Abstract

Two different models are proposed for the data produced in a Compton imaging device. A sequence of equations, which relate the model to the distribution of radioactivity that is being imaged, is developed for each of the two models. No series expansions are used in these developments. On the basis of these sequences of equations, a completeness condition is developed for each of the two models. These completeness conditions may prove useful in the future in determining appropriate shapes, configurations, and motions of the device’s detectors. A computer simulation is performed to verify one of these sequences of equations. A computer simulation is also performed to demonstrate that this sequence of equations can produce more accurate images than a backprojection reconstruction method. In addition, a procedure is proposed that could mitigate the effects of the Klein–Nishina distribution, the Doppler broadening, and the variability in the data due to the random generation of photons.

© 2005 Optical Society of America

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    [CrossRef]
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  12. S. E. King, G. W. Phillips, P. S. Haskins, J. E. McKission, “A solid-state Compton camera for three-dimensional imaging,” Nucl. Instrum. Methods Phys. Res. A 353, 320–323 (1994).
    [CrossRef]
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  14. P. Antich, R. Parkey, N. Slavin, E. Tsyganov, A. Zinchenko, “Compact Compton camera design: parameters and imaging algorithms,” in Nuclear Science Symposium, 2000. Conference Record (IEEE Press, Piscataway, N.J., 2000), Vol. 3, pp. 15–20.
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    [CrossRef]
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  27. C. H. Hua, N. H. Clinthorne, S. J. Wilderman, J. W. LeBlanc, W. L. Rogers, “Quantitative evaluation of information loss for Compton cameras,” IEEE Trans. Nucl. Sci. 45, (1999).
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    [CrossRef]
  30. M. J. Cree, P. J. Bones, “Towards direct reconstruction from a gamma camera based on Compton scattering,” IEEE Trans. Med. Imaging 13, 398–407 (1994).
    [CrossRef] [PubMed]
  31. L. A. Shepp, Y. Vardi, “Maximum likelihood reconstruction for emission tomography,” IEEE Trans. Med. Imaging MI-1, 113–122 (1982).
    [CrossRef]
  32. K. Lange, R. Carson, “EM reconstruction algorithms for emission and transmission tomography,” J. Comput. Assist. Tomogr. 8, 306–316 (1984).
    [PubMed]
  33. B. L. Evans, J. B. Martin, M. C. Roggemann, “Deconvolution of shift-variant broadening for Compton scatter imaging,” Nucl. Instrum. Methods Phys. Res. A 422, 661–666 (1999).
    [CrossRef]
  34. W. Feller, An Introduction to Probability Theory and Its Applications, 3rd ed. (Wiley, New York, 1968), Vol. 1.
  35. J. A. Fessler, “Penalized weighted least-square image re-construction for positron emission tomography,” IEEE Trans. Med. Imaging 13, 290–300 (1994).
    [CrossRef]
  36. I. M. Gel’fand, G. E. Shilov, Generalized Functions (Academic, New York, 1964), Vol. 1.
  37. R. N. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, New York, 1965).
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    [CrossRef]
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    [CrossRef]
  41. T. S. Pan, A. E. Yagle, “Numerical study of multigrid implementations of some iterative image reconstruction algorithms,” IEEE Trans. Med. Imaging 10, 572–588 (1991).
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  44. G. W. Phillips, “Applications of Compton imaging in nuclear waste characterization and treaty verification,” in Nuclear Science Symposium, 1996. Conference Record (IEEE Press, Piscataway, N.J., 1996), Vol. 1, pp. 362–364.
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    [CrossRef] [PubMed]
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    [CrossRef]

2004 (1)

S. Chelikani, J. Gore, G. Zuba, “Optimizing Compton camera geometries,” Phys. Med. Biol. 49, 1387–1408 (2004).
[CrossRef] [PubMed]

2003 (1)

M. Hirasawa, T. Tomitani, “An analytical image reconstruction algorithm to compensate for scattering angle broadening in Compton cameras,” Phys. Med. Biol. 48, 1009–1026 (2003).
[CrossRef] [PubMed]

2002 (2)

T. Tomitani, M. Hirasawa, “Image reconstruction from limited angle Compton camera data,” Phys. Med. Biol. 47, 1009–1026 (2002).
[CrossRef]

G. E. Fakhri, P. Maksud, M. F. Kijewski, R. E. Zimmerman, S. C. Moore, “Quantitative simultaneous  99mTc/123I SPECT: design study and validation with Monte Carlo simulations and physical acquisitions,” IEEE Trans. Nucl. Sci. 49, 2315–2321 (2002).
[CrossRef]

2001 (2)

Y. F. Yang, Y. Gono, S. Motomura, S. Enomoto, Y. Yano, “A Compton camera for multi-tracer imaging,” IEEE Trans. Nucl. Sci. 48, 656–661 (2001).
[CrossRef]

G. E. Fakhri, S. C. Moore, P. Maksud, A. Aurengo, M. F. Kijewski, “Absolute activity quantitation in simultaneous  123I/99mTc Brain SPECT,” J. Nucl. Med. 42, 300–308 (2001).
[PubMed]

2000 (1)

L. C. Parra, “Reconstruction of cone-beam projections from Compton scattered data,” IEEE Trans. Nucl. Sci. 47, 1543–1550 (2000).
[CrossRef]

1999 (4)

J. W. LeBlanc, N. H. Clinthorne, C.-H. Hua, E. Nygard, “Experimental results from the C-SPRINT prototype Compton camera,” IEEE Trans. Nucl. Sci. 46, 201–204 (1999).
[CrossRef]

A. C. Sauve, A. O. Hero, W. L. Rogers, S. J. Wilderman, N. H. Clinthorne, “3D image reconstruction for Compton SPECT camera model,” IEEE Trans. Nucl. Sci. 46, 2075–2084 (1999).
[CrossRef]

C. H. Hua, N. H. Clinthorne, S. J. Wilderman, J. W. LeBlanc, W. L. Rogers, “Quantitative evaluation of information loss for Compton cameras,” IEEE Trans. Nucl. Sci. 45, (1999).

B. L. Evans, J. B. Martin, M. C. Roggemann, “Deconvolution of shift-variant broadening for Compton scatter imaging,” Nucl. Instrum. Methods Phys. Res. A 422, 661–666 (1999).
[CrossRef]

1998 (1)

R. Basko, G. L. Zeng, G. T. Gullberg, “Application of spherical harmonics to image reconstruction for the Compton camera,” Phys. Med. Biol. 43, 887–894 (1998).
[CrossRef] [PubMed]

1996 (1)

R. C. Rohe, J. D. Valentine, “A novel Compton scatter camera design for in vivo medical imaging of radiopharmaceuticals. Part II,” IEEE Trans. Nucl. Sci. 43, 3256–3263 (1996).
[CrossRef]

1994 (3)

S. E. King, G. W. Phillips, P. S. Haskins, J. E. McKission, “A solid-state Compton camera for three-dimensional imaging,” Nucl. Instrum. Methods Phys. Res. A 353, 320–323 (1994).
[CrossRef]

J. A. Fessler, “Penalized weighted least-square image re-construction for positron emission tomography,” IEEE Trans. Med. Imaging 13, 290–300 (1994).
[CrossRef]

