Abstract

Positron emission tomography (PET) is one of the key molecular imaging modalities in medicine and biology. Penalized iterative image reconstruction algorithms frequently used in PET are based on maximum-likelihood (ML) and maximum a posterior (MAP) estimation techniques. The ML algorithm produces noisy artifacts whereas the MAP algorithm eliminates noisy artifacts by utilizing available prior information in the reconstruction process. The MAP-based algorithms fail to determine the density class in the reconstructed image and hence penalize the pixels irrespective of the density class and irrespective of the strength of interaction between the nearest neighbors. A Hebbian neural learning scheme is proposed to model the nature of interpixel interaction to reconstruct artifact-free edge preserving reconstruction. A key motivation of the proposed approach is to avoid oversmoothing across edges that is often the case with MAP algorithms. It is assumed that local correlation plays a significant role in PET image reconstruction, and proper modeling of correlation weight (which defines the strength of interpixel interaction) is essential to generate artifact-free reconstruction. The Hebbian learning-based approach modifies the interaction weight by adding a small correction that is proportional to the product of the input signal (neighborhood pixels) and output signal. Quantitative analysis shows that the Hebbian learning-based adaptive weight adjustment approach is capable of producing better reconstructed images compared with those reconstructed by conventional ML and MAP-based algorithms in PET image reconstruction.

