Abstract

This paper analyzes the tradeoff between spatial resolution and noise for simple pinhole imaging systems with position-sensitive photon-counting detectors. We consider image recovery algorithms based on density estimation methods using kernels that are based on apodized inverse filters. This approach allows a continuous-object, continuous-data treatment of the problem. The analysis shows that to minimize the variance of the emission-rate density estimate at a specified reconstructed spatial resolution, the pinhole size should be directly proportional to that spatial resolution. For a Gaussian pinhole, the variance-minimizing full-width half maximum (FWHM) of the pinhole equals the desired object spatial resolution divided by √2. Simulation results confirm this conclusion empirically. The general approach is a potentially useful addition to the collection of tools available for imaging system design.

© 1998 Optical Society of America

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  1. B M Tsui, C E Metz, F B Atkins, S J Starr, and R N Beck, “A comparison of optimum detector spatial resolution in nuclear imaging based on statistical theory and on observer performance,” Phys. Med. Biol. 23654–676 (1978).
    [Crossref] [PubMed]
  2. H H Barrett, J N Aarsvold, H B Barber, E B Cargill, R D Fiete, T S Hickernell, T D Milster, K J Myers, D D Patton, R K Rowe, R H Seacat, W E Smith, and J M Woolfenden, “Applications of statistical decision theory in nuclear medicine,” In C N de Graaf and M A Viergever, editors, Proc. Tenth Intl. Conf. on Information Processing in Medical Im., (Plenum Press, New York, 1987) pp. 151–166.
  3. K J Myers, J P Rolland, H H Barrett, and R F Wagner, “Aperture optimization for emission imaging: effect of a spatially varying background,” J. Opt. Soc. Am. A 71279–1293 (1990).
    [Crossref] [PubMed]
  4. H H Barrett, “Objective assessment of image quality: effects of quantum noise and object variability,” J. Opt. Soc. Am. A 71266–1278 (1990).
    [Crossref] [PubMed]
  5. J P Rolland, H H Barrett, and G W Seeley, “Ideal versus human observer for long-tailed point spread functions: does deconvolution help?” Phys. Med. Biol. 361091–1109 (1991).
    [Crossref] [PubMed]
  6. H H Barrett, J Yao, J P Rolland, and K J Myers, “Model observers for assessment of image quality,” Proc. Natl. Acad. Sci. 909758–65 (1993).
    [Crossref] [PubMed]
  7. C K Abbey and H H Barrett, “Linear iterative reconstruction algorithms: study of observer performance,” In Information Processing in Medical Imaging, (Kluwer, Dordrect, 1995) pp 65–76.
  8. H H Barrett, J L Denny, R F Wagner, and K J Myers, “Objective assessment of image quality. II. Fisher information, Fourier crosstalk, and figures of merit for task performance,” J. Opt. Soc. Am. A 12834–852 (1995).
    [Crossref]
  9. N E Hartsough, H H Barrett, H B Barber, and J M Woolfenden, “Intraoperative tumor detection: Relative performance of single-element, dual-element, and imaging probes with various collimators,” IEEE Trans. Med. Imaging 14259–265 (1995).
    [Crossref] [PubMed]
  10. T Kanungo, M Y Jaisimha, J Palmer, and R M Haralick, “A methodology for quantitative performance evaluation of detection algorithms,” IEEE Trans. Image Process. 41667–1674 (1995).
    [Crossref] [PubMed]
  11. E P Ficaro, J A Fessler, P D Shreve, J N Kritzman, P A Rose, and J R Corbett, “Simultaneous transmission/emission myocardial perfusion tomography: Diagnostic accuracy of attenuation-corrected 99m-Tc-Sestamibi SPECT,” Circulation 93463–473 (1996). http://www.eecs.umich.edu/?fessler
    [Crossref] [PubMed]
  12. A O Hero, J A Fessler, and M Usman, “Exploring estimator bias-variance tradeoffs using the uniform CR bound,” IEEE Trans. Signal Process. 442026–2041 (1996). http://www.eecs.umich.edu/?fessler
    [Crossref]
  13. J A Fessler and A O Hero, “Cramer-Rao lower bounds for biased image reconstruction,” In Proc. Midwest Symposium on Circuits and Systems, Vol. 1, (IEEE, New York, 1993) pp 253–256. http://www.eecs.umich.edu/~fessler
  14. Mohammad Usman, “Biased and unbiased Cramer-Rao bounds: computational issues and applications,” PhD thesis, Univ. of Michigan, Ann Arbor, MI, 48109-2122, Ann Arbor, MI., 1994.
  15. Chor-Yi Ng, “Preliminary studies on the feasibility of addition of vertex view to conventional brain SPECT imaging,” PhD thesis, Univ. of Michigan, Ann Arbor, MI, 48109-2122, Ann Arbor, MI., January 1997.
  16. F O’Sullivan and Y Pawitan, “Bandwidth selection for indirect density estimation based on corrupted histogram data,” J. Am. Stat. Assoc.,  91(434):610–26, June (1996).
    [Crossref]
  17. P P B Eggermont and V N LaRiccia, “Nonlinearly smoothed EM density estimation with automated smoothing parameter selection for nonparametric deconvolution problems,” J. Am. Stat. Assoc.,  92(440):1451–1458, December (1997).
    [Crossref]
  18. B W Silverman, Density estimation for statistics and data analysis, (Chapman and Hall, New York, 1986).
  19. I M Johnstone, “On singular value decompositions for the Radon Transform and smoothness classes of functions,” Technical Report 310, Dept. of Statistics, Stanford Univ., January 1989.
  20. I M Johnstone and B W Silverman, “Discretization effects in statistical inverse problems,” Technical Report 310, Dept of Statistics, Stanford Univ., August 1990.
  21. I M Johnstone and B W Silverman, “Speed of estimation in positron emission tomography,” Ann. Stat. 18251–280 (1990).
    [Crossref]
  22. P J Bickel and Y Ritov, “Estimating linear functionals of a PET image,” IEEE Trans. Med. Imaging 1481–87 (1995).
    [Crossref] [PubMed]
  23. B W Silverman, “Kernel density estimation using the fast Fourier transform,” Appl. Stat. 3193–99 (1982).
    [Crossref]
  24. B W Silverman, “On the estimation of a probability density function by the maximum penalized likelihood method,” Ann. Stat. 10795–810 (1982).
    [Crossref]
  25. D L Snyder and M I Miller, Random point processes in time and space, (Springer Verlag, New York, 1991).
    [Crossref]
  26. A Macovski, Medical imaging systems, (Prentice-Hall, New Jersey, 1983).
  27. M C Jones, J S Marron, and S J Sheather, “A brief survey of bandwidth selection for density estimation,” J. Am. Stat. Assoc.,  91(433):401–407, March (1996).
    [Crossref]
  28. P P B Eggermont and V N LaRiccia, “Maximum smoothed likelihood density estimation for inverse problems,” Ann. Stat. 23199–220 (1995).
    [Crossref]
  29. Y-C Tai, A Chatziioannou, M Dahlbom, and E J Hoffman, “Investigation on deadtime characteristics for simultaneous emission-transmission data acquisition in PET,” In Proc. IEEE Nuc. Sci. Symp. Med. Im. Conf., (IEEE, New York, 1997).
  30. D L Snyder and D G Politte, “Image reconstruction from list-mode data in emission tomography system having time-of-flight measurements,” IEEE Trans. Nucl. Sci. 201843–1849 (1983).
    [Crossref]
  31. H H Barrett, Timothy White, and Lucas C Parra, “List-mode likelihood,” J. Opt. Soc. Am. A 142914–2923 (1997).
    [Crossref]
  32. H H Barrett and W Swindell, Radiological imaging: the theory of image formation, detection, and processing, (Academic, New York, 1981).
  33. V Ochoa, R Mastrippolito, Y Charon, P Laniece, L Pinot, and L Valentin, “TOHR: Prototype design and characterization of an original small animal tomograph,” In Proc. IEEE Nuc. Sci. Symp. Med. Im. Conf., (IEEE, New York, 1997).
  34. S Geman and C R Hwang, “Nonparametric maximum likelihood estimation by the method of sieves,” Ann. Stat. 10401–414 (1982).
    [Crossref]
  35. R Bracewell, The Fourier transform and its applications, (McGraw-Hill, New York, 1978).

