Abstract

In a pure estimation task, an object of interest is known to be present, and we wish to determine numerical values for parameters that describe the object. This paper compares the theoretical framework, implementation method, and performance of two estimation procedures. We examined the performance of these estimators for tasks such as estimating signal location, signal volume, signal amplitude, or any combination of these parameters. The signal is embedded in a random background to simulate the effect of nuisance parameters. First, we explore the classical Wiener estimator, which operates linearly on the data and minimizes the ensemble mean-squared error. The results of our performance tests indicate that the Wiener estimator can estimate amplitude and shape once a signal has been located, but is fundamentally unable to locate a signal regardless of the quality of the image. Given these new results on the fundamental limitations of Wiener estimation, we extend our methods to include more complex data processing. We introduce and evaluate a scanning-linear estimator that performs impressively for location estimation. The scanning action of the estimator refers to seeking a solution that maximizes a linear metric, thereby requiring a global-extremum search. The linear metric to be optimized can be derived as a special case of maximum a posteriori (MAP) estimation when the likelihood is Gaussian and a slowly varying covariance approximation is made.

© 2008 Optical Society of America

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References

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  1. H. H. Barrett and K. J. Myers, Foundations of Image Science (Wiley-Interscience, 2004).
  2. S. C. Moore, M. F. Kijewski, and G. El Fakhri, “Collimator Optimization for Detection and Quantitation Tasks: Application to Gallium-67 Imaging,” IEEE Trans. Med. Imaging 24(10), 1347–1356 (2005).
    [Crossref] [PubMed]
  3. G. El Fakhri, S. C. Moore, and M. F. Kijewski, “Optimization of Ga-67 imaging for detection and estimation tasks: Dependence of imaging performance on spectral acquisition parameters,” Med. Phys. 29, 1859–1866 (2002).
    [Crossref] [PubMed]
  4. M. A. Kupinski, E. Clarkson, K. Gross, and J. W. Hoppin, “Optimizing imaging hardware for estimation tasks,” Proc. SPIE 5034, 309–313 (2003).
    [Crossref]
  5. H. H. Barrett, “Objective assessment of image quality: effects of quantum noise and object variability,” J. Opt. Soc. Am. A 7(7), 1266–1278 (1990).
    [Crossref] [PubMed]
  6. H. H. Barrett, K. J. Myers, N. Devaney, and J. C. Dainty, “Objective Assessment of Image Quality: IV. Application to Adaptive Optics,” J. Opt. Soc. Am. A 23(12), 3080–3105 (2006).
    [Crossref]
  7. Y. C. Eldar, “Comparing between estimation approaches: admissible and dominating linear estimators,” IEEE Trans. Signal Process. 54(5), 1689–1702 (2006).
    [Crossref]
  8. C. Caiafa, A. Proto, and E. Kuruoglu, “Long correlation Gaussian random fields: Parameter estimation and noise reduction,” Digital Signal Processing 17, 819–835 (2007).
    [Crossref]
  9. A. B. Hamza, H. Krim, and G. Unal, “Unifying probabilistic and variational estimation,” IEEE Signal Process Mag. 19, 37–47 (2002).
    [Crossref]
  10. R. E. Greenblatt, A. Ossadtchi, and M. E. Pflieger, “Local linear estimators for the bioelectromagnetic inverse problem,” IEEE Trans. Signal Process. 53(9), 3403–3412 (2005).
    [Crossref]
  11. F. O. Bochud, J.-F. Valley, and F. R. Verdun, “Estimation of the noisy component of anatomical backgrounds,” Med. Phys. 26(7), 1365–1370 (1999).
    [Crossref] [PubMed]
  12. S. P Mµller, M. F. Kijewski, S. C. Moore, and B. L. Holman, “Maximum-likelihood Estimation: A Mathematical Model for Quantitation in Nuclear Medicine,” J. Nucl. Med. 31, 1693–1701 (1989).
  13. S. P. Mµller, C. K. Abbey, F. J. Rybicki, S. C. Moore, and M. F. Kijewski, “Measures of performance in nonlinear estimation tasks: prediction of estimation performance at low signal-to-noise ratio,” Phys. Med. Biol. 50, 3697–3715 (2005).
    [Crossref]
  14. J. Shao, Mathematical Statistics (Springer, 1999).
  15. M. Kupinski, J.W. Hoppin, E. Clarkson, and H. H. Barrett, “Ideal-observer computation in medical imaging with use of Markov-chain Monte Carlo techniques,” J. Opt. Soc. Am. A 20, 430–438 (2003).
    [Crossref]
  16. N WienerExtrapolation, Interpolation, and Smoothing of Stationary Time Series with Engineering Applications (The MIT press, 1949).
    [PubMed]
  17. J. L. Melsa and D. L. Cohn, Decision and Estimation Theory (McGraw-Hill, 1978).
  18. J. P. Rolland and H. H. Barrett, “Effect of random background inhomogeneity on observer detection performance,” J. Opt. Soc. Am. A 9, 649–658 (1992).
    [Crossref] [PubMed]
  19. H. H. Barrett, K. J. Myers, B. Gallas, E. Clarkson, and H. Zhang, “Megalopinakophobia: Its Symptoms and Cures,” Proc. SPIE 4320, 299–307 (2001).
    [Crossref]

