Abstract

This paper derives the Cramer–Rao bound (CRB) on range separation estimation of two point sources interrogated by a three-dimensional flash laser detection and ranging (LADAR) system. An unbiased range separation estimator is also derived to compare against the bound. Additionally, the CRB can be expressed as a function of two LADAR design parameters (range sampling and transmitted pulse width), which can be selected in order to optimize the expected range resolution between two point sources. Given several range sampling capabilities, the CRB and simulation show agreement that there is an optimal pulse width where a shorter pulse width would increase estimation variance due to undersampling of the pulse and a longer pulse width would degrade the resolving capability. Finally, the optimal pulse-width concept is extended to more complex targets and a normalized pulse definition.

© 2011 Optical Society of America

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References

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  1. R. Richmond, R. Stettner, and H. Bailey, “Laser radar focal plane array for three-dimensional imaging (update),” Proc. SPIE 3380, 138–143 (1998).
    [CrossRef]
  2. H. L. van Trees, Detection, Estimation, and Modulation Theory (Wiley, 2001).
  3. M. I. Skolnik, Introduction to RADAR Systems, 3rd ed. (McGraw-Hill, 2002).
  4. A. Weiner, Ultrafast Optics (Wiley, 2009).
    [CrossRef]
  5. J. C. Dries, B. Miles, and R. Stettner., “A 32×32 pixel flash laser radar system incorporating InGaAs PIN and APD detectors,” Proc. SPIE 5412, 250–256 (2004).
    [CrossRef]
  6. S. C. Cain, R. Richmond, and E. Armstrong, “Flash light detection and ranging range accuracy limits for returns from single opaque surfaces via Cramer–Rao bounds,” Appl. Opt. 45, 6154–6162 (2006).
    [CrossRef] [PubMed]
  7. S. Johnson and S. C. Cain, “Bound on range precision for shot-noise limited ladar systems,” Appl. Opt. 47, 5147–5154(2008).
    [CrossRef] [PubMed]
  8. J. Khoury, C. Woods, J. Lorenzo, J. Kierstead, D. Pyburn, and S. Sengupta, “Resolution limits in imaging LADAR systems,” Appl. Opt. 45, 697–704 (2006).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  11. J. Eriksson and M. Viberg, “On Cramer–Rao bounds and optimal beamspace transformation in radar array processing,” in IEEE International Symposium on Phased Array Systems and Technology, 1996, (IEEE, 1996), pp. 301–306.
    [CrossRef]
  12. D. J. van der Laan, M. C. Maas, D. R. Schaart, P. Bruyndonckx, S. Leonard, and C. W. E. van Eijk, “Using Cramer–Rao theory combined with Monte Carlo simulations for the optimization of monolithic scintillator PET detectors,” IEEE Trans. Nucl. Sci. 53, 1063–1070 (2006).
    [CrossRef]
  13. J. Li, L. Xu, P. Stoica, K. Forsythe, and D. Bliss, “Range compression and waveform optimization for MIMO radar: a Cramer–Rao bound based study,” IEEE Trans. Signal Process. 56, 218–232 (2008).
    [CrossRef]
  14. R. Linnehan, D. Brady, J. Schindler, L. Perlovsky, and M. Rangaswamy, “On the design of SAR apertures using the Cramer–Rao bound,” IEEE Trans. Aerosp. Electron. Syst. 43, 344–355 (2007).
    [CrossRef]
  15. B. J. Rye and R. M. Hardesty, “Discrete spectral peak estimation in incoherent backscatter heterodyne lidar. I. Spectral accumulation and the Cramer–Rao lower bound,” IEEE Trans. Geosci. Remote Sens. 31, 16–27 (1993).
    [CrossRef]
  16. J. W. Goodman, Introduction to Fourier Optics (Roberts, 2005).
  17. A. K. Jain, Fundamentals of Digital Image Processing(Prentice-Hall, 1989).
  18. J. W. Goodman, Statistical Optics (McGraw-Hill, 1985).
  19. A. Papoulis and S. Pillai, Probability, Random Variables and Stochastic Processes (McGraw-Hill, 2002).
  20. C. R. Burris, “An estimation theory approach to detection and ranging of obscured targets in 3-D LADAR data,” Master’s thesis (Air Force Institute of Technology, 2006), http://handle.dtic.mil/100.2/ADA449928.
  21. R. Richmond and S. Cain, Direct-Detection LADAR Systems (SPIE, 2010).