© 2005 Optical Society of America

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References

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  1. Y. VardiL, A. Shepp, L. Kaufmann, “A statistical model for positron emission tomography,” J. Am. Stat. Assoc. 80, 8–37 (1985).
    [CrossRef]
  2. L. A. Shepp, Y. Vardi, “Maximum likelihood estimation for emission tomography,” IEEE Trans. Med. ImagingMI-1, 113–121 (1982).
    [CrossRef]
  3. T. Hebert, R. Leahy, “A generalized EM algorithm for 3-D Bayesian reconstruction from Poisson data using Gibbs priors,” IEEE Trans. Med. ImagingMI-8, 194–202 (1989).
    [CrossRef]
  4. E. Levitan, G. T. Herman, “A maximum a posteriori probability expectation maximization algorithm for image reconstruction in emission tomography,” IEEE Trans. Med. Imaging MI-6, 185–192 (1987).
    [CrossRef]
  5. P. J. Green, “Bayesian reconstruction from emission tomography data using a modified EM algorithm,” IEEE Trans. Med. Imaging 9, 84–93 (1990).
    [CrossRef]
  6. Z. Zhou, R. M. Leahy, J. Qi, “Approximate maximum likelihood hyperparameter estimation for Gibbs prior,” IEEE Trans. Image Process. 6, 844–861 (1997).
    [CrossRef]
  7. T. Hebert, R. Leahy, “Statistic based MAP image reconstruction from Poisson data using Gibbs proirs,” IEEE Trans. Signal Process. 40, 2290–2303 (1992).
    [CrossRef]
  8. J. Nuyts, D. Bequ, P. Dupont, L. Mortelmans, “A concave prior penalizing relative differences for maximum-a-posteriori reconstruction in emission tomography,” IEEE Trans. Nucl. Sci. 49, 56–60 (2002).
    [CrossRef]
  9. H. M. Hudson, R. S. Larkin, “Accelerated image reconstruction using ordered subsets of projection data,” IEEE Trans. Med. Imaging 13, 601–609 (1994).
    [CrossRef] [PubMed]
  10. L. A. Shepp, B. F. Logan, “The Fourier reconstruction of a head section,” IEEE Trans. Nucl. Sci. NS-21, 21–43 (1974).
    [CrossRef]
  11. J. A. Fessler, “Penalized weighted least-squares image reconstruction for positron emission tomography,” IEEE Trans. Med. Imaging 13, 290–300 (1994).
    [CrossRef] [PubMed]
  12. J. A. Fessler, “Mean and variance of implicitly defined biased estimators such as penalized maximum likelihood: applications to tomography,” IEEE Trans. Image Process. 5, 493–506 (1996).
    [CrossRef]
  13. C. M. Chen, S. Y. Lee, “Parallelization of the EM algorithm for 3-D PET image reconstruction,” IEEE Trans. Med. Imaging 10, 513–522 (1991).
    [CrossRef] [PubMed]
  14. K. Rajan, L. M. Patnaik, J. Ramakrishna, “High speed computation of the EM algorithm for PET image reconstruction,” IEEE Trans. Nucl. Sci. 41, 0–5 (1994).
    [CrossRef]
  15. J. A. Fessler, A. O. Hero, “Penalized maximum likelihood image reconstructionusing space-alternating generalized EM algorithms,” IEEE Trans. Image Process. 4, 1417–1429 (1995).
    [CrossRef]
  16. E. U. Mumcuoglu, R. Leahy, S. R. Cherry, Z. Zhou, “Fast gradient based methods for Bayesian reconstruction of transmission and emission PET images,” IEEE Trans. Med. Imaging 13, 687–701 (1994).
    [CrossRef]
  17. S. Alenius, U. Ruotsalainen, “Using local median as the location of prior distribution in iterative emission tomography reconstruction,” IEEE Trans. Nucl. Sci. 45, 3097–3104 (1998).
    [CrossRef]
  18. S. Alenius, U. Ruotsalainen, “Generalization of median root prior reconstruction,” IEEE Trans. Med. Imaging 21, 1413–1420 (2002).
    [CrossRef]
  19. P. P. Mondal, K. Rajan, “Image reconstruction by conditional entropy maximisation for PET system,” IEE Proc. Vision Image Signal Process. 151, 345–352 (2004).
    [CrossRef]
  20. D. Hebb, Organization of Behavior (Wiley, 1949).
  21. E. I. Papageorgiou, C. D. Stylios, P. P. Groumpos, “Active Hebbian learning algorithm to train fuzzy cognitive maps,” Int. J. Approx Reasoning 37, 219–249 (2004).
    [CrossRef]
  22. J. Besag, “Spatial interaction and the statistical analysis of lattice systems,” J. R. Stat. Soc. B 36, 192–236 (1974).
  23. P. P. Mondal, “Hebbian learning based image reconstruction for positron emission tomography,” in IEEE Instrumentation and measurement Technology Conference (IEEE Press, 2005).
    [CrossRef]
  24. N. Rajeevan, K. Rajgopal, G. Krishna, “Vector-extrapolated fast maximum likelihood estimation algorithms for emission tomography,” IEEE Trans. Med. Imaging 11, 9–20 (1992).
    [CrossRef] [PubMed]
  25. L. Kaufmann, “Implementing and accelerating the EM-algorithm for positron emission tomography,” IEEE Trans. Med. Imaging MI-6, 37–51 (1987).
    [CrossRef]
  26. S. J. Lee, “Accelerated deterministic annealing algorithms for transmission CT reconstruction using ordered subsets,” IEEE Trans. Nucl. Sci. 49, 2373–2380 (2002).
    [CrossRef]
  27. E. Veclerov, J. Llacer, “Stopping rule for MLE algorithm based on statistical hypothesis testing,” IEEE Trans. Med. Imaging MI-6, 313–319 (1987).
    [CrossRef]
  28. D. L. Snyder, M. I. Miller, “The use of sieves to stabilize images produced with the EM-algorithm for emission tomography,” IEEE Trans. Nucl. Sci. NS-32, 3864–3872 (1985).
    [CrossRef]

2004 (2)

P. P. Mondal, K. Rajan, “Image reconstruction by conditional entropy maximisation for PET system,” IEE Proc. Vision Image Signal Process. 151, 345–352 (2004).
[CrossRef]

E. I. Papageorgiou, C. D. Stylios, P. P. Groumpos, “Active Hebbian learning algorithm to train fuzzy cognitive maps,” Int. J. Approx Reasoning 37, 219–249 (2004).
[CrossRef]

2002 (3)

J. Nuyts, D. Bequ, P. Dupont, L. Mortelmans, “A concave prior penalizing relative differences for maximum-a-posteriori reconstruction in emission tomography,” IEEE Trans. Nucl. Sci. 49, 56–60 (2002).
[CrossRef]

S. J. Lee, “Accelerated deterministic annealing algorithms for transmission CT reconstruction using ordered subsets,” IEEE Trans. Nucl. Sci. 49, 2373–2380 (2002).
[CrossRef]

S. Alenius, U. Ruotsalainen, “Generalization of median root prior reconstruction,” IEEE Trans. Med. Imaging 21, 1413–1420 (2002).
[CrossRef]

1998 (1)

