Abstract

A theoretical framework for detection or discrimination tasks with list-mode data is developed. The object and imaging system are rigorously modeled via three random mechanisms: randomness of the object being imaged, randomness in the attribute vectors, and, finally, randomness in the attribute vector estimates due to noise in the detector outputs. By considering the list-mode data themselves, the theory developed in this paper yields a manageable expression for the likelihood of the list-mode data given the object being imaged. This, in turn, leads to an expression for the optimal Bayesian discriminant. Figures of merit for detection tasks via the ideal and optimal linear observers are derived. A concrete example discusses detection performance of the optimal linear observer for the case of a known signal buried in a random lumpy background.

© 2012 Optical Society of America

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2011 (3)

G. J. Gang, J. Lee, J. L. Prince, J. W. Stayman, D. J. Tward, W. Zbijewski, and J. H. Siewerdsen, “Analysis of Fourier-domain task-based detectability index in tomosynthesis and cone-beam CT in relation to human observer performance,” Med. Phys. 38, 1754–1768 (2011).
[CrossRef]

B. Guérin and G. E. Fakhri, “Novel scatter compensation of list-mode PET data using spatial and energy dependent corrections,” IEEE Trans. Med. Imag. 30, 759–773 (2011).
[CrossRef]

L. Platiša, B. Goossens, E. Vansteenkiste, S. Park, B. D. Gallas, A. Badano, and W. Philips, “Channelized Hotelling observers for the assessment of volumetric imaging data sets,” J. Opt. Soc. Am. A 28, 1145–1163 (2011).
[CrossRef]

2010 (1)

J. Y. Hesterman, L. Caucci, M. A. Kupinski, H. H. Barrett, and L. R. Furenlid, “Maximum-likelihood estimation with a contracting-grid search algorithm,” IEEE Trans. Nucl. Sci. 57, 1077–1084 (2010).
[CrossRef]

2009 (2)

E. C. Bellm, S. E. Boggs, M. S. Bandstra, J. D. Bowen, D. Perez-Becker, C. B. Wunderer, A. Zoglauer, M. Amman, P. N. Luke, H.-K. Chang, J.-L. Chiu, J.-S. Liang, Y.-H. Chang, Z.-K. Liu, W.-C. Hung, C.-H. Lin, M. A. Huang, and P. Jean, “Overview of the nuclear Compton telescope,” IEEE Trans. Nucl. Sci. 56, 1250–1256 (2009).
[CrossRef]

W. C. J. Hunter, H. H. Barrett, and L. R. Furenlid, “Calibration method for ML estimation of 3D interaction position in a thick gamma-ray detector,” IEEE Trans. Nucl. Sci. 56, 189–196 (2009).
[CrossRef]

2008 (2)

2007 (5)

2006 (1)

2005 (4)

2004 (4)

A. F. Abouraddy, P. R. Stone, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, “Entangled-photon imaging of a pure phase object,” Phys. Rev. Lett. 93, 213903 (2004).
[CrossRef]

P. Khurd, I.-T. Hsiao, A. Rangarajan, and G. Gindi, “A globally convergent regularized ordered-subset EM algorithm for list-mode reconstruction,” IEEE Trans. Nucl. Sci. 51, 719–725 (2004).
[CrossRef]

A. Zoglauer, R. Andritschke, and G. Kanbach, “Data analysis for the MEGA prototype,” New Astron. Rev. 48, 231–235 (2004).
[CrossRef]

X. He, E. C. Frey, J. M. Links, K. L. Gilland, W. P. Segars, and B. M. W. Tsui, “A mathematical observer study for the evaluation and optimization of compensation methods for myocardial SPECT using a phantom population that realistically models patient variability,” IEEE Trans. Nucl. Sci. 51, 218–224(2004).
[CrossRef]

2003 (1)

2002 (4)

E. C. Frey, K. L. Gilland, and B. M. W. Tsui, “Application of task-based measures of image quality to optimization and evaluation of three-dimensional reconstruction-based compensation methods in myocardial perfusion SPECT,” IEEE Trans. Med. Imag. 21, 1040–1050 (2002).
[CrossRef]

T. Takahashi, K. Nakazawa, T. Kamae, H. Tajima, Y. Fukazawa, M. Nomachi, and M. Kokubun, “High resolution CdTe detectors for the next-generation multi-Compton gamma-ray telescope,” Proc. SPIE 4851, 1228–1235 (2002).
[CrossRef]

P. C. Johns, J. Dubeau, D. G. Gobbi, M. Li, and M. S. Dixit, “Photon-counting detectors for digital radiography and x-ray computed tomography,” Proc. SPIE TD01, 367–369 (2002).