M. J. Cree, P. J. Bones, “Towards direct reconstruction from a gamma camera based on Compton scattering,” IEEE Trans. Med. Imaging 13, 398–407 (1994).
[CrossRef] [PubMed]

1991 (1)

T. S. Pan, A. E. Yagle, “Numerical study of multigrid implementations of some iterative image reconstruction algorithms,” IEEE Trans. Med. Imaging 10, 572–588 (1991).
[CrossRef] [PubMed]

1990 (2)

1988 (1)

M. V. Ranganath, A. P. Dhawan, N. Mullani, “A multigrid expectation maximization reconstruction algorithm for positron tomography,” IEEE Trans. Med. Imaging 7, 273–278 (1988).
[CrossRef]

1985 (1)

B. D. Smith, “Image reconstruction from cone-beam projections: necessary and sufficient conditions and reconstruction methods,” IEEE Trans. Med. Imaging MI-4, 14–28 (1985).
[CrossRef]

1984 (1)

K. Lange, R. Carson, “EM reconstruction algorithms for emission and transmission tomography,” J. Comput. Assist. Tomogr. 8, 306–316 (1984).
[PubMed]

1982 (1)

L. A. Shepp, Y. Vardi, “Maximum likelihood reconstruction for emission tomography,” IEEE Trans. Med. Imaging MI-1, 113–122 (1982).
[CrossRef]

1978 (1)

B. K. P. Horn, “Density reconstruction using arbitrary ray-sampling schemes,” Proc. IEEE 66, 551–562 (1978).
[CrossRef]

1977 (1)

K. T. Smith, D. C. Solomon, S. L. Wagner, “Mathematical aspects of divergent beam radiography,” Bull. Am. Math. Soc. 83, 1227–1270 (1977).
[CrossRef]

1974 (2)

E. Zeitler, “The reconstruction of objects from their projections,” Optik 39, 396–445 (1974).

R. W. Todd, J. M. Nightingale, D. B. Everett, “A proposed Gamma camera,” Nature 251, 132–134 (1974).
[CrossRef]

1964 (1)

A. M. Cormack, “Representation of a function by its line integrals, with some radiological applications: II,” J. Appl. Phys. 35, 2908–2917 (1964).
[CrossRef]

1963 (1)

A. M. Cormack, “Representation of a function by its line integrals, with some radiological applications,” J. Appl. Phys. 34, 2722–2727 (1963).
[CrossRef]

1923 (1)

A. H. Compton, “A quantum theory of the scattering of x-rays by light elements,” Phys. Rev. 21, 483–502 (1923).
[CrossRef]

Aarsvold, J. N.

L. Junqiang, J. D. Valentine, J. N. Aarsvold, M. Khamzin, “A rebinning technique for 3D reconstruction of Compton camera data,” in Nuclear Science Symposium, 2001. Conference Record (IEEE Press, Piscataway, N.J., 2001), Vol. 4, pp. 1877–1881.

Antich, P.

P. Antich, R. Parkey, N. Slavin, E. Tsyganov, A. Zinchenko, “Compact Compton camera design: parameters and imaging algorithms,” in Nuclear Science Symposium, 2000. Conference Record (IEEE Press, Piscataway, N.J., 2000), Vol. 3, pp. 15–20.

Arsenin, V. Y.

A. N. Tikhonov, V. Y. Arsenin, Solution of Ill-Prosed Problems (Winston, Washington, D.C., 1977).

Aurengo, A.

G. E. Fakhri, S. C. Moore, P. Maksud, A. Aurengo, M. F. Kijewski, “Absolute activity quantitation in simultaneous  123I/99mTc Brain SPECT,” J. Nucl. Med. 42, 300–308 (2001).
[PubMed]

Basko, R.

R. Basko, G. L. Zeng, G. T. Gullberg, “Application of spherical harmonics to image reconstruction for the Compton camera,” Phys. Med. Biol. 43, 887–894 (1998).
[CrossRef] [PubMed]

Bolozdynya, A.

C. E. Ordonez, A. Bolozdynya, W. Chang, “Doppler broadening of energy spectra in Compton cameras,” in Nuclear Science Symposium, 1997. Conference Record (IEEE Press, Piscataway, N.J., 1997), Vol. 2, pp. 1361–1365.

Bones, P. J.

M. J. Cree, P. J. Bones, “Towards direct reconstruction from a gamma camera based on Compton scattering,” IEEE Trans. Med. Imaging 13, 398–407 (1994).
[CrossRef] [PubMed]

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, New York, 1965).

Carson, R.

K. Lange, R. Carson, “EM reconstruction algorithms for emission and transmission tomography,” J. Comput. Assist. Tomogr. 8, 306–316 (1984).
[PubMed]

Chang, W.

C. E. Ordonez, A. Bolozdynya, W. Chang, “Doppler broadening of energy spectra in Compton cameras,” in Nuclear Science Symposium, 1997. Conference Record (IEEE Press, Piscataway, N.J., 1997), Vol. 2, pp. 1361–1365.

Chelikani, S.

S. Chelikani, J. Gore, G. Zuba, “Optimizing Compton camera geometries,” Phys. Med. Biol. 49, 1387–1408 (2004).
[CrossRef] [PubMed]

Clinthorne, N. H.

J. W. LeBlanc, N. H. Clinthorne, C.-H. Hua, E. Nygard, “Experimental results from the C-SPRINT prototype Compton camera,” IEEE Trans. Nucl. Sci. 46, 201–204 (1999).
[CrossRef]

C. H. Hua, N. H. Clinthorne, S. J. Wilderman, J. W. LeBlanc, W. L. Rogers, “Quantitative evaluation of information loss for Compton cameras,” IEEE Trans. Nucl. Sci. 45, (1999).

A. C. Sauve, A. O. Hero, W. L. Rogers, S. J. Wilderman, N. H. Clinthorne, “3D image reconstruction for Compton SPECT camera model,” IEEE Trans. Nucl. Sci. 46, 2075–2084 (1999).
[CrossRef]

N. H. Clinthorne, C. Ng, C. Hua, J. E. Gormley, “Theoretical performance comparison of a Compton-scatter aperture and parallel-hole collimator,” in Nuclear Science Symposium, 1996. Conference Record (IEEE Press, Piscataway, N.J., 1996), Vol. 2, pp. 788–792.

J. E. Gormley, W. L. Rogers, N. H. Clinthorne, D. K. Wehe, “Experimental Comparison of Mechanical and Electronic Collimation,” in Nuclear Science Symposium, 1996. Conference Record (IEEE Press, Piscataway, N.J., 1996), Vol. 2, pp. 798–802.

Compton, A. H.

A. H. Compton, “A quantum theory of the scattering of x-rays by light elements,” Phys. Rev. 21, 483–502 (1923).
[CrossRef]

Cormack, A. M.

A. M. Cormack, “Representation of a function by its line integrals, with some radiological applications: II,” J. Appl. Phys. 35, 2908–2917 (1964).
[CrossRef]

A. M. Cormack, “Representation of a function by its line integrals, with some radiological applications,” J. Appl. Phys. 34, 2722–2727 (1963).
[CrossRef]

Cree, M. J.

M. J. Cree, P. J. Bones, “Towards direct reconstruction from a gamma camera based on Compton scattering,” IEEE Trans. Med. Imaging 13, 398–407 (1994).
[CrossRef] [PubMed]

Deans, S. R.