2007 (1)

C. Caiafa, A. Proto, and E. Kuruoglu, “Long correlation Gaussian random fields: Parameter estimation and noise reduction,” Digital Signal Processing 17, 819–835 (2007).
[Crossref]

2006 (2)

H. H. Barrett, K. J. Myers, N. Devaney, and J. C. Dainty, “Objective Assessment of Image Quality: IV. Application to Adaptive Optics,” J. Opt. Soc. Am. A 23(12), 3080–3105 (2006).
[Crossref]

Y. C. Eldar, “Comparing between estimation approaches: admissible and dominating linear estimators,” IEEE Trans. Signal Process. 54(5), 1689–1702 (2006).
[Crossref]

2005 (3)

S. C. Moore, M. F. Kijewski, and G. El Fakhri, “Collimator Optimization for Detection and Quantitation Tasks: Application to Gallium-67 Imaging,” IEEE Trans. Med. Imaging 24(10), 1347–1356 (2005).
[Crossref] [PubMed]

R. E. Greenblatt, A. Ossadtchi, and M. E. Pflieger, “Local linear estimators for the bioelectromagnetic inverse problem,” IEEE Trans. Signal Process. 53(9), 3403–3412 (2005).
[Crossref]

S. P. Mµller, C. K. Abbey, F. J. Rybicki, S. C. Moore, and M. F. Kijewski, “Measures of performance in nonlinear estimation tasks: prediction of estimation performance at low signal-to-noise ratio,” Phys. Med. Biol. 50, 3697–3715 (2005).
[Crossref]

2003 (2)

M. Kupinski, J.W. Hoppin, E. Clarkson, and H. H. Barrett, “Ideal-observer computation in medical imaging with use of Markov-chain Monte Carlo techniques,” J. Opt. Soc. Am. A 20, 430–438 (2003).
[Crossref]

M. A. Kupinski, E. Clarkson, K. Gross, and J. W. Hoppin, “Optimizing imaging hardware for estimation tasks,” Proc. SPIE 5034, 309–313 (2003).
[Crossref]

2002 (2)

G. El Fakhri, S. C. Moore, and M. F. Kijewski, “Optimization of Ga-67 imaging for detection and estimation tasks: Dependence of imaging performance on spectral acquisition parameters,” Med. Phys. 29, 1859–1866 (2002).
[Crossref] [PubMed]

A. B. Hamza, H. Krim, and G. Unal, “Unifying probabilistic and variational estimation,” IEEE Signal Process Mag. 19, 37–47 (2002).
[Crossref]

2001 (1)

H. H. Barrett, K. J. Myers, B. Gallas, E. Clarkson, and H. Zhang, “Megalopinakophobia: Its Symptoms and Cures,” Proc. SPIE 4320, 299–307 (2001).
[Crossref]

1999 (1)

F. O. Bochud, J.-F. Valley, and F. R. Verdun, “Estimation of the noisy component of anatomical backgrounds,” Med. Phys. 26(7), 1365–1370 (1999).
[Crossref] [PubMed]

1992 (1)

1990 (1)

1989 (1)

S. P Mµller, M. F. Kijewski, S. C. Moore, and B. L. Holman, “Maximum-likelihood Estimation: A Mathematical Model for Quantitation in Nuclear Medicine,” J. Nucl. Med. 31, 1693–1701 (1989).

Abbey, C. K.

S. P. Mµller, C. K. Abbey, F. J. Rybicki, S. C. Moore, and M. F. Kijewski, “Measures of performance in nonlinear estimation tasks: prediction of estimation performance at low signal-to-noise ratio,” Phys. Med. Biol. 50, 3697–3715 (2005).
[Crossref]

Barrett, H. H.

Bochud, F. O.

F. O. Bochud, J.-F. Valley, and F. R. Verdun, “Estimation of the noisy component of anatomical backgrounds,” Med. Phys. 26(7), 1365–1370 (1999).
[Crossref] [PubMed]

Caiafa, C.