A. J. Reader, S. Ally, F. Bakatselos, R. Manavaki, R. J. Walledge, A. P. Jeavons, P. J. Julyan, S. Zhao, D. L. Hastings, and J. Zweit, “One-pass list-mode EM algorithm for high-resolution 3-D PET image reconstruction into large arrays,” IEEE Trans. Nucl. Sci. 49, 693–699 (2002).
[CrossRef]

2001 (2)

C. Byrne, “Likelihood maximization for list-mode emission tomographic image reconstruction,” IEEE Trans. Med. Imag. 20, 1084–1092 (2001).
[CrossRef]

R. Levkovitz, D. Falikman, M. Zibulevsky, A. Ben-Tal, and A. Nemirovski, “The design and implementation of COSEN, an iterative algorithm for fully 3-D listmode data,” IEEE Trans. Med. Imag. 20, 633–642 (2001).
[CrossRef]

2000 (5)

1999 (1)

A. J. Reader, K. Erlandsson, R. J. Ott, and M. A. Flower, “Attenuation and scatter correction of list-mode data driven iterative and analytic image reconstruction algorithms for rotating 3D PET systems,” IEEE Trans. Nucl. Sci. 46, 2218–2226 (1999).
[CrossRef]

1998 (3)

L. Parra and H. H. Barrett, “List-mode likelihood: EM algorithm and image quality estimation demonstrated on 2-D PET,” IEEE Trans. Med. Imag. 17, 228–235 (1998).
[CrossRef]

A. J. Reader, K. Erlandsson, M. A. Flower, and R. J. Ott, “Fast accurate iterative reconstruction for low-statistics positron volume imaging,” Phys. Med. Biol. 43, 835–846 (1998).
[CrossRef]

H. H. Barrett, C. K. Abbey, and E. Clarkson, “Objective assessment of image quality. III. ROC metrics, ideal observers, and likelihood-generating functions,” J. Opt. Soc. Am. A 15, 1520–1535 (1998).
[CrossRef]

1997 (2)

1995 (1)

1994 (1)

1992 (1)

1990 (2)

1989 (1)

J. G. Timothy, J. S. Morgan, D. C. Slater, D. B. Kasle, and R. L. Bybee, “MAMA detector systems: a status report,” Proc. SPIE 1158, 104–117 (1989).

1987 (2)

1986 (1)

1985 (3)

1984 (1)

D. L. Snyder, “Parameter estimation for dynamic studies in emission-tomography systems having list-mode data,” IEEE Trans. Nucl. Sci. 31, 925–931 (1984).
[CrossRef]

1983 (2)

D. L. Snyder, and D. G. Politte, “Image reconstruction from list-mode data in an emission tomography system having time-of-flight measurements,” IEEE Trans. Nucl. Sci. 30, 1843–1849 (1983).
[CrossRef]

J. G. Timothy, “Optical detectors for spectroscopy,” Pub. Astron. Soc. Pac. 95, 810–834 (1983).
[CrossRef]

1982 (1)

C. Papaliolios and L. Mertz, “New two-dimensional photon camera,” Proc. SPIE 331, 360–369 (1982).

1958 (1)

H. O. Anger, “Scintillation camera,” Rev. Sci. Instrum. 29, 27–33 (1958).
[CrossRef]

1931 (1)

H. Hotelling, “The generalization of student’s ratio,” Ann. Math. Stat. 2, 360–378 (1931).
[CrossRef]

Aarsvold, J. N.

M. N. Wernick and J. N. Aarsvold, Emission Tomography: The Fundamentals of PET and SPECT (Elsevier Academic, 2004).

Abbey, C. K.

Abouraddy, A. F.

A. F. Abouraddy, P. R. Stone, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, “Entangled-photon imaging of a pure phase object,” Phys. Rev. Lett. 93, 213903 (2004).
[CrossRef]

Akcay, C.

Ally, S.

A. J. Reader, S. Ally, F. Bakatselos, R. Manavaki, R. J. Walledge, A. P. Jeavons, P. J. Julyan, S. Zhao, D. L. Hastings, and J. Zweit, “One-pass list-mode EM algorithm for high-resolution 3-D PET image reconstruction into large arrays,” IEEE Trans. Nucl. Sci. 49, 693–699 (2002).
[CrossRef]

Aloisi, A.

K. A. Bostroem, A. Aloisi, R. Bohlin, R. Diaz, V. Dixon, P. Goudfrooij, P. Hodge, D. Lennon, C. Long, S. Niemi, R. Osten, C. Proffitt, N. Walborn, T. Wheeler, M. Wolfe, B. York, and W. Zheng, STIS Instrument Handbook, Version 10.0 (Space Telescope Science Institute, 2010).

Amman, M.

E. C. Bellm, S. E. Boggs, M. S. Bandstra, J. D. Bowen, D. Perez-Becker, C. B. Wunderer, A. Zoglauer, M. Amman, P. N. Luke, H.-K. Chang, J.-L. Chiu, J.-S. Liang, Y.-H. Chang, Z.-K. Liu, W.-C. Hung, C.-H. Lin, M. A. Huang, and P. Jean, “Overview of the nuclear Compton telescope,” IEEE Trans. Nucl. Sci. 56, 1250–1256 (2009).
[CrossRef]

Andrews, G. E.

G. E. Andrews, R. Askey, and R. Roy, Special Functions(Cambridge University, 2000).

Andritschke, R.

A. Zoglauer, R. Andritschke, and G. Kanbach, “Data analysis for the MEGA prototype,” New Astron. Rev. 48, 231–235 (2004).
[CrossRef]

Anger, H. O.

H. O. Anger, “Scintillation camera,” Rev. Sci. Instrum. 29, 27–33 (1958).
[CrossRef]

Apelblat, A.

A. Apelblat, Table of Definite and Infinite Integrals (Elsevier Scientific, 1983).

Askey, R.

G. E. Andrews, R. Askey, and R. Roy, Special Functions(Cambridge University, 2000).

Badano, A.

Bakatselos, F.

