计算机应用   2017, Vol. 37 Issue (4): 1193-1197  DOI: 10.11772/j.issn.1001-9081.2017.04.1193 0

### 引用本文

LIU Wei, CHEN Leiting. MRI image registration based on adaptive tangent space[J]. Journal of Computer Applications, 2017, 37(4): 1193-1197. DOI: 10.11772/j.issn.1001-9081.2017.04.1193.

### 文章历史

1. 电子科技大学 计算机科学与工程学院, 成都 611731;
2. 成都工业职业技术学院, 成都 610218

MRI image registration based on adaptive tangent space
LIU Wei1,2, CHEN Leiting1
1. School of Computer Science and Engineering, University of Electronic Science and Technology of China, Chengdu Sichuan 611731, China;
2. Chengdu Vocational and Technical College of Industry, Chengdu Sichuan 610218, China
Abstract: The diffeomorphism is a differential transformation with smooth and invertible properties, which leading to topology preservation between anatomic individuals while avoiding physically implausible phenomena during MRI image registration. In order to yield a more plausible diffeomorphism for spatial transformation, nonlinear structure of high-dimensional data was considered, and an MRI image registration using manifold learning based on adaptive tangent space was put forward. Firstly, Symmetric Positive Definite (SPD) covariance matrices were constructed by voxels from an MRI image, then to form a Lie group manifold. Secondly, tangent space on the Lie group was used to locally approximate nonlinear structure of the Lie group manifold. Thirdly, the local linear approximation was adaptively optimized by selecting appropriate neighborhoods for each sample voxel, therefore the linearization degree of tangent space was improved, the local nonlinearization structure of manifold was highly preserved, and the best optimal diffeomorphism could be obtained. Numerical comparative experiments were conducted on both synthetic data and clinical data. Experimental results show that compared with the existing algorithm, the proposed algorithm obtains a higher degree of topology preservation on a dense high-dimensional deformation field, and finally improves the registration accuracy.
Key words: diffeomorphism    tangent space    Lie group    neighborhood selection    MRI image registration
0 引言

Marsland等[1]把微分同胚变换看作时变速度场, 用测地插值样条基构造变换; Beg等[2]提出在速度场空间利用基于欧拉-拉格朗日方程的变分方法求解大形变的微分同胚变换, 采用梯度下降法更新形变参数; Ashburner等[3]对文献[2]进行改进, 任意时刻的速度场都用初始速度场来表示, 配准的每次迭代都利用初始形变值来计算, 减少了对内存和外存的要求; Ashburner[4]提出非时变速度场的配准方法, 同胚变换构成了复合运算下的李群, 将大形变同胚变换分解为一系列小形变来处理, 使可逆变换相对容易, 并降低计算代价; Janssens等[5]在微分同胚方法中利用累加位移场的可逆性配准对比增强度不同的图像; Arsigny等[6]借助李群理论提出Log-Euclidean框架, 在微分同胚空间里可以对向量场进行分析统计并且能保持变换的可逆性; Vercauteren等[7]基于Demons算法, 提出一种非参数微分同胚的Demons配准算法——DD-NP算法, 利用李群理论在连续域上进行空间变换, 使空间变换具有微分同胚性, 从而形变场具有拓扑保持性; 闫德勤等[8]在保持图像的局部和全局流形拓扑结构的基础上, 提出快速微分同胚变换速度场更新方法。

1 相关背景 1.1 李群和切空间

1)GL(m) 表示m×m矩阵构成的李群；

2)gl(m) 表示m×m矩阵构成的李代数 (线性向量空间)；

3) log () 表示对数映射；

4) exp () 表示指数映射。

 ${{\mathit{\boldsymbol{S}}}_{i}}=\exp ({{\mathit{\boldsymbol{s}}}_{i}});{{\mathit{\boldsymbol{s}}}_{i}}=\log ({{\mathit{\boldsymbol{S}}}_{i}})$ (1)

 图 1 李群和李代数之间的映射关系 Figure 1 Mapping between Lie group and Lie algebra
1.2 正定对称矩阵和李群

 ${{\mathit{\boldsymbol{S}}}_{i}}\odot {{\mathit{\boldsymbol{S}}}_{j}}=\exp (\log ({{\mathit{\boldsymbol{S}}}_{i}})+\log ({{\mathit{\boldsymbol{S}}}_{j}}))$ (2)
2 DD-NP模型

 $\min \mathit{\boldsymbol{E}}(u)=Sim({{I}_{r}}, {{I}_{f}}\circ \varphi )+Reg(\varphi )$ (3)