C. Caiafa, A. Proto, and E. Kuruoglu, “Long correlation Gaussian random fields: Parameter estimation and noise reduction,” Digital Signal Processing 17, 819–835 (2007).
[Crossref]

Clarkson, E.

M. A. Kupinski, E. Clarkson, K. Gross, and J. W. Hoppin, “Optimizing imaging hardware for estimation tasks,” Proc. SPIE 5034, 309–313 (2003).
[Crossref]

M. Kupinski, J.W. Hoppin, E. Clarkson, and H. H. Barrett, “Ideal-observer computation in medical imaging with use of Markov-chain Monte Carlo techniques,” J. Opt. Soc. Am. A 20, 430–438 (2003).
[Crossref]

H. H. Barrett, K. J. Myers, B. Gallas, E. Clarkson, and H. Zhang, “Megalopinakophobia: Its Symptoms and Cures,” Proc. SPIE 4320, 299–307 (2001).
[Crossref]

Cohn, D. L.

J. L. Melsa and D. L. Cohn, Decision and Estimation Theory (McGraw-Hill, 1978).

Dainty, J. C.

Devaney, N.

El Fakhri, G.

S. C. Moore, M. F. Kijewski, and G. El Fakhri, “Collimator Optimization for Detection and Quantitation Tasks: Application to Gallium-67 Imaging,” IEEE Trans. Med. Imaging 24(10), 1347–1356 (2005).
[Crossref] [PubMed]

G. El Fakhri, S. C. Moore, and M. F. Kijewski, “Optimization of Ga-67 imaging for detection and estimation tasks: Dependence of imaging performance on spectral acquisition parameters,” Med. Phys. 29, 1859–1866 (2002).
[Crossref] [PubMed]

Eldar, Y. C.

Y. C. Eldar, “Comparing between estimation approaches: admissible and dominating linear estimators,” IEEE Trans. Signal Process. 54(5), 1689–1702 (2006).
[Crossref]

Gallas, B.

H. H. Barrett, K. J. Myers, B. Gallas, E. Clarkson, and H. Zhang, “Megalopinakophobia: Its Symptoms and Cures,” Proc. SPIE 4320, 299–307 (2001).
[Crossref]

Greenblatt, R. E.

R. E. Greenblatt, A. Ossadtchi, and M. E. Pflieger, “Local linear estimators for the bioelectromagnetic inverse problem,” IEEE Trans. Signal Process. 53(9), 3403–3412 (2005).
[Crossref]

Gross, K.

M. A. Kupinski, E. Clarkson, K. Gross, and J. W. Hoppin, “Optimizing imaging hardware for estimation tasks,” Proc. SPIE 5034, 309–313 (2003).
[Crossref]

Hamza, A. B.

A. B. Hamza, H. Krim, and G. Unal, “Unifying probabilistic and variational estimation,” IEEE Signal Process Mag. 19, 37–47 (2002).
[Crossref]

Holman, B. L.

S. P Mµller, M. F. Kijewski, S. C. Moore, and B. L. Holman, “Maximum-likelihood Estimation: A Mathematical Model for Quantitation in Nuclear Medicine,” J. Nucl. Med. 31, 1693–1701 (1989).

Hoppin, J. W.

M. A. Kupinski, E. Clarkson, K. Gross, and J. W. Hoppin, “Optimizing imaging hardware for estimation tasks,” Proc. SPIE 5034, 309–313 (2003).
[Crossref]

Hoppin, J.W.

Kijewski, M. F.

S. P. Mµller, C. K. Abbey, F. J. Rybicki, S. C. Moore, and M. F. Kijewski, “Measures of performance in nonlinear estimation tasks: prediction of estimation performance at low signal-to-noise ratio,” Phys. Med. Biol. 50, 3697–3715 (2005).
[Crossref]

S. C. Moore, M. F. Kijewski, and G. El Fakhri, “Collimator Optimization for Detection and Quantitation Tasks: Application to Gallium-67 Imaging,” IEEE Trans. Med. Imaging 24(10), 1347–1356 (2005).
[Crossref] [PubMed]

G. El Fakhri, S. C. Moore, and M. F. Kijewski, “Optimization of Ga-67 imaging for detection and estimation tasks: Dependence of imaging performance on spectral acquisition parameters,” Med. Phys. 29, 1859–1866 (2002).
[Crossref] [PubMed]

S. P Mµller, M. F. Kijewski, S. C. Moore, and B. L. Holman, “Maximum-likelihood Estimation: A Mathematical Model for Quantitation in Nuclear Medicine,” J. Nucl. Med. 31, 1693–1701 (1989).