A. J. Reader, S. Ally, F. Bakatselos, R. Manavaki, R. J. Walledge, A. P. Jeavons, P. J. Julyan, S. Zhao, D. L. Hastings, and J. Zweit, “One-pass list-mode EM algorithm for high-resolution 3-D PET image reconstruction into large arrays,” IEEE Trans. Nucl. Sci. 49, 693–699 (2002).
[CrossRef]

Bandstra, M. S.

E. C. Bellm, S. E. Boggs, M. S. Bandstra, J. D. Bowen, D. Perez-Becker, C. B. Wunderer, A. Zoglauer, M. Amman, P. N. Luke, H.-K. Chang, J.-L. Chiu, J.-S. Liang, Y.-H. Chang, Z.-K. Liu, W.-C. Hung, C.-H. Lin, M. A. Huang, and P. Jean, “Overview of the nuclear Compton telescope,” IEEE Trans. Nucl. Sci. 56, 1250–1256 (2009).
[CrossRef]

Barrett, H. H.

J. Y. Hesterman, L. Caucci, M. A. Kupinski, H. H. Barrett, and L. R. Furenlid, “Maximum-likelihood estimation with a contracting-grid search algorithm,” IEEE Trans. Nucl. Sci. 57, 1077–1084 (2010).
[CrossRef]

W. C. J. Hunter, H. H. Barrett, and L. R. Furenlid, “Calibration method for ML estimation of 3D interaction position in a thick gamma-ray detector,” IEEE Trans. Nucl. Sci. 56, 189–196 (2009).
[CrossRef]

L. Caucci, H. H. Barrett, N. Devaney, and J. J. Rodríguez, “Application of the Hotelling and ideal observers to detection and localization of exoplanets,” J. Opt. Soc. Am. A 24, B13–B24 (2007).
[CrossRef]

H. H. Barrett, K. J. Myers, N. Devaney, and C. Dainty, “Objective assessment of image quality. IV. Application to adaptive optics,” J. Opt. Soc. Am. A 23, 3080–3105 (2006).
[CrossRef]

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M. A. Kupinski, J. W. Hoppin, E. Clarkson, and H. H. Barrett, “Ideal-observer computation in medical imaging with use of Markov-chain Monte Carlo techniques,” J. Opt. Soc. Am. A 20, 430–438 (2003).
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K. J. Myers and H. H. Barrett, “Addition of a channel mechanism to the ideal-observer model,” J. Opt. Soc. Am. A 4, 2447–2457 (1987).
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W. E. Smith and H. H. Barrett, “Hotelling trace criterion as a figure of merit for the optimization of imaging systems,” J. Opt. Soc. Am. A 3, 717–725 (1986).
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K. J. Myers, H. H. Barrett, M. C. Borgstrom, D. D. Patton, and G. W. Seeley, “Effect of noise correlation on detectability of disk signals in medical imaging,” J. Opt. Soc. Am. A 2, 1752–1759 (1985).
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K. A. Bostroem, A. Aloisi, R. Bohlin, R. Diaz, V. Dixon, P. Goudfrooij, P. Hodge, D. Lennon, C. Long, S. Niemi, R. Osten, C. Proffitt, N. Walborn, T. Wheeler, M. Wolfe, B. York, and W. Zheng, STIS Instrument Handbook, Version 10.0 (Space Telescope Science Institute, 2010).

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E. C. Bellm, S. E. Boggs, M. S. Bandstra, J. D. Bowen, D. Perez-Becker, C. B. Wunderer, A. Zoglauer, M. Amman, P. N. Luke, H.-K. Chang, J.-L. Chiu, J.-S. Liang, Y.-H. Chang, Z.-K. Liu, W.-C. Hung, C.-H. Lin, M. A. Huang, and P. Jean, “Overview of the nuclear Compton telescope,” IEEE Trans. Nucl. Sci. 56, 1250–1256 (2009).
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L. Caucci, H. H. Barrett, N. Devaney, and J. J. Rodríguez, “Application of the Hotelling and ideal observers to detection and localization of exoplanets,” J. Opt. Soc. Am. A 24, B13–B24 (2007).
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E. C. Bellm, S. E. Boggs, M. S. Bandstra, J. D. Bowen, D. Perez-Becker, C. B. Wunderer, A. Zoglauer, M. Amman, P. N. Luke, H.-K. Chang, J.-L. Chiu, J.-S. Liang, Y.-H. Chang, Z.-K. Liu, W.-C. Hung, C.-H. Lin, M. A. Huang, and P. Jean, “Overview of the nuclear Compton telescope,” IEEE Trans. Nucl. Sci. 56, 1250–1256 (2009).
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E. C. Bellm, S. E. Boggs, M. S. Bandstra, J. D. Bowen, D. Perez-Becker, C. B. Wunderer, A. Zoglauer, M. Amman, P. N. Luke, H.-K. Chang, J.-L. Chiu, J.-S. Liang, Y.-H. Chang, Z.-K. Liu, W.-C. Hung, C.-H. Lin, M. A. Huang, and P. Jean, “Overview of the nuclear Compton telescope,” IEEE Trans. Nucl. Sci. 56, 1250–1256 (2009).
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P. Bruyndonckx, C. Lemaître, D. Schaart, M. Maas, D. J. van der Laan, M. Krieguer, O. Devroede, and S. Tavernier, “Towards a continuous crystal APD-based PET detector design,” Nucl. Instrum. Methods Phys. Res. A 571, 182–186 (2007).
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P. C. Johns, J. Dubeau, D. G. Gobbi, M. Li, and M. S. Dixit, “Photon-counting detectors for digital radiography and x-ray computed tomography,” Proc. SPIE TD01, 367–369 (2002).

