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.

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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. 23 654{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 7 1279{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 7 1266{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. 36 1091{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. 90 9758{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 12 834{852 (1995).
    [CrossRef]
  9. N E Hartsough, H H Barrett, H B Barber, and J M Woolfenden, "Intraoperative tumor de- tection: Relative performance of single-element, dual-element, and imaging probes with various collimators," IEEE Trans. Med. Imaging 14 259{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. 4 1667{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 93 463{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. 44 2026{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 ap- plications." 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 OSullivan 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. 18 251{280 (1990).
    [CrossRef]
  22. P J Bickel and Y Ritov, "Estimating linear functionals of a PET image," IEEE Trans. Med. Imaging 14 81{87 (1995).
    [CrossRef] [PubMed]
  23. B W Silverman, "Kernel density estimation using the fast Fourier transform," Appl. Stat. 31 93{99 (1982).
    [CrossRef]
  24. B W Silverman, "On the estimation of a probability density function by the maximum penalized likelihood method," Ann. Stat. 10 795{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. 23 199{220 (1995).
    [CrossRef]
  29. Y-C Tai, A Chatziioannou, M Dahlbom, and E J Hoffman, "Investigation on deadtime charac- teristics 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- ight measurements," IEEE Trans. Nucl. Sci. 20 1843{1849 (1983).
    [CrossRef]
  31. H H Barrett, Timothy White, and Lucas C Parra, "List-mode likelihood," J. Opt. Soc. Am. A 14 2914{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. 10 401{414 (1982).
    [CrossRef]
  35. R Bracewell, The Fourier transform and its applications, (McGraw-Hill, New York, 1978).

Other

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. 23 654{676 (1978).
[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.

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 7 1279{1293 (1990).
[CrossRef] [PubMed]

H H Barrett, "Objective assessment of image quality: effects of quantum noise and object variability," J. Opt. Soc. Am. A 7 1266{1278 (1990).
[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. 36 1091{1109 (1991).
[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. 90 9758{65 (1993).
[CrossRef] [PubMed]

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, 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 12 834{852 (1995).
[CrossRef]

N E Hartsough, H H Barrett, H B Barber, and J M Woolfenden, "Intraoperative tumor de- tection: Relative performance of single-element, dual-element, and imaging probes with various collimators," IEEE Trans. Med. Imaging 14 259{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. 4 1667{1674 (1995).
[CrossRef] [PubMed]

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 93 463{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. 44 2026{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

Mohammad Usman, {Biased and unbiased Cramer-Rao bounds: computational issues and ap- plications." 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.

F OSullivan 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]

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.

I M Johnstone and B W Silverman, "Speed of estimation in positron emission tomography," Ann. Stat. 18 251{280 (1990).
[CrossRef]

P J Bickel and Y Ritov, "Estimating linear functionals of a PET image," IEEE Trans. Med. Imaging 14 81{87 (1995).
[CrossRef] [PubMed]

B W Silverman, "Kernel density estimation using the fast Fourier transform," Appl. Stat. 31 93{99 (1982).
[CrossRef]

B W Silverman, "On the estimation of a probability density function by the maximum penalized likelihood method," Ann. Stat. 10 795{810 (1982).
[CrossRef]

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).

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]

P P B Eggermont and V N LaRiccia, "Maximum smoothed likelihood density estimation for inverse problems," Ann. Stat. 23 199{220 (1995).
[CrossRef]

Y-C Tai, A Chatziioannou, M Dahlbom, and E J Hoffman, "Investigation on deadtime charac- teristics 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 D G Politte, "Image reconstruction from list-mode data in emission tomography system having time-of- ight measurements," IEEE Trans. Nucl. Sci. 20 1843{1849 (1983).
[CrossRef]

H H Barrett, Timothy White, and Lucas C Parra, "List-mode likelihood," J. Opt. Soc. Am. A 14 2914{2923 (1997).
[CrossRef]

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).

S Geman and C R Hwang, "Nonparametric maximum likelihood estimation by the method of sieves," Ann. Stat. 10 401{414 (1982).
[CrossRef]

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

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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)

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

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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