S. Alenius, U. Ruotsalainen, “Using local median as the location of prior distribution in iterative emission tomography reconstruction,” IEEE Trans. Nucl. Sci. 45, 3097–3104 (1998).
[CrossRef]

1997 (1)

Z. Zhou, R. M. Leahy, J. Qi, “Approximate maximum likelihood hyperparameter estimation for Gibbs prior,” IEEE Trans. Image Process. 6, 844–861 (1997).
[CrossRef]

1996 (1)

J. A. Fessler, “Mean and variance of implicitly defined biased estimators such as penalized maximum likelihood: applications to tomography,” IEEE Trans. Image Process. 5, 493–506 (1996).
[CrossRef]

1995 (1)

J. A. Fessler, A. O. Hero, “Penalized maximum likelihood image reconstructionusing space-alternating generalized EM algorithms,” IEEE Trans. Image Process. 4, 1417–1429 (1995).
[CrossRef]

1994 (4)

E. U. Mumcuoglu, R. Leahy, S. R. Cherry, Z. Zhou, “Fast gradient based methods for Bayesian reconstruction of transmission and emission PET images,” IEEE Trans. Med. Imaging 13, 687–701 (1994).
[CrossRef]

K. Rajan, L. M. Patnaik, J. Ramakrishna, “High speed computation of the EM algorithm for PET image reconstruction,” IEEE Trans. Nucl. Sci. 41, 0–5 (1994).
[CrossRef]

H. M. Hudson, R. S. Larkin, “Accelerated image reconstruction using ordered subsets of projection data,” IEEE Trans. Med. Imaging 13, 601–609 (1994).
[CrossRef] [PubMed]

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

1992 (2)

T. Hebert, R. Leahy, “Statistic based MAP image reconstruction from Poisson data using Gibbs proirs,” IEEE Trans. Signal Process. 40, 2290–2303 (1992).
[CrossRef]

N. Rajeevan, K. Rajgopal, G. Krishna, “Vector-extrapolated fast maximum likelihood estimation algorithms for emission tomography,” IEEE Trans. Med. Imaging 11, 9–20 (1992).
[CrossRef] [PubMed]

1991 (1)

C. M. Chen, S. Y. Lee, “Parallelization of the EM algorithm for 3-D PET image reconstruction,” IEEE Trans. Med. Imaging 10, 513–522 (1991).
[CrossRef] [PubMed]

1990 (1)

P. J. Green, “Bayesian reconstruction from emission tomography data using a modified EM algorithm,” IEEE Trans. Med. Imaging 9, 84–93 (1990).
[CrossRef]

1987 (3)

E. Levitan, G. T. Herman, “A maximum a posteriori probability expectation maximization algorithm for image reconstruction in emission tomography,” IEEE Trans. Med. Imaging MI-6, 185–192 (1987).
[CrossRef]

L. Kaufmann, “Implementing and accelerating the EM-algorithm for positron emission tomography,” IEEE Trans. Med. Imaging MI-6, 37–51 (1987).
[CrossRef]

E. Veclerov, J. Llacer, “Stopping rule for MLE algorithm based on statistical hypothesis testing,” IEEE Trans. Med. Imaging MI-6, 313–319 (1987).
[CrossRef]

1985 (2)

D. L. Snyder, M. I. Miller, “The use of sieves to stabilize images produced with the EM-algorithm for emission tomography,” IEEE Trans. Nucl. Sci. NS-32, 3864–3872 (1985).
[CrossRef]

Y. VardiL, A. Shepp, L. Kaufmann, “A statistical model for positron emission tomography,” J. Am. Stat. Assoc. 80, 8–37 (1985).
[CrossRef]

1974 (2)

L. A. Shepp, B. F. Logan, “The Fourier reconstruction of a head section,” IEEE Trans. Nucl. Sci. NS-21, 21–43 (1974).
[CrossRef]

J. Besag, “Spatial interaction and the statistical analysis of lattice systems,” J. R. Stat. Soc. B 36, 192–236 (1974).

Alenius, S.

S. Alenius, U. Ruotsalainen, “Generalization of median root prior reconstruction,” IEEE Trans. Med. Imaging 21, 1413–1420 (2002).
[CrossRef]

S. Alenius, U. Ruotsalainen, “Using local median as the location of prior distribution in iterative emission tomography reconstruction,” IEEE Trans. Nucl. Sci. 45, 3097–3104 (1998).
[CrossRef]

Bequ, D.