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K. A. Bostroem, A. Aloisi, R. Bohlin, R. Diaz, V. Dixon, P. Goudfrooij, P. Hodge, D. Lennon, C. Long, S. Niemi, R. Osten, C. Proffitt, N. Walborn, T. Wheeler, M. Wolfe, B. York, and W. Zheng, STIS Instrument Handbook, Version 10.0 (Space Telescope Science Institute, 2010).

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P. C. Johns, J. Dubeau, D. G. Gobbi, M. Li, and M. S. Dixit, “Photon-counting detectors for digital radiography and x-ray computed tomography,” Proc. SPIE TD01, 367–369 (2002).

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J. Y. Hesterman, L. Caucci, M. A. Kupinski, H. H. Barrett, and L. R. Furenlid, “Maximum-likelihood estimation with a contracting-grid search algorithm,” IEEE Trans. Nucl. Sci. 57, 1077–1084 (2010).
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A. Hayat, P. Ginzburg, and M. Orenstein, “Observation of two-photon emission from semiconductors,” Nat. Photon. 2, 238–241 (2008).
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X. He, E. C. Frey, J. M. Links, K. L. Gilland, W. P. Segars, and B. M. W. Tsui, “A mathematical observer study for the evaluation and optimization of compensation methods for myocardial SPECT using a phantom population that realistically models patient variability,” IEEE Trans. Nucl. Sci. 51, 218–224(2004).
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J. Y. Hesterman, L. Caucci, M. A. Kupinski, H. H. Barrett, and L. R. Furenlid, “Maximum-likelihood estimation with a contracting-grid search algorithm,” IEEE Trans. Nucl. Sci. 57, 1077–1084 (2010).
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K. A. Bostroem, A. Aloisi, R. Bohlin, R. Diaz, V. Dixon, P. Goudfrooij, P. Hodge, D. Lennon, C. Long, S. Niemi, R. Osten, C. Proffitt, N. Walborn, T. Wheeler, M. Wolfe, B. York, and W. Zheng, STIS Instrument Handbook, Version 10.0 (Space Telescope Science Institute, 2010).

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E. C. Bellm, S. E. Boggs, M. S. Bandstra, J. D. Bowen, D. Perez-Becker, C. B. Wunderer, A. Zoglauer, M. Amman, P. N. Luke, H.-K. Chang, J.-L. Chiu, J.-S. Liang, Y.-H. Chang, Z.-K. Liu, W.-C. Hung, C.-H. Lin, M. A. Huang, and P. Jean, “Overview of the nuclear Compton telescope,” IEEE Trans. Nucl. Sci. 56, 1250–1256 (2009).
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E. C. Bellm, S. E. Boggs, M. S. Bandstra, J. D. Bowen, D. Perez-Becker, C. B. Wunderer, A. Zoglauer, M. Amman, P. N. Luke, H.-K. Chang, J.-L. Chiu, J.-S. Liang, Y.-H. Chang, Z.-K. Liu, W.-C. Hung, C.-H. Lin, M. A. Huang, and P. Jean, “Overview of the nuclear Compton telescope,” IEEE Trans. Nucl. Sci. 56, 1250–1256 (2009).
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Hunter, W. C. J.

W. C. J. Hunter, H. H. Barrett, and L. R. Furenlid, “Calibration method for ML estimation of 3D interaction position in a thick gamma-ray detector,” IEEE Trans. Nucl. Sci. 56, 189–196 (2009).
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E. C. Bellm, S. E. Boggs, M. S. Bandstra, J. D. Bowen, D. Perez-Becker, C. B. Wunderer, A. Zoglauer, M. Amman, P. N. Luke, H.-K. Chang, J.-L. Chiu, J.-S. Liang, Y.-H. Chang, Z.-K. Liu, W.-C. Hung, C.-H. Lin, M. A. Huang, and P. Jean, “Overview of the nuclear Compton telescope,” IEEE Trans. Nucl. Sci. 56, 1250–1256 (2009).
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Jeavons, A. P.

A. J. Reader, S. Ally, F. Bakatselos, R. Manavaki, R. J. Walledge, A. P. Jeavons, P. J. Julyan, S. Zhao, D. L. Hastings, and J. Zweit, “One-pass list-mode EM algorithm for high-resolution 3-D PET image reconstruction into large arrays,” IEEE Trans. Nucl. Sci. 49, 693–699 (2002).
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P. C. Johns, J. Dubeau, D. G. Gobbi, M. Li, and M. S. Dixit, “Photon-counting detectors for digital radiography and x-ray computed tomography,” Proc. SPIE TD01, 367–369 (2002).

Judy, P. F.

Julyan, P. J.

A. J. Reader, S. Ally, F. Bakatselos, R. Manavaki, R. J. Walledge, A. P. Jeavons, P. J. Julyan, S. Zhao, D. L. Hastings, and J. Zweit, “One-pass list-mode EM algorithm for high-resolution 3-D PET image reconstruction into large arrays,” IEEE Trans. Nucl. Sci. 49, 693–699 (2002).
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Kamae, T.

