放射式斷層掃瞄 (Emission Computed Tomography)是一種非侵入式的醫學成像技術，它可以為全身生理功能的運作提供影像，為現今臨床診斷中最重要且應用範圍廣泛的影像工具之一。由於資料的不完整性，影像的重建可藉由統計的最大概似估計，得到較佳的影像品質。絕大多數的放射式斷層掃瞄統計影像重建法，是以Poisson分配來描述在每個像素點的放射計數量，並以此導出：在每個像素點Poisson參數給定的條件下，實際所收集到的資料為獨立的Poisson分配，然後利用統計方法，尋找每個像素點所對應的Poisson分配的平均數的估計值，具此重建影像。
雖然以條件獨立(conditional independent)的Poisson模型來描述光子對的放射偵收過程是適當的。但當我們重建影像的方法是以估計每個像素點的放射計數量，而非像素點所對應的Poisson分配的參數(即平均數)時，則發現資料模型不再是條件獨立的Poisson模型－ 相關性將被引入到資料中，導致資料模型為一組相依的隨機變數。
在本論文中，我們以估計每個像素點的放射計數量，而非像素點所對應的Poisson分配平均數的觀點，提出一個，結合相依的概似函數與具有局部變化的權重參數的quadratic pairwise difference事前機率(prior probability)階層貝式模型，來重建PET影像。另一方面，本文亦提出一個簡化相關性資料計算複雜度的作法，並搭配有效率的演算法，達到改善影像品質又保有最佳計算效率的雙重成效。
透過模擬與實際的PET資料，我們所提出的階層貝式模型在影像的對比與雜訊的抑制上，相較於MLEM演算法，有較佳的視覺影像品質。
Emission computed tomography is a non-invasive functional imaging technique that is nowadays widely applied in medical diagnostic imaging, especially to determine physiological function. The available set of measurements is, however, often incomplete and corrupted, and the quality of image reconstruction is enhanced by the computation of a statistically optimal estimate. Most statistical reconstruction methods for emission tomography use the conditionally independent Poisson model to measure fidelity to data.
Although the conditionally independent Poisson model is appropriate for a conceptual view of PET imaging, once the reconstruction problem is to estimate the number of emissions in each pixel, the stochastic nature of the emission process is no longer Poisson and the measurement data are no longer independent. Correlations are introduced into the measurement model, which result in dependent random variables.
In this dissertation, we propose a hierarchical Bayesian model which combines a dependent likelihood function and the quadratic pairwise difference prior with locally varying weighting hyerparameters to reconstruct image in terms of emission counts in pixels. This dissertation addresses the second concern, seeking to simplify the reconstruction of correlated data and provide a more precise image estimate than the conventional independent methods.
We apply and test the proposed hierarchical Bayesian model with two PET data sets. The reconstructed images reveal that the proposed hierarchical Bayesian model produces reconstruction with superior visual quality to MLEM in terms of contrast and noise artifact suppression.
