Abstract

The problem of image reconstruction with Compton-scattering spectral data is an ill-posed problem, and the measurement error may be seriously amplified in the reconstruction result. For a stable solution, some kinds of a priori models of the problem should be incorporated into the process of reconstruction. Lee et al. [IEEE. Trans. Nucl. Sci. 40, 2049 (1993)] have proposed a continuous model with binary line processes. Owing to the coexistence of the continuous variable and the binary variable, the commonly used optimization methods for problems with continuous variables cannot be used in this case, and therefore a coupled-gradient artificial neural network was proposed for this mixed-integer problem. By introducing two interacting parts (with one part for the continuous variable and the other for the binary line processes) into the network, and by defining the appropriate energy function and dynamics, high-quality solutions were obtained upon convergence of the dynamics. Some simulated results are presented.

© 1998 Optical Society of America

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  1. P. G. Lale, “The examination of internal tissues, using gamma ray scatter with a possible extension to megavoltage radiography,” Phys. Med. Biol. 4, 159–166 (1959).
    [CrossRef] [PubMed]
  2. G. Harding, “On the sensitivity and application possibilities of a novel Compton scatter imaging,” IEEE Trans. Nucl. Sci. 29, 1260–1265 (1982).
    [CrossRef]
  3. J. J. Battista, M. J. Bronskill, “Compton scatter imaging of transverse section: corrections for multiple scattering and attenuation,” Phys. Med. Biol. 22, 229–244 (1977).
    [CrossRef] [PubMed]
  4. J. R. Larmarsh, Introduction to Nuclear Engineering (Addison-Wesley, Reading, Mass., 1983).
  5. F. T. Farmer, M. P. Collins, “A new approach to the determination of anatomical cross sections of the body by Compton scattering of gamma rays,” Phys. Med. Biol. 16, 577 (1970).
    [CrossRef]
  6. F. T. Farmer, M. P. Collins, “A further appraisal of the Compton scattering method for determining anatomical cross-sections of the body,” Phys. Med. Biol. 19, 808 (1974).
    [CrossRef] [PubMed]
  7. R. Guzzardi, M. Mey, S. Solfanelli, C. Giuntini, “The 90° Compton scattering tomography of the lung” J. Nucl. Meth. All. Sci. 22, 11, 1978.
  8. M. Lee, A. Rangarajan, I. G. Zubal, G. Gindi, “A continuation method for emission tomography,” IEEE Trans. Nucl. Sci. 40, 2049–2058 (1993).
    [CrossRef]
  9. R. Cesareo, A. L. Hanson, G. E. Gigante, L. J. Pedraza, S. Q. G. Mahtaboally, “Interaction of Key Photons with matter and new applications,” Phys. Rep. 213, 117–178 (1992).
    [CrossRef]
  10. R. Guzzardi, G. Licitra, “A critical review of Compton imaging,” CRC Rep. 15, 237–268 (1988).
  11. N. V. Arendtsz, E. M. A. Hussein, “Energy spectral Compton scatter imaging—part 1: theory and mathematics,” IEEE Trans. Nucl. Sci. 42, 2155–2165 (1995).
    [CrossRef]
  12. T. Hebert, R. Leahy, “A generalized EM algorithm for 3-D Bayesian reconstruction from Poisson data using Gibbs priors,” IEEE. Trans. Med. Imaging 8, 194–202 (1989).
    [CrossRef] [PubMed]
  13. J. Besag, “On the statistical analysis of dirty pictures,” J. R. Statist. Soc. B 48, 259–302 (1986).
  14. S. Geman, D. Geman, “Stochastic relaxation, Gibbs distribution, and Bayesian restoration of images,” IEEE Trans. Patt. Anal. Mach. Intell. PAMI-6, 721–741 (1984).
    [CrossRef]
  15. G. Wilson, G. Pawley, “On the stability of the traveling salesman problem algorithm of Hopfield and Tank,” Biol. Cybern. 58, 63–70 (1988).
    [CrossRef]

1995 (1)

N. V. Arendtsz, E. M. A. Hussein, “Energy spectral Compton scatter imaging—part 1: theory and mathematics,” IEEE Trans. Nucl. Sci. 42, 2155–2165 (1995).
[CrossRef]