T. Takahashi, K. Nakazawa, T. Kamae, H. Tajima, Y. Fukazawa, M. Nomachi, and M. Kokubun, “High resolution CdTe detectors for the next-generation multi-Compton gamma-ray telescope,” Proc. SPIE 4851, 1228–1235 (2002).
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Kasle, D. B.

J. G. Timothy, J. S. Morgan, D. C. Slater, D. B. Kasle, and R. L. Bybee, “MAMA detector systems: a status report,” Proc. SPIE 1158, 104–117 (1989).

Khurd, P.

P. Khurd, I.-T. Hsiao, A. Rangarajan, and G. Gindi, “A globally convergent regularized ordered-subset EM algorithm for list-mode reconstruction,” IEEE Trans. Nucl. Sci. 51, 719–725 (2004).
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Kokubun, M.

T. Takahashi, K. Nakazawa, T. Kamae, H. Tajima, Y. Fukazawa, M. Nomachi, and M. Kokubun, “High resolution CdTe detectors for the next-generation multi-Compton gamma-ray telescope,” Proc. SPIE 4851, 1228–1235 (2002).
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S. Korpar, P. Križan, R. Pestotnik, A. Gorišek, A. Stanovnik, M. Starič, and D. Škrk, “Multianode photomultipliers as position-sensitive detectors of single photons,” Nucl. Instrum. Methods Phys. Res. A 442, 316–321 (2000).
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Krieguer, M.

P. Bruyndonckx, C. Lemaître, D. Schaart, M. Maas, D. J. van der Laan, M. Krieguer, O. Devroede, and S. Tavernier, “Towards a continuous crystal APD-based PET detector design,” Nucl. Instrum. Methods Phys. Res. A 571, 182–186 (2007).
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Križan, P.

S. Korpar, P. Križan, R. Pestotnik, A. Gorišek, A. Stanovnik, M. Starič, and D. Škrk, “Multianode photomultipliers as position-sensitive detectors of single photons,” Nucl. Instrum. Methods Phys. Res. A 442, 316–321 (2000).
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Kupinski, M. A.

J. Y. Hesterman, L. Caucci, M. A. Kupinski, H. H. Barrett, and L. R. Furenlid, “Maximum-likelihood estimation with a contracting-grid search algorithm,” IEEE Trans. Nucl. Sci. 57, 1077–1084 (2010).
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G. J. Gang, J. Lee, J. L. Prince, J. W. Stayman, D. J. Tward, W. Zbijewski, and J. H. Siewerdsen, “Analysis of Fourier-domain task-based detectability index in tomosynthesis and cone-beam CT in relation to human observer performance,” Med. Phys. 38, 1754–1768 (2011).
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P. Bruyndonckx, C. Lemaître, D. Schaart, M. Maas, D. J. van der Laan, M. Krieguer, O. Devroede, and S. Tavernier, “Towards a continuous crystal APD-based PET detector design,” Nucl. Instrum. Methods Phys. Res. A 571, 182–186 (2007).
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Lennon, D.

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Figures (1)

Fig. 1.
Fig. 1.

Plots of SNR2 for three different cases. For all plots, Δf0=100s1, f0=1000s1, p=20000m2, σ=0.001m, and rb=0.005m. For the solid curves, rs=0.0045m; for the dashed curves, rs=rb=0.0050m; and, finally, for the dotted–dashed curves, rs=0.0055m. Thick curves correspond to the SNR2 for the Hotelling observer, while thin curves correspond to the MCMC-calculated SNR2 for the ideal observer.

Equations (110)