Krim, H.

A. B. Hamza, H. Krim, and G. Unal, “Unifying probabilistic and variational estimation,” IEEE Signal Process Mag. 19, 37–47 (2002).
[Crossref]

Kupinski, M.

Kupinski, M. A.

M. A. Kupinski, E. Clarkson, K. Gross, and J. W. Hoppin, “Optimizing imaging hardware for estimation tasks,” Proc. SPIE 5034, 309–313 (2003).
[Crossref]

Kuruoglu, E.

C. Caiafa, A. Proto, and E. Kuruoglu, “Long correlation Gaussian random fields: Parameter estimation and noise reduction,” Digital Signal Processing 17, 819–835 (2007).
[Crossref]

Mµller, S. P

S. P Mµller, M. F. Kijewski, S. C. Moore, and B. L. Holman, “Maximum-likelihood Estimation: A Mathematical Model for Quantitation in Nuclear Medicine,” J. Nucl. Med. 31, 1693–1701 (1989).

Mµller, S. P.

S. P. Mµller, C. K. Abbey, F. J. Rybicki, S. C. Moore, and M. F. Kijewski, “Measures of performance in nonlinear estimation tasks: prediction of estimation performance at low signal-to-noise ratio,” Phys. Med. Biol. 50, 3697–3715 (2005).
[Crossref]

Melsa, J. L.

J. L. Melsa and D. L. Cohn, Decision and Estimation Theory (McGraw-Hill, 1978).

Moore, S. C.

S. P. Mµller, C. K. Abbey, F. J. Rybicki, S. C. Moore, and M. F. Kijewski, “Measures of performance in nonlinear estimation tasks: prediction of estimation performance at low signal-to-noise ratio,” Phys. Med. Biol. 50, 3697–3715 (2005).
[Crossref]

S. C. Moore, M. F. Kijewski, and G. El Fakhri, “Collimator Optimization for Detection and Quantitation Tasks: Application to Gallium-67 Imaging,” IEEE Trans. Med. Imaging 24(10), 1347–1356 (2005).
[Crossref] [PubMed]

G. El Fakhri, S. C. Moore, and M. F. Kijewski, “Optimization of Ga-67 imaging for detection and estimation tasks: Dependence of imaging performance on spectral acquisition parameters,” Med. Phys. 29, 1859–1866 (2002).
[Crossref] [PubMed]

S. P Mµller, M. F. Kijewski, S. C. Moore, and B. L. Holman, “Maximum-likelihood Estimation: A Mathematical Model for Quantitation in Nuclear Medicine,” J. Nucl. Med. 31, 1693–1701 (1989).

Myers, K. J.

H. H. Barrett, K. J. Myers, N. Devaney, and J. C. Dainty, “Objective Assessment of Image Quality: IV. Application to Adaptive Optics,” J. Opt. Soc. Am. A 23(12), 3080–3105 (2006).
[Crossref]

H. H. Barrett, K. J. Myers, B. Gallas, E. Clarkson, and H. Zhang, “Megalopinakophobia: Its Symptoms and Cures,” Proc. SPIE 4320, 299–307 (2001).
[Crossref]

H. H. Barrett and K. J. Myers, Foundations of Image Science (Wiley-Interscience, 2004).

Ossadtchi, A.

R. E. Greenblatt, A. Ossadtchi, and M. E. Pflieger, “Local linear estimators for the bioelectromagnetic inverse problem,” IEEE Trans. Signal Process. 53(9), 3403–3412 (2005).
[Crossref]

Pflieger, M. E.

R. E. Greenblatt, A. Ossadtchi, and M. E. Pflieger, “Local linear estimators for the bioelectromagnetic inverse problem,” IEEE Trans. Signal Process. 53(9), 3403–3412 (2005).
[Crossref]

Proto, A.

C. Caiafa, A. Proto, and E. Kuruoglu, “Long correlation Gaussian random fields: Parameter estimation and noise reduction,” Digital Signal Processing 17, 819–835 (2007).
[Crossref]

Rolland, J. P.

Rybicki, F. J.

S. P. Mµller, C. K. Abbey, F. J. Rybicki, S. C. Moore, and M. F. Kijewski, “Measures of performance in nonlinear estimation tasks: prediction of estimation performance at low signal-to-noise ratio,” Phys. Med. Biol. 50, 3697–3715 (2005).
[Crossref]

Shao, J.

J. Shao, Mathematical Statistics (Springer, 1999).

Unal, G.