S. R. Deans, The Radon Transform and Some of Its Applications (Wiley-Interscience, New York, 1983).

Dhawan, A. P.

M. V. Ranganath, A. P. Dhawan, N. Mullani, “A multigrid expectation maximization reconstruction algorithm for positron tomography,” IEEE Trans. Med. Imaging 7, 273–278 (1988).
[CrossRef]

Earnhart, J. R. D.

J. R. D. Earnhart, “A Compton camera for spectroscopic imaging from 100 keV to 1 MeV,” Ph.D. thesis (North Carolina State University, Raleigh, N.C., 1999).

Enomoto, S.

Y. F. Yang, Y. Gono, S. Motomura, S. Enomoto, Y. Yano, “A Compton camera for multi-tracer imaging,” IEEE Trans. Nucl. Sci. 48, 656–661 (2001).
[CrossRef]

Evans, B. L.

B. L. Evans, J. B. Martin, M. C. Roggemann, “Deconvolution of shift-variant broadening for Compton scatter imaging,” Nucl. Instrum. Methods Phys. Res. A 422, 661–666 (1999).
[CrossRef]

Everett, D. B.

R. W. Todd, J. M. Nightingale, D. B. Everett, “A proposed Gamma camera,” Nature 251, 132–134 (1974).
[CrossRef]

Fakhri, G. E.

G. E. Fakhri, P. Maksud, M. F. Kijewski, R. E. Zimmerman, S. C. Moore, “Quantitative simultaneous  99mTc/123I SPECT: design study and validation with Monte Carlo simulations and physical acquisitions,” IEEE Trans. Nucl. Sci. 49, 2315–2321 (2002).
[CrossRef]

G. E. Fakhri, S. C. Moore, P. Maksud, A. Aurengo, M. F. Kijewski, “Absolute activity quantitation in simultaneous  123I/99mTc Brain SPECT,” J. Nucl. Med. 42, 300–308 (2001).
[PubMed]

Feller, W.

W. Feller, An Introduction to Probability Theory and Its Applications, 3rd ed. (Wiley, New York, 1968), Vol. 1.

Fessler, J. A.

J. A. Fessler, “Penalized weighted least-square image re-construction for positron emission tomography,” IEEE Trans. Med. Imaging 13, 290–300 (1994).
[CrossRef]

Gehrels, N.

R. A. Kroeger, W. N. Johnson, R. L. Kinzer, J. D. Kurfess, S. E. Inderhees, B. Phlips, N. Gehrels, “Spatial and Spectral Resolution of a Germanium Strip Detector,” in Imaging in High Energy Astronomy, L. Bassani, G. di Cocco, eds. (Kluwer, Dordrecht, The Netherlands, 1995), pp. 325–329.

Gel’fand, I. M.

I. M. Gel’fand, G. E. Shilov, Generalized Functions (Academic, New York, 1964), Vol. 1.

I. M. Gel’fand, M. I. Graev, N. Y. Valenkin, Generalized Functions: Integral Geometry and Representation Theory (Academic, New York, 1966), Vol. 2. Translated by Eugene Saletan.

Gono, Y.

Y. F. Yang, Y. Gono, S. Motomura, S. Enomoto, Y. Yano, “A Compton camera for multi-tracer imaging,” IEEE Trans. Nucl. Sci. 48, 656–661 (2001).
[CrossRef]

Gore, J.

S. Chelikani, J. Gore, G. Zuba, “Optimizing Compton camera geometries,” Phys. Med. Biol. 49, 1387–1408 (2004).
[CrossRef] [PubMed]

Gormley, J. E.

J. E. Gormley, W. L. Rogers, N. H. Clinthorne, D. K. Wehe, “Experimental Comparison of Mechanical and Electronic Collimation,” in Nuclear Science Symposium, 1996. Conference Record (IEEE Press, Piscataway, N.J., 1996), Vol. 2, pp. 798–802.

N. H. Clinthorne, C. Ng, C. Hua, J. E. Gormley, “Theoretical performance comparison of a Compton-scatter aperture and parallel-hole collimator,” in Nuclear Science Symposium, 1996. Conference Record (IEEE Press, Piscataway, N.J., 1996), Vol. 2, pp. 788–792.

Graev, M. I.

I. M. Gel’fand, M. I. Graev, N. Y. Valenkin, Generalized Functions: Integral Geometry and Representation Theory (Academic, New York, 1966), Vol. 2. Translated by Eugene Saletan.

Gullberg, G. T.

R. Basko, G. L. Zeng, G. T. Gullberg, “Application of spherical harmonics to image reconstruction for the Compton camera,” Phys. Med. Biol. 43, 887–894 (1998).
[CrossRef] [PubMed]

Haskins, P. S.

S. E. King, G. W. Phillips, P. S. Haskins, J. E. McKission, “A solid-state Compton camera for three-dimensional imaging,” Nucl. Instrum. Methods Phys. Res. A 353, 320–323 (1994).
[CrossRef]

Hebert, T.

Hero, A. O.

A. C. Sauve, A. O. Hero, W. L. Rogers, S. J. Wilderman, N. H. Clinthorne, “3D image reconstruction for Compton SPECT camera model,” IEEE Trans. Nucl. Sci. 46, 2075–2084 (1999).
[CrossRef]

Hirasawa, M.

M. Hirasawa, T. Tomitani, “An analytical image reconstruction algorithm to compensate for scattering angle broadening in Compton cameras,” Phys. Med. Biol. 48, 1009–1026 (2003).
[CrossRef] [PubMed]

T. Tomitani, M. Hirasawa, “Image reconstruction from limited angle Compton camera data,” Phys. Med. Biol. 47, 1009–1026 (2002).
[CrossRef]

Horn, B. K. P.

B. K. P. Horn, “Density reconstruction using arbitrary ray-sampling schemes,” Proc. IEEE 66, 551–562 (1978).
[CrossRef]

Hua, C.

N. H. Clinthorne, C. Ng, C. Hua, J. E. Gormley, “Theoretical performance comparison of a Compton-scatter aperture and parallel-hole collimator,” in Nuclear Science Symposium, 1996. Conference Record (IEEE Press, Piscataway, N.J., 1996), Vol. 2, pp. 788–792.

Hua, C. H.

C. H. Hua, N. H. Clinthorne, S. J. Wilderman, J. W. LeBlanc, W. L. Rogers, “Quantitative evaluation of information loss for Compton cameras,” IEEE Trans. Nucl. Sci. 45, (1999).

Hua, C.-H.

J. W. LeBlanc, N. H. Clinthorne, C.-H. Hua, E. Nygard, “Experimental results from the C-SPRINT prototype Compton camera,” IEEE Trans. Nucl. Sci. 46, 201–204 (1999).
[CrossRef]

Inderhees, S. E.

R. A. Kroeger, W. N. Johnson, R. L. Kinzer, J. D. Kurfess, S. E. Inderhees, B. Phlips, N. Gehrels, “Spatial and Spectral Resolution of a Germanium Strip Detector,” in Imaging in High Energy Astronomy, L. Bassani, G. di Cocco, eds. (Kluwer, Dordrecht, The Netherlands, 1995), pp. 325–329.