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A^{A^j,j=1,,J},
pr(G|Hk)=pr(A^,J|Hk)=pr(A^|J,Hk)Pr(J|Hk).
pr(G|Hk)=dfpr(G|f)pr(f|Hk)=dfpr(A^|J,f)Pr(J|f)pr(f|Hk)pr(G|f)f|Hk,
pr(A^|J,f)=pr({A^j}|J,f)=j=1Jpr(A^j|f),
pr(A^|J,Hk)=pr(A^|J,f)f|Hk=df[j=1Jpr(A^j|f)]pr(f|Hk).
pr(A^|J,Hk)=df[j=1JdAjpr(A^j|Aj)pr(Aj|f)]pr(f|Hk)=df{j=1JdAjpr(A^j|Aj)[drpr(Aj|r)pr(r|f)]}pr(f|Hk),
pr(A^|J,f)=j=1JdAjpr(A^j|Aj)[drpr(Aj|r)pr(r|f)]
pr(G|Hk)=df{j=1JdAjpr(A^j|Aj)[drpr(Aj|r)pr(r|f)]}Pr(J|f)pr(f|Hk).
pr(r|f)=f(r)s(r)drf(r)s(r),
pr(A^j|f)=dAjpr(A^j|Aj)drpr(Aj|r)f(r)s(r)drf(r)s(r)=drpr(A^j|r)f(r)s(r)drf(r)s(r).
pr(G|Hk)=df{j=1JdAjpr(A^j|Aj)drpr(Aj|r)f(r)s(r)drf(r)s(r)}Pr(J|f)pr(f|Hk).
Pr(J|f)=exp[J¯(f)][J¯(f)]JJ!,
J¯(f)=τdrf(r)s(r).
pr(G|Hk)=τJJ!df{j=1JdAjpr(A^j|Aj)[drpr(Aj|r)f(r)s(r)]}exp[J¯(f)]pr(f|Hk).
pr(G|Hk)=df{j=1Jdrpr(A^j|r)f(r)s(r)drf(r)s(r)}Pr(J|f)pr(f|Hk)={j=1Jdrpr(A^j|r)f(r)s(r)drf(r)s(r)}Pr(J|f)f|Hk,
pr(G|Hk)=τJJ!df{j=1Jdrpr(A^j|r)f(r)s(r)}exp[J¯(f)]pr(f|Hk)=df{j=1Jpr(A^j|f)}exp[J¯(f)][J¯(f)]JJ!pr(f|Hk).
pr(G|Hk)=pr(G|fk)=τJJ!{j=1Jdrpr(A^j|r)fk(r)s(r)}exp[J¯(fk)].
log[pr(G|fk)]=JlogτlogJ!J¯(fk)+j=1J{log[drpr(A^j|r)fk(r)s(r)]}.
Λ(G|f0)=pr(G|f0+Δf)pr(G|f0)=exp[J¯(f0+Δf)+J¯(f0)]j=1Jdrpr(A^j|r)[f0(r)+Δf(r)]s(r)drpr(A^j|r)f0(r)s(r),
λ(G|f0)=log[pr(G|f0+Δf)pr(G|f0)]=J¯(f0+Δf)+J¯(f0)+logj=1Jdrpr(A^j|r)[f0(r)+Δf(r)]s(r)drpr(A^j|r)f0(r)s(r)=τdrΔf(r)s(r)+j=1Jlog[1+drpr(A^j|r)Δf(r)s(r)drpr(A^j|r)f0(r)s(r)].
λ(G|f0)=τdrΔf(r)s(r)+j=1Jdrpr(A^j|r)Δf(r)s(r)drpr(A^j|r)f0(r)s(r)12j=1J[drpr(A^j|r)Δf(r)s(r)drpr(A^j|r)f0(r)s(r)]2+.
λ1=λ0,var1(λ)=var0(λ)=2λ1,
SNR2[λ1λ0]212[var0(λ)+var1(λ)]=2λ1=2λ0.
j=1Jdrpr(A^j|r)Δf(r)s(r)drpr(A^j|r)f0(r)s(r)G|H0=J=0Pr(J|f0)j=1JdA^jpr(A^j|f0)drpr(A^j|r)Δf(r)s(r)drpr(A^j|r)f0(r)s(r)=J=0Pr(J|f0)j=1JdA^jdrpr(A^j|r)f0(r)s(r)drf0(r)s(r)drpr(A^j|r)Δf(r)s(r)drpr(A^j|r)f0(r)s(r)=J=0Pr(J|f0)j=1JdrΔf(r)s(r)dA^jpr(A^j|r)drf0(r)s(r)=J=0Pr(J|f0)j=1JdrΔf(r)s(r)drf0(r)s(r),
j=1Jdrpr(A^j|r)Δf(r)s(r)drpr(A^j|r)f0(r)s(r)G|H0=drΔf(r)s(r)drf0(r)s(r)J=0Pr(J|f0)J=τdrΔf(r)s(r),
λ012J=0Pr(J|f0)j=1JdA^jpr(A^j|f0)[drpr(A^j|r)Δf(r)s(r)drpr(A^j|r)f0(r)s(r)]2=12τdA^[drpr(A^|r)Δf(r)s(r)]2drpr(A^|r)f0(r)s(r).
SNR2τdA^[drpr(A^|r)Δf(r)s(r)]2drpr(A^|r)f0(r)s(r).
SNR2J¯(f0)dA^pr(A^|f0)[drpr(A^|r)Δf(r)s(r)drpr(A^|r)f0(r)s(r)]2=J¯(f0)[drpr(A^|r)Δf(r)s(r)drpr(A^|r)f0(r)s(r)]2A^|f0.
