Abstract

We extend recent results for estimating the parameters of a one-dimensional Gaussian profile to two-dimensional profiles, deriving the exact covariance matrix of the estimated parameters. While the exact form is easy to compute, we provide a set of close approximations that allow the covariance to take on a simple analytic form. This not only provides new insight into the behavior of the estimation parameters, but also lays a foundation for clarifying previously published work. We also show how to calculate the parameter variances for the case of truncated sampling, where the profile lies near the edge of the array detector. Finally, we calculate expressions for the bias in the classical formulation of the problem and provide an approach for its removal. This allows us to show how the bias affects the problem of choosing an optimal pixel size for minimizing parameter variances.

© 2008 Optical Society of America

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References

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  1. N. Hagen, M. Kupinski, and E. L. Dereniak, “Gaussian profile estimation in one dimension,” Appl. Opt. 46, 5374-5383(2007).
    [CrossRef] [PubMed]
  2. L. G. Kazovsky, “Beam position estimation by means of detector arrays,” Opt. Quantum Electron. 13, 201-208(1981).
    [CrossRef]
  3. L. H. Auer and W. F. van Altena, “Digital image centering II,” Astron. J. 83, 531-537 (1978).
    [CrossRef]
  4. R. Irwan and R. G. Lane, “Analysis of optimal centroid estimation applied to Shack-Hartmann sensing,” Appl. Opt. 38, 6737-6743 (1999).
    [CrossRef]
  5. M. K. Cheezum, W. F. Walker, and W. H. Guilford, “Quantitative comparison of algorithms for tracking single fluorescent particles,” Biophys. J. 81, 2378-2388 (2001).
    [CrossRef] [PubMed]
  6. R. E. Thompson, D. R. Larson, and W. W. Webb, “Precise nanometer localization analysis for individual fluorescent probes,” Biophys. J. 82, 2775-2783 (2002).
    [CrossRef] [PubMed]
  7. Y.-C. Chen, L. R. Furenlid, D. W. Wilson, and H. H. Barrett, “Calibration of scintillation cameras and pinhole SPECT imaging systems,” in Small-Animal SPECT Imaging, M. A. Kupinski and H. H. Barrett, eds. (Springer, 2005), Chap. 12, pp. 195-202.
    [CrossRef]
  8. S. V. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory (Prentice Hall, 1993), p. 34.
  9. What we call “flat noise” here was labeled “uniform noise” in . This has been changed in recognition that this can easily be mistaken as referring to noise obtained from the uniform probability distribution.
  10. It may seem that σ2mgm/Qm is the appropriate substitution here, but since the raw measurements gm are scaled versions of the Poisson-distributed photoelectrons, the variance increases as the square of the scale parameter, Qm.
  11. S. Brandt, Data Analysis, 3rd ed., (Springer1999), p. 113-114.
  12. K. A. Winick, “Cramér-Rao lower bounds on the performance of charge-coupled-device optical position estimators,” J. Opt. Soc. Am. A 3, 1809-1815 (1986).
    [CrossRef]
  13. H. H. Barrett, C. Dainty, and D. Lara, “Maximum-likelihood methods in wavefront sensing: stochastic models and likelihood functions,” J. Opt. Soc. Am. A 24, 391-414 (2007).
    [CrossRef]
  14. The Interactive Data Language software is developed by ITT Visual Information Systems, http://www.ittvis.com/index.asp.

2007 (2)

2002 (1)

R. E. Thompson, D. R. Larson, and W. W. Webb, “Precise nanometer localization analysis for individual fluorescent probes,” Biophys. J. 82, 2775-2783 (2002).
[CrossRef] [PubMed]

2001 (1)

M. K. Cheezum, W. F. Walker, and W. H. Guilford, “Quantitative comparison of algorithms for tracking single fluorescent particles,” Biophys. J. 81, 2378-2388 (2001).
[CrossRef] [PubMed]

1999 (1)

1986 (1)

1981 (1)

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

1978 (1)

L. H. Auer and W. F. van Altena, “Digital image centering II,” Astron. J. 83, 531-537 (1978).
[CrossRef]

Auer, L. H.