Johnson, W. N.

R. A. Kroeger, W. N. Johnson, R. L. Kinzer, J. D. Kurfess, S. E. Inderhees, B. Phlips, N. Gehrels, “Spatial and Spectral Resolution of a Germanium Strip Detector,” in Imaging in High Energy Astronomy, L. Bassani, G. di Cocco, eds. (Kluwer, Dordrecht, The Netherlands, 1995), pp. 325–329.

Junqiang, L.

L. Junqiang, J. D. Valentine, J. N. Aarsvold, M. Khamzin, “A rebinning technique for 3D reconstruction of Compton camera data,” in Nuclear Science Symposium, 2001. Conference Record (IEEE Press, Piscataway, N.J., 2001), Vol. 4, pp. 1877–1881.

Khamzin, M.

L. Junqiang, J. D. Valentine, J. N. Aarsvold, M. Khamzin, “A rebinning technique for 3D reconstruction of Compton camera data,” in Nuclear Science Symposium, 2001. Conference Record (IEEE Press, Piscataway, N.J., 2001), Vol. 4, pp. 1877–1881.

Kijewski, M. F.

G. E. Fakhri, P. Maksud, M. F. Kijewski, R. E. Zimmerman, S. C. Moore, “Quantitative simultaneous  99mTc/123I SPECT: design study and validation with Monte Carlo simulations and physical acquisitions,” IEEE Trans. Nucl. Sci. 49, 2315–2321 (2002).
[CrossRef]

G. E. Fakhri, S. C. Moore, P. Maksud, A. Aurengo, M. F. Kijewski, “Absolute activity quantitation in simultaneous  123I/99mTc Brain SPECT,” J. Nucl. Med. 42, 300–308 (2001).
[PubMed]

King, S. E.

S. E. King, G. W. Phillips, P. S. Haskins, J. E. McKission, “A solid-state Compton camera for three-dimensional imaging,” Nucl. Instrum. Methods Phys. Res. A 353, 320–323 (1994).
[CrossRef]

Kinzer, R. L.

R. A. Kroeger, W. N. Johnson, R. L. Kinzer, J. D. Kurfess, S. E. Inderhees, B. Phlips, N. Gehrels, “Spatial and Spectral Resolution of a Germanium Strip Detector,” in Imaging in High Energy Astronomy, L. Bassani, G. di Cocco, eds. (Kluwer, Dordrecht, The Netherlands, 1995), pp. 325–329.

Kroeger, R. A.

R. A. Kroeger, W. N. Johnson, R. L. Kinzer, J. D. Kurfess, S. E. Inderhees, B. Phlips, N. Gehrels, “Spatial and Spectral Resolution of a Germanium Strip Detector,” in Imaging in High Energy Astronomy, L. Bassani, G. di Cocco, eds. (Kluwer, Dordrecht, The Netherlands, 1995), pp. 325–329.

Kurfess, J. D.

R. A. Kroeger, W. N. Johnson, R. L. Kinzer, J. D. Kurfess, S. E. Inderhees, B. Phlips, N. Gehrels, “Spatial and Spectral Resolution of a Germanium Strip Detector,” in Imaging in High Energy Astronomy, L. Bassani, G. di Cocco, eds. (Kluwer, Dordrecht, The Netherlands, 1995), pp. 325–329.

Lange, K.

K. Lange, R. Carson, “EM reconstruction algorithms for emission and transmission tomography,” J. Comput. Assist. Tomogr. 8, 306–316 (1984).
[PubMed]

Leahy, R.

LeBlanc, J. W.

J. W. LeBlanc, N. H. Clinthorne, C.-H. Hua, E. Nygard, “Experimental results from the C-SPRINT prototype Compton camera,” IEEE Trans. Nucl. Sci. 46, 201–204 (1999).
[CrossRef]

C. H. Hua, N. H. Clinthorne, S. J. Wilderman, J. W. LeBlanc, W. L. Rogers, “Quantitative evaluation of information loss for Compton cameras,” IEEE Trans. Nucl. Sci. 45, (1999).

J. W. Leblanc, “A Compton camera for low energy gamma ray imaging in nuclear medicine applications,” Ph.D. thesis (University of Michigan, Ann Arbor, Mich., 1999).

Maksud, P.

G. E. Fakhri, P. Maksud, M. F. Kijewski, R. E. Zimmerman, S. C. Moore, “Quantitative simultaneous  99mTc/123I SPECT: design study and validation with Monte Carlo simulations and physical acquisitions,” IEEE Trans. Nucl. Sci. 49, 2315–2321 (2002).
[CrossRef]

G. E. Fakhri, S. C. Moore, P. Maksud, A. Aurengo, M. F. Kijewski, “Absolute activity quantitation in simultaneous  123I/99mTc Brain SPECT,” J. Nucl. Med. 42, 300–308 (2001).
[PubMed]

Martin, J. B.

B. L. Evans, J. B. Martin, M. C. Roggemann, “Deconvolution of shift-variant broadening for Compton scatter imaging,” Nucl. Instrum. Methods Phys. Res. A 422, 661–666 (1999).
[CrossRef]

McKission, J. E.

S. E. King, G. W. Phillips, P. S. Haskins, J. E. McKission, “A solid-state Compton camera for three-dimensional imaging,” Nucl. Instrum. Methods Phys. Res. A 353, 320–323 (1994).
[CrossRef]

Moore, S. C.

G. E. Fakhri, P. Maksud, M. F. Kijewski, R. E. Zimmerman, S. C. Moore, “Quantitative simultaneous  99mTc/123I SPECT: design study and validation with Monte Carlo simulations and physical acquisitions,” IEEE Trans. Nucl. Sci. 49, 2315–2321 (2002).
[CrossRef]

G. E. Fakhri, S. C. Moore, P. Maksud, A. Aurengo, M. F. Kijewski, “Absolute activity quantitation in simultaneous  123I/99mTc Brain SPECT,” J. Nucl. Med. 42, 300–308 (2001).
[PubMed]

Motomura, S.

Y. F. Yang, Y. Gono, S. Motomura, S. Enomoto, Y. Yano, “A Compton camera for multi-tracer imaging,” IEEE Trans. Nucl. Sci. 48, 656–661 (2001).
[CrossRef]

Mullani, N.

M. V. Ranganath, A. P. Dhawan, N. Mullani, “A multigrid expectation maximization reconstruction algorithm for positron tomography,” IEEE Trans. Med. Imaging 7, 273–278 (1988).
[CrossRef]

Natterer, F.

F. Natterer, The Mathematics of Computerized Tomography (Wiley, New York, 1986).

Ng, C.

N. H. Clinthorne, C. Ng, C. Hua, J. E. Gormley, “Theoretical performance comparison of a Compton-scatter aperture and parallel-hole collimator,” in Nuclear Science Symposium, 1996. Conference Record (IEEE Press, Piscataway, N.J., 1996), Vol. 2, pp. 788–792.

Nightingale, J. M.

R. W. Todd, J. M. Nightingale, D. B. Everett, “A proposed Gamma camera,” Nature 251, 132–134 (1974).
[CrossRef]

Nygard, E.