pr(A^j,A^j|Hk)=dfpr(A^j|f)pr(A^j|f)pr(f|Hk)[dfpr(A^j|f)pr(f|Hk)][dfpr(A^|f)pr(f|Hk)]=pr(A^j|Hk)pr(A^j|Hk).
pr(G|H0)=dfpr(G|f)prb(f)=dfpr(A^|J,f)Pr(J|f)prb(f),pr(G|H1)=dfpr(G|f+Δf)prb(f)=dfpr(A^|J,f+Δf)Pr(J|f+Δf)prb(f).
Λ(G)pr(G|H1)pr(G|H0)=dfpr(G|f+Δf)prb(f)dfpr(G|f)prb(f)=pr(G|f+Δf)f|H0pr(G|f)f|H0.
Λ(G)=1+Δfapr(G|H0)+12ΔfBΔfpr(G|H0)+,
a=df[pr(G|f)f]prb(f)=pr(G|f)ff|H0,
B=df[2pr(G|f)ff]prb(f)=2pr(G|f)fff|H0.
λ(G)=Δfapr(G|H0)+12ΔfBΔfpr(G|H0)12[Δfapr(G|H0)+12ΔfBΔfpr(G|H0)]2+.
λ(G)Δfapr(G|H0)+12ΔfBΔfpr(G|H0)12[Δfapr(G|H0)]2.
Δfapr(G|H0)=Δfdf[fpr(G|f)]prb(f)dfpr(G|f)prb(f).
pr(G|f)=τJJ!exp[J¯(f)]j=1Jdrpr(A^j|r)f(r)s(r).
Δffpr(G|f)=ΔfJ¯(f)fpr(G|f)+τJJ!exp[J¯(f)]j=1J{[Δffdrpr(A^j|r)f(r)s(r)]j=1jjjdrpr(A^j|r)f(r)s(r)}=τpr(G|f)drΔf(r)s(r)+τJJ!exp[J¯(f)]j=1J{drpr(A^j|r)Δf(r)s(r)j=1jjJdrpr(A^j|r)f(r)s(r)},
Δfa=τpr(G|H0)drΔf(r)s(r)+τJJ!exp[J¯(f)]j=1J{drpr(A^j|r)Δf(r)s(r)drpr(A^j|r)f(r)s(r)j=1Jdrpr(A^j|r)f(r)s(r)}f|H0.
Δfa=τpr(G|H0)drΔf(r)s(r)+1J!exp[J¯(f)][J¯(f)]Jj=1J{drpr(A^j|r)Δf(r)s(r)drpr(A^j|r)f(r)s(r)j=1Jdrpr(A^j|r)f(r)s(r)drf(r)s(r)}f|H0.
Δfapr(G|H0)=τdrΔf(r)s(r)+exp[J¯(f)][J¯(f)]Jj=1Jdrpr(A^j|r)Δf(r)s(r)drpr(A^j|r)f(r)s(r)j=1Jdrpr(A^j|r)f(r)s(r)drf(r)s(r)f|H0exp[J¯(f)][J¯(f)]Jj=1Jdrpr(A^j|r)f(r)s(r)drf(r)s(r)f|H0.
j=1Jdrpr(A^j|r)f(r)s(r)drf(r)s(r)=pr(A^|J,f).
pr(A^|J,f)Cδ(fb^ML).
Δfapr(G|H0)=τdrΔf(r)s(r)+j=1Jdrpr(A^j|r)Δf(r)s(r)drpr(A^j|r)b^ML(r)s(r).
Φ(A^j,b^ML)drpr(A^j|r)Δf(r)s(r)drpr(A^j|r)b^ML(r)s(r).
Δfapr(G|H0)G|H0=Δf{dGdf[fpr(G|f)]prb(f)}=Δf{df[fdGpr(G|f)]prb(f)}=0;
ΔfBΔfpr(G|H0)G|H0=Δf{dGdf[2ffpr(G|f)]prb(f)}Δf=Δf{df[2ffdGpr(G|f)]prb(f)}=0,
SNR2j=1Jj=1JΦ(A^j,b^ML)Φ(A^j,b^ML)G|H0[τdrΔf(r)s(r)]2.
SNR2J[Φ(A^j,b^ML)]2G|H0+(J2J)Φ(A^j,b^ML)Φ(A^j,b^ML)G|H0[τdrΔf(r)s(r)]2,
J[Φ(A^j,b^ML)]2G|H0=J[Φ(A^j,b^ML)]2J|b^MLA^j|b^MLb^ML|H0=J¯(b^ML)[Φ(A^j,b^ML)]2A^j|b^MLb^ML|H0=τdA^j[drpr(A^j|r)Δf(r)s(r)]2drpr(A^j|r)b^ML(r)s(r)b^ML|H0,
(J2J)Φ(A^j,b^ML)Φ(A^j,b^ML)G|H0=J2JJ|b^MLΦ(A^j,b^ML)Φ(A^j,b^ML)A^j,A^j|b^MLb^ML|H0.
J2JJ|b^ML=[J¯(b^ML)]2.
(J2J)Φ(A^j,b^ML)Φ(A^j,b^ML)G|H0=[τdrΔf(r)s(r)]2.
SNR2τdA^[drpr(A^|r)Δf(r)s(r)]2drpr(A^|r)b^ML(r)s(r)b^ML|H0.
SNR2J¯(b^ML)[drpr(A^|r)Δf(r)s(r)drpr(A^|r)b^ML(r)s(r)]2A^|b^MLb^ML|H0.
Λ(G)1Nn=1NΛ(G|fΦn),
Λ(G|fΦn)=pr(G|fΦn+Δf)pr(G|fΦn),
u(A^)j=1Jδ(A^A^j).
u¯(A^|f)=u(A^|f)G|f=J=0Pr(J|f)JdA^jpr(A^j|f)δ(A^A^j)=J¯(f)pr(A^|f).
u¯(A^|f)=τdrpr(A^|r)f(r)s(r).
u¯(A^|f)=τ[Lf](A^),
u¯=τLf.
Ku|f(A^,A^)[u(A^)u¯(A^|f)][u(A^)u¯(A^|f)]u|f,Ku|Hk(A^,A^)[u(A^)u¯¯k(A^)][u(A^)u¯¯k(A^)]u|ff|Hk,
