Spatial resolution and noise tradeoffs in pinhole imaging system design: a density estimation approach
Optics Express, Vol. 2, Issue 6, pp. 237-253 (1998)
http://dx.doi.org/10.1364/OE.2.000237
Acrobat PDF (565 KB)
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.
© Optical Society of America
1. Introduction
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]
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]
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]
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]
2. Problem
25. D L Snyder and M I Miller, Random point processes in time and space, (Springer Verlag, New York, 1991). [CrossRef]
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]
2.1 The estimation problem
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]
3. Kernel-based indirect density estimator
3.1 Mean function
3.2 Second-moment functions
4. Shift-invariant case
4.1 Spatially smooth objects λ(x̲)
4.2 Resolution-noise tradeoffs
4.3 Apodized inverse filter
4.4 Relationship to sieves
4.5 Gaussian pinhole example
5. Laplacian pinhole example
6. Simulation results
7. Discussion
Acknowledgement
Footnotes
1 | This intuition is somewhat consistent with the findings of Myers et al.[3 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] |
2 | All integrals over d x̲ are d-dimensional. |
3 | Strictly speaking this is a conditional pdf, conditioned on the event that the emission is detected. We consider only the detected emissions, so for simplicity we omit the notation for conditioning on detection. With such notation, (1) is just a form of Bayes rule. |
4 | Without loss of generality, one can rescale the time axis exponentially to account for radioactive decay. |
5 | Silverman [18, p. 27] refers to such methods as general weight function estimates in the context of direct density estimation. |
6 | Our purpose here is to analyze such estimators for the goal of system design, not to argue the merits of such estimators over alternatives. |
7 | Equation (8) is closely related to equation (3.6) on on p. 36 of [18] for direct density estimation; the remainder of the derivation is distinct to indirect density estimation. |
8 | Neglecting edge effects at the boundaries of the field-of-view, and assuming that any magnification factor has already been accounted for in the V̲_{n} ’s [26]. |
9 | As a further approximation, one can assume |
10 | The transmissivity of the 1D version of this pinhole has the form of the Laplacian pdf ½e ^{-|x|}, hence the name—for lack of a better name. |
References
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). |
OCIS Codes
(110.0110) Imaging systems : Imaging systems
(110.2990) Imaging systems : Image formation theory
(110.6960) Imaging systems : Tomography
ToC Category:
Focus Issue: Tomographic image reconstruction
History
Original Manuscript: December 14, 1997
Published: March 16, 1998
Citation
Jeffrey Fessler, "Spatial Resolution and Noise Tradeoffs in Pinhole Imaging System Design: A Density Estimation Approach," Opt. Express 2, 237-253 (1998)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-2-6-237
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References
- 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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