J. W. LeBlanc, N. H. Clinthorne, C.-H. Hua, E. Nygard, “Experimental results from the C-SPRINT prototype Compton camera,” IEEE Trans. Nucl. Sci. 46, 201–204 (1999).
[CrossRef]

Ordonez, C. E.

C. E. Ordonez, A. Bolozdynya, W. Chang, “Doppler broadening of energy spectra in Compton cameras,” in Nuclear Science Symposium, 1997. Conference Record (IEEE Press, Piscataway, N.J., 1997), Vol. 2, pp. 1361–1365.

Pan, T. S.

T. S. Pan, A. E. Yagle, “Numerical study of multigrid implementations of some iterative image reconstruction algorithms,” IEEE Trans. Med. Imaging 10, 572–588 (1991).
[CrossRef] [PubMed]

Parkey, R.

P. Antich, R. Parkey, N. Slavin, E. Tsyganov, A. Zinchenko, “Compact Compton camera design: parameters and imaging algorithms,” in Nuclear Science Symposium, 2000. Conference Record (IEEE Press, Piscataway, N.J., 2000), Vol. 3, pp. 15–20.

Parra, L. C.

L. C. Parra, “Reconstruction of cone-beam projections from Compton scattered data,” IEEE Trans. Nucl. Sci. 47, 1543–1550 (2000).
[CrossRef]

Phillips, G. W.

S. E. King, G. W. Phillips, P. S. Haskins, J. E. McKission, “A solid-state Compton camera for three-dimensional imaging,” Nucl. Instrum. Methods Phys. Res. A 353, 320–323 (1994).
[CrossRef]

G. W. Phillips, “Applications of Compton imaging in nuclear waste characterization and treaty verification,” in Nuclear Science Symposium, 1996. Conference Record (IEEE Press, Piscataway, N.J., 1996), Vol. 1, pp. 362–364.

Phlips, B.

R. A. Kroeger, W. N. Johnson, R. L. Kinzer, J. D. Kurfess, S. E. Inderhees, B. Phlips, N. Gehrels, “Spatial and Spectral Resolution of a Germanium Strip Detector,” in Imaging in High Energy Astronomy, L. Bassani, G. di Cocco, eds. (Kluwer, Dordrecht, The Netherlands, 1995), pp. 325–329.

Ranganath, M. V.

M. V. Ranganath, A. P. Dhawan, N. Mullani, “A multigrid expectation maximization reconstruction algorithm for positron tomography,” IEEE Trans. Med. Imaging 7, 273–278 (1988).
[CrossRef]

Rogers, W. L.

A. C. Sauve, A. O. Hero, W. L. Rogers, S. J. Wilderman, N. H. Clinthorne, “3D image reconstruction for Compton SPECT camera model,” IEEE Trans. Nucl. Sci. 46, 2075–2084 (1999).
[CrossRef]

C. H. Hua, N. H. Clinthorne, S. J. Wilderman, J. W. LeBlanc, W. L. Rogers, “Quantitative evaluation of information loss for Compton cameras,” IEEE Trans. Nucl. Sci. 45, (1999).

J. E. Gormley, W. L. Rogers, N. H. Clinthorne, D. K. Wehe, “Experimental Comparison of Mechanical and Electronic Collimation,” in Nuclear Science Symposium, 1996. Conference Record (IEEE Press, Piscataway, N.J., 1996), Vol. 2, pp. 798–802.

Roggemann, M. C.

B. L. Evans, J. B. Martin, M. C. Roggemann, “Deconvolution of shift-variant broadening for Compton scatter imaging,” Nucl. Instrum. Methods Phys. Res. A 422, 661–666 (1999).
[CrossRef]

Rohe, R. C.

R. C. Rohe, J. D. Valentine, “A novel Compton scatter camera design for in vivo medical imaging of radiopharmaceuticals. Part II,” IEEE Trans. Nucl. Sci. 43, 3256–3263 (1996).
[CrossRef]

Sauve, A. C.

A. C. Sauve, A. O. Hero, W. L. Rogers, S. J. Wilderman, N. H. Clinthorne, “3D image reconstruction for Compton SPECT camera model,” IEEE Trans. Nucl. Sci. 46, 2075–2084 (1999).
[CrossRef]

Shepp, L. A.

L. A. Shepp, Y. Vardi, “Maximum likelihood reconstruction for emission tomography,” IEEE Trans. Med. Imaging MI-1, 113–122 (1982).
[CrossRef]

Shilov, G. E.

I. M. Gel’fand, G. E. Shilov, Generalized Functions (Academic, New York, 1964), Vol. 1.

Singh, M.

Slavin, N.

P. Antich, R. Parkey, N. Slavin, E. Tsyganov, A. Zinchenko, “Compact Compton camera design: parameters and imaging algorithms,” in Nuclear Science Symposium, 2000. Conference Record (IEEE Press, Piscataway, N.J., 2000), Vol. 3, pp. 15–20.

Smith, B. D.

B. D. Smith, “Cone beam tomography: recent advances and a tutorial,” Opt. Eng. 29, 524–534 (1990).
[CrossRef]

B. D. Smith, “Image reconstruction from cone-beam projections: necessary and sufficient conditions and reconstruction methods,” IEEE Trans. Med. Imaging MI-4, 14–28 (1985).
[CrossRef]

B. D. Smith, “Computer-aided tomography imaging from cone-beam data,” Ph.D. thesis (University of Rhode Island, Kingston, R.I., 1987).

Smith, K. T.

K. T. Smith, D. C. Solomon, S. L. Wagner, “Mathematical aspects of divergent beam radiography,” Bull. Am. Math. Soc. 83, 1227–1270 (1977).
[CrossRef]

Solomon, D. C.

K. T. Smith, D. C. Solomon, S. L. Wagner, “Mathematical aspects of divergent beam radiography,” Bull. Am. Math. Soc. 83, 1227–1270 (1977).
[CrossRef]

Tikhonov, A. N.

A. N. Tikhonov, V. Y. Arsenin, Solution of Ill-Prosed Problems (Winston, Washington, D.C., 1977).

Todd, R. W.

R. W. Todd, J. M. Nightingale, D. B. Everett, “A proposed Gamma camera,” Nature 251, 132–134 (1974).
[CrossRef]

Tomitani, T.

M. Hirasawa, T. Tomitani, “An analytical image reconstruction algorithm to compensate for scattering angle broadening in Compton cameras,” Phys. Med. Biol. 48, 1009–1026 (2003).
[CrossRef] [PubMed]

T. Tomitani, M. Hirasawa, “Image reconstruction from limited angle Compton camera data,” Phys. Med. Biol. 47, 1009–1026 (2002).
[CrossRef]

Tsyganov, E.

P. Antich, R. Parkey, N. Slavin, E. Tsyganov, A. Zinchenko, “Compact Compton camera design: parameters and imaging algorithms,” in Nuclear Science Symposium, 2000. Conference Record (IEEE Press, Piscataway, N.J., 2000), Vol. 3, pp. 15–20.

Valenkin, N. Y.

I. M. Gel’fand, M. I. Graev, N. Y. Valenkin, Generalized Functions: Integral Geometry and Representation Theory (Academic, New York, 1966), Vol. 2. Translated by Eugene Saletan.

Valentine, J. D.