L. H. Auer and W. F. van Altena, “Digital image centering II,” Astron. J. 83, 531-537 (1978).
[CrossRef]

Barrett, H. H.

H. H. Barrett, C. Dainty, and D. Lara, “Maximum-likelihood methods in wavefront sensing: stochastic models and likelihood functions,” J. Opt. Soc. Am. A 24, 391-414 (2007).
[CrossRef]

Y.-C. Chen, L. R. Furenlid, D. W. Wilson, and H. H. Barrett, “Calibration of scintillation cameras and pinhole SPECT imaging systems,” in Small-Animal SPECT Imaging, M. A. Kupinski and H. H. Barrett, eds. (Springer, 2005), Chap. 12, pp. 195-202.
[CrossRef]

Brandt, S.

S. Brandt, Data Analysis, 3rd ed., (Springer1999), p. 113-114.

Cheezum, M. K.

M. K. Cheezum, W. F. Walker, and W. H. Guilford, “Quantitative comparison of algorithms for tracking single fluorescent particles,” Biophys. J. 81, 2378-2388 (2001).
[CrossRef] [PubMed]

Chen, Y.-C.

Y.-C. Chen, L. R. Furenlid, D. W. Wilson, and H. H. Barrett, “Calibration of scintillation cameras and pinhole SPECT imaging systems,” in Small-Animal SPECT Imaging, M. A. Kupinski and H. H. Barrett, eds. (Springer, 2005), Chap. 12, pp. 195-202.
[CrossRef]

Dainty, C.

Dereniak, E. L.

Furenlid, L. R.

Y.-C. Chen, L. R. Furenlid, D. W. Wilson, and H. H. Barrett, “Calibration of scintillation cameras and pinhole SPECT imaging systems,” in Small-Animal SPECT Imaging, M. A. Kupinski and H. H. Barrett, eds. (Springer, 2005), Chap. 12, pp. 195-202.
[CrossRef]

Guilford, W. H.

M. K. Cheezum, W. F. Walker, and W. H. Guilford, “Quantitative comparison of algorithms for tracking single fluorescent particles,” Biophys. J. 81, 2378-2388 (2001).
[CrossRef] [PubMed]

Hagen, N.

Irwan, R.

Kay,

S. V. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory (Prentice Hall, 1993), p. 34.

Kazovsky, L. G.

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

Kupinski, M.

Lane, R. G.

Lara, D.

Larson, D. R.

R. E. Thompson, D. R. Larson, and W. W. Webb, “Precise nanometer localization analysis for individual fluorescent probes,” Biophys. J. 82, 2775-2783 (2002).
[CrossRef] [PubMed]

Thompson, R. E.

R. E. Thompson, D. R. Larson, and W. W. Webb, “Precise nanometer localization analysis for individual fluorescent probes,” Biophys. J. 82, 2775-2783 (2002).
[CrossRef] [PubMed]

van Altena, W. F.

L. H. Auer and W. F. van Altena, “Digital image centering II,” Astron. J. 83, 531-537 (1978).
[CrossRef]

Walker, W. F.

M. K. Cheezum, W. F. Walker, and W. H. Guilford, “Quantitative comparison of algorithms for tracking single fluorescent particles,” Biophys. J. 81, 2378-2388 (2001).
[CrossRef] [PubMed]

Webb, W. W.

R. E. Thompson, D. R. Larson, and W. W. Webb, “Precise nanometer localization analysis for individual fluorescent probes,” Biophys. J. 82, 2775-2783 (2002).
[CrossRef] [PubMed]

Wilson, D. W.

