乳房斷層合成屬於欠定系統，在影像重建上，傳統的濾波反投影法並不適用。使用最佳化方法加入我們對影像的先備知識，並且搭配疊代式演算法逐步調整、優化影像品質的特性，變成了現今影像重建的趨勢。
因為感知壓縮理論的出現，使得低劑量或是投影角度不足的X-ray成像系統得以發展，減少了人類接受醫學造影檢查時，輻射這把雙面刃所帶來的危害。
交替方向乘數法(alternating direction method of multiplier, ADMM)自1974被發表以來，並沒有受到廣泛的應用。直到近20年來平行運算的發展，有關此算法的應用如雨後春筍般蓬勃發展，又因為此演算法框架的可調整性高，諸如:影像處理、機器學習、資料探勘…等各大不同領域的研究，都可以發現其身影。此論文中，基於交替方向乘數法的框架，我們提出一個乳房斷層影像重建演算法，並加入零範數(l0-norm)影像平滑當作演算法的正則化限制。為了加快演算法的收斂速度，我們加入簡潔的回溯追蹤法於ADMM中，且確保了演算法的收斂。
觀察電腦假體模擬實驗的結果，若待重建的物體具備強稀疏性，我們提出的演算法能得到良好的影像品質。為此，本研究示範了零範數平滑影像的效果，對提升乳房斷成合成重建影像品質所帶來的可能性。

Breast tomosynthesis is different from Computer Tomographic(CT) with its limit projection angle. In other words, breast tomosynthesis can be viewed as the undetermined system. Applying conventional CT reconstruction technique like “Filter Back Projection(FBP)” to tomosynthesis will lead to undesirable strip and phantom artifacts. Thus, the model based iterative algorithms, which carefully adjust the image in each iteration would be more suitable for tomosynthesis image reconstruction.
Thanks for the booming development of compressive sensing, new class of image reconstruction method shows more promise in reconstructing a three dimensional image from limit angular, or low dose computer tomography. This kind of method such as total variation, dictionary learning also benefit the breast tomosynthesis.
Alternating directional method of multiplier(ADMM) is an algorithm that can handle many kinds of optimization problem. The wide variety of uses is one of its big attractions so far. In this thesis, we propose an image reconstruction algorithm based on ADMM framework and use l0-norm smoothing image as prior, which impose the sparsity constraint to the image. To bridge the gap of convergence rate of existing state-of-art iterative algorithms, we use the adaptive parameter in the gradient descent part of our algorithm and backtracking to ensure the convergence.
The results from the simulation experiment shows that our proposed algorithm gives the good image quality under the assumption of strong sparsity of the image to be reconstructed.

1. 前言....................1
2. 乳房斷層合成(Tomosynthesis)..........3
2.1 基本儀器架構....................3
2.2 影像重建物理模型..........5
3. 疊代型影像重建演算法..........9
3.1 相關數學回顧....................9
3.2 最陡下降法(Steepest Descent, SD)..........11
3.3 代數重建法(Algebraic Reconstruction Technique, ART)..........12
3.4 同步代數重建法(Simultaneous Algebraic Reconstruction Technique, SART)..........13
3.5 交替方向乘數法(Alternative Direction Method of Multiplier, ADMM)..........14
4. 稀疏近似(Sparse Approximation)..........19
4.1 離散梯度轉換....................19
4.2 稀疏編碼(Sparse Coding)..........20
4.3 卷積神經網路(Convolution Neuron Network)..........24
5. 零範數交替方向乘數法(l0-Guided ADMM)....................25
5.1 零範數梯度最小化[28]....................25
5.2 l0-Guided ADMM....................28
6. 實驗設計....................33
6.1 儀器硬體設定....................33
6.2 電腦模擬假體....................33
6.3 影像評估指標....................37
6.4 實驗相關細節設定....................38
6.5 演算法相關細節設定....................40
7. 結果與討論....................44
7.1 無雜訊無正則化模擬實驗....................44
7.2 無雜訊模擬實驗....................50
7.3 雜訊模擬實驗....................57
8. 結論與未來方向....................65

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