R. C. Rohe, J. D. Valentine, “A novel Compton scatter camera design for in vivo medical imaging of radiopharmaceuticals. Part II,” IEEE Trans. Nucl. Sci. 43, 3256–3263 (1996).
[CrossRef]

L. Junqiang, J. D. Valentine, J. N. Aarsvold, M. Khamzin, “A rebinning technique for 3D reconstruction of Compton camera data,” in Nuclear Science Symposium, 2001. Conference Record (IEEE Press, Piscataway, N.J., 2001), Vol. 4, pp. 1877–1881.

Vardi, Y.

L. A. Shepp, Y. Vardi, “Maximum likelihood reconstruction for emission tomography,” IEEE Trans. Med. Imaging MI-1, 113–122 (1982).
[CrossRef]

Wagner, S. L.

K. T. Smith, D. C. Solomon, S. L. Wagner, “Mathematical aspects of divergent beam radiography,” Bull. Am. Math. Soc. 83, 1227–1270 (1977).
[CrossRef]

Wehe, D. K.

J. E. Gormley, W. L. Rogers, N. H. Clinthorne, D. K. Wehe, “Experimental Comparison of Mechanical and Electronic Collimation,” in Nuclear Science Symposium, 1996. Conference Record (IEEE Press, Piscataway, N.J., 1996), Vol. 2, pp. 798–802.

Wilderman, S. J.

C. H. Hua, N. H. Clinthorne, S. J. Wilderman, J. W. LeBlanc, W. L. Rogers, “Quantitative evaluation of information loss for Compton cameras,” IEEE Trans. Nucl. Sci. 45, (1999).

A. C. Sauve, A. O. Hero, W. L. Rogers, S. J. Wilderman, N. H. Clinthorne, “3D image reconstruction for Compton SPECT camera model,” IEEE Trans. Nucl. Sci. 46, 2075–2084 (1999).
[CrossRef]

Yagle, A. E.

T. S. Pan, A. E. Yagle, “Numerical study of multigrid implementations of some iterative image reconstruction algorithms,” IEEE Trans. Med. Imaging 10, 572–588 (1991).
[CrossRef] [PubMed]

Yang, Y. F.

Y. F. Yang, Y. Gono, S. Motomura, S. Enomoto, Y. Yano, “A Compton camera for multi-tracer imaging,” IEEE Trans. Nucl. Sci. 48, 656–661 (2001).
[CrossRef]

Yano, Y.

Y. F. Yang, Y. Gono, S. Motomura, S. Enomoto, Y. Yano, “A Compton camera for multi-tracer imaging,” IEEE Trans. Nucl. Sci. 48, 656–661 (2001).
[CrossRef]

Zeitler, E.

E. Zeitler, “The reconstruction of objects from their projections,” Optik 39, 396–445 (1974).

Zeng, G. L.

R. Basko, G. L. Zeng, G. T. Gullberg, “Application of spherical harmonics to image reconstruction for the Compton camera,” Phys. Med. Biol. 43, 887–894 (1998).
[CrossRef] [PubMed]

Zimmerman, R. E.

G. E. Fakhri, P. Maksud, M. F. Kijewski, R. E. Zimmerman, S. C. Moore, “Quantitative simultaneous  99mTc/123I SPECT: design study and validation with Monte Carlo simulations and physical acquisitions,” IEEE Trans. Nucl. Sci. 49, 2315–2321 (2002).
[CrossRef]

Zinchenko, A.

P. Antich, R. Parkey, N. Slavin, E. Tsyganov, A. Zinchenko, “Compact Compton camera design: parameters and imaging algorithms,” in Nuclear Science Symposium, 2000. Conference Record (IEEE Press, Piscataway, N.J., 2000), Vol. 3, pp. 15–20.

Zuba, G.

S. Chelikani, J. Gore, G. Zuba, “Optimizing Compton camera geometries,” Phys. Med. Biol. 49, 1387–1408 (2004).
[CrossRef] [PubMed]

Bull. Am. Math. Soc. (1)

K. T. Smith, D. C. Solomon, S. L. Wagner, “Mathematical aspects of divergent beam radiography,” Bull. Am. Math. Soc. 83, 1227–1270 (1977).
[CrossRef]

IEEE Trans. Med. Imaging (6)

M. V. Ranganath, A. P. Dhawan, N. Mullani, “A multigrid expectation maximization reconstruction algorithm for positron tomography,” IEEE Trans. Med. Imaging 7, 273–278 (1988).
[CrossRef]

T. S. Pan, A. E. Yagle, “Numerical study of multigrid implementations of some iterative image reconstruction algorithms,” IEEE Trans. Med. Imaging 10, 572–588 (1991).
[CrossRef] [PubMed]

B. D. Smith, “Image reconstruction from cone-beam projections: necessary and sufficient conditions and reconstruction methods,” IEEE Trans. Med. Imaging MI-4, 14–28 (1985).
[CrossRef]

M. J. Cree, P. J. Bones, “Towards direct reconstruction from a gamma camera based on Compton scattering,” IEEE Trans. Med. Imaging 13, 398–407 (1994).
[CrossRef] [PubMed]

L. A. Shepp, Y. Vardi, “Maximum likelihood reconstruction for emission tomography,” IEEE Trans. Med. Imaging MI-1, 113–122 (1982).
[CrossRef]

J. A. Fessler, “Penalized weighted least-square image re-construction for positron emission tomography,” IEEE Trans. Med. Imaging 13, 290–300 (1994).
[CrossRef]

IEEE Trans. Nucl. Sci. (7)

C. H. Hua, N. H. Clinthorne, S. J. Wilderman, J. W. LeBlanc, W. L. Rogers, “Quantitative evaluation of information loss for Compton cameras,” IEEE Trans. Nucl. Sci. 45, (1999).

A. C. Sauve, A. O. Hero, W. L. Rogers, S. J. Wilderman, N. H. Clinthorne, “3D image reconstruction for Compton SPECT camera model,” IEEE Trans. Nucl. Sci. 46, 2075–2084 (1999).
[CrossRef]

G. E. Fakhri, P. Maksud, M. F. Kijewski, R. E. Zimmerman, S. C. Moore, “Quantitative simultaneous  99mTc/123I SPECT: design study and validation with Monte Carlo simulations and physical acquisitions,” IEEE Trans. Nucl. Sci. 49, 2315–2321 (2002).
[CrossRef]

Y. F. Yang, Y. Gono, S. Motomura, S. Enomoto, Y. Yano, “A Compton camera for multi-tracer imaging,” IEEE Trans. Nucl. Sci. 48, 656–661 (2001).
[CrossRef]

J. W. LeBlanc, N. H. Clinthorne, C.-H. Hua, E. Nygard, “Experimental results from the C-SPRINT prototype Compton camera,” IEEE Trans. Nucl. Sci. 46, 201–204 (1999).
[CrossRef]

R. C. Rohe, J. D. Valentine, “A novel Compton scatter camera design for in vivo medical imaging of radiopharmaceuticals. Part II,” IEEE Trans. Nucl. Sci. 43, 3256–3263 (1996).
[CrossRef]

L. C. Parra, “Reconstruction of cone-beam projections from Compton scattered data,” IEEE Trans. Nucl. Sci. 47, 1543–1550 (2000).
[CrossRef]