A. B. Hamza, H. Krim, and G. Unal, “Unifying probabilistic and variational estimation,” IEEE Signal Process Mag. 19, 37–47 (2002).
[Crossref]

Valley, J.-F.

F. O. Bochud, J.-F. Valley, and F. R. Verdun, “Estimation of the noisy component of anatomical backgrounds,” Med. Phys. 26(7), 1365–1370 (1999).
[Crossref] [PubMed]

Verdun, F. R.

F. O. Bochud, J.-F. Valley, and F. R. Verdun, “Estimation of the noisy component of anatomical backgrounds,” Med. Phys. 26(7), 1365–1370 (1999).
[Crossref] [PubMed]

Wiener, N

N WienerExtrapolation, Interpolation, and Smoothing of Stationary Time Series with Engineering Applications (The MIT press, 1949).
[PubMed]

Zhang, H.

H. H. Barrett, K. J. Myers, B. Gallas, E. Clarkson, and H. Zhang, “Megalopinakophobia: Its Symptoms and Cures,” Proc. SPIE 4320, 299–307 (2001).
[Crossref]

Digital Signal Processing (1)

C. Caiafa, A. Proto, and E. Kuruoglu, “Long correlation Gaussian random fields: Parameter estimation and noise reduction,” Digital Signal Processing 17, 819–835 (2007).
[Crossref]

IEEE Signal Process Mag. (1)

A. B. Hamza, H. Krim, and G. Unal, “Unifying probabilistic and variational estimation,” IEEE Signal Process Mag. 19, 37–47 (2002).
[Crossref]

IEEE Trans. Med. Imaging (1)

S. C. Moore, M. F. Kijewski, and G. El Fakhri, “Collimator Optimization for Detection and Quantitation Tasks: Application to Gallium-67 Imaging,” IEEE Trans. Med. Imaging 24(10), 1347–1356 (2005).
[Crossref] [PubMed]

IEEE Trans. Signal Process. (2)

R. E. Greenblatt, A. Ossadtchi, and M. E. Pflieger, “Local linear estimators for the bioelectromagnetic inverse problem,” IEEE Trans. Signal Process. 53(9), 3403–3412 (2005).
[Crossref]

Y. C. Eldar, “Comparing between estimation approaches: admissible and dominating linear estimators,” IEEE Trans. Signal Process. 54(5), 1689–1702 (2006).
[Crossref]

J. Nucl. Med. (1)

S. P Mµller, M. F. Kijewski, S. C. Moore, and B. L. Holman, “Maximum-likelihood Estimation: A Mathematical Model for Quantitation in Nuclear Medicine,” J. Nucl. Med. 31, 1693–1701 (1989).

J. Opt. Soc. Am. A (4)

Med. Phys. (2)

G. El Fakhri, S. C. Moore, and M. F. Kijewski, “Optimization of Ga-67 imaging for detection and estimation tasks: Dependence of imaging performance on spectral acquisition parameters,” Med. Phys. 29, 1859–1866 (2002).
[Crossref] [PubMed]

F. O. Bochud, J.-F. Valley, and F. R. Verdun, “Estimation of the noisy component of anatomical backgrounds,” Med. Phys. 26(7), 1365–1370 (1999).
[Crossref] [PubMed]

Phys. Med. Biol. (1)

S. P. Mµller, C. K. Abbey, F. J. Rybicki, S. C. Moore, and M. F. Kijewski, “Measures of performance in nonlinear estimation tasks: prediction of estimation performance at low signal-to-noise ratio,” Phys. Med. Biol. 50, 3697–3715 (2005).
[Crossref]

Proc. SPIE (2)

H. H. Barrett, K. J. Myers, B. Gallas, E. Clarkson, and H. Zhang, “Megalopinakophobia: Its Symptoms and Cures,” Proc. SPIE 4320, 299–307 (2001).
[Crossref]

M. A. Kupinski, E. Clarkson, K. Gross, and J. W. Hoppin, “Optimizing imaging hardware for estimation tasks,” Proc. SPIE 5034, 309–313 (2003).
[Crossref]

Other (4)

N WienerExtrapolation, Interpolation, and Smoothing of Stationary Time Series with Engineering Applications (The MIT press, 1949).
[PubMed]

J. L. Melsa and D. L. Cohn, Decision and Estimation Theory (McGraw-Hill, 1978).

J. Shao, Mathematical Statistics (Springer, 1999).

H. H. Barrett and K. J. Myers, Foundations of Image Science (Wiley-Interscience, 2004).

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Figures (14)

Fig. 1.
Fig. 1.

The probability density function (red line) is superimposed on a histogram of 10,000 samples.

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Fig. 2.
Fig. 2.