J. Nuyts, D. Bequ, P. Dupont, L. Mortelmans, “A concave prior penalizing relative differences for maximum-a-posteriori reconstruction in emission tomography,” IEEE Trans. Nucl. Sci. 49, 56–60 (2002).
[CrossRef]

Besag, J.

J. Besag, “Spatial interaction and the statistical analysis of lattice systems,” J. R. Stat. Soc. B 36, 192–236 (1974).

Chen, C. M.

C. M. Chen, S. Y. Lee, “Parallelization of the EM algorithm for 3-D PET image reconstruction,” IEEE Trans. Med. Imaging 10, 513–522 (1991).
[CrossRef] [PubMed]

Cherry, S. R.

E. U. Mumcuoglu, R. Leahy, S. R. Cherry, Z. Zhou, “Fast gradient based methods for Bayesian reconstruction of transmission and emission PET images,” IEEE Trans. Med. Imaging 13, 687–701 (1994).
[CrossRef]

Dupont, P.

J. Nuyts, D. Bequ, P. Dupont, L. Mortelmans, “A concave prior penalizing relative differences for maximum-a-posteriori reconstruction in emission tomography,” IEEE Trans. Nucl. Sci. 49, 56–60 (2002).
[CrossRef]

Fessler, J. A.

J. A. Fessler, “Mean and variance of implicitly defined biased estimators such as penalized maximum likelihood: applications to tomography,” IEEE Trans. Image Process. 5, 493–506 (1996).
[CrossRef]

J. A. Fessler, A. O. Hero, “Penalized maximum likelihood image reconstructionusing space-alternating generalized EM algorithms,” IEEE Trans. Image Process. 4, 1417–1429 (1995).
[CrossRef]

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

Green, P. J.

P. J. Green, “Bayesian reconstruction from emission tomography data using a modified EM algorithm,” IEEE Trans. Med. Imaging 9, 84–93 (1990).
[CrossRef]

Groumpos, P. P.

E. I. Papageorgiou, C. D. Stylios, P. P. Groumpos, “Active Hebbian learning algorithm to train fuzzy cognitive maps,” Int. J. Approx Reasoning 37, 219–249 (2004).
[CrossRef]

Hebb, D.

D. Hebb, Organization of Behavior (Wiley, 1949).

Hebert, T.

T. Hebert, R. Leahy, “Statistic based MAP image reconstruction from Poisson data using Gibbs proirs,” IEEE Trans. Signal Process. 40, 2290–2303 (1992).
[CrossRef]

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

Herman, G. T.

E. Levitan, G. T. Herman, “A maximum a posteriori probability expectation maximization algorithm for image reconstruction in emission tomography,” IEEE Trans. Med. Imaging MI-6, 185–192 (1987).
[CrossRef]

Hero, A. O.

J. A. Fessler, A. O. Hero, “Penalized maximum likelihood image reconstructionusing space-alternating generalized EM algorithms,” IEEE Trans. Image Process. 4, 1417–1429 (1995).
[CrossRef]

Hudson, H. M.

H. M. Hudson, R. S. Larkin, “Accelerated image reconstruction using ordered subsets of projection data,” IEEE Trans. Med. Imaging 13, 601–609 (1994).
[CrossRef] [PubMed]

Kaufmann, L.

L. Kaufmann, “Implementing and accelerating the EM-algorithm for positron emission tomography,” IEEE Trans. Med. Imaging MI-6, 37–51 (1987).
[CrossRef]

Y. VardiL, A. Shepp, L. Kaufmann, “A statistical model for positron emission tomography,” J. Am. Stat. Assoc. 80, 8–37 (1985).
[CrossRef]

Krishna, G.

N. Rajeevan, K. Rajgopal, G. Krishna, “Vector-extrapolated fast maximum likelihood estimation algorithms for emission tomography,” IEEE Trans. Med. Imaging 11, 9–20 (1992).
[CrossRef] [PubMed]

Larkin, R. S.

H. M. Hudson, R. S. Larkin, “Accelerated image reconstruction using ordered subsets of projection data,” IEEE Trans. Med. Imaging 13, 601–609 (1994).
[CrossRef] [PubMed]

Leahy, R.