J. Appl. Phys. (2)

A. M. Cormack, “Representation of a function by its line integrals, with some radiological applications,” J. Appl. Phys. 34, 2722–2727 (1963).
[CrossRef]

A. M. Cormack, “Representation of a function by its line integrals, with some radiological applications: II,” J. Appl. Phys. 35, 2908–2917 (1964).
[CrossRef]

J. Comput. Assist. Tomogr. (1)

K. Lange, R. Carson, “EM reconstruction algorithms for emission and transmission tomography,” J. Comput. Assist. Tomogr. 8, 306–316 (1984).
[PubMed]

J. Nucl. Med. (1)

G. E. Fakhri, S. C. Moore, P. Maksud, A. Aurengo, M. F. Kijewski, “Absolute activity quantitation in simultaneous  123I/99mTc Brain SPECT,” J. Nucl. Med. 42, 300–308 (2001).
[PubMed]

J. Opt. Soc. Am. A (1)

Nature (1)

R. W. Todd, J. M. Nightingale, D. B. Everett, “A proposed Gamma camera,” Nature 251, 132–134 (1974).
[CrossRef]

Nucl. Instrum. Methods Phys. Res. A (2)

S. E. King, G. W. Phillips, P. S. Haskins, J. E. McKission, “A solid-state Compton camera for three-dimensional imaging,” Nucl. Instrum. Methods Phys. Res. A 353, 320–323 (1994).
[CrossRef]

B. L. Evans, J. B. Martin, M. C. Roggemann, “Deconvolution of shift-variant broadening for Compton scatter imaging,” Nucl. Instrum. Methods Phys. Res. A 422, 661–666 (1999).
[CrossRef]

Opt. Eng. (1)

B. D. Smith, “Cone beam tomography: recent advances and a tutorial,” Opt. Eng. 29, 524–534 (1990).
[CrossRef]

Optik (1)

E. Zeitler, “The reconstruction of objects from their projections,” Optik 39, 396–445 (1974).

Phys. Med. Biol. (4)

R. Basko, G. L. Zeng, G. T. Gullberg, “Application of spherical harmonics to image reconstruction for the Compton camera,” Phys. Med. Biol. 43, 887–894 (1998).
[CrossRef] [PubMed]

T. Tomitani, M. Hirasawa, “Image reconstruction from limited angle Compton camera data,” Phys. Med. Biol. 47, 1009–1026 (2002).
[CrossRef]

M. Hirasawa, T. Tomitani, “An analytical image reconstruction algorithm to compensate for scattering angle broadening in Compton cameras,” Phys. Med. Biol. 48, 1009–1026 (2003).
[CrossRef] [PubMed]

S. Chelikani, J. Gore, G. Zuba, “Optimizing Compton camera geometries,” Phys. Med. Biol. 49, 1387–1408 (2004).
[CrossRef] [PubMed]

Phys. Rev. (1)

A. H. Compton, “A quantum theory of the scattering of x-rays by light elements,” Phys. Rev. 21, 483–502 (1923).
[CrossRef]

Proc. IEEE (1)

B. K. P. Horn, “Density reconstruction using arbitrary ray-sampling schemes,” Proc. IEEE 66, 551–562 (1978).
[CrossRef]

Other (17)

B. D. Smith, “Computer-aided tomography imaging from cone-beam data,” Ph.D. thesis (University of Rhode Island, Kingston, R.I., 1987).

I. M. Gel’fand, G. E. Shilov, Generalized Functions (Academic, New York, 1964), Vol. 1.

R. N. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, New York, 1965).

W. Feller, An Introduction to Probability Theory and Its Applications, 3rd ed. (Wiley, New York, 1968), Vol. 1.

L. Junqiang, J. D. Valentine, J. N. Aarsvold, M. Khamzin, “A rebinning technique for 3D reconstruction of Compton camera data,” in Nuclear Science Symposium, 2001. Conference Record (IEEE Press, Piscataway, N.J., 2001), Vol. 4, pp. 1877–1881.

P. Antich, R. Parkey, N. Slavin, E. Tsyganov, A. Zinchenko, “Compact Compton camera design: parameters and imaging algorithms,” in Nuclear Science Symposium, 2000. Conference Record (IEEE Press, Piscataway, N.J., 2000), Vol. 3, pp. 15–20.

R. A. Kroeger, W. N. Johnson, R. L. Kinzer, J. D. Kurfess, S. E. Inderhees, B. Phlips, N. Gehrels, “Spatial and Spectral Resolution of a Germanium Strip Detector,” in Imaging in High Energy Astronomy, L. Bassani, G. di Cocco, eds. (Kluwer, Dordrecht, The Netherlands, 1995), pp. 325–329.

C. E. Ordonez, A. Bolozdynya, W. Chang, “Doppler broadening of energy spectra in Compton cameras,” in Nuclear Science Symposium, 1997. Conference Record (IEEE Press, Piscataway, N.J., 1997), Vol. 2, pp. 1361–1365.

J. W. Leblanc, “A Compton camera for low energy gamma ray imaging in nuclear medicine applications,” Ph.D. thesis (University of Michigan, Ann Arbor, Mich., 1999).

N. H. Clinthorne, C. Ng, C. Hua, J. E. Gormley, “Theoretical performance comparison of a Compton-scatter aperture and parallel-hole collimator,” in Nuclear Science Symposium, 1996. Conference Record (IEEE Press, Piscataway, N.J., 1996), Vol. 2, pp. 788–792.

J. E. Gormley, W. L. Rogers, N. H. Clinthorne, D. K. Wehe, “Experimental Comparison of Mechanical and Electronic Collimation,” in Nuclear Science Symposium, 1996. Conference Record (IEEE Press, Piscataway, N.J., 1996), Vol. 2, pp. 798–802.

S. R. Deans, The Radon Transform and Some of Its Applications (Wiley-Interscience, New York, 1983).

F. Natterer, The Mathematics of Computerized Tomography (Wiley, New York, 1986).

J. R. D. Earnhart, “A Compton camera for spectroscopic imaging from 100 keV to 1 MeV,” Ph.D. thesis (North Carolina State University, Raleigh, N.C., 1999).

A. N. Tikhonov, V. Y. Arsenin, Solution of Ill-Prosed Problems (Winston, Washington, D.C., 1977).

I. M. Gel’fand, M. I. Graev, N. Y. Valenkin, Generalized Functions: Integral Geometry and Representation Theory (Academic, New York, 1966), Vol. 2. Translated by Eugene Saletan.

G. W. Phillips, “Applications of Compton imaging in nuclear waste characterization and treaty verification,” in Nuclear Science Symposium, 1996. Conference Record (IEEE Press, Piscataway, N.J., 1996), Vol. 1, pp. 362–364.

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Figures (10)

Fig. 1
Fig. 1

Geometry for defining hollow cone regions.

Fig. 2
Fig. 2

Cross section containing the z axis of the reconstruction resulting from the sequence of equations.

Fig. 3
Fig. 3

Middle transaxial cross section of the reconstruction resulting from the sequence of equations.

Fig. 4
Fig. 4

Transaxial cross section, that is higher than the middle cross section, of the reconstruction resulting from the sequence of equations.

Fig. 5
Fig. 5

The cross section containing the z axis of the backprojection reconstruction method.

Fig. 6
Fig. 6

Middle transaxial cross section of the backprojection reconstruction.