1997 (2)

P P B Eggermont and V N LaRiccia, “Nonlinearly smoothed EM density estimation with automated smoothing parameter selection for nonparametric deconvolution problems,” J. Am. Stat. Assoc.,  92(440):1451–1458, December (1997).
[Crossref]

H H Barrett, Timothy White, and Lucas C Parra, “List-mode likelihood,” J. Opt. Soc. Am. A 142914–2923 (1997).
[Crossref]

1996 (4)

E P Ficaro, J A Fessler, P D Shreve, J N Kritzman, P A Rose, and J R Corbett, “Simultaneous transmission/emission myocardial perfusion tomography: Diagnostic accuracy of attenuation-corrected 99m-Tc-Sestamibi SPECT,” Circulation 93463–473 (1996). http://www.eecs.umich.edu/?fessler
[Crossref] [PubMed]

A O Hero, J A Fessler, and M Usman, “Exploring estimator bias-variance tradeoffs using the uniform CR bound,” IEEE Trans. Signal Process. 442026–2041 (1996). http://www.eecs.umich.edu/?fessler
[Crossref]

F O’Sullivan and Y Pawitan, “Bandwidth selection for indirect density estimation based on corrupted histogram data,” J. Am. Stat. Assoc.,  91(434):610–26, June (1996).
[Crossref]

M C Jones, J S Marron, and S J Sheather, “A brief survey of bandwidth selection for density estimation,” J. Am. Stat. Assoc.,  91(433):401–407, March (1996).
[Crossref]

1995 (5)

P P B Eggermont and V N LaRiccia, “Maximum smoothed likelihood density estimation for inverse problems,” Ann. Stat. 23199–220 (1995).
[Crossref]

P J Bickel and Y Ritov, “Estimating linear functionals of a PET image,” IEEE Trans. Med. Imaging 1481–87 (1995).
[Crossref] [PubMed]

H H Barrett, J L Denny, R F Wagner, and K J Myers, “Objective assessment of image quality. II. Fisher information, Fourier crosstalk, and figures of merit for task performance,” J. Opt. Soc. Am. A 12834–852 (1995).
[Crossref]

N E Hartsough, H H Barrett, H B Barber, and J M Woolfenden, “Intraoperative tumor detection: Relative performance of single-element, dual-element, and imaging probes with various collimators,” IEEE Trans. Med. Imaging 14259–265 (1995).
[Crossref] [PubMed]

T Kanungo, M Y Jaisimha, J Palmer, and R M Haralick, “A methodology for quantitative performance evaluation of detection algorithms,” IEEE Trans. Image Process. 41667–1674 (1995).
[Crossref] [PubMed]

1993 (1)

H H Barrett, J Yao, J P Rolland, and K J Myers, “Model observers for assessment of image quality,” Proc. Natl. Acad. Sci. 909758–65 (1993).
[Crossref] [PubMed]

1991 (1)

J P Rolland, H H Barrett, and G W Seeley, “Ideal versus human observer for long-tailed point spread functions: does deconvolution help?” Phys. Med. Biol. 361091–1109 (1991).
[Crossref] [PubMed]

1990 (3)

1987 (1)

H H Barrett, J N Aarsvold, H B Barber, E B Cargill, R D Fiete, T S Hickernell, T D Milster, K J Myers, D D Patton, R K Rowe, R H Seacat, W E Smith, and J M Woolfenden, “Applications of statistical decision theory in nuclear medicine,” In C N de Graaf and M A Viergever, editors, Proc. Tenth Intl. Conf. on Information Processing in Medical Im., (Plenum Press, New York, 1987) pp. 151–166.

1983 (1)

D L Snyder and D G Politte, “Image reconstruction from list-mode data in emission tomography system having time-of-flight measurements,” IEEE Trans. Nucl. Sci. 201843–1849 (1983).
[Crossref]

1982 (3)

B W Silverman, “Kernel density estimation using the fast Fourier transform,” Appl. Stat. 3193–99 (1982).
[Crossref]

B W Silverman, “On the estimation of a probability density function by the maximum penalized likelihood method,” Ann. Stat. 10795–810 (1982).
[Crossref]

S Geman and C R Hwang, “Nonparametric maximum likelihood estimation by the method of sieves,” Ann. Stat. 10401–414 (1982).
[Crossref]

1978 (1)

B M Tsui, C E Metz, F B Atkins, S J Starr, and R N Beck, “A comparison of optimum detector spatial resolution in nuclear imaging based on statistical theory and on observer performance,” Phys. Med. Biol. 23654–676 (1978).
[Crossref] [PubMed]

Aarsvold, J N

H H Barrett, J N Aarsvold, H B Barber, E B Cargill, R D Fiete, T S Hickernell, T D Milster, K J Myers, D D Patton, R K Rowe, R H Seacat, W E Smith, and J M Woolfenden, “Applications of statistical decision theory in nuclear medicine,” In C N de Graaf and M A Viergever, editors, Proc. Tenth Intl. Conf. on Information Processing in Medical Im., (Plenum Press, New York, 1987) pp. 151–166.

Abbey, C K

C K Abbey and H H Barrett, “Linear iterative reconstruction algorithms: study of observer performance,” In Information Processing in Medical Imaging, (Kluwer, Dordrect, 1995) pp 65–76.

Atkins, F B

B M Tsui, C E Metz, F B Atkins, S J Starr, and R N Beck, “A comparison of optimum detector spatial resolution in nuclear imaging based on statistical theory and on observer performance,” Phys. Med. Biol. 23654–676 (1978).
[Crossref] [PubMed]

Barber, H B

N E Hartsough, H H Barrett, H B Barber, and J M Woolfenden, “Intraoperative tumor detection: Relative performance of single-element, dual-element, and imaging probes with various collimators,” IEEE Trans. Med. Imaging 14259–265 (1995).
[Crossref] [PubMed]

H H Barrett, J N Aarsvold, H B Barber, E B Cargill, R D Fiete, T S Hickernell, T D Milster, K J Myers, D D Patton, R K Rowe, R H Seacat, W E Smith, and J M Woolfenden, “Applications of statistical decision theory in nuclear medicine,” In C N de Graaf and M A Viergever, editors, Proc. Tenth Intl. Conf. on Information Processing in Medical Im., (Plenum Press, New York, 1987) pp. 151–166.