Y.-C. Chen, L. R. Furenlid, D. W. Wilson, and H. H. Barrett, “Calibration of scintillation cameras and pinhole SPECT imaging systems,” in Small-Animal SPECT Imaging, M. A. Kupinski and H. H. Barrett, eds. (Springer, 2005), Chap. 12, pp. 195-202.
[CrossRef]

Winick, K. A.

Appl. Opt. (2)

Astron. J. (1)

L. H. Auer and W. F. van Altena, “Digital image centering II,” Astron. J. 83, 531-537 (1978).
[CrossRef]

Biophys. J. (2)

M. K. Cheezum, W. F. Walker, and W. H. Guilford, “Quantitative comparison of algorithms for tracking single fluorescent particles,” Biophys. J. 81, 2378-2388 (2001).
[CrossRef] [PubMed]

R. E. Thompson, D. R. Larson, and W. W. Webb, “Precise nanometer localization analysis for individual fluorescent probes,” Biophys. J. 82, 2775-2783 (2002).
[CrossRef] [PubMed]

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

Opt. Quantum Electron. (1)

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

Other (6)

The Interactive Data Language software is developed by ITT Visual Information Systems, http://www.ittvis.com/index.asp.

Y.-C. Chen, L. R. Furenlid, D. W. Wilson, and H. H. Barrett, “Calibration of scintillation cameras and pinhole SPECT imaging systems,” in Small-Animal SPECT Imaging, M. A. Kupinski and H. H. Barrett, eds. (Springer, 2005), Chap. 12, pp. 195-202.
[CrossRef]

S. V. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory (Prentice Hall, 1993), p. 34.

What we call “flat noise” here was labeled “uniform noise” in . This has been changed in recognition that this can easily be mistaken as referring to noise obtained from the uniform probability distribution.

It may seem that σ2mgm/Qm is the appropriate substitution here, but since the raw measurements gm are scaled versions of the Poisson-distributed photoelectrons, the variance increases as the square of the scale parameter, Qm.

S. Brandt, Data Analysis, 3rd ed., (Springer1999), p. 113-114.

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Figures (7)

Fig. 1
Fig. 1

Example noiseless symmetric 2D Gaussian profile, sampled on a pixel grid.

Fig. 2
Fig. 2

Example noiseless asymmetric 2D Gaussian profile, sampled on a pixel grid.

Fig. 3
Fig. 3

Example noiseless general 2D Gaussian profile, sampled on a pixel grid. (Note that the axis of the Gaussian is not aligned to the grid.)

Fig. 4
Fig. 4

Truncated sampling of a Gaussian profile, showing a case in which 60% of the volume under the profile lies outside of the region falling on the array detector.

Fig. 5
Fig. 5

The change in variance of estimation parameters as the sampling of the profile is increasingly truncated. The truncation parameter t is defined simply as the fraction of volume under the Gaussian profile which lies outside of the sampling region. In the figures, lines indicate the parameter variances as calculated numerically from the exact Fisher information matrix. (The analytic approximations are not valid for the case of a truncated profile.) Dots indicate variance estimates from a Monte Carlo simulation. The variances have been normalized to the completely sampled data’s parameter variances. In both figures, the value of x ¯ is allowed to vary while all other model parameters are held constant.

Fig. 6
Fig. 6

A comparison of var ( x ¯ ^ ) and bias 2 ( x ¯ ^ ) , given in units of ( pixels ) 2 , for an example symmetric Gaussian profile as a function of the profile width w. Keeping a constant value for U, we increase the profile width (and thus reduce the profile height A) while maintaining the same sampling rate. The approximate variance is calculated by Eq. (9) and the exact variance by taking the inverse of Eq. (5). The two curves shown for the variance are for two different values of U.

Fig. 7
Fig. 7

Contour maps showing slices through the 4D likelihood function for (a) a well-behaved model and (b) a poorly behaved one. The parameters are ( A , x ¯ , y ¯ , w ) = ( 100 , 0.25 , 0 , 3 ) and (10,0.25,0,0.5) respectively, where x ¯ , y ¯ , and w are given in units of pixel widths.

