正子斷層掃描為現今重要醫學診斷工具之一，而其影像需利用統計方法重建。
本篇論文延續以往利用類似EM 演算法求取最大概似估計量的方式來進行影
像重建。我們提出了一種新的影像重建方法，將階層式貝氏架構與一種改良的
EM 演算法-OSL 演算法相結合，其不僅考慮了相近亮度部位應平滑的概念，更
加入影像之間的權重以避免過度平滑，使影像邊緣得以保留。此種重建法會以
模擬的方式來進行說明並與其他方法比較。

Positron Emission Tomography (PET) is one of the most important techniques for
medical diagnosis. The statistical methods are needed in image reconstruction for
PET. In this dissertation we develop a new approach to the reconstruction of the
image. We use hierarchical Bayesian model to describe how the observations are
obtained and apply a modied EM-type approach, the one-step-late algorithm, to
calculate the maximum likelihood estimate for the emission intensity at each pixel.
This method take care of both smoothness and edge eects of the image simulta-
neously. Finally the performance of this method is demonstrated and compared
with other methods by computer simulations.

[1] L. A. Shepp and Y. Vardi, "Maximum Likelihood Reconstruction for Emis-
sion Tomography," Medical Imaging, IEEE Transactions on, vol. 1, pp. 113 -122,
oct. 1982.
[2] S. Geman and D. Geman, "Stochastic Relaxation, Gibbs Distributions, and
the Bayesian Restoration of Images," Pattern Analysis and Machine Intelligence,
IEEE Transactions on, vol. PAMI-6, pp. 721 -741, nov. 1984.
[3] P. J. Green, "Bayesian reconstructions from emission tomography data using
a modied EM algorithm," Medical Imaging, IEEE Transactions on, vol. 9, pp.
84 -93, mar 1990.
[4] JIA-HUNG,TANG, "Hierarchical Bayesian Image Reconstruction for Emis-
sion Tomography Using Dependent Likelihood," Ph.D Dissertation, Institute of
statistics, NTHU Hsinchu Taiwan, 2012.
[5] J. Besag, "On the Statistical Analysis of Dirty Pictures," Journal of the
Royal Statistical Society. Series B (Methodological), vol. 48, pp. pp. 259-302,
1986.
[6] J. G. Colsher, "Fully 3-dimensional positron emission tomography," Physics
in Medicine and Biology, vol. 25, pp. 103-115, 1980.
[7] M. I. Miller, D. L. Snyder, and S. M. Moore, "An Evaluation of the Use
of Sieves for Producing Estimates. Of Radioactivity Distributions with the EM
Algorithm for PET," Nuclear Science, IEEE Transactions on, vol. 33, pp. 492
-495, feb. 1986.
[8] K. Lange, "Convergence of EM image reconstruction algorithms with Gibbs
smoothing," Medical Imaging, IEEE Transactions on, vol. 9, pp. 439 -446, dec
1990.
[9] T. Hebert and R. Leahy, "A generalized EM algorithm for 3-D Bayesian
reconstruction from Poisson data using Gibbs priors," Medical Imaging, IEEE
Transactions on, vol. 8, pp. 194 -202, jun 1989.
[10] P. J. Green, "On Use of the EM for Penalized Likelihood Estimation,"
Journal of the Royal Statistical Society. Series B (Methodological), vol. 52, pp.
pp. 443-452, 1990.
[11] B. A. Mair and J. Zahnen, "A generalization of Green's one-step-late
36
algorithm for penalized ML reconstruction of PET images," in Nuclear Science
Symposium Conference Record, 2006. IEEE vol. 5, ed, 2006, pp. 2775 -2777.