Barrett, H H

H H Barrett, Timothy White, and Lucas C Parra, “List-mode likelihood,” J. Opt. Soc. Am. A 142914–2923 (1997).
[Crossref]

H H Barrett, J L Denny, R F Wagner, and K J Myers, “Objective assessment of image quality. II. Fisher information, Fourier crosstalk, and figures of merit for task performance,” J. Opt. Soc. Am. A 12834–852 (1995).
[Crossref]

N E Hartsough, H H Barrett, H B Barber, and J M Woolfenden, “Intraoperative tumor detection: Relative performance of single-element, dual-element, and imaging probes with various collimators,” IEEE Trans. Med. Imaging 14259–265 (1995).
[Crossref] [PubMed]

H H Barrett, J Yao, J P Rolland, and K J Myers, “Model observers for assessment of image quality,” Proc. Natl. Acad. Sci. 909758–65 (1993).
[Crossref] [PubMed]

J P Rolland, H H Barrett, and G W Seeley, “Ideal versus human observer for long-tailed point spread functions: does deconvolution help?” Phys. Med. Biol. 361091–1109 (1991).
[Crossref] [PubMed]

K J Myers, J P Rolland, H H Barrett, and R F Wagner, “Aperture optimization for emission imaging: effect of a spatially varying background,” J. Opt. Soc. Am. A 71279–1293 (1990).
[Crossref] [PubMed]

H H Barrett, “Objective assessment of image quality: effects of quantum noise and object variability,” J. Opt. Soc. Am. A 71266–1278 (1990).
[Crossref] [PubMed]

H H Barrett, J N Aarsvold, H B Barber, E B Cargill, R D Fiete, T S Hickernell, T D Milster, K J Myers, D D Patton, R K Rowe, R H Seacat, W E Smith, and J M Woolfenden, “Applications of statistical decision theory in nuclear medicine,” In C N de Graaf and M A Viergever, editors, Proc. Tenth Intl. Conf. on Information Processing in Medical Im., (Plenum Press, New York, 1987) pp. 151–166.

C K Abbey and H H Barrett, “Linear iterative reconstruction algorithms: study of observer performance,” In Information Processing in Medical Imaging, (Kluwer, Dordrect, 1995) pp 65–76.

H H Barrett and W Swindell, Radiological imaging: the theory of image formation, detection, and processing, (Academic, New York, 1981).

Beck, R N

B M Tsui, C E Metz, F B Atkins, S J Starr, and R N Beck, “A comparison of optimum detector spatial resolution in nuclear imaging based on statistical theory and on observer performance,” Phys. Med. Biol. 23654–676 (1978).
[Crossref] [PubMed]

Bickel, P J

P J Bickel and Y Ritov, “Estimating linear functionals of a PET image,” IEEE Trans. Med. Imaging 1481–87 (1995).
[Crossref] [PubMed]

Bracewell, R

R Bracewell, The Fourier transform and its applications, (McGraw-Hill, New York, 1978).

Cargill, E B

H H Barrett, J N Aarsvold, H B Barber, E B Cargill, R D Fiete, T S Hickernell, T D Milster, K J Myers, D D Patton, R K Rowe, R H Seacat, W E Smith, and J M Woolfenden, “Applications of statistical decision theory in nuclear medicine,” In C N de Graaf and M A Viergever, editors, Proc. Tenth Intl. Conf. on Information Processing in Medical Im., (Plenum Press, New York, 1987) pp. 151–166.

Charon, Y

V Ochoa, R Mastrippolito, Y Charon, P Laniece, L Pinot, and L Valentin, “TOHR: Prototype design and characterization of an original small animal tomograph,” In Proc. IEEE Nuc. Sci. Symp. Med. Im. Conf., (IEEE, New York, 1997).

Chatziioannou, A

Y-C Tai, A Chatziioannou, M Dahlbom, and E J Hoffman, “Investigation on deadtime characteristics for simultaneous emission-transmission data acquisition in PET,” In Proc. IEEE Nuc. Sci. Symp. Med. Im. Conf., (IEEE, New York, 1997).

Corbett, J R

E P Ficaro, J A Fessler, P D Shreve, J N Kritzman, P A Rose, and J R Corbett, “Simultaneous transmission/emission myocardial perfusion tomography: Diagnostic accuracy of attenuation-corrected 99m-Tc-Sestamibi SPECT,” Circulation 93463–473 (1996). http://www.eecs.umich.edu/?fessler
[Crossref] [PubMed]

Dahlbom, M

Y-C Tai, A Chatziioannou, M Dahlbom, and E J Hoffman, “Investigation on deadtime characteristics for simultaneous emission-transmission data acquisition in PET,” In Proc. IEEE Nuc. Sci. Symp. Med. Im. Conf., (IEEE, New York, 1997).

Denny, J L

Eggermont, P P B

P P B Eggermont and V N LaRiccia, “Nonlinearly smoothed EM density estimation with automated smoothing parameter selection for nonparametric deconvolution problems,” J. Am. Stat. Assoc.,  92(440):1451–1458, December (1997).
[Crossref]

P P B Eggermont and V N LaRiccia, “Maximum smoothed likelihood density estimation for inverse problems,” Ann. Stat. 23199–220 (1995).
[Crossref]

Fessler, J A

A O Hero, J A Fessler, and M Usman, “Exploring estimator bias-variance tradeoffs using the uniform CR bound,” IEEE Trans. Signal Process. 442026–2041 (1996). http://www.eecs.umich.edu/?fessler
[Crossref]

E P Ficaro, J A Fessler, P D Shreve, J N Kritzman, P A Rose, and J R Corbett, “Simultaneous transmission/emission myocardial perfusion tomography: Diagnostic accuracy of attenuation-corrected 99m-Tc-Sestamibi SPECT,” Circulation 93463–473 (1996). http://www.eecs.umich.edu/?fessler
[Crossref] [PubMed]

J A Fessler and A O Hero, “Cramer-Rao lower bounds for biased image reconstruction,” In Proc. Midwest Symposium on Circuits and Systems, Vol. 1, (IEEE, New York, 1993) pp 253–256. http://www.eecs.umich.edu/~fessler

Ficaro, E P

E P Ficaro, J A Fessler, P D Shreve, J N Kritzman, P A Rose, and J R Corbett, “Simultaneous transmission/emission myocardial perfusion tomography: Diagnostic accuracy of attenuation-corrected 99m-Tc-Sestamibi SPECT,” Circulation 93463–473 (1996). http://www.eecs.umich.edu/?fessler
[Crossref] [PubMed]

Fiete, R D

H H Barrett, J N Aarsvold, H B Barber, E B Cargill, R D Fiete, T S Hickernell, T D Milster, K J Myers, D D Patton, R K Rowe, R H Seacat, W E Smith, and J M Woolfenden, “Applications of statistical decision theory in nuclear medicine,” In C N de Graaf and M A Viergever, editors, Proc. Tenth Intl. Conf. on Information Processing in Medical Im., (Plenum Press, New York, 1987) pp. 151–166.