2008 (2)

J. Li, L. Xu, P. Stoica, K. Forsythe, and D. Bliss, “Range compression and waveform optimization for MIMO radar: a Cramer–Rao bound based study,” IEEE Trans. Signal Process. 56, 218–232 (2008).
[CrossRef]

S. Johnson and S. C. Cain, “Bound on range precision for shot-noise limited ladar systems,” Appl. Opt. 47, 5147–5154(2008).
[CrossRef] [PubMed]

2007 (2)

N. Hagen, M. Kupinski, and E. L. Dereniak, “Gaussian profile estimation in one dimension,” Appl. Opt. 46, 5374–5383(2007).
[CrossRef] [PubMed]

R. Linnehan, D. Brady, J. Schindler, L. Perlovsky, and M. Rangaswamy, “On the design of SAR apertures using the Cramer–Rao bound,” IEEE Trans. Aerosp. Electron. Syst. 43, 344–355 (2007).
[CrossRef]

2006 (3)

J. Khoury, C. Woods, J. Lorenzo, J. Kierstead, D. Pyburn, and S. Sengupta, “Resolution limits in imaging LADAR systems,” Appl. Opt. 45, 697–704 (2006).
[CrossRef] [PubMed]

S. C. Cain, R. Richmond, and E. Armstrong, “Flash light detection and ranging range accuracy limits for returns from single opaque surfaces via Cramer–Rao bounds,” Appl. Opt. 45, 6154–6162 (2006).
[CrossRef] [PubMed]

D. J. van der Laan, M. C. Maas, D. R. Schaart, P. Bruyndonckx, S. Leonard, and C. W. E. van Eijk, “Using Cramer–Rao theory combined with Monte Carlo simulations for the optimization of monolithic scintillator PET detectors,” IEEE Trans. Nucl. Sci. 53, 1063–1070 (2006).
[CrossRef]

2004 (1)

J. C. Dries, B. Miles, and R. Stettner., “A 32×32 pixel flash laser radar system incorporating InGaAs PIN and APD detectors,” Proc. SPIE 5412, 250–256 (2004).
[CrossRef]

1998 (1)

R. Richmond, R. Stettner, and H. Bailey, “Laser radar focal plane array for three-dimensional imaging (update),” Proc. SPIE 3380, 138–143 (1998).
[CrossRef]

1993 (1)

B. J. Rye and R. M. Hardesty, “Discrete spectral peak estimation in incoherent backscatter heterodyne lidar. I. Spectral accumulation and the Cramer–Rao lower bound,” IEEE Trans. Geosci. Remote Sens. 31, 16–27 (1993).
[CrossRef]

1986 (1)

Armstrong, E.

Bailey, H.

R. Richmond, R. Stettner, and H. Bailey, “Laser radar focal plane array for three-dimensional imaging (update),” Proc. SPIE 3380, 138–143 (1998).
[CrossRef]

Bliss, D.

J. Li, L. Xu, P. Stoica, K. Forsythe, and D. Bliss, “Range compression and waveform optimization for MIMO radar: a Cramer–Rao bound based study,” IEEE Trans. Signal Process. 56, 218–232 (2008).
[CrossRef]

Brady, D.

R. Linnehan, D. Brady, J. Schindler, L. Perlovsky, and M. Rangaswamy, “On the design of SAR apertures using the Cramer–Rao bound,” IEEE Trans. Aerosp. Electron. Syst. 43, 344–355 (2007).
[CrossRef]

Bruyndonckx, P.

D. J. van der Laan, M. C. Maas, D. R. Schaart, P. Bruyndonckx, S. Leonard, and C. W. E. van Eijk, “Using Cramer–Rao theory combined with Monte Carlo simulations for the optimization of monolithic scintillator PET detectors,” IEEE Trans. Nucl. Sci. 53, 1063–1070 (2006).
[CrossRef]

Burris, C. R.

C. R. Burris, “An estimation theory approach to detection and ranging of obscured targets in 3-D LADAR data,” Master’s thesis (Air Force Institute of Technology, 2006), http://handle.dtic.mil/100.2/ADA449928.

Cain, S.

R. Richmond and S. Cain, Direct-Detection LADAR Systems (SPIE, 2010).

Cain, S. C.

Dereniak, E. L.

Dries, J. C.

J. C. Dries, B. Miles, and R. Stettner., “A 32×32 pixel flash laser radar system incorporating InGaAs PIN and APD detectors,” Proc. SPIE 5412, 250–256 (2004).
[CrossRef]

Eriksson, J.