1993 (1)

M. Lee, A. Rangarajan, I. G. Zubal, G. Gindi, “A continuation method for emission tomography,” IEEE Trans. Nucl. Sci. 40, 2049–2058 (1993).
[CrossRef]

1992 (1)

R. Cesareo, A. L. Hanson, G. E. Gigante, L. J. Pedraza, S. Q. G. Mahtaboally, “Interaction of Key Photons with matter and new applications,” Phys. Rep. 213, 117–178 (1992).
[CrossRef]

1989 (1)

T. Hebert, R. Leahy, “A generalized EM algorithm for 3-D Bayesian reconstruction from Poisson data using Gibbs priors,” IEEE. Trans. Med. Imaging 8, 194–202 (1989).
[CrossRef] [PubMed]

1988 (2)

R. Guzzardi, G. Licitra, “A critical review of Compton imaging,” CRC Rep. 15, 237–268 (1988).

G. Wilson, G. Pawley, “On the stability of the traveling salesman problem algorithm of Hopfield and Tank,” Biol. Cybern. 58, 63–70 (1988).
[CrossRef]

1986 (1)

J. Besag, “On the statistical analysis of dirty pictures,” J. R. Statist. Soc. B 48, 259–302 (1986).

1984 (1)

S. Geman, D. Geman, “Stochastic relaxation, Gibbs distribution, and Bayesian restoration of images,” IEEE Trans. Patt. Anal. Mach. Intell. PAMI-6, 721–741 (1984).
[CrossRef]

1982 (1)

G. Harding, “On the sensitivity and application possibilities of a novel Compton scatter imaging,” IEEE Trans. Nucl. Sci. 29, 1260–1265 (1982).
[CrossRef]

1978 (1)

R. Guzzardi, M. Mey, S. Solfanelli, C. Giuntini, “The 90° Compton scattering tomography of the lung” J. Nucl. Meth. All. Sci. 22, 11, 1978.

1977 (1)

J. J. Battista, M. J. Bronskill, “Compton scatter imaging of transverse section: corrections for multiple scattering and attenuation,” Phys. Med. Biol. 22, 229–244 (1977).
[CrossRef] [PubMed]

1974 (1)

F. T. Farmer, M. P. Collins, “A further appraisal of the Compton scattering method for determining anatomical cross-sections of the body,” Phys. Med. Biol. 19, 808 (1974).
[CrossRef] [PubMed]

1970 (1)

F. T. Farmer, M. P. Collins, “A new approach to the determination of anatomical cross sections of the body by Compton scattering of gamma rays,” Phys. Med. Biol. 16, 577 (1970).
[CrossRef]

1959 (1)

P. G. Lale, “The examination of internal tissues, using gamma ray scatter with a possible extension to megavoltage radiography,” Phys. Med. Biol. 4, 159–166 (1959).
[CrossRef] [PubMed]

Arendtsz, N. V.

N. V. Arendtsz, E. M. A. Hussein, “Energy spectral Compton scatter imaging—part 1: theory and mathematics,” IEEE Trans. Nucl. Sci. 42, 2155–2165 (1995).
[CrossRef]

Battista, J. J.

J. J. Battista, M. J. Bronskill, “Compton scatter imaging of transverse section: corrections for multiple scattering and attenuation,” Phys. Med. Biol. 22, 229–244 (1977).
[CrossRef] [PubMed]

Besag, J.

J. Besag, “On the statistical analysis of dirty pictures,” J. R. Statist. Soc. B 48, 259–302 (1986).

Bronskill, M. J.

J. J. Battista, M. J. Bronskill, “Compton scatter imaging of transverse section: corrections for multiple scattering and attenuation,” Phys. Med. Biol. 22, 229–244 (1977).
[CrossRef] [PubMed]

Cesareo, R.

R. Cesareo, A. L. Hanson, G. E. Gigante, L. J. Pedraza, S. Q. G. Mahtaboally, “Interaction of Key Photons with matter and new applications,” Phys. Rep. 213, 117–178 (1992).
[CrossRef]

Collins, M. P.