Geman, S

S Geman and C R Hwang, “Nonparametric maximum likelihood estimation by the method of sieves,” Ann. Stat. 10401–414 (1982).
[Crossref]

Haralick, R M

T Kanungo, M Y Jaisimha, J Palmer, and R M Haralick, “A methodology for quantitative performance evaluation of detection algorithms,” IEEE Trans. Image Process. 41667–1674 (1995).
[Crossref] [PubMed]

Hartsough, N E

N E Hartsough, H H Barrett, H B Barber, and J M Woolfenden, “Intraoperative tumor detection: Relative performance of single-element, dual-element, and imaging probes with various collimators,” IEEE Trans. Med. Imaging 14259–265 (1995).
[Crossref] [PubMed]

Hero, A O

A O Hero, J A Fessler, and M Usman, “Exploring estimator bias-variance tradeoffs using the uniform CR bound,” IEEE Trans. Signal Process. 442026–2041 (1996). http://www.eecs.umich.edu/?fessler
[Crossref]

J A Fessler and A O Hero, “Cramer-Rao lower bounds for biased image reconstruction,” In Proc. Midwest Symposium on Circuits and Systems, Vol. 1, (IEEE, New York, 1993) pp 253–256. http://www.eecs.umich.edu/~fessler

Hickernell, T S

H H Barrett, J N Aarsvold, H B Barber, E B Cargill, R D Fiete, T S Hickernell, T D Milster, K J Myers, D D Patton, R K Rowe, R H Seacat, W E Smith, and J M Woolfenden, “Applications of statistical decision theory in nuclear medicine,” In C N de Graaf and M A Viergever, editors, Proc. Tenth Intl. Conf. on Information Processing in Medical Im., (Plenum Press, New York, 1987) pp. 151–166.

Hoffman, E J

Y-C Tai, A Chatziioannou, M Dahlbom, and E J Hoffman, “Investigation on deadtime characteristics for simultaneous emission-transmission data acquisition in PET,” In Proc. IEEE Nuc. Sci. Symp. Med. Im. Conf., (IEEE, New York, 1997).

Hwang, C R

S Geman and C R Hwang, “Nonparametric maximum likelihood estimation by the method of sieves,” Ann. Stat. 10401–414 (1982).
[Crossref]

Jaisimha, M Y

T Kanungo, M Y Jaisimha, J Palmer, and R M Haralick, “A methodology for quantitative performance evaluation of detection algorithms,” IEEE Trans. Image Process. 41667–1674 (1995).
[Crossref] [PubMed]

Johnstone, I M

I M Johnstone and B W Silverman, “Speed of estimation in positron emission tomography,” Ann. Stat. 18251–280 (1990).
[Crossref]

I M Johnstone, “On singular value decompositions for the Radon Transform and smoothness classes of functions,” Technical Report 310, Dept. of Statistics, Stanford Univ., January 1989.

I M Johnstone and B W Silverman, “Discretization effects in statistical inverse problems,” Technical Report 310, Dept of Statistics, Stanford Univ., August 1990.

Jones, M C

M C Jones, J S Marron, and S J Sheather, “A brief survey of bandwidth selection for density estimation,” J. Am. Stat. Assoc.,  91(433):401–407, March (1996).
[Crossref]

Kanungo, T

T Kanungo, M Y Jaisimha, J Palmer, and R M Haralick, “A methodology for quantitative performance evaluation of detection algorithms,” IEEE Trans. Image Process. 41667–1674 (1995).
[Crossref] [PubMed]

Kritzman, J N

E P Ficaro, J A Fessler, P D Shreve, J N Kritzman, P A Rose, and J R Corbett, “Simultaneous transmission/emission myocardial perfusion tomography: Diagnostic accuracy of attenuation-corrected 99m-Tc-Sestamibi SPECT,” Circulation 93463–473 (1996). http://www.eecs.umich.edu/?fessler
[Crossref] [PubMed]

Laniece, P

V Ochoa, R Mastrippolito, Y Charon, P Laniece, L Pinot, and L Valentin, “TOHR: Prototype design and characterization of an original small animal tomograph,” In Proc. IEEE Nuc. Sci. Symp. Med. Im. Conf., (IEEE, New York, 1997).

LaRiccia, V N

P P B Eggermont and V N LaRiccia, “Nonlinearly smoothed EM density estimation with automated smoothing parameter selection for nonparametric deconvolution problems,” J. Am. Stat. Assoc.,  92(440):1451–1458, December (1997).
[Crossref]

P P B Eggermont and V N LaRiccia, “Maximum smoothed likelihood density estimation for inverse problems,” Ann. Stat. 23199–220 (1995).
[Crossref]

Macovski, A

A Macovski, Medical imaging systems, (Prentice-Hall, New Jersey, 1983).

Marron, J S

M C Jones, J S Marron, and S J Sheather, “A brief survey of bandwidth selection for density estimation,” J. Am. Stat. Assoc.,  91(433):401–407, March (1996).
[Crossref]

Mastrippolito, R

V Ochoa, R Mastrippolito, Y Charon, P Laniece, L Pinot, and L Valentin, “TOHR: Prototype design and characterization of an original small animal tomograph,” In Proc. IEEE Nuc. Sci. Symp. Med. Im. Conf., (IEEE, New York, 1997).

Metz, C E

B M Tsui, C E Metz, F B Atkins, S J Starr, and R N Beck, “A comparison of optimum detector spatial resolution in nuclear imaging based on statistical theory and on observer performance,” Phys. Med. Biol. 23654–676 (1978).
[Crossref] [PubMed]

Miller, M I

D L Snyder and M I Miller, Random point processes in time and space, (Springer Verlag, New York, 1991).
[Crossref]

Milster, T D

H H Barrett, J N Aarsvold, H B Barber, E B Cargill, R D Fiete, T S Hickernell, T D Milster, K J Myers, D D Patton, R K Rowe, R H Seacat, W E Smith, and J M Woolfenden, “Applications of statistical decision theory in nuclear medicine,” In C N de Graaf and M A Viergever, editors, Proc. Tenth Intl. Conf. on Information Processing in Medical Im., (Plenum Press, New York, 1987) pp. 151–166.

Myers, K J

H H Barrett, J L Denny, R F Wagner, and K J Myers, “Objective assessment of image quality. II. Fisher information, Fourier crosstalk, and figures of merit for task performance,” J. Opt. Soc. Am. A 12834–852 (1995).
[Crossref]

H H Barrett, J Yao, J P Rolland, and K J Myers, “Model observers for assessment of image quality,” Proc. Natl. Acad. Sci. 909758–65 (1993).
[Crossref] [PubMed]

K J Myers, J P Rolland, H H Barrett, and R F Wagner, “Aperture optimization for emission imaging: effect of a spatially varying background,” J. Opt. Soc. Am. A 71279–1293 (1990).
[Crossref] [PubMed]

H H Barrett, J N Aarsvold, H B Barber, E B Cargill, R D Fiete, T S Hickernell, T D Milster, K J Myers, D D Patton, R K Rowe, R H Seacat, W E Smith, and J M Woolfenden, “Applications of statistical decision theory in nuclear medicine,” In C N de Graaf and M A Viergever, editors, Proc. Tenth Intl. Conf. on Information Processing in Medical Im., (Plenum Press, New York, 1987) pp. 151–166.

Ng, Chor-Yi

Chor-Yi Ng, “Preliminary studies on the feasibility of addition of vertex view to conventional brain SPECT imaging,” PhD thesis, Univ. of Michigan, Ann Arbor, MI, 48109-2122, Ann Arbor, MI., January 1997.

O’Sullivan, F

F O’Sullivan and Y Pawitan, “Bandwidth selection for indirect density estimation based on corrupted histogram data,” J. Am. Stat. Assoc.,  91(434):610–26, June (1996).
[Crossref]

Ochoa, V

V Ochoa, R Mastrippolito, Y Charon, P Laniece, L Pinot, and L Valentin, “TOHR: Prototype design and characterization of an original small animal tomograph,” In Proc. IEEE Nuc. Sci. Symp. Med. Im. Conf., (IEEE, New York, 1997).