J. Eriksson and M. Viberg, “On Cramer–Rao bounds and optimal beamspace transformation in radar array processing,” in IEEE International Symposium on Phased Array Systems and Technology, 1996, (IEEE, 1996), pp. 301–306.
[CrossRef]

Forsythe, K.

J. Li, L. Xu, P. Stoica, K. Forsythe, and D. Bliss, “Range compression and waveform optimization for MIMO radar: a Cramer–Rao bound based study,” IEEE Trans. Signal Process. 56, 218–232 (2008).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (Roberts, 2005).

J. W. Goodman, Statistical Optics (McGraw-Hill, 1985).

Hagen, N.

Hardesty, R. M.

B. J. Rye and R. M. Hardesty, “Discrete spectral peak estimation in incoherent backscatter heterodyne lidar. I. Spectral accumulation and the Cramer–Rao lower bound,” IEEE Trans. Geosci. Remote Sens. 31, 16–27 (1993).
[CrossRef]

Jain, A. K.

A. K. Jain, Fundamentals of Digital Image Processing(Prentice-Hall, 1989).

Johnson, S.

Khoury, J.

Kierstead, J.

Kupinski, M.

Leonard, S.

D. J. van der Laan, M. C. Maas, D. R. Schaart, P. Bruyndonckx, S. Leonard, and C. W. E. van Eijk, “Using Cramer–Rao theory combined with Monte Carlo simulations for the optimization of monolithic scintillator PET detectors,” IEEE Trans. Nucl. Sci. 53, 1063–1070 (2006).
[CrossRef]

Li, J.

J. Li, L. Xu, P. Stoica, K. Forsythe, and D. Bliss, “Range compression and waveform optimization for MIMO radar: a Cramer–Rao bound based study,” IEEE Trans. Signal Process. 56, 218–232 (2008).
[CrossRef]

Linnehan, R.

R. Linnehan, D. Brady, J. Schindler, L. Perlovsky, and M. Rangaswamy, “On the design of SAR apertures using the Cramer–Rao bound,” IEEE Trans. Aerosp. Electron. Syst. 43, 344–355 (2007).
[CrossRef]

Lorenzo, J.

Maas, M. C.

D. J. van der Laan, M. C. Maas, D. R. Schaart, P. Bruyndonckx, S. Leonard, and C. W. E. van Eijk, “Using Cramer–Rao theory combined with Monte Carlo simulations for the optimization of monolithic scintillator PET detectors,” IEEE Trans. Nucl. Sci. 53, 1063–1070 (2006).
[CrossRef]

Miles, B.

J. C. Dries, B. Miles, and R. Stettner., “A 32×32 pixel flash laser radar system incorporating InGaAs PIN and APD detectors,” Proc. SPIE 5412, 250–256 (2004).
[CrossRef]

Papoulis, A.

A. Papoulis and S. Pillai, Probability, Random Variables and Stochastic Processes (McGraw-Hill, 2002).

Perlovsky, L.

R. Linnehan, D. Brady, J. Schindler, L. Perlovsky, and M. Rangaswamy, “On the design of SAR apertures using the Cramer–Rao bound,” IEEE Trans. Aerosp. Electron. Syst. 43, 344–355 (2007).
[CrossRef]

Pillai, S.

A. Papoulis and S. Pillai, Probability, Random Variables and Stochastic Processes (McGraw-Hill, 2002).

Pyburn, D.

Rangaswamy, M.

R. Linnehan, D. Brady, J. Schindler, L. Perlovsky, and M. Rangaswamy, “On the design of SAR apertures using the Cramer–Rao bound,” IEEE Trans. Aerosp. Electron. Syst. 43, 344–355 (2007).
[CrossRef]

Richmond, R.

S. C. Cain, R. Richmond, and E. Armstrong, “Flash light detection and ranging range accuracy limits for returns from single opaque surfaces via Cramer–Rao bounds,” Appl. Opt. 45, 6154–6162 (2006).
[CrossRef] [PubMed]

R. Richmond, R. Stettner, and H. Bailey, “Laser radar focal plane array for three-dimensional imaging (update),” Proc. SPIE 3380, 138–143 (1998).
[CrossRef]

R. Richmond and S. Cain, Direct-Detection LADAR Systems (SPIE, 2010).

Rye, B. J.