F. T. Farmer, M. P. Collins, “A further appraisal of the Compton scattering method for determining anatomical cross-sections of the body,” Phys. Med. Biol. 19, 808 (1974).
[CrossRef] [PubMed]

F. T. Farmer, M. P. Collins, “A new approach to the determination of anatomical cross sections of the body by Compton scattering of gamma rays,” Phys. Med. Biol. 16, 577 (1970).
[CrossRef]

Farmer, F. T.

F. T. Farmer, M. P. Collins, “A further appraisal of the Compton scattering method for determining anatomical cross-sections of the body,” Phys. Med. Biol. 19, 808 (1974).
[CrossRef] [PubMed]

F. T. Farmer, M. P. Collins, “A new approach to the determination of anatomical cross sections of the body by Compton scattering of gamma rays,” Phys. Med. Biol. 16, 577 (1970).
[CrossRef]

Geman, D.

S. Geman, D. Geman, “Stochastic relaxation, Gibbs distribution, and Bayesian restoration of images,” IEEE Trans. Patt. Anal. Mach. Intell. PAMI-6, 721–741 (1984).
[CrossRef]

Geman, S.

S. Geman, D. Geman, “Stochastic relaxation, Gibbs distribution, and Bayesian restoration of images,” IEEE Trans. Patt. Anal. Mach. Intell. PAMI-6, 721–741 (1984).
[CrossRef]

Gigante, G. E.

R. Cesareo, A. L. Hanson, G. E. Gigante, L. J. Pedraza, S. Q. G. Mahtaboally, “Interaction of Key Photons with matter and new applications,” Phys. Rep. 213, 117–178 (1992).
[CrossRef]

Gindi, G.

M. Lee, A. Rangarajan, I. G. Zubal, G. Gindi, “A continuation method for emission tomography,” IEEE Trans. Nucl. Sci. 40, 2049–2058 (1993).
[CrossRef]

Giuntini, C.

R. Guzzardi, M. Mey, S. Solfanelli, C. Giuntini, “The 90° Compton scattering tomography of the lung” J. Nucl. Meth. All. Sci. 22, 11, 1978.

Guzzardi, R.

R. Guzzardi, G. Licitra, “A critical review of Compton imaging,” CRC Rep. 15, 237–268 (1988).

R. Guzzardi, M. Mey, S. Solfanelli, C. Giuntini, “The 90° Compton scattering tomography of the lung” J. Nucl. Meth. All. Sci. 22, 11, 1978.

Hanson, A. L.

R. Cesareo, A. L. Hanson, G. E. Gigante, L. J. Pedraza, S. Q. G. Mahtaboally, “Interaction of Key Photons with matter and new applications,” Phys. Rep. 213, 117–178 (1992).
[CrossRef]

Harding, G.

G. Harding, “On the sensitivity and application possibilities of a novel Compton scatter imaging,” IEEE Trans. Nucl. Sci. 29, 1260–1265 (1982).
[CrossRef]

Hebert, T.

T. Hebert, R. Leahy, “A generalized EM algorithm for 3-D Bayesian reconstruction from Poisson data using Gibbs priors,” IEEE. Trans. Med. Imaging 8, 194–202 (1989).
[CrossRef] [PubMed]

Hussein, E. M. A.

N. V. Arendtsz, E. M. A. Hussein, “Energy spectral Compton scatter imaging—part 1: theory and mathematics,” IEEE Trans. Nucl. Sci. 42, 2155–2165 (1995).
[CrossRef]

Lale, P. G.

P. G. Lale, “The examination of internal tissues, using gamma ray scatter with a possible extension to megavoltage radiography,” Phys. Med. Biol. 4, 159–166 (1959).
[CrossRef] [PubMed]

Larmarsh, J. R.

J. R. Larmarsh, Introduction to Nuclear Engineering (Addison-Wesley, Reading, Mass., 1983).

Leahy, R.

T. Hebert, R. Leahy, “A generalized EM algorithm for 3-D Bayesian reconstruction from Poisson data using Gibbs priors,” IEEE. Trans. Med. Imaging 8, 194–202 (1989).
[CrossRef] [PubMed]

Lee, M.