Equations (44)

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θ ( k + 1 ) = θ ( k ) ( H ( k ) ) 1 ( k ) ,
f ( r ) = A e ( r r ¯ ) 2 / 2 w 2 ,
g ¯ m = δ x δ y Q m f ( x ) δ ( x x m ) δ ( y y m ) d x d y = δ x δ y Q m A e ( r m r ¯ ) 2 / 2 w 2 .
pr ( g m ) = 1 2 π σ m 2 e ( g m g ¯ m ) 2 / 2 σ m 2 ,
L ( θ | g ) = pr ( g | θ ) .
θ i = 1 σ m 2 ( g m g ¯ m ) g ¯ m θ i .
A = γ m , x ¯ = A w 2 γ m ξ m , y ¯ = A w 2 γ m η m , w = A w 3 γ m ρ m 2 ,
ξ m = ( x m x ¯ ) , η m = ( y m y ¯ ) , ρ m = ξ m 2 + η m 2 , γ m = δ x δ y σ m 2 ( g m g ¯ m ) Q m E m , E m = e ( ξ m 2 + η m 2 ) / 2 w 2 .
H = 2 A 2 2 A x ¯ 2 A y ¯ 2 A w 2 x ¯ A 2 x ¯ 2 2 x ¯ y ¯ 2 x ¯ w 2 y ¯ A 2 y ¯ x ¯ 2 y ¯ 2 2 y ¯ w 2 w A 2 w x ¯ 2 w y ¯ 2 w 2 .
H 11 = δ x 2 δ y 2 1 σ m 2 Q m 2 E m 2 H 12 = 1 w 2 Γ m ξ m H 13 = 1 w 2 Γ m η m H 14 = 1 w 3 Γ m ρ m 2 H 22 = A w 4 Γ m ξ m 2 γ m w 2 H 23 = A w 4 Γ m ξ m η m H 24 = A w 5 Γ m ξ m ρ m 2 2 γ m ξ m w 2 H 33 = A w 4 Γ m η m 2 γ m w 2 H 34 = A w 5 Γ m η m ρ m 2 2 γ m η m w 2 H 44 = A w 6 Γ m ρ m 4 3 γ m ρ m 2 w 2 ,
F i j = E { H i j } = ( θ ) ( 2 θ i θ j ) d M g ,
F 11 = R m F 23 = A 2 w 4 R m ξ m η m F 12 = A w 2 R m ξ m F 24 = A 2 w 5 R m ρ m 2 ξ m F 13 = A w 2 R m η m F 33 = A 2 w 4 R m η m 2 F 14 = A w 3 R m ρ m 2 F 34 = A 2 w 5 R m ρ m 2 η m F 22 = A 2 w 4 R m ξ m 2 F 44 = A 2 w 6 R m ρ m 4 }
E m 2 = 1 δ x δ y e ( ξ m 2 + η m 2 ) / w 2 δ ξ δ η 1 δ x δ y e ξ 2 / w 2 d ξ e η 2 / w 2 d η = π w 2 δ x δ y
E m 2 ξ m 0 E m 2 ξ m η m 0 E m 2 ξ m 2 π w 4 2 δ x δ y E m 2 ρ m 2 π w 4 δ x δ y E m 2 ρ m 4 2 π w 6 δ x δ y } .
K Flat σ 2 π δ x δ y Q 2 ( 2 w 2 0 0 1 Aw 0 2 A 2 0 0 0 0 2 A 2 0 1 Aw 0 0 1 A 2 ) .
K Poisson 1 2 π ( 2 A w 2 0 0 1 2 w 0 1 A 0 0 0 0 1 A 0 1 2 w 0 0 1 4 A ) .
var ( A ^ ) { 2 β / w 2 ( Flat ) A / ( 2 π w 2 ) ( Poisson ) , var ( x ¯ ^ ) = var ( y ¯ ^ ) { 2 β / A 2 ( Flat ) 1 / ( 2 π A ) ( Poisson) , var ( w ^ ) { β / A 2 ( Flat ) 1 / ( 8 π A ) ( Poisson ) ,