u¯¯k(A^)u¯(A^|f)f|Hk=τdrpr(A^|r)f¯k(r)s(r),
Ku|f(A^,A^)=u¯(A^|f)δ(A^A^);Ku|Hk(A^,A^)=u¯¯k(A^)δ(A^A^)+τ2[LKf|HkL](A^,A^),
[LKf|HkL](A^,A^)=drdrpr(A^|r)s(r)Kf|Hk(r,r)pr(A^|r)s(r),
u¯¯k=τLf¯k,Ku|Hk=τ(Lf¯k)IA^+τ2LKf|HkL,
t(G)=wudA^w(A^)u(A^).
t¯¯k=wu¯¯k=τdA^w(A^)drpr(A^|r)f¯k(r)s(r).
σt2¯12var{t(G)|H1)}+12var{t(G)|H2)}=dA^dA^w(A^)K¯u(A^,A^)w(A^)=wK¯uw,
K¯u(A^,A^)=12Ku|H0(A^,A^)+12Ku|H1(A^,A^),
K¯u=12Ku|H1+12Ku|H2.
SNRw2=(wΔu¯¯)2wK¯uw,
Δu¯¯=u¯¯1u¯¯0=τ(LΔf).
K¯uw=Δu¯¯orw=K¯u1Δu¯¯.
SNRHot2=Δu¯¯K¯u1Δu¯¯,
K¯u(A^,A^)=[τdrpr(A^|r)f¯¯(r)s(r)]δ(A^A^)+τ2drdrpr(A^|r)s(r)K¯f(r,r)pr(A^|r)s(r),
K¯u1={τB¯+τ2LK¯fL}1={τ[B¯+τLK¯fL]}1={τB¯[IA^+τB¯1LK¯fL]}1=1τ[IA^+τB¯1LK¯fL]1B¯1,
B¯=(Lf¯¯)IA^,andK¯f=12Kf|H0+12Kf|H1.
A^={R^j,j=1,,J}.
pr(R|r)=popt(Rr).
pr(R^|R)=pdet(R^R).
popt(Rr)=12πσopt2exp[|Rr|22σopt2],pdet(R^R)=12πσdet2exp[|R^R|22σdet2].
Δf(r)=Δf02πrs2exp[|r|22rs2],
f(r)=f0k=1K(rrk),
(rrk)=12πrb2exp[|rrk|22rb2].
Kf|H0(r,r)=Kf|H1(r,r)=K¯f(r,r)=pf024πrb2exp[|rr|24rb2],
pr(R^|r)=12πσ2exp[|R^r|22σ2],
[LK¯fL](R^,R^)=pf024π(rb2+σ2)exp[|R^R^|24(rb2+σ2)].
[Lf¯0](R^)=pf0,[Lf¯1](R^)=pf0+Δf02π(rs2+σ2)exp[|R^|22(rs2+σ2)].
K¯u(R^,R^)=τpf0{1+Δf04πpf0(rs2+σ2)exp[|R^|22(rs2+σ2)]}δ(R^R^)+τ2pf024π(rb2+σ2)exp[|R^R^|24(rb2+σ2)],
K¯u(R^,R^)τpf0{δ(R^R^)+τf04π(rs2+σ2)exp[|R^R^|24(rb2+σ2)]}.
[B¯1LK¯fL](R^,R^)f04π(rb2+σ2)exp[|R^R^|24(rb2+σ2)].
B¯1LK¯fL=UDU,
[U](R^,ρ)=ei2πR^·ρ,[D](ρ,ρ)=f0δ(ρρ)e4π2(rb2+σ2)|ρ|2.
{IR^+τB¯1LKfL}1={IR^+τUDU}1={UU+τUDU}1={U[Iρ+τD]U}1=U[Iρ+τD]1U,
SNRHot2τ(LΔf)U{Iρ+τD}1UB¯1(LΔf),
((LΔf)U)(ρ)=Δf0e2π2(rs2+σ2)|ρ|2,(UB¯1LΔf)(ρ)=Δf0pf0e2π2(rs2+σ2)|ρ|2.
SNRHot2(Δf0)2pf02R2d2ρf0τe4π2(rs2+σ2)|ρ|21+f0τe4π2(rb2+σ2)|ρ|2.
SNRHot22πβ0dραρeaρ21+αebρ2=πβb01dyyγ11/α+y=πβaα2F1(1,γ;1+γ;α)=πβaα1+α2F1(1,1;1+γ;α1+α),
2F1(a,b;c;z)=n=0(a)n(b)n(c)nznn!
SNRHot2(τ,rs=rb)=(Δf0)2pf0214π(rs2+σ2)log(1+f0τ).
SNRHot2=πβan=0(1)n(1+γ)n(α1+α)n+1,
(1)n(1+γ)n=1×2××n(1+γ)×(2+γ)××(n+γ).
SNRHot2(τ,rs<rb)>(Δf0)2pf0214π(rs2+σ2)n=11n+1(α1+α)n+1=(Δf0)2pf0214π(rs2+σ2)log(1+f0τ).
SNRHot2(τ,rs<rb)(Δf0)2pf02csc(πrs2+σ2rb2+σ2)14(rb2+σ2)(f0τ)1rs2+σ2rb2+σ2,τ1.
SNRHot2(τ,rs>rb)<(Δf0)2pf0214π(rs2+σ2)n=11n+1(α1+α)n+1=(Δf0)2pf0214π(rs2+σ2)log(1+f0τ),
limτSNRHot2(τ,rs>rb)=limτπβaα1+α2F1(1,1;1+γ;α1+α)=πβa[Γ(γ+1)Γ(γ)][Γ(γ)Γ(γ1)]1=πβaγγ1,
SNRHot2(τ,rs>rb)(Δf0)2pf0214π(rs2rb2),τ1.

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