E. U. Mumcuoglu, R. Leahy, S. R. Cherry, Z. Zhou, “Fast gradient based methods for Bayesian reconstruction of transmission and emission PET images,” IEEE Trans. Med. Imaging 13, 687–701 (1994).
[CrossRef]

T. Hebert, R. Leahy, “Statistic based MAP image reconstruction from Poisson data using Gibbs proirs,” IEEE Trans. Signal Process. 40, 2290–2303 (1992).
[CrossRef]

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

Leahy, R. M.

Z. Zhou, R. M. Leahy, J. Qi, “Approximate maximum likelihood hyperparameter estimation for Gibbs prior,” IEEE Trans. Image Process. 6, 844–861 (1997).
[CrossRef]

Lee, S. J.

S. J. Lee, “Accelerated deterministic annealing algorithms for transmission CT reconstruction using ordered subsets,” IEEE Trans. Nucl. Sci. 49, 2373–2380 (2002).
[CrossRef]

Lee, S. Y.

C. M. Chen, S. Y. Lee, “Parallelization of the EM algorithm for 3-D PET image reconstruction,” IEEE Trans. Med. Imaging 10, 513–522 (1991).
[CrossRef] [PubMed]

Levitan, E.

E. Levitan, G. T. Herman, “A maximum a posteriori probability expectation maximization algorithm for image reconstruction in emission tomography,” IEEE Trans. Med. Imaging MI-6, 185–192 (1987).
[CrossRef]

Llacer, J.

E. Veclerov, J. Llacer, “Stopping rule for MLE algorithm based on statistical hypothesis testing,” IEEE Trans. Med. Imaging MI-6, 313–319 (1987).
[CrossRef]

Logan, B. F.

L. A. Shepp, B. F. Logan, “The Fourier reconstruction of a head section,” IEEE Trans. Nucl. Sci. NS-21, 21–43 (1974).
[CrossRef]

Miller, M. I.

D. L. Snyder, M. I. Miller, “The use of sieves to stabilize images produced with the EM-algorithm for emission tomography,” IEEE Trans. Nucl. Sci. NS-32, 3864–3872 (1985).
[CrossRef]

Mondal, P. P.

P. P. Mondal, K. Rajan, “Image reconstruction by conditional entropy maximisation for PET system,” IEE Proc. Vision Image Signal Process. 151, 345–352 (2004).
[CrossRef]

P. P. Mondal, “Hebbian learning based image reconstruction for positron emission tomography,” in IEEE Instrumentation and measurement Technology Conference (IEEE Press, 2005).
[CrossRef]

Mortelmans, L.

J. Nuyts, D. Bequ, P. Dupont, L. Mortelmans, “A concave prior penalizing relative differences for maximum-a-posteriori reconstruction in emission tomography,” IEEE Trans. Nucl. Sci. 49, 56–60 (2002).
[CrossRef]

Mumcuoglu, E. U.

E. U. Mumcuoglu, R. Leahy, S. R. Cherry, Z. Zhou, “Fast gradient based methods for Bayesian reconstruction of transmission and emission PET images,” IEEE Trans. Med. Imaging 13, 687–701 (1994).
[CrossRef]

Nuyts, J.

J. Nuyts, D. Bequ, P. Dupont, L. Mortelmans, “A concave prior penalizing relative differences for maximum-a-posteriori reconstruction in emission tomography,” IEEE Trans. Nucl. Sci. 49, 56–60 (2002).
[CrossRef]

Papageorgiou, E. I.

E. I. Papageorgiou, C. D. Stylios, P. P. Groumpos, “Active Hebbian learning algorithm to train fuzzy cognitive maps,” Int. J. Approx Reasoning 37, 219–249 (2004).
[CrossRef]

Patnaik, L. M.

K. Rajan, L. M. Patnaik, J. Ramakrishna, “High speed computation of the EM algorithm for PET image reconstruction,” IEEE Trans. Nucl. Sci. 41, 0–5 (1994).
[CrossRef]

Qi, J.

Z. Zhou, R. M. Leahy, J. Qi, “Approximate maximum likelihood hyperparameter estimation for Gibbs prior,” IEEE Trans. Image Process. 6, 844–861 (1997).
[CrossRef]

Rajan, K.