Palmer, J

T Kanungo, M Y Jaisimha, J Palmer, and R M Haralick, “A methodology for quantitative performance evaluation of detection algorithms,” IEEE Trans. Image Process. 41667–1674 (1995).
[Crossref] [PubMed]

Parra, Lucas C

Patton, D D

H H Barrett, J N Aarsvold, H B Barber, E B Cargill, R D Fiete, T S Hickernell, T D Milster, K J Myers, D D Patton, R K Rowe, R H Seacat, W E Smith, and J M Woolfenden, “Applications of statistical decision theory in nuclear medicine,” In C N de Graaf and M A Viergever, editors, Proc. Tenth Intl. Conf. on Information Processing in Medical Im., (Plenum Press, New York, 1987) pp. 151–166.

Pawitan, Y

F O’Sullivan and Y Pawitan, “Bandwidth selection for indirect density estimation based on corrupted histogram data,” J. Am. Stat. Assoc.,  91(434):610–26, June (1996).
[Crossref]

Pinot, L

V Ochoa, R Mastrippolito, Y Charon, P Laniece, L Pinot, and L Valentin, “TOHR: Prototype design and characterization of an original small animal tomograph,” In Proc. IEEE Nuc. Sci. Symp. Med. Im. Conf., (IEEE, New York, 1997).

Politte, D G

D L Snyder and D G Politte, “Image reconstruction from list-mode data in emission tomography system having time-of-flight measurements,” IEEE Trans. Nucl. Sci. 201843–1849 (1983).
[Crossref]

Ritov, Y

P J Bickel and Y Ritov, “Estimating linear functionals of a PET image,” IEEE Trans. Med. Imaging 1481–87 (1995).
[Crossref] [PubMed]

Rolland, J P

H H Barrett, J Yao, J P Rolland, and K J Myers, “Model observers for assessment of image quality,” Proc. Natl. Acad. Sci. 909758–65 (1993).
[Crossref] [PubMed]

J P Rolland, H H Barrett, and G W Seeley, “Ideal versus human observer for long-tailed point spread functions: does deconvolution help?” Phys. Med. Biol. 361091–1109 (1991).
[Crossref] [PubMed]

K J Myers, J P Rolland, H H Barrett, and R F Wagner, “Aperture optimization for emission imaging: effect of a spatially varying background,” J. Opt. Soc. Am. A 71279–1293 (1990).
[Crossref] [PubMed]

Rose, P A

E P Ficaro, J A Fessler, P D Shreve, J N Kritzman, P A Rose, and J R Corbett, “Simultaneous transmission/emission myocardial perfusion tomography: Diagnostic accuracy of attenuation-corrected 99m-Tc-Sestamibi SPECT,” Circulation 93463–473 (1996). http://www.eecs.umich.edu/?fessler
[Crossref] [PubMed]

Rowe, R K

H H Barrett, J N Aarsvold, H B Barber, E B Cargill, R D Fiete, T S Hickernell, T D Milster, K J Myers, D D Patton, R K Rowe, R H Seacat, W E Smith, and J M Woolfenden, “Applications of statistical decision theory in nuclear medicine,” In C N de Graaf and M A Viergever, editors, Proc. Tenth Intl. Conf. on Information Processing in Medical Im., (Plenum Press, New York, 1987) pp. 151–166.

Seacat, R H

H H Barrett, J N Aarsvold, H B Barber, E B Cargill, R D Fiete, T S Hickernell, T D Milster, K J Myers, D D Patton, R K Rowe, R H Seacat, W E Smith, and J M Woolfenden, “Applications of statistical decision theory in nuclear medicine,” In C N de Graaf and M A Viergever, editors, Proc. Tenth Intl. Conf. on Information Processing in Medical Im., (Plenum Press, New York, 1987) pp. 151–166.

Seeley, G W

J P Rolland, H H Barrett, and G W Seeley, “Ideal versus human observer for long-tailed point spread functions: does deconvolution help?” Phys. Med. Biol. 361091–1109 (1991).
[Crossref] [PubMed]

Sheather, S J

M C Jones, J S Marron, and S J Sheather, “A brief survey of bandwidth selection for density estimation,” J. Am. Stat. Assoc.,  91(433):401–407, March (1996).
[Crossref]

Shreve, P D

E P Ficaro, J A Fessler, P D Shreve, J N Kritzman, P A Rose, and J R Corbett, “Simultaneous transmission/emission myocardial perfusion tomography: Diagnostic accuracy of attenuation-corrected 99m-Tc-Sestamibi SPECT,” Circulation 93463–473 (1996). http://www.eecs.umich.edu/?fessler
[Crossref] [PubMed]

Silverman, B W

I M Johnstone and B W Silverman, “Speed of estimation in positron emission tomography,” Ann. Stat. 18251–280 (1990).
[Crossref]

B W Silverman, “Kernel density estimation using the fast Fourier transform,” Appl. Stat. 3193–99 (1982).
[Crossref]

B W Silverman, “On the estimation of a probability density function by the maximum penalized likelihood method,” Ann. Stat. 10795–810 (1982).
[Crossref]

B W Silverman, Density estimation for statistics and data analysis, (Chapman and Hall, New York, 1986).

I M Johnstone and B W Silverman, “Discretization effects in statistical inverse problems,” Technical Report 310, Dept of Statistics, Stanford Univ., August 1990.

Smith, W E

H H Barrett, J N Aarsvold, H B Barber, E B Cargill, R D Fiete, T S Hickernell, T D Milster, K J Myers, D D Patton, R K Rowe, R H Seacat, W E Smith, and J M Woolfenden, “Applications of statistical decision theory in nuclear medicine,” In C N de Graaf and M A Viergever, editors, Proc. Tenth Intl. Conf. on Information Processing in Medical Im., (Plenum Press, New York, 1987) pp. 151–166.

Snyder, D L

D L Snyder and D G Politte, “Image reconstruction from list-mode data in emission tomography system having time-of-flight measurements,” IEEE Trans. Nucl. Sci. 201843–1849 (1983).
[Crossref]

D L Snyder and M I Miller, Random point processes in time and space, (Springer Verlag, New York, 1991).
[Crossref]

Starr, S J

B M Tsui, C E Metz, F B Atkins, S J Starr, and R N Beck, “A comparison of optimum detector spatial resolution in nuclear imaging based on statistical theory and on observer performance,” Phys. Med. Biol. 23654–676 (1978).
[Crossref] [PubMed]

Swindell, W

H H Barrett and W Swindell, Radiological imaging: the theory of image formation, detection, and processing, (Academic, New York, 1981).

Tai, Y-C

Y-C Tai, A Chatziioannou, M Dahlbom, and E J Hoffman, “Investigation on deadtime characteristics for simultaneous emission-transmission data acquisition in PET,” In Proc. IEEE Nuc. Sci. Symp. Med. Im. Conf., (IEEE, New York, 1997).

Tsui, B M

B M Tsui, C E Metz, F B Atkins, S J Starr, and R N Beck, “A comparison of optimum detector spatial resolution in nuclear imaging based on statistical theory and on observer performance,” Phys. Med. Biol. 23654–676 (1978).
[Crossref] [PubMed]

Usman, M

A O Hero, J A Fessler, and M Usman, “Exploring estimator bias-variance tradeoffs using the uniform CR bound,” IEEE Trans. Signal Process. 442026–2041 (1996). http://www.eecs.umich.edu/?fessler
[Crossref]

Usman, Mohammad

Mohammad Usman, “Biased and unbiased Cramer-Rao bounds: computational issues and applications,” PhD thesis, Univ. of Michigan, Ann Arbor, MI, 48109-2122, Ann Arbor, MI., 1994.