M. Lee, A. Rangarajan, I. G. Zubal, G. Gindi, “A continuation method for emission tomography,” IEEE Trans. Nucl. Sci. 40, 2049–2058 (1993).
[CrossRef]

Licitra, G.

R. Guzzardi, G. Licitra, “A critical review of Compton imaging,” CRC Rep. 15, 237–268 (1988).

Mahtaboally, S. Q. G.

R. Cesareo, A. L. Hanson, G. E. Gigante, L. J. Pedraza, S. Q. G. Mahtaboally, “Interaction of Key Photons with matter and new applications,” Phys. Rep. 213, 117–178 (1992).
[CrossRef]

Mey, M.

R. Guzzardi, M. Mey, S. Solfanelli, C. Giuntini, “The 90° Compton scattering tomography of the lung” J. Nucl. Meth. All. Sci. 22, 11, 1978.

Pawley, G.

G. Wilson, G. Pawley, “On the stability of the traveling salesman problem algorithm of Hopfield and Tank,” Biol. Cybern. 58, 63–70 (1988).
[CrossRef]

Pedraza, L. J.

R. Cesareo, A. L. Hanson, G. E. Gigante, L. J. Pedraza, S. Q. G. Mahtaboally, “Interaction of Key Photons with matter and new applications,” Phys. Rep. 213, 117–178 (1992).
[CrossRef]

Rangarajan, A.

M. Lee, A. Rangarajan, I. G. Zubal, G. Gindi, “A continuation method for emission tomography,” IEEE Trans. Nucl. Sci. 40, 2049–2058 (1993).
[CrossRef]

Solfanelli, S.

R. Guzzardi, M. Mey, S. Solfanelli, C. Giuntini, “The 90° Compton scattering tomography of the lung” J. Nucl. Meth. All. Sci. 22, 11, 1978.

Wilson, G.

G. Wilson, G. Pawley, “On the stability of the traveling salesman problem algorithm of Hopfield and Tank,” Biol. Cybern. 58, 63–70 (1988).
[CrossRef]

Zubal, I. G.

M. Lee, A. Rangarajan, I. G. Zubal, G. Gindi, “A continuation method for emission tomography,” IEEE Trans. Nucl. Sci. 40, 2049–2058 (1993).
[CrossRef]

Biol. Cybern. (1)

G. Wilson, G. Pawley, “On the stability of the traveling salesman problem algorithm of Hopfield and Tank,” Biol. Cybern. 58, 63–70 (1988).
[CrossRef]

CRC Rep. (1)

R. Guzzardi, G. Licitra, “A critical review of Compton imaging,” CRC Rep. 15, 237–268 (1988).

IEEE Trans. Nucl. Sci. (3)

N. V. Arendtsz, E. M. A. Hussein, “Energy spectral Compton scatter imaging—part 1: theory and mathematics,” IEEE Trans. Nucl. Sci. 42, 2155–2165 (1995).
[CrossRef]

G. Harding, “On the sensitivity and application possibilities of a novel Compton scatter imaging,” IEEE Trans. Nucl. Sci. 29, 1260–1265 (1982).
[CrossRef]

M. Lee, A. Rangarajan, I. G. Zubal, G. Gindi, “A continuation method for emission tomography,” IEEE Trans. Nucl. Sci. 40, 2049–2058 (1993).
[CrossRef]

IEEE Trans. Patt. Anal. Mach. Intell. (1)

S. Geman, D. Geman, “Stochastic relaxation, Gibbs distribution, and Bayesian restoration of images,” IEEE Trans. Patt. Anal. Mach. Intell. PAMI-6, 721–741 (1984).
[CrossRef]

IEEE. Trans. Med. Imaging (1)

T. Hebert, R. Leahy, “A generalized EM algorithm for 3-D Bayesian reconstruction from Poisson data using Gibbs priors,” IEEE. Trans. Med. Imaging 8, 194–202 (1989).
[CrossRef] [PubMed]

J. Nucl. Meth. All. Sci. (1)

R. Guzzardi, M. Mey, S. Solfanelli, C. Giuntini, “The 90° Compton scattering tomography of the lung” J. Nucl. Meth. All. Sci. 22, 11, 1978.