U = A d ξ d η e ( ξ 2 + η 2 ) / 2 w 2 = 2 π A w 2 ,
var ( U ^ ) = U θ T K θ U θ { 8 π 2 β w 2 ( Flat ) 2 π A w 2 ( Poisson ) .
f ( r ) = A e ( x x ¯ ) 2 / 2 w x 2 e ( y y ¯ ) 2 / 2 w y 2 ,
[ ] 1 = A = Σ γ m [ ] 2 = x ¯ = A w x 2 Σ γ m ξ m [ ] 3 = y ¯ = A w y 2 Σ γ m η m [ ] 4 = w x = A w x 3 Σ γ m ξ m 2 [ ] 5 = w y = A w y 3 Σ γ m η m 2 ,
H 11 = δ x 2 δ y 2 Q m 2 E m 2 σ m 2 H 12 = 1 w x 2 Γ m ξ m H 13 = 1 w y 2 Γ m η m H 14 = 1 w x 3 Γ m ξ m 2 H 15 = 1 w y 3 Γ m η m 2 H 22 = A w x 4 Γ m ξ m 2 γ m w x 2 H 23 = A w x 2 w y 2 Γ m ξ m η m H 24 = A w x 5 Γ m ξ m 3 2 γ m ξ m w x 2 H 25 = A w x 2 w y 3 Γ m ξ m η m 2 H 33 = A w y 4 Γ m η m 2 γ m w y 2 H 34 = A w x 3 w y 2 Γ m ξ m 2 η m H 35 = A w y 5 Γ m η m 3 2 γ m η m w y 2 H 44 = A w x 6 Γ m ξ m 4 3 γ m ξ m 2 w x 2 H 45 = A w x 3 w y 3 Γ m ξ m 2 η m 2 H 55 = A w y 6 Γ m η m 4 3 γ m η m 2 w y 2 ,
F 11 = R m F 12 = A w x 2 R m ξ m F 25 = A 2 w x 2 w y 3 R m ξ m η m 2 F 13 = A w y 2 R m η m F 33 = A 2 w y 4 R m η m 2 F 14 = A w x 3 R m ξ m 2 F 34 = A 2 w x 3 w y 2 R m ξ 2 η m F 15 = A w y 3 R m η m 2 F 35 = A 2 w y 5 R m η m 3 F 22 = A 2 w x 4 R m ξ m 2 F 44 = A 2 w x 6 R m ξ m 4 F 23 = A 2 w x 2 w y 2 R m ξ m η m F 45 = A 2 w x 3 w y 3 R m ξ m 2 η m 2 F 24 = A 2 w x 5 R m ξ m 3 F 55 = A 2 w y 6 R m η m 4 ,
E m 2 π w x w y δ x δ y E m 2 ξ m 2 π w x 3 w y 2 δ x δ y E m 2 η m 2 π w x w y 3 2 δ x δ y E m 2 ξ m 2 η m 2 π w x 3 w y 3 4 δ x δ y E m 2 ξ m 4 3 π w x 5 w y 4 δ x δ y E m 2 η m 4 3 π w x w y 5 4 δ x δ y } .
K Flat σ 2 π δ x δ y Q 2 ( 2 w x w y 0 0 1 A w y 1 A w x 0 2 w x A 2 w y 0 0 0 0 0 2 w y A 2 w x 0 0 1 A w y 0 0 2 w x A 2 w y 0 1 A w x 0 0 0 2 w y A 2 w x ) , K Poisson 1 2 π ( 3 A w x w y 0 0 1 w y 1 w x 0 w x A w y 0 0 0 0 0 w y A w x 0 0 1 w y 0 0 2 w x 3 A w y 1 3 A 1 w x 0 0 1 3 A 2 w y 3 A w x ) .