Valentin, L

V Ochoa, R Mastrippolito, Y Charon, P Laniece, L Pinot, and L Valentin, “TOHR: Prototype design and characterization of an original small animal tomograph,” In Proc. IEEE Nuc. Sci. Symp. Med. Im. Conf., (IEEE, New York, 1997).

Wagner, R F

White, Timothy

Woolfenden, J M

N E Hartsough, H H Barrett, H B Barber, and J M Woolfenden, “Intraoperative tumor detection: Relative performance of single-element, dual-element, and imaging probes with various collimators,” IEEE Trans. Med. Imaging 14259–265 (1995).
[Crossref] [PubMed]

H H Barrett, J N Aarsvold, H B Barber, E B Cargill, R D Fiete, T S Hickernell, T D Milster, K J Myers, D D Patton, R K Rowe, R H Seacat, W E Smith, and J M Woolfenden, “Applications of statistical decision theory in nuclear medicine,” In C N de Graaf and M A Viergever, editors, Proc. Tenth Intl. Conf. on Information Processing in Medical Im., (Plenum Press, New York, 1987) pp. 151–166.

Yao, J

H H Barrett, J Yao, J P Rolland, and K J Myers, “Model observers for assessment of image quality,” Proc. Natl. Acad. Sci. 909758–65 (1993).
[Crossref] [PubMed]

Ann. Stat. (4)

I M Johnstone and B W Silverman, “Speed of estimation in positron emission tomography,” Ann. Stat. 18251–280 (1990).
[Crossref]

B W Silverman, “On the estimation of a probability density function by the maximum penalized likelihood method,” Ann. Stat. 10795–810 (1982).
[Crossref]

P P B Eggermont and V N LaRiccia, “Maximum smoothed likelihood density estimation for inverse problems,” Ann. Stat. 23199–220 (1995).
[Crossref]

S Geman and C R Hwang, “Nonparametric maximum likelihood estimation by the method of sieves,” Ann. Stat. 10401–414 (1982).
[Crossref]

Appl. Stat. (1)

B W Silverman, “Kernel density estimation using the fast Fourier transform,” Appl. Stat. 3193–99 (1982).
[Crossref]

Circulation (1)

E P Ficaro, J A Fessler, P D Shreve, J N Kritzman, P A Rose, and J R Corbett, “Simultaneous transmission/emission myocardial perfusion tomography: Diagnostic accuracy of attenuation-corrected 99m-Tc-Sestamibi SPECT,” Circulation 93463–473 (1996). http://www.eecs.umich.edu/?fessler
[Crossref] [PubMed]

IEEE Trans. Image Process. (1)

T Kanungo, M Y Jaisimha, J Palmer, and R M Haralick, “A methodology for quantitative performance evaluation of detection algorithms,” IEEE Trans. Image Process. 41667–1674 (1995).
[Crossref] [PubMed]

IEEE Trans. Med. Imaging (2)

P J Bickel and Y Ritov, “Estimating linear functionals of a PET image,” IEEE Trans. Med. Imaging 1481–87 (1995).
[Crossref] [PubMed]

N E Hartsough, H H Barrett, H B Barber, and J M Woolfenden, “Intraoperative tumor detection: Relative performance of single-element, dual-element, and imaging probes with various collimators,” IEEE Trans. Med. Imaging 14259–265 (1995).
[Crossref] [PubMed]

IEEE Trans. Nucl. Sci. (1)

D L Snyder and D G Politte, “Image reconstruction from list-mode data in emission tomography system having time-of-flight measurements,” IEEE Trans. Nucl. Sci. 201843–1849 (1983).
[Crossref]

IEEE Trans. Signal Process. (1)

A O Hero, J A Fessler, and M Usman, “Exploring estimator bias-variance tradeoffs using the uniform CR bound,” IEEE Trans. Signal Process. 442026–2041 (1996). http://www.eecs.umich.edu/?fessler
[Crossref]

J. Am. Stat. Assoc. (3)

F O’Sullivan and Y Pawitan, “Bandwidth selection for indirect density estimation based on corrupted histogram data,” J. Am. Stat. Assoc.,  91(434):610–26, June (1996).
[Crossref]

P P B Eggermont and V N LaRiccia, “Nonlinearly smoothed EM density estimation with automated smoothing parameter selection for nonparametric deconvolution problems,” J. Am. Stat. Assoc.,  92(440):1451–1458, December (1997).
[Crossref]

M C Jones, J S Marron, and S J Sheather, “A brief survey of bandwidth selection for density estimation,” J. Am. Stat. Assoc.,  91(433):401–407, March (1996).
[Crossref]

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

Phys. Med. Biol. (2)

J P Rolland, H H Barrett, and G W Seeley, “Ideal versus human observer for long-tailed point spread functions: does deconvolution help?” Phys. Med. Biol. 361091–1109 (1991).
[Crossref] [PubMed]

B M Tsui, C E Metz, F B Atkins, S J Starr, and R N Beck, “A comparison of optimum detector spatial resolution in nuclear imaging based on statistical theory and on observer performance,” Phys. Med. Biol. 23654–676 (1978).
[Crossref] [PubMed]

Proc. Natl. Acad. Sci. (1)

H H Barrett, J Yao, J P Rolland, and K J Myers, “Model observers for assessment of image quality,” Proc. Natl. Acad. Sci. 909758–65 (1993).
[Crossref] [PubMed]

Proc. Tenth Intl. Conf. on Information Processing in Medical Im. (1)

H H Barrett, J N Aarsvold, H B Barber, E B Cargill, R D Fiete, T S Hickernell, T D Milster, K J Myers, D D Patton, R K Rowe, R H Seacat, W E Smith, and J M Woolfenden, “Applications of statistical decision theory in nuclear medicine,” In C N de Graaf and M A Viergever, editors, Proc. Tenth Intl. Conf. on Information Processing in Medical Im., (Plenum Press, New York, 1987) pp. 151–166.

Other (13)

C K Abbey and H H Barrett, “Linear iterative reconstruction algorithms: study of observer performance,” In Information Processing in Medical Imaging, (Kluwer, Dordrect, 1995) pp 65–76.

B W Silverman, Density estimation for statistics and data analysis, (Chapman and Hall, New York, 1986).

I M Johnstone, “On singular value decompositions for the Radon Transform and smoothness classes of functions,” Technical Report 310, Dept. of Statistics, Stanford Univ., January 1989.

I M Johnstone and B W Silverman, “Discretization effects in statistical inverse problems,” Technical Report 310, Dept of Statistics, Stanford Univ., August 1990.

J A Fessler and A O Hero, “Cramer-Rao lower bounds for biased image reconstruction,” In Proc. Midwest Symposium on Circuits and Systems, Vol. 1, (IEEE, New York, 1993) pp 253–256. http://www.eecs.umich.edu/~fessler

Mohammad Usman, “Biased and unbiased Cramer-Rao bounds: computational issues and applications,” PhD thesis, Univ. of Michigan, Ann Arbor, MI, 48109-2122, Ann Arbor, MI., 1994.

Chor-Yi Ng, “Preliminary studies on the feasibility of addition of vertex view to conventional brain SPECT imaging,” PhD thesis, Univ. of Michigan, Ann Arbor, MI, 48109-2122, Ann Arbor, MI., January 1997.

