乳房斷層合成由於限制掃描角度，是屬於不完整取樣的系統，在影像重建上會
使用最佳化方法來取得品質較好的重建影像。近期針對稀疏影像(Sparse image) 在
不完整取樣的系統中進行影像重建時，在最佳化目標函數的設計上常利用l1(總變
數) 限制給與影像限制能。然而，l1 限制由於並非平滑函數，演算法的選擇上需能
處理部分不可微分的特性。
Chambolle 與Pock 提出的主要對偶(Primal-Dual) 最佳化演算框架能處理廣泛
的凸最佳化問題，其中包含總變數限制的部分。因此本研究將利用主要對偶最佳化
框架來處理乳房斷層合成所使用的凸最佳化範疇，並利用CP 演算法進行影像重
建。
在結果呈現上，線性模型中使用CP 演算法相較於其他傳統的演算法能取得較
好的重建結果。而在針對總變數限制的最佳化部分，在影像上也能得到邊緣明顯且
分布較連續的影像結果，並且CP 演算有收斂檢查的機制可以避免取得發散的影像
結果。整體而言，CP 演算法使用於乳房斷層合成的影像重建上有較好且較穩的表
現。
關鍵字：乳房斷層合成、迭代式影像重建、總變數限制最小化、主要對偶最佳化、
CP 演算法

Breast tomosynthesis is an underdetermined system due to the limit angular
range. As a result, we tends to use optimization to get better quality of reconstructed
image. Recently, for the reconstruction of the sparse image under an undersampled
system, we tend to use the l1 norm (Total variation) constraint for the image in
designing of the objective function. However, l1 norm constraint is not a smooth
function, we need to choose an algorithm that can handle not everywhere-differentiable
property.
The primal-dual optimization framework proposed by Chambolle and Pock (CP)
can handle generic convex optimization problem, including the usage of total variation
constraint. In this study, we use the primal-dual framework to solve the convex
optimization using in the reconstruction of breast tomosynthesis and reconstruct the
breast image by corresponding CP algorithm.
The results show that, we can have better reconstructed image by using CP algorithm
while comparing with other algorithm that used in the linear model. For the
total variation optimization problem, we can easily use the CP algorithm to get edgepreserved
and smoothly distributed images. CP algorithm can also avoid getting the
divergence reconstructed results by checking the convergence check. In conclusion,
using CP algorithm is robust and can perform well for the image reconstruction of
the breast tomosynthesis.
Keywords: Breast tomosynthesis, iterative image reconstruction, total variation minimization,
primal-dual optimization, CP algorithm

1 前言1
2 乳房斷層合成(Breast Tomosynthesis) 4
2.1 成像原理. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1.1 X 光儀. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1.2 斷層合成. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 硬體設備. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2.1 一般性系統設計. . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2.2 偵測器. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 硬體參數最佳化. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3.1 幾何最佳化. . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3.2 方法最佳化. . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.4 使用乳房斷層合成儀器的優勢. . . . . . . . . . . . . . . . . . . . . . 12
3 影像重建14
3.1 物理模型. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.2 系統矩陣. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.3 資料模型. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.3.1 普松模型. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.3.2 最大相似度函數評估(Maximum Likelihood Estimation, MLE) 16
3.3.2.1 期望值最大演算法(Expectation Maximization algorithm,
EM) . . . . . . . . . . . . . . . . . . . . . . . 17
3.3.2.2 梯度演算法(Gradient Ascent Algorithm, GD) . . . 18
3.3.2.3 凸演算法(Convex Algorithm, CV) . . . . . . . . . . 19
i
3.3.3 線性模型. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.3.4 最小平方誤差(Least Square Error) . . . . . . . . . . . . . . . 20
3.3.4.1 代數重建法(Algebraic Reconstruction Technique,
ART) . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.3.4.2 同步代數重建法(Simultaneous Algebraic Reconstruction
Technique, SART) . . . . . . . . . . . . . 22
3.3.4.3 最陡下降演算法(Steepest Descent Algorithm, SD) . 23
3.4 總變數限制最小化(Total Variation Minimization) . . . . . . . . . . . 24
4 主要-對偶最佳化(Priaml-Dual Optimization) 25
4.1 Chambolle-Pock 演算法(CP algorithm) . . . . . . . . . . . . . . . . 26
4.2 主要-對偶原形(Primal-Dual Prototype) . . . . . . . . . . . . . . . . 27
4.2.1 參數計算. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.2.2 乳房斷層合成中使用的CP 演算法. . . . . . . . . . . . . . . 28
5 實驗設計38
5.1 實驗平台. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
5.2 假體設計. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5.3 演算法. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
5.4 投影資料. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
5.5 影像指標. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
6 實驗結果與討論42
6.1 最大相似度函數評估. . . . . . . . . . . . . . . . . . . . . . . . . . . 43
6.1.1 無高斯雜訊模擬實驗. . . . . . . . . . . . . . . . . . . . . . . 43
6.1.2 加入高斯雜訊模擬實驗. . . . . . . . . . . . . . . . . . . . . 49
ii
6.2 最小平方誤差. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
6.2.1 無高斯雜訊模擬實驗. . . . . . . . . . . . . . . . . . . . . . . 52
6.2.2 加入高斯雜訊模擬實驗. . . . . . . . . . . . . . . . . . . . . 57
6.3 主要對偶最佳化. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
6.3.1 最小平方誤差與非負限制. . . . . . . . . . . . . . . . . . . . 60
6.3.1.1 無高斯雜訊模擬實驗. . . . . . . . . . . . . . . . . 60
6.3.1.2 加入高斯雜訊模擬實驗. . . . . . . . . . . . . . . . 64
6.3.2 最小平方誤差與非負限制及總變數限制. . . . . . . . . . . . 67
6.3.2.1 無高斯雜訊模擬實驗. . . . . . . . . . . . . . . . . 67
6.3.2.2 加入高斯雜訊模擬實驗. . . . . . . . . . . . . . . . 72
7 結論與未來方向76

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