J. R. Statist. Soc. B (1)

J. Besag, “On the statistical analysis of dirty pictures,” J. R. Statist. Soc. B 48, 259–302 (1986).

Phys. Med. Biol. (4)

P. G. Lale, “The examination of internal tissues, using gamma ray scatter with a possible extension to megavoltage radiography,” Phys. Med. Biol. 4, 159–166 (1959).
[CrossRef] [PubMed]

J. J. Battista, M. J. Bronskill, “Compton scatter imaging of transverse section: corrections for multiple scattering and attenuation,” Phys. Med. Biol. 22, 229–244 (1977).
[CrossRef] [PubMed]

F. T. Farmer, M. P. Collins, “A new approach to the determination of anatomical cross sections of the body by Compton scattering of gamma rays,” Phys. Med. Biol. 16, 577 (1970).
[CrossRef]

F. T. Farmer, M. P. Collins, “A further appraisal of the Compton scattering method for determining anatomical cross-sections of the body,” Phys. Med. Biol. 19, 808 (1974).
[CrossRef] [PubMed]

Phys. Rep. (1)

R. Cesareo, A. L. Hanson, G. E. Gigante, L. J. Pedraza, S. Q. G. Mahtaboally, “Interaction of Key Photons with matter and new applications,” Phys. Rep. 213, 117–178 (1992).
[CrossRef]

Other (1)

J. R. Larmarsh, Introduction to Nuclear Engineering (Addison-Wesley, Reading, Mass., 1983).

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Figures (6)

Fig. 1
Fig. 1

Experimental configuration of Compton-scattering imaging.

Fig. 2
Fig. 2

Structure of the coupled-gradient neural network: (a) input and output structure, (b) coupled structure.

Fig. 3
Fig. 3

Computer-simulated phantom.

Fig. 4
Fig. 4

Reconstructed result under the condition λ=0.005, NRMS=0.0705.

Fig. 5
Fig. 5

Reconstructed result under the condition λ=0.1, NRMS=0.6643.

Fig. 6
Fig. 6

Reconstructed results without priors: λ=0, NRMS=0.9319.

Equations (19)

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ψ(Ei)=j=1N2aijρj=Φ2πj=1N2δijf1(E0)f2(Ei)dσdΩ(E0, Ei)ρjΔVj/Rj2,
f1=exp-jρjσc(E0)lj,
f2=exp-jρjσc(Ei)lj.
ψ=A(ρ)ρ,
ln P(ρ|ψ)=ln P(ψ|ρ)+ln P(ρ)-ln P(ψ).
P(ψ|ρ)=i=1M(ψ¯i)ψi exp(-ψ¯i)ψi!,
P(ρ)=1Zexp[-EP(ρ, l)],
EP(ρ, l)=λj[ρv2(j)(1-ljh)+αljh]+λj[ρh2(j)(1-ljv)+αljv]
ρ=arg min(ρ0, l){E(ρ, l)=ED(ρ)+EP(ρ, l)},
ED(ρ)=-ln P(ψ|ρ)=i=1M[ψ¯i-ψi ln(ψ¯i)]+termsindependentofρ.
Ω=i=1N2[0, ρM]×i=1N2{0, 1}×i=1N2{0, 1}.
ρj=fjρ(hjρ)=ρM1+exp(-hjρ),
ljx=fjl(hjx)=11+exp(-hjx),x=horv.
(ρ, l)=E(ρ, l)+βj=1N2ljh(1-ljh)+j=1N2ljv(1-ljv),
hjρ=-ηρ(ρ, l)ρj,ρj=fjρ(hjρ),
hjx=-ηl(ρ, l)ljx,ljx=fjl(hjx),x=horv,
ddt=j=1N2ρjρj+j=1N2ljhljh+j=1N2ljvljv=-j=1N2ηρ-1(hjρ)2[fjρ(hjρ)]-j=1N2ηl-1(hjh)2[fjl(hjh)]-j=1N2ηl-1(hjv)2[fjl(hjv)]0.
NRMS=j=1N2(ρj-ρˆj)2j=1N2(ρj-ρ¯)21/2,
ρ¯=1N2j=1N2ρˆj,

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