B. J. Rye and R. M. Hardesty, “Discrete spectral peak estimation in incoherent backscatter heterodyne lidar. I. Spectral accumulation and the Cramer–Rao lower bound,” IEEE Trans. Geosci. Remote Sens. 31, 16–27 (1993).
[CrossRef]

Schaart, D. R.

D. J. van der Laan, M. C. Maas, D. R. Schaart, P. Bruyndonckx, S. Leonard, and C. W. E. van Eijk, “Using Cramer–Rao theory combined with Monte Carlo simulations for the optimization of monolithic scintillator PET detectors,” IEEE Trans. Nucl. Sci. 53, 1063–1070 (2006).
[CrossRef]

Schindler, J.

R. Linnehan, D. Brady, J. Schindler, L. Perlovsky, and M. Rangaswamy, “On the design of SAR apertures using the Cramer–Rao bound,” IEEE Trans. Aerosp. Electron. Syst. 43, 344–355 (2007).
[CrossRef]

Sengupta, S.

Skolnik, M. I.

M. I. Skolnik, Introduction to RADAR Systems, 3rd ed. (McGraw-Hill, 2002).

Stettner, R.

J. C. Dries, B. Miles, and R. Stettner., “A 32×32 pixel flash laser radar system incorporating InGaAs PIN and APD detectors,” Proc. SPIE 5412, 250–256 (2004).
[CrossRef]

R. Richmond, R. Stettner, and H. Bailey, “Laser radar focal plane array for three-dimensional imaging (update),” Proc. SPIE 3380, 138–143 (1998).
[CrossRef]

Stoica, P.

J. Li, L. Xu, P. Stoica, K. Forsythe, and D. Bliss, “Range compression and waveform optimization for MIMO radar: a Cramer–Rao bound based study,” IEEE Trans. Signal Process. 56, 218–232 (2008).
[CrossRef]

van der Laan, D. J.

D. J. van der Laan, M. C. Maas, D. R. Schaart, P. Bruyndonckx, S. Leonard, and C. W. E. van Eijk, “Using Cramer–Rao theory combined with Monte Carlo simulations for the optimization of monolithic scintillator PET detectors,” IEEE Trans. Nucl. Sci. 53, 1063–1070 (2006).
[CrossRef]

van Eijk, C. W. E.

D. J. van der Laan, M. C. Maas, D. R. Schaart, P. Bruyndonckx, S. Leonard, and C. W. E. van Eijk, “Using Cramer–Rao theory combined with Monte Carlo simulations for the optimization of monolithic scintillator PET detectors,” IEEE Trans. Nucl. Sci. 53, 1063–1070 (2006).
[CrossRef]

van Trees, H. L.

H. L. van Trees, Detection, Estimation, and Modulation Theory (Wiley, 2001).

Viberg, M.

J. Eriksson and M. Viberg, “On Cramer–Rao bounds and optimal beamspace transformation in radar array processing,” in IEEE International Symposium on Phased Array Systems and Technology, 1996, (IEEE, 1996), pp. 301–306.
[CrossRef]

Weiner, A.

A. Weiner, Ultrafast Optics (Wiley, 2009).
[CrossRef]

Winick, K.

Woods, C.

Xu, L.

J. Li, L. Xu, P. Stoica, K. Forsythe, and D. Bliss, “Range compression and waveform optimization for MIMO radar: a Cramer–Rao bound based study,” IEEE Trans. Signal Process. 56, 218–232 (2008).
[CrossRef]

Appl. Opt. (4)

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

R. Linnehan, D. Brady, J. Schindler, L. Perlovsky, and M. Rangaswamy, “On the design of SAR apertures using the Cramer–Rao bound,” IEEE Trans. Aerosp. Electron. Syst. 43, 344–355 (2007).
[CrossRef]

IEEE Trans. Geosci. Remote Sens. (1)

B. J. Rye and R. M. Hardesty, “Discrete spectral peak estimation in incoherent backscatter heterodyne lidar. I. Spectral accumulation and the Cramer–Rao lower bound,” IEEE Trans. Geosci. Remote Sens. 31, 16–27 (1993).
[CrossRef]

IEEE Trans. Nucl. Sci. (1)

D. J. van der Laan, M. C. Maas, D. R. Schaart, P. Bruyndonckx, S. Leonard, and C. W. E. van Eijk, “Using Cramer–Rao theory combined with Monte Carlo simulations for the optimization of monolithic scintillator PET detectors,” IEEE Trans. Nucl. Sci. 53, 1063–1070 (2006).
[CrossRef]

IEEE Trans. Signal Process. (1)