P. P. Mondal, K. Rajan, “Image reconstruction by conditional entropy maximisation for PET system,” IEE Proc. Vision Image Signal Process. 151, 345–352 (2004).
[CrossRef]

K. Rajan, L. M. Patnaik, J. Ramakrishna, “High speed computation of the EM algorithm for PET image reconstruction,” IEEE Trans. Nucl. Sci. 41, 0–5 (1994).
[CrossRef]

Rajeevan, N.

N. Rajeevan, K. Rajgopal, G. Krishna, “Vector-extrapolated fast maximum likelihood estimation algorithms for emission tomography,” IEEE Trans. Med. Imaging 11, 9–20 (1992).
[CrossRef] [PubMed]

Rajgopal, K.

N. Rajeevan, K. Rajgopal, G. Krishna, “Vector-extrapolated fast maximum likelihood estimation algorithms for emission tomography,” IEEE Trans. Med. Imaging 11, 9–20 (1992).
[CrossRef] [PubMed]

Ramakrishna, J.

K. Rajan, L. M. Patnaik, J. Ramakrishna, “High speed computation of the EM algorithm for PET image reconstruction,” IEEE Trans. Nucl. Sci. 41, 0–5 (1994).
[CrossRef]

Ruotsalainen, U.

S. Alenius, U. Ruotsalainen, “Generalization of median root prior reconstruction,” IEEE Trans. Med. Imaging 21, 1413–1420 (2002).
[CrossRef]

S. Alenius, U. Ruotsalainen, “Using local median as the location of prior distribution in iterative emission tomography reconstruction,” IEEE Trans. Nucl. Sci. 45, 3097–3104 (1998).
[CrossRef]

Shepp, A.

Y. VardiL, A. Shepp, L. Kaufmann, “A statistical model for positron emission tomography,” J. Am. Stat. Assoc. 80, 8–37 (1985).
[CrossRef]

Shepp, L. A.

L. A. Shepp, B. F. Logan, “The Fourier reconstruction of a head section,” IEEE Trans. Nucl. Sci. NS-21, 21–43 (1974).
[CrossRef]

L. A. Shepp, Y. Vardi, “Maximum likelihood estimation for emission tomography,” IEEE Trans. Med. ImagingMI-1, 113–121 (1982).
[CrossRef]

Snyder, D. L.

D. L. Snyder, M. I. Miller, “The use of sieves to stabilize images produced with the EM-algorithm for emission tomography,” IEEE Trans. Nucl. Sci. NS-32, 3864–3872 (1985).
[CrossRef]

Stylios, C. D.

E. I. Papageorgiou, C. D. Stylios, P. P. Groumpos, “Active Hebbian learning algorithm to train fuzzy cognitive maps,” Int. J. Approx Reasoning 37, 219–249 (2004).
[CrossRef]

Vardi, Y.

L. A. Shepp, Y. Vardi, “Maximum likelihood estimation for emission tomography,” IEEE Trans. Med. ImagingMI-1, 113–121 (1982).
[CrossRef]

VardiL, Y.

Y. VardiL, A. Shepp, L. Kaufmann, “A statistical model for positron emission tomography,” J. Am. Stat. Assoc. 80, 8–37 (1985).
[CrossRef]

Veclerov, E.

E. Veclerov, J. Llacer, “Stopping rule for MLE algorithm based on statistical hypothesis testing,” IEEE Trans. Med. Imaging MI-6, 313–319 (1987).
[CrossRef]

Zhou, Z.

Z. Zhou, R. M. Leahy, J. Qi, “Approximate maximum likelihood hyperparameter estimation for Gibbs prior,” IEEE Trans. Image Process. 6, 844–861 (1997).
[CrossRef]

E. U. Mumcuoglu, R. Leahy, S. R. Cherry, Z. Zhou, “Fast gradient based methods for Bayesian reconstruction of transmission and emission PET images,” IEEE Trans. Med. Imaging 13, 687–701 (1994).
[CrossRef]

IEE Proc. Vision Image Signal Process. (1)

P. P. Mondal, K. Rajan, “Image reconstruction by conditional entropy maximisation for PET system,” IEE Proc. Vision Image Signal Process. 151, 345–352 (2004).
[CrossRef]

IEEE Trans. Image Process. (3)

J. A. Fessler, A. O. Hero, “Penalized maximum likelihood image reconstructionusing space-alternating generalized EM algorithms,” IEEE Trans. Image Process. 4, 1417–1429 (1995).
[CrossRef]