H H Barrett and W Swindell, Radiological imaging: the theory of image formation, detection, and processing, (Academic, New York, 1981).

V Ochoa, R Mastrippolito, Y Charon, P Laniece, L Pinot, and L Valentin, “TOHR: Prototype design and characterization of an original small animal tomograph,” In Proc. IEEE Nuc. Sci. Symp. Med. Im. Conf., (IEEE, New York, 1997).

R Bracewell, The Fourier transform and its applications, (McGraw-Hill, New York, 1978).

Y-C Tai, A Chatziioannou, M Dahlbom, and E J Hoffman, “Investigation on deadtime characteristics for simultaneous emission-transmission data acquisition in PET,” In Proc. IEEE Nuc. Sci. Symp. Med. Im. Conf., (IEEE, New York, 1997).

D L Snyder and M I Miller, Random point processes in time and space, (Springer Verlag, New York, 1991).
[Crossref]

A Macovski, Medical imaging systems, (Prentice-Hall, New Jersey, 1983).

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Figures (6)

Figure 1.
Figure 1.

Profile through an approximate Gaussian pinhole.

Figure 2.
Figure 2.

Standard deviation of estimate versus Gaussian pinhole width.

Figure 3.
Figure 3.

Profile through an approximate Laplacian pinhole.

Figure 4.
Figure 4.

Sample means of 4000 realizations of the estimates of λ(x), for 21 Gaussian pinhole sizes ranging from 0.9 to 2.9 mm FWHM. The 21 mean curves are virtually indistinguishable since the reconstructed spatial resolution has been held fixed at 3mm FWHM.

Figure 5.
Figure 5.

Sample standard deviations of 4000 realizations of the estimates of λ(x), for 3 of the 21 Gaussian pinhole sizes: 1, 2.1, and 2.9 mm FWHM.

Figure 6.
Figure 6.

Normalized sample standard deviations of 4000 realizations of the estimates of λ(x), versus the FWHM of the Gaussian pinhole sizes. There are 119 plots, one for each of the x positions for which λ(x) > 0. The minimum is consistently near the theoretical prediction of 3/√2 ≈ 2.1.

Equations (86)