var ( U ^ ) { 8 π σ 2 w x w y δ x δ y Q 2 ( Flat ) 2 π A w x w y ( Poisson ) .
g ¯ m = A δ x δ y Q m exp [ 1 2 ( r m r ¯ ) T C 1 ( r m r ¯ ) ] A δ x δ y E m ,
C 1 = ( c 1 c 3 c 3 c 2 ) ,
[ ] 1 = A = Σ γ m [ ] 2 = x ¯ = A Σ γ m ( c 1 ξ m + c 3 η m ) [ ] 3 = y ¯ = A Σ γ m ( c 2 η m + c 3 ξ m ) [ ] 4 = c 1 = A 2 Σ γ m ξ m 2 [ ] 5 = c 2 = A 2 Σ γ m η m 2 [ ] 6 = c 3 = A Σ γ m ξ m η m ,
H 11 = δ x 2 δ y 2 Q m 2 E m 2 σ m 2 H 12 = Γ m ( c 1 ξ m + c 3 η m ) H 13 = Γ m ( c 2 η m + c 3 ξ m ) H 14 = 1 2 Γ m ξ m 2 H 15 = 1 2 Γ m η m 2 H 16 = Γ m ξ m η m H 22 = A Γ m ( c 1 ξ m + c 3 η m ) 2 c 1 γ m H 23 = A Γ m ( c 1 ξ m + c 3 η m ) ( c 2 η m + c 3 ξ m ) c 3 γ m H 24 = 1 2 A Γ m ( c 1 ξ m + c 3 η m ) ξ m 2 2 γ m ξ m H 25 = 1 2 A Γ m ( c 1 ξ m + c 3 η m ) η m 2 H 26 = A Γ m ( c 1 ξ m + c 3 η m ) ξ m η m γ m η m H 33 = A Γ m ( c 2 η m + c 3 ξ m ) 2 c 2 γ m H 34 = 1 2 A Γ m ( c 2 η m + c 3 ξ m ) ξ m 2 H 35 = 1 2 A Γ m ( c 2 η m + c 3 ξ m ) η m 2 2 γ m η m H 36 = A Γ m ( c 2 η m + c 3 ξ m ) ξ m η m γ m ξ m H 44 = A 4 Γ m ξ m 4 H 45 = A 4 Γ m ξ m 2 η m 2 H 46 = A 2 Γ m ξ m 3 η m H 55 = A 4 Γ m η m 4 H 56 = A 2 Γ m ξ m η m 3 H 66 = A Γ m ξ m 2 η m 2 ,
F 11 = R m F 12 = A R m ( c 1 ξ m + c 3 η m ) F 13 = A R m ( c 2 η m + c 3 ξ m ) F 14 = A 2 R m ξ m 2 F 15 = A 2 R m η m 2 F 16 = A R m ξ m η m F 22 = A 2 R m ( c 1 ξ m + c 3 η m ) 2 F 23 = A 2 R m ( c 1 ξ m + c 3 η m ) ( c 2 η m + c 3 ξ m ) F 24 = A 2 2 R m ( c 1 ξ m + c 3 η m ) ξ m 2 F 25 = A 2 2 R m ( c 1 ξ m + c 3 η m ) η m 2 F 26 = A 2 R m ( c 1 ξ m + c 3 η m ) ξ m η m F 33 = A 2 R m ( c 2 η m + c 3 ξ m ) 2 F 34 = A 2 2 R m ( c 2 η m + c 3 ξ m ) ξ m 2 F 35 = A 2 2 R m ( c 2 η m + c 3 ξ m ) η m 2 F 36 = A 2 R m ( c 2 η m + c 3 ξ m ) ξ m η m F 44 = A 2 4 R m ξ m 4 F 45 = A 2 4 R m ξ m 2 η m 2 F 46 = A 2 2 R m ξ m 3 η m F 55 = A 2 4 R m η m 4 F 56 = A 2 2 R m ξ m η m 3 F 66 = A 2 R m ξ m 2 η m 2 ,