 $\frac{\rm{d}\varphi ({{\mathit{\boldsymbol{S}}}_{i}}, \mathit{t})}{\rm{d}\mathit{t}}=u(\varphi ({{\mathit{\boldsymbol{S}}}_{i}}, t), t);\varphi ({{\mathit{\boldsymbol{S}}}_{i}}, 0)={{\mathit{\boldsymbol{S}}}_{i}}$ (4)

 ${{S}_{i}}^{j}={{\mathit{\boldsymbol{S}}}_{i}}\circ \varphi ={{\mathit{\boldsymbol{S}}}_{i}}\circ \exp (u)$ (5)

3 基于自适应切空间的配准算法 3.1 自适应邻域选择

 $\mathit{\boldsymbol{u}}=\sum\limits_{i=1}^{d}{{{u}^{i}}}{{\tau }_{i}}$ (6)

Eε是单位元Id的邻域, 且Eεk个最近邻点{ε1, ε2, …, εk}组成, εjSym+(m), j=1, 2, …, k, 邻域点的线性逼近用线性拟合$\bar{\varepsilon }+\mathit{\boldsymbol{T}}{{\mathit{\boldsymbol{u}}}^{i}}$来近似,

 $\sum\limits_{j=1}^{k}{||{{\varepsilon }_{j}}}-(\bar{\varepsilon }+\mathit{\boldsymbol{T}}{{u}^{j}})||=||{{\mathit{\boldsymbol{E}}}_{\varepsilon }}-(\bar{\varepsilon }{{\mathit{\boldsymbol{e}}}^{\rm{T}}}+\mathit{\boldsymbol{TU}})||$ (7)

$\mathit{\boldsymbol{U}}=[{{u}^{1}}, {{u}^{2}}, \cdots, {{u}^{k}}]$, 且URd×k, 是对应正交基矩阵的局部坐标系:

 $\mathit{\boldsymbol{U}}={{\mathit{\boldsymbol{T}}}^{\rm{T}}}({{\mathit{\boldsymbol{E}}}_{\varepsilon }}-\bar{\varepsilon }{{\mathit{\boldsymbol{e}}}^{\rm{T}}})$ (8)

 $\mathit{\boldsymbol{U}}=\rm{diag}({{\sigma }_{1}}, {{\sigma }_{2}}, \cdots, {{\sigma }_{d}}){{\mathit{\boldsymbol{V}}}^{\rm{T}}}$ (9)

 $\left\| \mathit{\boldsymbol{U}} \right\|=\sqrt{\sum\limits_{j\le d}{{{({{\sigma }_{j}})}^{2}}}}$ (10)

 $\left\| {{\mathit{\boldsymbol{E}}}_{\varepsilon }}-(\bar{\varepsilon }{{\mathit{\boldsymbol{e}}}^{\rm{T}}}+\mathit{\boldsymbol{T}}\mathit{\Theta }) \right\|=\sqrt{\sum\limits_{j>d}{{{({{\sigma }_{j}})}^{2}}}}$ (11)

 $r=\sqrt{\sum\limits_{j\ge d}{{{({{\sigma }_{j}})}^{2}}}}/\sqrt{\sum\limits_{j>d}{{{({{\sigma }_{j}})}^{2}}}}$ (12)
3.2 基于自适应切空间配准算法

4 实验与分析

4.1 构造正定对称的图像特征

 \begin{align} & {{\mathit{\boldsymbol{f}}}_{i}}=[SIFT, x, y, {{I}_{i}}(x y), |\frac{\partial }{\partial x}{{I}_{i}}(x y)|, |\frac{\partial }{\partial y}{{I}_{i}}(x y)| \\ & \sqrt{ |\frac{\partial }{\partial x}{{I}_{i}}(x y){{|}^{^{2}}}+ |\frac{\partial }{\partial y}{{I}_{i}}(x y){{|}^{^{2}}}} |\frac{{{\partial }^{2}}}{\partial x\partial x}{{I}_{i}}(x y)| \\ & |\frac{{{\partial }^{2}}}{\partial y\partial y}{{I}_{i}}(x y)| {{]}^{\rm{T}}} \\ \end{align} (13)

 ${{\mathit{\boldsymbol{S}}}_{i}}={{\mathit{\boldsymbol{f}}}_{i}}{{\mathit{\boldsymbol{f}}}_{i}}^{\rm{T}}$ (14)

4.2 仿真数据实验及分析

 图 2 仿真数据在大脑白质图像上的配准结果 Figure 2 Synthetic results on white matter images
 图 3 仿真数据在大脑灰质图像上的配准结果 Figure 3 Synthetic results on gray matter images

4.3 临床数据实验及分析

 图 4 临床数据的配准结果对比 Figure 4 Comparison of clinical data
 图 5 临床数据中RMSE的对比 Figure 5 RMSE comparison of clinical data
5 结语

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