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 南方医科大学学报  2018, Vol. 38Issue (1): 48-54  DOI: 10.3969/j.issn.1673-4254.2018.01.08. 0

### 引用本文 [复制中英文]

CHEN Meiling, TAO Xi, LI Huayong, CHEN Wufan, ZHANG Hua. Low-dose digital breast tomosynthsis imaging via noise correlation based penalized weighted least-squares algorithm[J]. Journal of Southern Medical University, 2018, 38(1): 48-54. DOI: 10.3969/j.issn.1673-4254.2018.01.08.

### 文章历史

1. 南方医科大学 生物医学工程学院，广东 广州 510515;
2. 南方医科大学 广东省医学图像处理重点实验室，广东 广州 510515

Low-dose digital breast tomosynthsis imaging via noise correlation based penalized weighted least-squares algorithm
CHEN Meiling1,2, TAO Xi1,2, LI Huayong1,2, CHEN Wufan1,2, ZHANG Hua1,2
1. Department of Biomedical Engineering, Southern Medical University, Guangzhou, 510515, China;
2. Guangdong Provincial Key Laboratory of Medical Image Processing, Southern Medical University, Guangzhou, 510515, China
Supported by National Natural Science Foundation of China (81501466)
Abstract: Objective To achieve low-dose digital breast tomosynthsis (DBT) projection recovery using penalized weighted least square algorithm incorporating accurate modeling of the variance of the projection data and noise correlation in the flat panel detector. Methods Models were established for the quantal noise and electronic noise in the DBT system to construct the penalized weighted least squares algorithm based on noise correlation for projection data restoration. The filter back projection algorithm was then used for DBT image reconstruction. Results The reconstruction results of the ACR phantom data at different dose levels showed a good performance of the proposed method in noise suppression and detail preservation. CNRs and LSNRs of the reconstructed images from the restored projections were increased by about 3.6 times compared to those of reconstructed images from the original projections. Conclusions The proposed method can significantly reduce noise and improve the quality of DBT images.
Key words: digital breast tomosynthesis    low-dose    noise correlation    weighted least squares algorithm

X线乳腺摄影技术[4]对微小病灶具有高度敏感性，但传统的全视野数字乳腺X摄影（FFDM）由于平面二维成像的组织重叠效应在乳腺肿瘤的检测中存在一定程度的检出失败率[5-6]，同时，由正常组织重叠而产生的假阳性肿瘤会误导医生的诊断。近年来，数字乳腺层析成像（DBT）逐渐在临床中得到广泛应用[7-8]，DBT成像利用传统的X线源，以平板探测器为旋转中心采集一定数量的投影数据，并通过有限数量的投影数据进行三维重建，实现了乳腺的三维成像。数字乳腺层析成像很大程度上解决了二维成像组织重叠所导致的检出失败和假阳性误诊的问题[9]。同时，相对于CT成像技术，DBT成像很大程度上降低了患者的辐射剂量[10]，但乳腺作为一个对X射线敏感器官[11]，其辐射危害仍然不容小觑。为进一步降低辐射剂量，可以通过降低X射线球管电流（mAs）来实现超低剂量DBT成像[12]，但超低剂量扫描条件下的重建图像噪声水平较高，导致低对比度病灶及微小钙化点难以识别。

1 方法 1.1 投影数据噪声建模

 ${\tilde I_i} \sim \sum\limits_k {G({E_k})Pois\;{\rm{son}}\{ {{\tilde \lambda }_i}({E_k})\} + Normal\{ {d_i}, {\sigma _{e, i}}^2\} }$ (1)

i表示探测单元序列编号，参数k表示X光能量级，G(.)表示能谱概率，Possion{.}表征数据的泊松统计特性，其中λ代表穿过测量体后的光子平均数，即探测器实际接收到的信号。公式（1）的第二项表征电子噪声，服从均值为di，方差为σe, i2的高斯分布，电子噪声的参数可以通过暗电流数据测量。对于复合泊松分布，目前还还缺闭合、准确的似然函数，本文忽略X光多能特性，将X线光子分布函数简化为：

 ${{\tilde I}_i} \sim Pois\;{\rm{son}}\{ {\lambda _i}\} + Normal\{ {d_i}, {\sigma _{e, i}}^2\}$ (2)

 ${p_i} = \ln \frac{{{I_{{o_i}}}}}{{{{\tilde I}_i}}} = \ln {I_{{o_i}}}-\ln {{\tilde I}_i}$ (3)

 $\sigma _{pi}^2 = {\mathop{\rm var}} ({p_i}) = \frac{1}{{{\lambda _i}}}(1 + \frac{{{\sigma _{e, i}}^2-1.25}}{{{\lambda _i}}})$ (4)