J. Li, L. Xu, P. Stoica, K. Forsythe, and D. Bliss, “Range compression and waveform optimization for MIMO radar: a Cramer–Rao bound based study,” IEEE Trans. Signal Process. 56, 218–232 (2008).
[CrossRef]

J. Opt. Soc. Am. A (1)

Proc. SPIE (2)

J. C. Dries, B. Miles, and R. Stettner., “A 32×32 pixel flash laser radar system incorporating InGaAs PIN and APD detectors,” Proc. SPIE 5412, 250–256 (2004).
[CrossRef]

R. Richmond, R. Stettner, and H. Bailey, “Laser radar focal plane array for three-dimensional imaging (update),” Proc. SPIE 3380, 138–143 (1998).
[CrossRef]

Other (10)

H. L. van Trees, Detection, Estimation, and Modulation Theory (Wiley, 2001).

M. I. Skolnik, Introduction to RADAR Systems, 3rd ed. (McGraw-Hill, 2002).

A. Weiner, Ultrafast Optics (Wiley, 2009).
[CrossRef]

J. Eriksson and M. Viberg, “On Cramer–Rao bounds and optimal beamspace transformation in radar array processing,” in IEEE International Symposium on Phased Array Systems and Technology, 1996, (IEEE, 1996), pp. 301–306.
[CrossRef]

J. W. Goodman, Introduction to Fourier Optics (Roberts, 2005).

A. K. Jain, Fundamentals of Digital Image Processing(Prentice-Hall, 1989).

J. W. Goodman, Statistical Optics (McGraw-Hill, 1985).

A. Papoulis and S. Pillai, Probability, Random Variables and Stochastic Processes (McGraw-Hill, 2002).

C. R. Burris, “An estimation theory approach to detection and ranging of obscured targets in 3-D LADAR data,” Master’s thesis (Air Force Institute of Technology, 2006), http://handle.dtic.mil/100.2/ADA449928.

R. Richmond and S. Cain, Direct-Detection LADAR Systems (SPIE, 2010).

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Figures (14)

Fig. 1
Fig. 1

(a) For illustrative purposes, this figure is a range image of the truth data where the reference target is in the center of the array at 1000 m with the unknown target placed at Δ m = 2 pixels and Δ k = 1.7 m . (b) Defined by Eq. (5), this shows the ideal waveforms of the unknown p ( r k K t ) and reference target p ( r k K r ) from Fig. 1a with a pulse-width standard deviation σ p t = 0.88 ns .

Fig. 2
Fig. 2

This plot shows an example CRB with Δ m = 1 pixel, σ h = 3 pixels, σ p t = 3 ns , A t = 0.5 × 10 4 photons, and A r = 2 × 10 5 photons. The bound behaves appropriately considering the variance goes up as the separation becomes smaller, corresponding to the notion that close-in targets are tougher to resolve. The peak of the bound occurs when the range and spatial coupling are at their maximum. Further, when the range separation is near zero, the range coupling is diminished, but the bound does not go to exactly zero because the spatial coupling is still present.

Fig. 3
Fig. 3

Effects on CRB ( Δ k ) when changing the target amplitudes. (a)  A t —inversely proportional to bound. (b)  A r —proportional to bound.

Fig. 4
Fig. 4

Effects on CRB ( Δ k ) when changing blurring severity and spatial separation. (a)  σ h —proportional to bound. (b)  Δ m —inversely proportional to bound.

Fig. 5
Fig. 5

Range separation estimation results of a two-point target data model simulation. (a) Each curve is a bias calculation for a different Δ k over many trials. At each trial, the estimated range is an average of the previous estimated ranges (i.e., a running average). (b) Bias results taken from the last trial from (a).

Fig. 6
Fig. 6

Range separation estimation results of a two-point target data model simulation. (a) MSE between the truth data and the estimate. (b) Range separation estimate variance.

Fig. 7
Fig. 7

Taking the results from Figs. 2, 6b, this plot compares the CRB and the simulated range variance showing agreement both in the Cramer–Rao inequality and in the curve shapes.

Fig. 8
Fig. 8

Example plot of how the range-resolution metric is determined. The circled value is the range resolution and corresponds to the location where the square root of the CRB is equal to the range separation. At smaller range separations, the square of the CRB is greater than the separation and vice versa.

