PET掃描為提供人體功能性影像之醫學利器，利用統計影像重建法（statistical image reconstruction）可以提昇核醫重建影像的品質，然而這類重建法需要精確估算光子被偵測機率並涉及大量的正向及反向投影（forward and backward projections）運算。傳統即時計算（on-the-fly）方法在每次疊代（iteration）運算的正向及反向投影步驟中皆需計算偵測機率值（detection probability）。隨著光子偵測機率計算模型趨於複雜，投影運算所需時間會大幅提高。本研究對統計影像重建法提出較有校執行方法的建議，將投影的幾何模型以機率矩陣（p矩陣）形式表示，則正向及反向投影運算轉為矩陣相乘形式，可避免重複前述之偵測機率值的重複計算。此外，p矩陣亦可結合非均勻取樣(non-uniform sampling)，重建時不需額外考慮幾何弧形校正(geometric arc correction)，避免不必要的修正計算。由於p矩陣為稀疏矩陣，針對其非零值作處理，進一步藉由環狀偵檢器與正方形影像間的八方對稱關係，大幅減少重建影像所需處理的資料量至原來的0.18 %。如此不僅節省p矩陣所需的儲存空間，亦有效提高疊代運算的速度。搭配面積法的幾何模型，以較精確的方式估算偵檢機率值，使統計影像重建法的重建影像品質較佳。本研究並對八方對稱提出新的計算方法，使其能自然應用於加速演算法OSEM（ordered subsets expectation maximization）中，在臨床上能快速重建出核醫影像。

Statistical image reconstruction methods can improve the quality of PET image results by using accurate probability model of photon detection. However, these statistical methods usually require repetitive forward and backward projections, which are computationally intensive. Implementation variations of the projection operations can greatly affect the reconstruction efficiency. The traditional on-the-fly method directly computes probability of the forward and backward projections during image iterations. As the probability model of photon detection become more complex, this approach will become less applicable due to the heavier computational load. In order to effectively compute projection operations, we suggest a matrix-based approach that each element of the matrix represents the probability of detecting a coincidence event from a voxel to a detector pair based on scanner’s geometry. Consequently, a forward or backward projection can be transformed into a simple matrix multiplication without repeated computation of probability during image reconstruction. In addition, probability matrix can incorporate non-uniform sampling distance, so that the PET data needs not to be pre-processed for geometric arc correction additionally. Because most PET scanners adopt cylindrical structure, there exist several geometric symmetries that can be used to reduce the numerical computation as well as the matrix storage by a factor of eight as suggested by Kaufman. Moreover, by integration of the symmetry and the sparseness of the probability matrix, the storage space can be further downsized to 0.18% of its original magnitude. In this work, we also examine two types of probabilistic model for coincidence detection: area-based and interpolative. From the experimental results, the area-based model shows better quality of the reconstructive image compared to interpolative one. In this thesis, we have shown that statistical image reconstructions with probability matrix and area-based detection model can generate more effective and accurate results for PET imaging.

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