J. A. Fessler, “Mean and variance of implicitly defined biased estimators such as penalized maximum likelihood: applications to tomography,” IEEE Trans. Image Process. 5, 493–506 (1996).
[CrossRef]

Z. Zhou, R. M. Leahy, J. Qi, “Approximate maximum likelihood hyperparameter estimation for Gibbs prior,” IEEE Trans. Image Process. 6, 844–861 (1997).
[CrossRef]

IEEE Trans. Med. Imaging (10)

H. M. Hudson, R. S. Larkin, “Accelerated image reconstruction using ordered subsets of projection data,” IEEE Trans. Med. Imaging 13, 601–609 (1994).
[CrossRef] [PubMed]

E. Levitan, G. T. Herman, “A maximum a posteriori probability expectation maximization algorithm for image reconstruction in emission tomography,” IEEE Trans. Med. Imaging MI-6, 185–192 (1987).
[CrossRef]

P. J. Green, “Bayesian reconstruction from emission tomography data using a modified EM algorithm,” IEEE Trans. Med. Imaging 9, 84–93 (1990).
[CrossRef]

C. M. Chen, S. Y. Lee, “Parallelization of the EM algorithm for 3-D PET image reconstruction,” IEEE Trans. Med. Imaging 10, 513–522 (1991).
[CrossRef] [PubMed]

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

E. U. Mumcuoglu, R. Leahy, S. R. Cherry, Z. Zhou, “Fast gradient based methods for Bayesian reconstruction of transmission and emission PET images,” IEEE Trans. Med. Imaging 13, 687–701 (1994).
[CrossRef]

N. Rajeevan, K. Rajgopal, G. Krishna, “Vector-extrapolated fast maximum likelihood estimation algorithms for emission tomography,” IEEE Trans. Med. Imaging 11, 9–20 (1992).
[CrossRef] [PubMed]

L. Kaufmann, “Implementing and accelerating the EM-algorithm for positron emission tomography,” IEEE Trans. Med. Imaging MI-6, 37–51 (1987).
[CrossRef]

E. Veclerov, J. Llacer, “Stopping rule for MLE algorithm based on statistical hypothesis testing,” IEEE Trans. Med. Imaging MI-6, 313–319 (1987).
[CrossRef]

S. Alenius, U. Ruotsalainen, “Generalization of median root prior reconstruction,” IEEE Trans. Med. Imaging 21, 1413–1420 (2002).
[CrossRef]

IEEE Trans. Nucl. Sci. (6)

D. L. Snyder, M. I. Miller, “The use of sieves to stabilize images produced with the EM-algorithm for emission tomography,” IEEE Trans. Nucl. Sci. NS-32, 3864–3872 (1985).
[CrossRef]

S. J. Lee, “Accelerated deterministic annealing algorithms for transmission CT reconstruction using ordered subsets,” IEEE Trans. Nucl. Sci. 49, 2373–2380 (2002).
[CrossRef]

S. Alenius, U. Ruotsalainen, “Using local median as the location of prior distribution in iterative emission tomography reconstruction,” IEEE Trans. Nucl. Sci. 45, 3097–3104 (1998).
[CrossRef]

J. Nuyts, D. Bequ, P. Dupont, L. Mortelmans, “A concave prior penalizing relative differences for maximum-a-posteriori reconstruction in emission tomography,” IEEE Trans. Nucl. Sci. 49, 56–60 (2002).
[CrossRef]

K. Rajan, L. M. Patnaik, J. Ramakrishna, “High speed computation of the EM algorithm for PET image reconstruction,” IEEE Trans. Nucl. Sci. 41, 0–5 (1994).
[CrossRef]

L. A. Shepp, B. F. Logan, “The Fourier reconstruction of a head section,” IEEE Trans. Nucl. Sci. NS-21, 21–43 (1974).
[CrossRef]

IEEE Trans. Signal Process. (1)

T. Hebert, R. Leahy, “Statistic based MAP image reconstruction from Poisson data using Gibbs proirs,” IEEE Trans. Signal Process. 40, 2290–2303 (1992).
[CrossRef]

Int. J. Approx Reasoning (1)

E. I. Papageorgiou, C. D. Stylios, P. P. Groumpos, “Active Hebbian learning algorithm to train fuzzy cognitive maps,” Int. J. Approx Reasoning 37, 219–249 (2004).
[CrossRef]