Samples of 2D Gaussian lumpy backgrounds.

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Fig. 3.
Fig. 3.

Noise-free images generated by sampling from the ensemble of possible signals and backgrounds.

Download Full Size | PPT Slide | PDF

Fig. 4.
Fig. 4.

Noisy images generated by sampling from the ensemble of possible signals and backgrounds, and adding Poisson noise.

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Fig. 5.
Fig. 5.

Rows of the matrix K θ,g , the cross-covariance between the data and the parameters to be estimated.

Download Full Size | PPT Slide | PDF

Fig. 6.
Fig. 6.

The three terms that contribute to the triply stochastic data covariance Kg .

Download Full Size | PPT Slide | PDF

Fig. 7.
Fig. 7.

Scatter plots of the Wiener estimates versus the true parameter values for each element of θ=[A,R,c x ,c y ]. The red line indicates perfect estimation performance, the actual estimates are blue points, and the black line shows θ ¯ , the prior mean.

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Fig. 8.
Fig. 8.

The objective functions scanned over solutions for signal locations are paired with the noisy image data.

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Fig. 9.
Fig. 9.

Scatter plots of the scanning linear estimates versus the true parameter values for each element of θ=[A,R,c x ,c y ]. The red line indicates perfect estimation performance and the actual estimates are blue points.

Download Full Size | PPT Slide | PDF

Fig. 10.
Fig. 10.

(a): The Wiener EMSE of the estimated signal amplitude as dependent upon the standard deviation of the pdf on signal location. (b): The Wiener EMSE of the estimated location divided by the variance of the pdf on signal location as dependent upon the width of the pdf on signal location.

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Fig. 11.
Fig. 11.

Scatter plots of the Wiener estimates versus the true parameter values for each element of θ=[A,c x ,c y ] when the radius is fixed at 2 mm (4 pixels). The red line indicates perfect estimation performance, and the actual estimates are blue points.

Download Full Size | PPT Slide | PDF

Fig. 12.
Fig. 12.

Scatter plot of true versus Wiener estimates of amplitude when the signal shape and location are known a priori. The red line indicates perfect estimation performance, and the actual estimates are blue points.

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Fig. 13.
Fig. 13.

The linear template used on image data (black line) and the average signal (red line) are normalized by their respective maximum values to accentuate their regions of overlap.

Download Full Size | PPT Slide | PDF

Fig. 14.
Fig. 14.

Scatter plot of true versus scanning linear estimates of amplitude when the signal shape and location are known a priori. The red line indicates perfect estimation performance, and the actual estimates are blue points.

Download Full Size | PPT Slide | PDF

Equations (75)

