Abstract

This paper analyzes the tradeo between spatial resolution and noise for simple pinhole imaging systems with 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 the pinhole size that minimizes the estimate variance for a specied reconstructed spatial resolution is 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 p2. Simulation results confirm this conclusion empirically. The general approach is a potentially useful addition to the collection of tools available for imaging system design.

© Optical Society of America

PDF Article

References

  • View by:
  • |

  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).
  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).
  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).
  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).
  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).
  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).
  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).
  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).
  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
  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
  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).
  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).
  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).
  22. P J Bickel and Y Ritov, "Estimating linear functionals of a PET image," IEEE Trans. Med. Imaging 14 81{87 (1995).
  23. B W Silverman, "Kernel density estimation using the fast Fourier transform," Appl. Stat. 31 93{99 (1982).
  24. B W Silverman, "On the estimation of a probability density function by the maximum penalized likelihood method," Ann. Stat. 10 795{810 (1982).
  25. D L Snyder and M I Miller, Random point processes in time and space, (Springer Verlag, New York, 1991).
  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).
  28. P P B Eggermont and V N LaRiccia, "Maximum smoothed likelihood density estimation for inverse problems," Ann. Stat. 23 199{220 (1995).
  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).
  31. H H Barrett, Timothy White, and Lucas C Parra, "List-mode likelihood," J. Opt. Soc. Am. A 14 2914{2923 (1997).
  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).
  35. R Bracewell, The Fourier transform and its applications, (McGraw-Hill, New York, 1978).

Other (35)

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).

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).

H H Barrett, "Objective assessment of image quality: effects of quantum noise and object variability," J. Opt. Soc. Am. A 7 1266{1278 (1990).

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).

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).

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).

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).

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).

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

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

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).

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).

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).

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

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

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

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

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).

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

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).

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

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

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

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Metrics