Fig. 9
Fig. 9

(a) For differing range sampling cases, the range resolution derived from the CRB is plotted versus the pulse width. As the range sampling t s becomes either faster ( 0.6 t s o and 0.8 t s o ) or slower ( 1.2 t s o and 1.4 t s o ), the optimal pulse width becomes narrower or wider, respectively, with a corresponding improvement or degradation in the range resolution. (b) The simulation range resolution determined from the range separation variance is plotted versus the pulse width. As expected, the resolution values are larger than those predicted by CRB theory. Also, the optimal pulse width trends in a similar manner as the CRB results.

Fig. 10
Fig. 10

Utilizing the optimal σ p t values from Table 1, this plot shows the near-exact percentage change of the CRB and simulation optimal pulse widths with respect to the percentage change in range sampling.

Fig. 11
Fig. 11

(a) True target scene with the color bar representing range in meters. (b) Optimal pulse results against a complex target with t s = t s o .

Fig. 12
Fig. 12

(a) True target scene. (b) Optimal pulse results for a three-bar target with t s = t s o .

Fig. 13
Fig. 13

(a) True target scene. (b) Optimal pulse for a connected blocks target with t s = t s o .

Fig. 14
Fig. 14

(a) Using the normalized pulse model, this graph shows the CRB optimal pulse standard deviation referring to Eq. (15). (b) CRB optimal pulse versus range sampling. The optimal pulse width changes proportionally as the sampling changes.

Tables (2)

Tables Icon

Table 1 Optimal Pulse-Width Results: Two-Point Target

Tables Icon

Table 2 Optimal Pulse-Width Results: Complex Targets

Equations (16)

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

i k ( x , y ) = m = 1 M n = 1 N o k ( m , n ) h ( x m , y n ) + B ( x , y ) ,
r k = K s + ( k · t s · c 2 ) ,
o k ( m , n ) = A t p ( r k ( K r Δ k ) ) δ ( m Δ m , n ) + ... A r p ( r k K r ) δ ( m , n ) ,
i k ( x , y ) = A t p ( r k ( K r Δ k ) ) h ( x Δ m , y ) + ... A r p ( r k K r ) h ( x , y ) + B ( x , y ) ,
p ( r k ) = 1 2 π σ p d exp { ( r k ) 2 2 σ p d 2 } ,
h ( x , y ) = 1 2 π σ h 2 exp { ( x 2 + y 2 ) 2 σ h 2 } ,
var [ θ ^ i ( D ) θ i ] [ J 1 ] i i ,
J i j = E [ 2 ln P ( D k ( x , y ) = d k ( x , y ) k , x , y ) θ i θ j ] ,
P [ D k ( x , y ) = d k ( x , y ) k , x , y ] = k , x , y i k ( x , y ) ( x , y ) exp { i k ( x , y ) } d k ( x , y ) ! ,
J i j = k = 1 K x = 1 X y = 1 Y 1 i k ( x , y ) i k ( x , y ) θ i i k ( x , y ) θ j .
i k ( x , y ) Δ m = A t p ( r k K t ) Δ m h ( x Δ m , y ) , i k ( x , y ) Δ k = A t h ( x Δ m , y ) Δ k p ( r k K t ) , i k ( x , y ) A t = p ( r k K t ) h ( x Δ m , y ) , i k ( x , y ) A r = p ( r k K r ) h ( x , y ) , Δ m h ( x Δ m , y ) = ( x Δ m ) σ h 2 h ( x Δ m , y ) , Δ k p ( r k K t ) = ( r k K t ) σ p d 2 p ( r k K t ) .
E ( Δ k ) = k x y ( d k ( x , y ) i k ( x , y ) ) 2 ,
C 11 A t + C 12 A r = D 1 , C 21 A t + C 22 A r = D 2 ,
C 11 = k , x , y [ p ( k ( K r Δ k ) ) h ( x Δ m , y ) ] 2 , C 12 = k , x , y p ( k ( K r Δ k ) ) h ( x Δ m , y ) × ... p ( k K r ) h ( x , y ) , C 22 = k , x , y [ p ( k K r ) h ( x , y ) ] 2 , D 1 = k , x , y ( B ( x , y ) d k ( x , y ) ) p ( k ( K r Δ k ) ) × ... h ( x Δ m , y ) , D 2 = k , x , y ( B ( x , y ) d k ( x , y ) ) p ( k K r ) h ( x , y ) ,
p ( n ) = 1 2 π σ n exp { ( n ) 2 2 σ n 2 } ,
σ n = σ p d c t s = c σ p t 2 c t s = σ p t 2 t s ,

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