J. Am. Stat. Assoc. (1)

Y. VardiL, A. Shepp, L. Kaufmann, “A statistical model for positron emission tomography,” J. Am. Stat. Assoc. 80, 8–37 (1985).
[CrossRef]

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

J. Besag, “Spatial interaction and the statistical analysis of lattice systems,” J. R. Stat. Soc. B 36, 192–236 (1974).

Other (4)

P. P. Mondal, “Hebbian learning based image reconstruction for positron emission tomography,” in IEEE Instrumentation and measurement Technology Conference (IEEE Press, 2005).
[CrossRef]

L. A. Shepp, Y. Vardi, “Maximum likelihood estimation for emission tomography,” IEEE Trans. Med. ImagingMI-1, 113–121 (1982).
[CrossRef]

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

D. Hebb, Organization of Behavior (Wiley, 1949).

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Figures (9)

Fig. 1
Fig. 1

Linear neural network unit with eight neighborhood synaptic connections.

Fig. 2
Fig. 2

(a) Original test phantom. (b)–(f) are the reconstructed images after 50 iterations of the proposed algorithm for μ = 0.1, 0.3, 0.5, 0.7, and 1.0, respectively.

Fig. 3
Fig. 3

(a) and (f) are the OSEM reconstructed images with eight subsets after 5 and 15 iterations. (b)–(e) and (g)–(j) are the reconstructed images after 16 and 50 iterations by ML, MAP, MRP, and the proposed algorithm, respectively.

Fig. 4
Fig. 4

Percent error versus the iteration plot for ML, MAP, MRP, and the proposed algorithm.

Fig. 5
Fig. 5

NMSE versus the iteration plot for ML, MAP, MRP, and the proposed algorithm.

Fig. 6
Fig. 6

(a) and (b) Line plots along the line segment ab through two different regions in the original image and the reconstructed images by ML, MAP, and the proposed algorithm.

Fig. 7
Fig. 7

(a) Original test image; (b)–(e) are the real MRI reconstructed images after 50 iterations by ML, MAP, MRP, and the proposed algorithm.

Fig. 8
Fig. 8

(a)–(e) and (f)–(j) are the reconstructed images after 50 iterations by ML, MAP, MRP, and the proposed algorithm with Gaussian noise G(0, 0.01) and G(0, 0.005), respectively.

Fig. 9
Fig. 9

Log-likelihood test for noisy and noise-free projections.

Equations (14)

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λ MAP = max λ 0 [ log P ( y / λ ) + log P ( λ ) ] .
P ( λ ) = 1 Z exp [ - 1 β i j N i w i , j V ( λ i , λ j ) ] ,
w i j = { 1 for orthogonal nearest neighbors 1 2 , for diagonal nearest neighbors 0 , otherwise .
λ i k + 1 = λ i k [ j = 1 M p i j + 1 β j N i w i j ( V ( λ i , λ j ) λ i ) λ i = λ i k ] × j = 1 M y j p i j o = 1 N λ o k p o j .
λ i k + 1 = λ i k [ j = 1 M p i j + ( λ i k - M b ) β M b ] j = 1 M y j p i j o = 1 N λ o k p o j .
λ i k + 1 = λ i k [ j = 1 M p i j + B i k ] j = 1 M y j p i j o = 1 N λ o k p o j . B i k = 1 β j N i γ i j k ( V ( ) λ i ) λ i = λ i k .
B i k = 2 β j N i γ i j k ( λ i k - λ j k ) = 2 β [ λ i k j N i γ i j k - j N i γ i j k λ j k ] .
B i k = 2 β [ λ i k - j N i γ i j k λ j k ] .
η i k = 1 8 j N i γ i j k λ j k .
γ i , j k + 1 = γ i , j k + μ ( λ j k η i k max j N i ( λ j k η i k ) ) ,             j N i ,
λ i k + 1 = λ i k [ j = 1 M p i j + 2 β ( λ i k - j N i γ i j k λ j k ) ] × j = 1 M y j p i j o = 1 N λ o k p o j ,
ξ k = λ k - λ ˜ λ k × 100 % ,
ζ k = i = 1 N ( λ i k - λ ˜ i ) 2 i = 1 N λ ˜ i 2 ,
l ( λ k ) = j = 1 M [ - ϕ j k + y j log ϕ j k - log ( y j ! ) ] .

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