Fig. 7
Fig. 7

Transaxial cross section of the backprojection reconstruction that is higher than the middle cross section.

Fig. 8
Fig. 8

Line graph along the z axis of the values obtained from the sequence of expressions presented in condition (17).

Fig. 9
Fig. 9

Line graph along the z axis of the values obtained from the backprojection method.

Fig. 10
Fig. 10

Geometry for defining the joint probability density function.

Equations (59)

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

Y ( j ,   l ,   k ) = m = 1 N ψ W ( j ,   l ,   k ,   m ) , k = 1 , ,   N e .
E { Y ( j ,   l ,   k ) } = m = 1 N ψ p ( j ,   l ,   k | m ) α ( j ,   l ,   m ) ,
k = 1 , ,   N e .
n ( j ,   l ,   k ) = m = 1 N ψ p ( j ,   l ,   k | m ) α ( j ,   l ,   m ) ,
k = 1 , ,   N e .
α     α ( ϕ ,   ψ )     ( cos ϕ sin ψ ,   sin ϕ sin ψ ,   cos ψ ) T ,
S ( Φ ,   β ,   ψ )
= sin ψ ϕ = 0 2 π r = 0 f ( Φ + rM T α ( ϕ ,   ψ ) ) r d r d ϕ ,
M T = [ β 1 | β 2 | β ]
C ( β ,   l )     lim 0   - f ˇ ( β ,   t ) p ( l - t ) d t ,
p ( t )     1 / t   for | t | > 0 otherwise ,
f ˇ ( β ,   l )     - - f ( l β + s β 1 + t β 2 ) d s d t .
C ( β ,   Φ     β ) = - lim 0   0 π S ( Φ ,   β ,   ψ ) p ( cos ψ ) d ψ .
F ( β ,   l )     1 π   lim 0   - + H ( l - t ) f ˇ ( β ,   t ) d t ,
H ( t ) = 1 / 2 for | t | < - 1 / t 2 for | t | .
P F     ( β ,   l )   :   β S 2 2 | l | R ,
φ = ( cos φ ,   sin φ ,   0 ) T , φ = ( sin φ ,   - cos φ ,   0 ) T .
P ( x ,   φ )     - f ( x + s φ ) d s .
P P     ( x ,   φ )   :   x R ,   φ S 1 2 ,
P ( x ,   φ ) = 1 2 π   0 π F ( β ( θ ,   φ ) ,   x     β ( θ ,   φ ) ) d θ ,
β ( θ ,   φ ) = ( cos φ sin θ ,   sin φ sin θ ,   cos θ ) T .
F ( β ,   l ) = 1 π l   C ( β ,   l ) .
S ( Φ ,   β ,   ψ ) on P S Theorem 1 C ( β ,   l ) on P C
Theorem 2 F ( β ,   l ) on P F
Smith ( 5.2 ) ( Ref . 28 )     f ( x ) on B R .
g ( Φ ,   α )     0 f ( Φ + t α ) d t .
S CB ( Φ ,   β ,   ψ )     0 2 π g ( Φ ,   M T α ( ϕ ,   ψ ) ) d ϕ .
F ( β ,   Φ     β ) = lim 0   0 π S CB ( Φ ,   β ,   ψ ) H ( cos ψ ) sin ψ d ψ .
S CB ( Φ ,   β ,   ψ ) on P SCB Theorem 3 F ( β ,   l ) on P F
Smith ( 5.2 ) ( Ref . 28 ) f ( x ) on B R .
p ( j ,   l ,   k | m ) = p ( l | j ,   k ,   m ) p ( j | m ) p ( k ) ,
p ( l | j ,   k ,   m ) = t = 1 N t s = 1 N s p ( l | j ,   k ,   s ,   t ) p ( j ,   k ,   s ,   t ) p ( j ,   k ,   m ) ,
p ( l | j ,   k ,   m ) = p ( j | s ,   t ) p ( s ,   t ) p ( j | m ) p ( m )   t = 1 N t s = 1 N s p ( l | j ,   k ,   s ,   t ) .
p ( j ,   l ,   k | m )
= p ( k ) p ( j | s ,   t )   p ( s ,   t ) p ( m )   t = 1 N t s = 1 N s p ( l | j ,   k ,   s ,   t ) ,
p ( k ) = e k e k + 1 f KN ( ϱ ) d ϱ ,
f ( θ ,   φ ) = 1 2 π   f D θ ,   e k + e k + 1 2 for 0 θ π , 0 φ 2 π 0 otherwise .
p ( l | j ,   k ,   s ,   t ) = ( θ , φ ) Ω ( j , l , s , t ) f ( θ ,   φ ) d θ d φ .
- f ˇ ( β ,   t ) g ( t ) d t = R 3 f ( x ) g ( x     β ) d x .
- f ˇ ( β ,   t ) p ( Φ     β - t ) d t = R 3 f ( Φ - x ) p ( x     β ) d x .
S 2 0 + f ( Φ - r α ) p ( r α     β ) r 2 d r d α ,
0 + 0 π 0 2 π f ( Φ - rM T α ( ϕ ,   ψ ) ) d ϕ p ( r cos ψ )
× sin ψ d ψ r 2 d r .
0 + - 1 + 1 0 2 π f ( Φ - rM T α ( ϕ ,   cos - 1   z ) ) d ϕ p ( rz ) d zr 2 d r .
lim 0   0 + - 1 + 1 0 2 π f ( Φ - rM T α ( ϕ ,   cos - 1   z ) ) d ϕ p ( z ) d zr d r .
lim 0   0 π 0 2 π 0 + f ( Φ - rM T α ( ϕ ,   ψ ) )
×   r d r d ϕ p ( cos ψ ) sin ψ d ψ .
lim 0   g ( t ) p ( kt ) d t = lim 0   1 k   g ( t ) p ( t ) d t .
LHS = lim 0 - - / | k | + / | k |   1 t   g ( t ) d t .
LHS = lim 0 - - +   1 t   g ( t ) d t .
- f ˇ ( β ,   t ) H ( Φ     β - t ) d t = R 3 f ( Φ - x ) H ( x     β ) d x .
0 S 2 f ( Φ - r α ) H ( r α     β ) d α r 2 d r .
0 0 2 π 0 π f ( Φ - rM T α ( ϕ ,   ψ ) )
×   H ( r cos ψ ) sin ψ d ψ d ϕ r 2 d r .
0 0 2 π - 1 + 1 f ( Φ - rM T α ( ϕ ,   cos - 1   z ) ) H ( rz ) d z d ϕ r 2 d r .
F ( β ,   Φ     β ) = lim 0   0 π 0 2 π 0 f ( Φ - rM T α ( ϕ ,   ψ ) ) × d r d ϕ H ( cos ψ ) sin ψ d ψ .
lim 0   g ( t ) H ( kt ) d t = lim 0   1 k 2   g ( t ) H ( t ) d t .
LHS = lim 0 1 k 2   - / | k | / | k | k 2 2 g ( t ) d t + 1 k 2   - - / | k | + / | k |   ( - 1 ) t 2   g ( t ) d t .
LHS = lim 0   1 k 2   -   g ( t ) 2   d t + - - +   ( - 1 ) t 2   g ( t ) d t .

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