ) y = i j c i j λ s g i ( x ) g j ( y ) λ s g i ( x ) g j ( y ) + λ N ,
g i ( x ) Δ ¯ x g i ( x )
g j ( y ) Δ ¯ y g j ( y ) .
E [ Q x 2 ] = i j [ λ s g i ( x ) g j ( y ) λ s g i ( x ) g j ( y ) + λ N ] 2 E [ c i j 2 ] + i j n m n m i j λ s g i ( x ) g j ( y ) λ s g i ( x ) g j ( y ) + λ N × λ s g n ( x ) g m ( y ) λ s g n ( x ) g m ( y ) + λ N E [ c i j ] E [ c n m ] ,
E [ Q x Q y ] = i j λ s g i ( x ) g j ( y ) λ s g i ( x ) g j ( y ) + λ N λ s g i ( x ) g j ( y ) λ s g i ( x ) g j ( y ) + λ N E [ c i j 2 ] + i j n m n m i j λ s g i ( x ) g j ( y ) λ s g i ( x ) g j ( y ) + λ N × λ s g n ( x ) g m ( y ) λ s g n ( x ) g m ( y ) + λ N E [ c i j ] E [ c n m ] ,
E [ Q y 2 ] = i j [ λ s g i ( x ) g j ( y ) λ s g i ( x ) g j ( y ) + λ N ] 2 E [ c i j 2 ] + i j n m n m i j λ s g i ( x ) g j ( y ) λ s g i ( x ) g j ( y ) + λ N + λ s g n ( x ) g m ( y ) λ s g n ( x ) g m ( y ) + λ N E [ c i j ] E [ c n m ] .
E [ c i j ] = λ s g i ( x ) g j ( y ) + λ N ,
E [ c i j 2 ] = [ λ s g i ( x ) g j ( y ) + λ N ] [ 1 + λ s g i ( x ) g j ( y ) + λ N ] .
E [ Q x 2 ] = [ i j λ s g i ( x ) g j ( y ) ] 2 + i j [ λ s g i ( x ) g j ( y ) ] 2 λ s g i ( x ) g j ( y ) + λ N ,
E [ Q x Q y ] = i j λ s g i ( x ) g j ( y ) n m λ s g n ( x ) g m ( y ) + i j λ s 2 g i ( x ) g j ( y ) g i ( x ) g j ( y ) λ s g i ( x ) g j ( y ) + λ N ,
E [ Q y 2 ] = [ i j λ s g i ( x ) g j ( y ) ] 2 + i j [ λ s g i ( x ) g j ( y ) ] 2 λ s g i ( x ) g j ( y ) + λ N .
i j g i ( x ) g j ( y ) = i j g i ( x ) g j ( y ) = 0 ,
E [ ( ˆ x ( c ) x ) 2 ] E [ Q y 2 ] E [ Q x 2 ] E [ Q y 2 ] ( E [ Q x Q y ] ) 2 = { i j [ λ s g i ( x ) g j ( y ) ] 2 λ s g i ( x ) g j ( y ) + λ N } / { i j [ λ s g i ( x ) g j ( y ) ] 2 λ s g i ( x ) g j ( y ) + λ N i j [ λ s g j ( x ) g j ( y ) ] 2 λ s g i ( x ) g j ( y ) + λ N [ i j λ s 2 g i ( x ) g j ( y ) g i ( x ) g j ( y ) λ s g i ( x ) g j ( y ) + λ N ] 2 } = λ s 1 i j [ g i ( x ) g j ( y ) ] 2 g i ( x ) g j ( y ) + ( λ s / λ N ) 1 [ i j g i ( x ) g j ( y ) g i ( x ) g j ( y ) g i ( x ) g j ( y ) + ( λ s / λ N ) 1 ] 2 i j g i ( x ) g j ( y ) g i ( x ) g j ( y ) + ( λ s / λ N ) 1 .
Δ x 2 x Δ x 2 , Δ x 2 y Δ x 2 .
= [ 1 ( Δ x ) 2 Δ x / 2 Δ x / 2 Δ x / 2 Δ x / 2 E [ ˆ x ( c ) x ) 2 ] d x d y ] 1 / 2 σ s .

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