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

〈 y ( x ) 〉 x = ∫ d p x pr ( x ) y ( x ) ,
EMSE ( θ ̂ ( g ) ) = 〈 〈 ∥ θ ̂ ( g ) − θ ∥ 2 〉 g | θ 〉 θ = tr 〈 〈 ( θ ̂ ( g ) − θ ) ( θ ̂ ( g ) − θ ) t 〉 g | θ 〉 θ .
θ ̂ ( g ) PM = argmin θ ̂ ( g ) [ EMSE ( θ ̂ ( g ) ) ] = ∫ d N θ pr ( θ ∣ g ) θ ,
θ ̂ W ( g ) = θ ¯ + K θ , g K g − 1 ( g − g = ) ,
EMSE W = tr [ K θ ] − tr [ K θ , g K g − 1 K θ , g t ] .
θ ̂ MAP = argmax θ { pr ( θ ∣ g ) }
= argmax θ { pr ( g ∣ θ ) pr ( θ ) pr ( g ) }
= argmax θ { ln pr ( g ∣ θ ) + ln pr ( θ ) } ,
pr ( g | θ ) ≈ 1 ( 2 π ) M det ( K g | θ ) exp [ − 1 2 ( g − g = ( θ ) ) t K g | θ − 1 ( g − g = ( θ ) ) ] ,
K g ∣ θ ≈ 〈 K g ∣ θ 〉 θ = K ¯ g .
ln [ pr ( θ | g ) ] ≈ − 1 2 ( g − g = ( θ ) ) t K ¯ g − 1 ( g − g = ( θ ) ) + ln [ pr ( θ ) ] + const .
θ ̂ SL ( g ) = argmax θ { g = ( θ ) t K ¯ g − 1 g − 1 2 g = ( θ ) t K ¯ g − 1 g = ( θ ) + ln [ pr ( θ ) ] } .
∣ g ¯ ∣ m = ∫ f ( r ) h m ( r ) d q r ,
g ¯ ( f ) = ℋ f .
f ( r ; θ ) = f sig ( r ; θ ) + f bkgnd ( r ) .
g = ℋ f + n .
K n = 〈 ( g − g ¯ ( f ) ) ( g − g ¯ ( f ) ) t 〉 g | f = Diag ( g ¯ ( f ) ) .
K ¯ n ≡ 〈 K n 〉 f = Diag ( g = ) .
g ¯ ( f ) ≡ 〈 g 〉 g ∣ f = ℋ f sig ( r ; θ ) + ℋ f bkgnd ( r ) ,
g = ≡ 〈 g ¯ ( f ) 〉 f ≡ 〈 g ¯ ( f ) 〉 θ , f bkgnd = ℋ f ¯ sig ( r ) + ℋ f ¯ bkgnd ( r ) .
g = ( θ ) ≡ 〈 g ¯ ( f ) 〉 f bkgnd = ℋ f sig ( r ; θ ) + ℋ f ¯ bkgnd ( r ) .
K g ≡ 〈 〈 ( g − g = ) ( g − g = ) t 〉 g | f 〉 θ , f bkgnd
K g = 〈 ( [ g − g ¯ ( f ) ] + [ g ¯ ( f ) − g = ] ) ( [ g − g ¯ ( f ) ] + [ g ¯ ( f ) − g = ] ) t 〉 g , f
= 〈 [ g − g ¯ ( f ) ] [ g − g ¯ ( f ) ] t 〉 g , f + 〈 [ g ¯ ( f ) − g = ] [ g ¯ ( f ) − g = ] t 〉 g , f
= K ¯ n + 〈 [ g ¯ ( f ) − g = ] [ g ¯ ( f ) − g = ] t 〉 g , f ,
〈 〈 [ g − g ¯ ( f ) ] [ g ¯ ( f ) − g = ] t 〉 g | f 〉 f = 〈 [ 〈 g 〉 g | f − g ¯ ( f ) ] [ g ¯ ( f ) − g = ] t 〉 f
= 〈 [ g ¯ ( f ) − g ¯ ( f ) ] [ g ¯ ( f ) − g = ] t 〉 f
= 0 .
K g = K ¯ n + K ¯ g ¯ bkgnd + K g = sig .
K ¯ g ¯ bkgnd = 〈 [ g ¯ ( f ) − g = ( θ ) ] [ g ¯ ( f ) − g = ( θ ) ] t 〉 f bkgnd , θ
= ℋ 〈 [ f bkgnd − f ¯ bkgnd ] [ f bkgnd − f ¯ bkgnd ] † 〉 f bkgnd ℋ †
= ℋ K f bkgnd ℋ †
K g = sig = 〈 [ g = ( θ ) − g = ] [ g = ( θ ) − g = ] t 〉 f bkgnd , θ
= ℋ 〈 [ f sig − f ¯ sig ] [ f sig − f ¯ sig ] † 〉 θ ℋ †
= ℋ K f sig ℋ † .
K ¯ n = 〈 [ g − g ¯ ( f ) ] [ g − g ¯ ( f ) ] t 〉 g , f = Diag ( g = ) .
K θ , g = 〈 ( θ − θ ¯ ) ( g − g = ) t 〉 θ , g ,
K θ , g = 〈 〈 ( θ − θ ¯ ) ( g − g = ) t 〉 g | θ 〉 θ
= 〈 ( θ − θ ¯ ) 〈 ( g − g = ) t 〉 g | θ 〉 θ
= 〈 ( θ − θ ¯ ) ( g = ( θ ) − g = ) t 〉 θ
K θ , g = 〈 ( θ − θ ¯ ) ( g = ( θ ) ) t 〉 θ .
K θ , g = 〈 ( θ − θ ¯ ) ( f sig ( θ ) + f ¯ bkgnd ) t 〉 θ ℋ † .
