VARIATIONAL POSTERIOR DISTRIBUTION APPROXIMATION IN BAYESIAN EMISSION TOMOGRAPHY RECONSTRUCTION USING A GAMMA MIXTURE PRIOR

Rafael Molina, Antonio López, José Manuel Martín, Aggelos K. Katsaggelos

Abstract

Following the Bayesian framework we propose a method to reconstruct emission tomography images which uses gamma mixture prior and variational methods to approximate the posterior distribution of the unknown parameters and image instead of estimating them by using the Evidence Analysis or alternating between the estimation of parameters and image (Iterated Conditional Mode (ICM)) approach. By analyzing the posterior distribution approximation we can examine the quality of the proposed estimates. The method is tested on real Single Positron Emission Tomography (SPECT) images.

References

  1. Andrieu, C., de Freitras, N., Doucet, A., and Jordan, M. (2003). An introduction to MCMC for machine learning. Machine Learning, 50:5-43.
  2. Beal, M. (2003). Variational algorithms for approximate Bayesian inference. PhD thesis, The Gatsby Computational Neuroscience Unit, University College London.
  3. Bishop, C. and Tipping, M. (2000). Variational relevance vector machine. In Proceedings of the 16th Conference on Uncertainty in Articial Intelligence, pages 46-53. Morgan Kaufmann Publishers.
  4. Hsiao, I.-T., Rangarajan, A., and Gini, G. (1998). Joint-map reconstruction/segmentation for transmission tomography using mixture-models as priors. In Proceedings of EEE Nuclear Science Symposium and Medical Imaging Conference, volume II, pages 1689-1693.
  5. Galatsanos, N. P., Mesarovic, V. Z., Molina, R., and Katsaggelos, A. K. (2000). Hierarchical bayesian image restoration for partially-known blur. IEEE Trans Image Process, 9(10):1784-1797.
  6. Galatsanos, N. P., Mesarovic, V. Z., Molina, R., Katsaggelos, A. K., and Mateos, J. (2002). Hyperparameter estimation in image restoration problems with partiallyknown blurs. Optical Eng., 41(8):1845-1854.
  7. Jordan, M. I., Ghahramani, Z., Jaakola, T. S., and Saul, L. K. (1998). An introduction to variational methods for graphical models. In Learning in Graphical Models, pages 105-162. MIT Press.
  8. Kullback, S. (1959). Information Theory and Statistics. New York, Dover Publications.
  9. Hsiao, I.-T., Rangarajan, A., and Gini, G. (2002). Joint-map Bayesian tomographic reconstruction with a gammamixture prior. IEEE Trans Image Process, 11:1466- 1477.
  10. L ópez, A., Molina, R., Katsaggelos, A. K., Rodriguez, A., L ópez, J. M., and Llamas, J. M. (2004). Parameter estimation in bayesian reconstruction of SPECT images: An aid in nuclear medicine diagnosis. Int J Imaging Syst Technol, 14:21-27.
  11. L ópez, A., Molina, R., Mateos, J., and Katsaggelos, A. K. (2002). SPECT image reconstruction using compound prior models. Int J Pattern Recognit Artif Intell, 16:317-330.
  12. Miskin, J. (2000). Ensemble Learning for Independent Component Analysis. PhD thesis, Astrophysics Group, University of Cambridge.
  13. Miskin, J. W. and MacKay, D. J. C. (2000). Ensemble learning for blind image separation and deconvolution. In Girolami, M., editor, Advances in Independent Component Analysis. Springer-Verlag.
  14. Mohammad-Djafari, A. (1995). A full bayesian approach for inverse problems. In in Maximum Entropy and Bayesian Methods, Kluwer Academic Publishers, K. Hanson and R.N. Silver eds. (MaxEnt95).
  15. Mohammad-Djafari, A. (1996). Joint estimation of parameters and hyperparameters in a bayesian approach of solving inverse problems. In Proceedings of the International Conference on Image Processing, pages 473-477.
  16. Molina, R., Katsaggelos, A. K., and Mateos, J. (1999). Bayesian and regularization methods for hyperparameter estimation in image restoration. IEEE Trans Image Process, 8(2):231-246.
  17. Molina, R., Mateos, J., and Katsaggelos, A. K. (2006). Blind deconvolution using a variational approach to parameter, image, and blur estimation. IEEE Trans Image Process, 15(12):3715-3727.
  18. In algorithm 3 we need to calculate the quantities E [x j]qk(x), E [log(pc pG(x j | ßc, ac))]qk+1(x),qk+1(ß),qk(p),
  19. E [1/ßc]qk+1(ß) and E [log Ai, jx j]qk+1(x).
  20. To calculate E [x j]qk(x) we note that (see Eq. 8)
Download


Paper Citation


in Harvard Style

Molina R., López A., Manuel Martín J. and K. Katsaggelos A. (2007). VARIATIONAL POSTERIOR DISTRIBUTION APPROXIMATION IN BAYESIAN EMISSION TOMOGRAPHY RECONSTRUCTION USING A GAMMA MIXTURE PRIOR . In Proceedings of the Second International Conference on Computer Vision Theory and Applications - Volume 3: Bayesian Approach for Inverse Problems in Computer Vision, (VISAPP 2007) ISBN 978-972-8865-75-7, pages 165-173. DOI: 10.5220/0002066001650173


in Bibtex Style

@conference{bayesian approach for inverse problems in computer vision07,
author={Rafael Molina and Antonio López and José Manuel Martín and Aggelos K. Katsaggelos},
title={VARIATIONAL POSTERIOR DISTRIBUTION APPROXIMATION IN BAYESIAN EMISSION TOMOGRAPHY RECONSTRUCTION USING A GAMMA MIXTURE PRIOR},
booktitle={Proceedings of the Second International Conference on Computer Vision Theory and Applications - Volume 3: Bayesian Approach for Inverse Problems in Computer Vision, (VISAPP 2007)},
year={2007},
pages={165-173},
publisher={SciTePress},
organization={INSTICC},
doi={10.5220/0002066001650173},
isbn={978-972-8865-75-7},
}


in EndNote Style

TY - CONF
JO - Proceedings of the Second International Conference on Computer Vision Theory and Applications - Volume 3: Bayesian Approach for Inverse Problems in Computer Vision, (VISAPP 2007)
TI - VARIATIONAL POSTERIOR DISTRIBUTION APPROXIMATION IN BAYESIAN EMISSION TOMOGRAPHY RECONSTRUCTION USING A GAMMA MIXTURE PRIOR
SN - 978-972-8865-75-7
AU - Molina R.
AU - López A.
AU - Manuel Martín J.
AU - K. Katsaggelos A.
PY - 2007
SP - 165
EP - 173
DO - 10.5220/0002066001650173