正子斷層掃瞄(Positron Emission Tomography：PET)是利用人體中正子放射和斷層掃瞄的原理，可以研究人體內的新陳代謝現象，提供活體的生化造影給醫生作為進一步診斷的參考，是近幾年來在核子醫學中一門發展相當快速且嶄新的影像診斷技術。由於造影所收集到的正子放射資料和人體中的生化現象是間接的關係，必須經過逆推的計算方式才能重建人體內的生化造影，目前最常見的影像重建方法有兩種，一種為濾鏡式反投影（Filtered Backprojection：FBP），是一種快速方便的影像重組方式，但是有造成斑紋狀的假影（Streak Artifacts）的缺點；第二種為疊代式重組（Iterative Reconstruction）方法，如ML-EM(Maximum Likelihood-Expectation Maximization)，能消除假影改善影像品質，但是因為其反覆分析疊代式之運算需要冗長的計算時間，造成過多記憶體空間的耗費。
本論文提出一種非線性之濾除雜訊技術，平均曲度濾波（Mean Curvature Diffusion：MCD）方法，應用於正子斷層掃瞄之濾鏡式反投影影像重建的處理過程。平均曲度濾波方法在去除雜訊過程中，不但能有效地去除雜訊，且能同時保留影像中組織輪廓的特性，加上濾鏡式反投影之技術，不僅能有效的抑制影像雜訊，更節省了龐大的運算時間。
本論文主要從事於平均曲度濾波方法對於正子斷層掃瞄影像雜訊的處理研究，我們分別於正向投射影像（Sinogram）與濾鏡式反投影之重組影像進行平均曲度濾波方法的處理，我們已於實驗的結果中得到比傳統方法更快速有效的重建影像品質。

Positron Emission Tomography (PET) is a tomographic technique to display metabolic activity in slices through a patient''s body. The popular reconstruction methods today in PET are Filtered Backprojection (FBP) and iterative reconstruction algorithm. FBP is based on a Fourier Transform algorithm and is extremely fast, but the reconstructed image may suffer from annoying streak artifacts. Iterative reconstruction, like Maximum Likelihood-Expectation Maximization (ML-EM) algorithm, depresses the noise problem, but the algorithm is iterated too long, such that the reconstructed image starts to degrade.
In this paper, Mean Curvature Diffusion (MCD), a nonlinear filtering technique, will be applied in the processing of the Filtered Backprojection reconstruction. The Mean Curvature Diffusion approach not only can depress noise but can also reserve the outlines of tissues. Using the combination of Mean Curvature Diffusion and Filtered Backprojection methods, a reconstructed PET image of good quality can be obtained quickly.
In our study, the effect of MCD filtering in depressing the noise of PET image was investigated. The filtering of Mean Curvature Diffusion is applied to both the projection image (sinogram) prior to reconstruction and to the FBP reconstructed image. Preliminary studies on simulating target (phantom) of “sphere-ellipsoid”, we can rapidly get a good reconstructed image by using FBP reconstruction technique and the later MCD filtering method.

目錄
第一章 簡介
1-1. 背景1-1
1-2. 動機1-3
1-3. 全文架構1-4
第二章 平均曲度濾波方法
2-1. 曲面之表示與擴散2-1
2-2. MCD的特性2-2
2-2-1. 邊界之不變性2-2
2-2-2. 穩定性2-2
2-3. 尺度參數與演化速率2-3
2-3-1. 尺度參數2-3
2-3-2. 演化速率2-3
2-4. 影像邊界之維護2-4
2-4-1. 高斯誤差函數2-5
2-4-2. 測試結果2-6
第三章 影像重建
3-1. FBP重建法3-1
3-2. MLEM重建法3-6
3-3. 實驗架構3-10
3-3-1. 卜松雜訊3-11
3-3-2. 雜訊圖譜3-12
第四章 實驗結果
4-1. MCD於卜松雜訊圖譜上之應用4-1
4-2. MCD與FBP於卜松雜訊圖譜之應用4-5
4-3. MLEM於卜松雜訊圖譜之應用4-7
4-4. 結果比較4-11
第五章 結論
Contents
Chapter 1 Introduction
1-1. Background1-1
1-2. Motivation1-3
1-3. Organization1-4
Chapter 2 Mean Curvature Diffusion
2-1. Surface Representation and Diffusion2-1
2-2. Properties of MCD2-2
2-2-1. Edge Invariance2-2
2-2-2. Stability2-2
2-3. The Scaling Parameter and Evolution Speed2-3
2-3-1. Scaling Parameter2-3
2-3-2. Evolution Speed2-3
2-4. Image Edge Preservation2-4
2-4-1. Gaussian Error Function2-5
2-4-2. Performance2-6
Chapter 3 Image Reconstruction
3-1. Filtered Backprojeciton Reconstruction3-1
3-2. Maximum Likelihood-Expectation Maximization Reconstruction3-6
3-3. Experiment Procedure3-10
3-3-1. Poisson Noise3-11
3-3-2. Noisy Sinogram3-12
Chapter 4 Experiment Results
4-1. MCD on Poisson Noisy Sinogram4-1
4-2. MCD with FBP Reconstruction on Poisson Noisy Sinogram4-5
4-3. MLEM Reconstruction on Poisson Noisy Sinogram4-7
4-4. Performance Comparison4-11
Chapter 5 Conclusion

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