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P [ n = 1 n 0 { X ¯ n B n } ] = n = 1 n 0 P [ X ¯ n B n ] = n = 1 n 0 B n λ ( x ¯ ) d x ¯ Ω λ ( x ¯ ) d x ¯ .
f ( x ¯ ) = λ ( x ¯ ) s ( x ¯ ) λ ( x ¯ ) s ( x ¯ ) d x ¯ = λ ( x ¯ ) s ( x ¯ ) r ,
r λ ( x ¯ ) s ( x ¯ ) d x ¯
f ̂ ( x ¯ ) = 1 N n = 1 N 1 β k ( x ¯ V ¯ n m β ) = 1 N n = 1 N 1 β k ( x ¯ X ¯ n β ) ,
f ( v ¯ | x ¯ )
f ( v ¯ 1 , v ¯ 2 , x ¯ 1 , x ¯ 2 , ) = Π n f ( v ¯ n x ¯ n ) .
E [ N ] = t 0 λ ( x ¯ ) s ( x ¯ ) d x ¯ = t 0 r .
f V ( v ¯ ) = f ( v ¯ x ¯ ) f ( x ¯ ) d x ¯
λ ( x ¯ ) = f ( x ¯ ) s ( x ¯ ) r = f ( x ¯ ) s ( x ¯ ) E [ N ] t 0 .
λ ̂ ( x ¯ ) = f ̂ ( x ¯ ) s ( x ¯ ) N t 0 .
f ̂ ( x ¯ ) = 1 N n = 1 N g β ( x ¯ , V ¯ n ) ,
1 = g β ( x ¯ , v ¯ ) d x ¯ .
λ ̂ ( x ¯ ) = f ̂ ( x ¯ ) s ( x ¯ ) N t 0 = 1 t 0 s ( x ¯ ) n = 1 N g β ( x ¯ , V ¯ n ) .
μ ( x ¯ ) = E [ λ ̂ ( x ¯ ) ]
= E N [ E V [ λ ̂ ( x ¯ ) | N ] ]
= E N [ N t 0 s ( x ¯ ) E V [ g β ( x ¯ , V ¯ ) ] ]
= r s ( x ¯ ) E V [ g β ( x ¯ , V ¯ ) ]
= r s ( x ¯ ) g β ( x ¯ , v ¯ ) f V ( v ¯ ) d v ¯
= r s ( x ¯ ) g β ( x ¯ , v ¯ ) [ f ( v ¯ x ¯ ) f ( x ¯ ) d x ¯ ] d v ¯
= r s ( x ¯ ) [ g β ( x ¯ , v ¯ ) f ( v ¯ x ¯ ) d v ¯ ] f ( x ¯ ) d x ¯
= [ s ( x ¯ ) s ( x ¯ ) g β ( x ¯ , v ¯ ) f ( v ¯ x ¯ ) d v ¯ ] λ ( x ¯ ) d x ¯ .
μ ( x ¯ ) = psf ( x ¯ , x ¯ ) λ ( x ¯ ) d x ¯
psf ( x ¯ , x ¯ ) s ( x ¯ ) s ( x ¯ ) g β ( x ¯ , v ¯ ) f ( v ¯ | x ¯ ) d v ¯
E [ N 2 ] = Var { N } + ( E [ N ] ) 2 = E [ N ] + ( E [ N ] ) 2 = t 0 r + ( t 0 r ) 2
R λ ̂ ( x ¯ 1 , x ¯ 2 ) = E [ λ ̂ ( x ¯ 1 ) λ ̂ ( x ¯ 2 ) ]
= E N [ E V [ λ ̂ ( x ¯ 1 ) λ ̂ ( x ¯ 2 ) N ] ]
= 1 t 0 2 s ( x ¯ 1 ) s ( x ¯ 2 ) E N [ E V [ n = 1 N m = 1 N g β ( x ¯ 1 , V ¯ n ) g β ( x ¯ 2 , V ¯ m ) N ] ]
= 1 t 0 2 s ( x ¯ 1 ) s ( x ¯ 2 ) E N [ ( N 2 N ) E V [ g β ( x ¯ 1 , V ¯ ) ] E V [ g β ( x ¯ 2 , V ¯ ) ]
+ N E V [ g β ( x ¯ 1 , V ¯ ) g β ( x ¯ 2 , V ¯ ) ] ]
= ( r 2 s ( x ¯ 1 ) s ( x ¯ 2 ) ) E V [ g β ( x ¯ 1 , V ¯ ) ] E V [ g β ( x ¯ 2 , V ¯ ) ]
+ r t 0 s ( x ¯ 1 ) s ( x ¯ 2 ) E V [ g β ( x ¯ 1 , V ¯ ) g β ( x ¯ 2 , V ¯ ) ]
= μ ( x ¯ 1 ) μ ( x ¯ 2 ) + r t 0 s ( x ¯ 1 ) s ( x ¯ 2 ) E V [ g β ( x ¯ 1 , V ¯ ) g β ( x ¯ 2 , V ¯ ) ] .
K λ ̂ ( x ¯ 1 , x ¯ 2 ) = E [ λ ̂ ( x ¯ 1 ) λ ̂ ( x ¯ 2 ) ] μ ( x ¯ 1 ) μ ( x ¯ 2 )
= r t 0 s ( x ¯ 1 ) s ( x ¯ 2 ) E [ g β ( x ¯ 1 , V ¯ ) g β ( x ¯ 2 , V ¯ ) ] .
E [ g β ( x ¯ 1 , V ¯ ) g β ( x ¯ 2 , V ¯ ) ] = g β ( x ¯ 1 , v ¯ ) g β ( x ¯ 2 , v ¯ ) f V ( v ¯ ) d v ¯
= g β ( x ¯ 1 , v ¯ ) g β ( x ¯ 2 , v ¯ ) [ f ( v ¯ x ¯ ) f ( x ¯ ) d x ¯ ] d v ¯
= [ g β ( x ¯ 1 , v ¯ ) g β ( x ¯ 2 , v ¯ ) f ( v ¯ x ¯ ) d v ¯ ] f ( x ¯ ) d x ¯ ,
K λ ̂ ( x ¯ 1 , x ¯ 2 ) = r t 0 s ( x ¯ 1 ) s ( x ¯ 2 ) [ g β ( x ¯ 1 , v ¯ ) g β ( x ¯ 2 , v ¯ ) f ( v ¯ | x ¯ ) d v ¯ ] s ( x ¯ ) λ ( x ¯ ) d x ¯ .
σ 2 ( x ¯ ) Var { λ ̂ ( x ¯ ) } = K λ ̂ ( x ¯ , x ¯ ) = 1 t 0 s 2 ( x ¯ ) [ g β 2 ( x ¯ , v ¯ ) f ( v ¯ | x ¯ ) d v ¯ ] s ( x ¯ ) λ ( x ¯ ) d x ¯ .
μ ( x ¯ ) = [ g β ( x ¯ , v ¯ ) f ( v ¯ x ¯ ) d v ¯ ] λ ( x ¯ ) d x ¯
= [ g β ( x ¯ ν ¯ ) h ( v ¯ x ¯ ) d v ¯ ] λ ( x ¯ ) d x ¯
= [ g β ( x ¯ x ¯ x ¯ ) h ( x ¯ ) d x ¯ ] λ ( x ¯ ) d x ¯
= ( g β * h ) ( x ¯ x ¯ ) λ ( x ¯ ) d x ¯ ,
μ = g β * h * λ ,
psf ( x ¯ ) ( g β * h ) ( x ¯ ) ,
PSF ( u ¯ ) G β ( u ¯ ) H ( u ¯ ) ,
g β 2 ( x ¯ , v ¯ ) f ( v ¯ x ¯ ) d v ¯ = g β 2 ( x ¯ v ¯ ) h ( v ¯ x ¯ ) d v ¯
= g β 2 ( x ¯ x ¯ x ¯ ) h ( x ¯ ) d x ¯
= ( g β 2 * h ) ( x ¯ x ¯ ) ,
σ 2 ( x ¯ ) = 1 t 0 s 0 ( g β 2 * h ) ( x ¯ x ¯ ) λ ( x ¯ ) d x ¯
= 1 t 0 s 0 ( g β 2 * h * λ ) ( x ¯ ) .
t 0 s 0 σ 2 ( x ¯ ) = ( g β 2 * h * λ ) ( x ¯ )
= g β 2 ( x ¯ ) ( h * λ ) ( x ¯ x ¯ ) d x ¯
( h * λ ) ( x ¯ ) g β 2 ( x ¯ ) d x ¯
= λ ˜ ( x ¯ ) g β 2 ( x ¯ ) d x ¯ ,
σ 2 ( x ¯ ) λ ˜ ( x ¯ ) t 0 s 0 g β 2 ( x ¯ ) d x ¯ = λ ˜ ( x ¯ ) t 0 s 0 G β ( u ¯ ) 2 d u ¯ .
G β ( u ¯ ) A ( β u ¯ ) H ( u ¯ ) = A ( β u ¯ ) T ( w u ¯ ) ,
PSF ( u ¯ ) = G β ( u ¯ ) H ( u ¯ ) = A ( β u ¯ ) ,
σ 2 ( x ¯ ) λ ˜ ( x ¯ ) t 0 c 0 w p G β ( u ¯ ) 2 d u ¯
= c 1 w p T ( w u ¯ ) 2 A 2 ( β u ¯ ) d u ¯
= c 1 w p + d T ( z ¯ ) 2 A 2 ( z ¯ β w ) d z ¯
0 = ( p + d ) c 1 w p + d + 1 T ( z ¯ ) 2 A 2 ( z ¯ β w ) d z ¯
+ c 1 w p + d T ( z ¯ ) 2 2 A ( z ¯ β w ) A ̇ ( z ¯ β w ) ( z ¯ β w 2 ) d z ¯
0 = p + d 2 A 2 ( α z ¯ ) + A ( α z ¯ ) A ̇ ( α z ¯ ) α z ¯ T ( z ¯ ) 2 d z ¯ .
τ w ( r ) = { e μl ( r r b ) 2 , r r b e μl , r r b .
τ w ( r ) = exp ( π ( k w r ) 2 )
κ 2 ln 2 π .
s w = τ w ( x ¯ ) d x ¯ = ( w κ ) d ,
t ( r ) = τ 1 ( r ) s 1 = κ d e π ( kr ) 2 ,
T ( ρ ) = e π ( ρ κ ) 2 .
G β ( u ¯ ) = A ( βρ ) T ( ) = e π ( βρ κ ) 2 e π ( κ ) 2 = exp ( π ( ρ β 2 w 2 κ ) 2 ) ,
g β ( x ¯ ) = κ 2 β 2 w 2 exp ( π ( β 2 w 2 ) 2 ) .
σ 2 ( x ¯ ) λ ˜ ( x ¯ ) κ d t 0 w 2 d e π 2 ( ρ κ ) 2 exp ( π 2 ( ρβ ) 2 ) d u ¯
= λ ˜ ( x ¯ ) κ d t 0 w 2 d exp ( π ρ 2 2 κ 2 ( ( β w ) 2 1 ) ) d u ¯
= λ ˜ ( x ¯ ) κ d t 0 w 2 d [ 2 κ 2 ( ( β w ) 2 1 ) ] d 2
= λ ˜ ( x ¯ ) κ 2 d 2 d 2 t 0 ( β w 2 w 4 ) d 2 ,
w min = β 2 .
τ ω ( r ) = { e μ l r r b , r r b e μ l , r r b .
τ w ( r ) = e γr w ,
s w = τ w ( x ¯ ) d x ¯ = 2 π ( w γ ) 2 ,
t ( r ) = τ 1 ( r ) s 1 = γ 2 2 π e γr
T ( ρ ) = γ 3 [ ( 2 πρ ) 2 + γ 2 ] 3 2 .
σ 2 ( x ¯ ) λ ˜ ( x ¯ ) t 0 2 π w 2 γ 2 0 2 π 0 1 γ 6 [ ( 2 πρ ) 2 + γ 2 ] 3 exp ( π 2 ( ρβ ) 2 ) ρ
= λ ˜ ( x ¯ ) t 0 w 2 1 γ 4 0 ρ [ ( 2 πρ ) 2 + γ 2 ] 3 exp ( π 2 ( ρβ ) 2 ) .
σ 2 ( x ¯ ) λ ˜ ( x ¯ ) t 0 1 2 ( βκ ) 4 [ 6 y 2 + 6 y + 3 + y 1 ] where y = 2 π ( w κλβ ) 2 .
λ ( x ¯ ) 9 δ ( x 146 ) + rect ( ( x 208 ) 64 + 2 Λ ( x 64 ) 44 ) ,

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