1.2 基于噪声相关性的惩罚加权最小二乘算法在DBT投影数据恢复中的应用

 ${\rho _{ij}} = \frac{{{\mathop{\rm cov}} ({p_i}, {p_j})}}{{{\sigma _i}{\sigma _j}}} = \frac{{{\mathop{\rm cov}} ({p_i}, {p_j})}}{{\sqrt {{\mathop{\rm var}} ({p_i})} \sqrt {{\mathop{\rm var}} ({p_j})} }}$ (5)

 $\Phi (p) = (\hat y-p)'{\sum ^{-1}}(\hat y-p) + \beta R(p)$ (6)

 $R(p) = \sum\limits_j {R({p_j}) = \sum\limits_j {\sum\limits_{k \in {S_j}} {w(k, j){{({p_j}-{p_k})}^2}} } }$ (7)

Sj指在二维投影数据中与像素j最相邻的4个像素，本文中4个像素的权w(k, j)取相同值。通过最小化代价函数（6）可以得到优化的线积分投影数据：

 $\begin{array}{l} \hat p = \arg \;\mathop {\min }\limits_{\mu \ge 0} (\hat y-p)'{\sum ^{-1}}(\hat y-p) + \\ \beta \sum\limits_j {\sum\limits_{k \in {S_j}} {w(k, j){{({p_j} - {p_k})}^2}} } \end{array}$ (8)

 ${A_{ij}} = \left\{ \begin{array}{l} 4, \;\;\;\;\;\;\;\;\;\;\;i = j\\ -1, \;\;\;\;\;\;\;i-j = \pm 1\;or\;i-j = \pm B\\ 0, \;\;\;\;\;\;\;\;\;\;o\;{\rm{th}}\;erwise \end{array} \right.$ (9)

 $\begin{array}{l} \nabla \Phi (p) = \nabla ({(y-p)^T}{\sum ^{-1}}(y-p)) + \beta \nabla R(p) = 0\\ \Rightarrow {\sum ^{ - 1}}(p - y)) + \beta Ap = 0 \end{array}$ (10)

0.初始化p(0)，并设定迭代终止条件ε;

1.求解线性方程∑b = y -p(n)，以更新b;

2.对步骤1求解得到的b值，求解线性方程βAp(n + 1) = b，得到p(n + 1);

3.当满足|p(n + 1) -p(n)| < ε停止迭代，否则返回步骤1;

*其中b为中间变量

1.3 恢复投影数据重建

2 结果 2.1 实验数据获取

 图 1 乳腺成像标准体膜ACR体模内部结构信息及实物图 Figure 1 Internal structure and appearance of an ACR standard phantom.

2.2 评价参数

 $LSNR = \bar f/{\sigma _f}$ (11)
 $CNR = \frac{{2\left| {\bar f-\bar b} \right|}}{{{\sigma _f} + {\sigma _b}}}$ (12)

2.3 投影图像恢复前后比较

 图 2 30 kVp、40 mA和30 kVp、60 mA剂量下0角度投影数据恢复前后的图像. Figure 2 Original and restored projection images at angle 0 with different scan settings. A: Original projection image obtained at 30 kVp and 40 mA; B: Restored projection image obtained at 30 kV and 40 mA; C: Original projection image obtained at 30 kV and 60 mA; D: Restored projection image obtained at 30 kV and 60 mA.

 图 3 0角度实验数据在30 kVp、40 mA剂量下，参数β = 1200时，PWLS算法的收敛测试图 Figure 3 Convergence of PWLS algorithm at 30 kVp and 40 mA at the angle of 0 with β=1200.

 图 4 0角度下40、60、80、100 mA四个剂量的投影数据恢复前后局部信噪比折线图 Figure 4 Local SNRs calculated from projection data at angle 0 and at the dose levels of 40, 60, 80 and 100 mA before and after denoising.
2.4 重建结果比较

 图 5 不同剂量下投影数据恢复前后的重建结果 Figure 5 Reconstruction results before and after projection data recovery at different doses. A: The image reconstructed from original projections at 30 kVp and 40 mA; B: The image reconstructed from the restored projections at 30 kVp and 40 mA; C: The image reconstructed from original projections at 30 kVp and 60 mA; D: The image reconstructed from the restored projections at 30 kVp and 60 mA; E: The image reconstructed from original projections at 30 kVp and 80 mA; F: The image reconstructed from the restored projections at 30 kVp and 80 mA; G: The image reconstructed from original projections at 30 kVp and 100 mA; H: The image reconstructed from the restored projections at 30 kVp and 100 mA.

 图 6 不同算法对40 mA投影数据处理后的重建图像. Figure 6 Reconstructed images from restored projection data (40 mA) using different algorithms. A: BM3D; B: the proposed method.

 图 7 局部信噪比（LSNR）和对比度信噪比（CNR）感兴趣区域图 Figure 7 Analysis of LSNR and CNR in the region of interest (ROI). The red and blue boxes indicate the ROIs for LSNR and CNR analysis, respectively

3 讨论