K θ , g = 〈 ( θ − θ ¯ ) f sig ( θ ) t 〉 θ ℋ † .
f sig ( r ; θ ) = A I cyl ( R ) cyl ( r − c R )
cyl ( r − c R ) = { 0 , for ∣ r − c ∣ > R 1 , otherwise .
f ¯ sig ( r ) = 〈 f sig ( r ; θ ) 〉 θ
K f sig ( r , r ′ ) = 〈 ( f sig ( r ; θ ) − f ¯ sig ( r ; θ ) ) ( f sig ( r ′ ; θ ) − f ¯ sig ( r ′ ; θ ) ) 〉 θ .
pr A ( A ) = { ( A − A 0 ) α A − 1 exp ( − ( A − A 0 ) β A ) β A α A Γ ( α A ) ; for A > A 0 0 ; otherwise .
pr R ( R ) = { ( R − R 0 ) α R − 1 exp ( − ( R − R 0 ) β R ) β R α R Γ ( α R ) ; for R > R 0 0 ; otherwise .
f bkgnd ( r ) = S ( r ) ( A b + ∑ n = 1 L l ( r − r n ) ) .
f ¯ bkgnd ( r ) = 〈 f bkgnd ( r ) 〉 r n , L = S ( r ) ( A b + L ¯ I l V FOV ) .
K f bkgnd ( r , r ′ ) = 〈 f bkgnd ( r ) f bkgnd ( r ′ ) 〉 r n , L − f ¯ bkgnd ( r ) f ¯ bkgnd ( r ′ ) .
K f bkgnd ( r , r ′ ) = S ( r ) S ( r ′ ) ( L ¯ V FOV [ l ⊗ l ] ( r − r ′ ) ) ,
f ¯ bkgnd ( r ) = S ( r ) ( A b + L ¯ b 0 ( 2 π ) 3 2 σ 3 V FOV )
K f bkgnd ( r , r ′ ) = S ( r ) S ( r ′ ) ( L ¯ b 0 2 π 3 2 σ 3 exp ( − ( r − r ′ ) 2 4 σ 2 ) V FOV ) .
[ g ¯ ] m = f ( r m ) ,
〈 ( θ − θ ¯ ) f sig ( θ ) t 〉 θ = [ 〈 ( A − A ¯ ) f sig ( θ ) t 〉 θ 〈 ( R − R ¯ ) f sig ( θ ) t 〉 θ 〈 ( c x − c ¯ x ) f sig ( θ ) t 〉 θ 〈 ( c y − c ¯ y ) f sig ( θ ) t 〉 θ ] .
〈 ( A − A ¯ ) f sig ( r ; θ ) 〉 θ = ∫ d c p r ( c ) ∫ d R p r ( R ) ×
∫ d A p r ( A ) ( A − A ¯ ) A I cyl ( R ) cyl ( r − c R )
= σ A 2 A ¯ f ¯ sig ( r ) .
〈 ( R − R ¯ ) f sig ( r ; θ ) 〉 θ = A ¯ 〈 ∫ d c Gaus ( c ) ( R − R ¯ ) I cyl ( R ) cyl ( r − c R ) 〉 R ,
〈 ( c i − c ¯ i ) f sig ( r ; θ ) t 〉 θ = A ¯ 〈 ∫ d c Gaus ( c ) ( c i − c ¯ i ) I cyl ( R ) cyl ( r − c R ) 〉 R .
[ K f sig ] m , m ′ = 〈 A 2 〉 A 〈 1 I cyl ( R ) 2 ∫ d 2 c Gaus ( c ) cyl ( r m − c R ) cyl ( r m ′ − c R ) 〉 R ,
pr ( g ∣ θ ) = ∫ d M b pr ( g ∣ θ ; b ) pr ( b )
g = ( θ ) = ℋ f sig ( θ ) + ℋ f ¯ bkgnd
≡ s ( θ ) + b ¯
K g ∣ θ = 〈 ( g ( θ ) − g = ( θ ) ) ( g ( θ ) − g = ( θ ) ) t 〉 g ∣ θ
= Diag ( g = ( θ ) ) + K ¯ g ¯ bkgnd .
K g ∣ θ ≈ 〈 K g ∣ θ 〉 θ = Diag ( g = ) + K ¯ g ¯ bkgnd .
θ ̂ SL ( g ) = argmax θ { s ( θ ) t ( Diag ( g = ) + K ¯ g ¯ bkgnd ) − 1 ( g − b ¯ − 1 2 s ( θ ) ) + ln [ pr ( θ ) ] } .
A ̂ SL ( g ) = argmax A { s ( A ) t ( Diag ( g = ) + K ¯ g ¯ bkgnd ) − 1 ( g − b ¯ − 1 2 s ( A ) ) + ln [ pr ( A ) ] }
s ( A ) = A s ~ ,
A ̂ SL ( g ) = argmax A { A s ~ t ( Diag ( g = ) + K ¯ g ¯ bkgnd ) − 1 ( g − b ¯ − 1 2 A s ~ ) + ln [ pr ( A ) ] } .
A ̂ ( g ) = s ~ t ( Diag ( g = ) + K ¯ g ¯ bkgnd ) − 1 ( g − b ¯ ) s ~ t ( Diag ( g = ) + K ¯ g ¯ bkgnd ) − 1 s ~
〈 A ̂ ( g ) 〉 g ∣ A = A

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