K F 2 σ 2 π δ x δ y Q 2 ( 0 0 A A A 0 c 2 A 2 μ c 3 A 2 μ 0 0 0 0 c 3 A 2 μ c 1 A 2 μ 0 0 0 A 0 0 A 2 A 2 A 2 A 0 0 A 2 A 2 A 2 A 0 0 A 2 A 2 A 2 ) , K P μ 2 π ( 2 A 0 0 c 1 c 2 c 3 0 c 2 A μ 2 c 3 A μ 2 0 0 0 0 c 3 A μ 2 c 1 A μ 2 0 0 0 c 1 0 0 2 c 1 2 A 2 c 3 2 A 2 c 1 c 3 A c 2 0 0 2 c 3 2 A 2 c 2 2 A 2 c 2 c 3 A c 3 0 0 2 c 1 c 3 A 2 c 2 c 3 A c 1 c 2 + c 3 2 A ) ,
var ( U ^ ) { σ 2 δ x δ y Q 2 ( Flat ) A ( Poisson ) .
f ^ unbiased ( x , y ) = f ^ biased ( x , y ) + b ( x , y ) .
g ¯ m = Q m f ( x , y ) × rect ( x x m δ x ) rect ( y y m δ y ) d x d y ,
rect ( x / L ) = { 1 | x | < L / 2 0 | x | > L / 2 .
f ( x , y ) f ( x m , y m ) + ( x x m ) [ f x ] x = x m y = y m + ( y y m ) [ f y ] x = x m y = y m + 1 2 ( ( x x m ) 2 [ 2 f x 2 ] x = x m y = y m + 2 ( x x m ) ( y y m ) [ 2 f x y ] x = x m y = y m + ( y y m ) 2 [ 2 f y 2 ] x = x m y = y m ) .
b ( r m ) x m 1 2 δ x x m + 1 2 δ x y m 1 2 δ y y m + 1 2 δ y [ second - order   term ] d x d y .
b ( r m ) A Q m E ( r m ) 24 w 2 ( [ ( x m x ¯ ) 2 w 2 1 ] δ x 2 + [ ( y m y ¯ ) 2 w 2 1 ] δ y 2 ) .
b ( r m ) A Q m E ( r m ) 24 ( [ ( x m x ¯ ) 2 w x 2 1 ] δ x 2 w x 2 + [ ( y m y ¯ ) 2 w y 2 1 ] δ y 2 w y 2 ) .
b ( r m ) A Q m E ( r m ) 24 [ ( [ c 1 ( x x ¯ ) + c 3 ( y y ¯ ) ] 2 c 1 ) δ x 2 + ( [ c 2 ( y y ¯ ) + c 3 ( x x ¯ ) ] 2 c 2 ) δ y 2 ] .
U = { 2 π A w 2 ( 4   params ) 2 π A w x w y ( 5   params ) 2 π A / μ ( 6   params ) ,
( 4   params ) : var ( x ¯ ^ ) { 8 π σ 2 w 4 δ x δ y Q 2 U 2 ( Flat ) 4 w 2 U ( Poisson ) ( 5   params ) : var ( x ¯ ^ ) { 8 π σ 2 w x 3 w y δ x δ y Q 2 U 2 ( Flat ) 4 w x 2 U ( Poisson ) ( 6   params ) : var ( x ¯ ^ ) { 2 π σ 2 c 2 δ x δ y μ 3 Q 2 U 2 ( Flat ) 2 c 1 U ( Poisson ) .
F i j m = 1 M 1 σ G 2 + g ¯ m Q m g ¯ m θ i g ¯ m θ j ,

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