A second order variational approach for diffeomorphic matching of 3D surfaces
A second order variational approach for diffeomorphic matching of 3D surfaces
复制标题
DOI：
--
复制
发表时间：
2013-08
期刊：
影响因子：
--
通讯作者：
Yue Qin
中科院分区：
文献类型：
--
作者：
Yue Qin
在医学3D成像中，图像配准的主要目标之一是准确地比较两个观察到的3D形状。在本文中，我们利用基于微分同胚变换的Hilbert空间的变分方法来研究曲面的最优匹配问题。我们首先在抽象的环境下将最优匹配问题描述为最优控制问题，其中向量场流动被寻求最小化由动能和匹配质量组成的成本泛函。为了便于计算，我们将再生核Hilbert空间与Dirac测度的高斯核和加权和结合起来。提出了一种基于Bellman最优性原理的二阶方法，并给出了一种动态规划算法。我们成功地将二阶方法应用于前叶和后叶快照的差同胚匹配。我们得到了由数百个点组成的数据集的二次收敛。为了进一步提高大数据集的计算效率，我们引入了新的形状表示方法，并开发了一种多尺度方法。最后，我们在代价函数中加入了拉伸分数来探索弹性模型，并提供了一个计算上可行的包含弹性能的算法。以医学二尖瓣三维成像减少过度收缩和拉伸为例的数值结果说明了该算法的性能。
In medical 3D-imaging, one of the main goals of image registration is to accurately compare two observed 3D-shapes. In this dissertation, we consider optimal matching of surfaces by a variational approach based on Hilbert spaces of diffeomorphic transformations. We first formulate, in an abstract setting, the optimal matching as an optimal control problem, where a vector field flow is sought to minimize a cost functional that consists of the kinetic energy and the matching quality. To make the problem computationally accessible, we then incorporate reproducing kernel Hilbert spaces with the Gaussian kernels and weighted sums of Dirac measures. We propose a second order method based the Bellman’s optimality principle and develop a dynamic programming algorithm. We apply successfully the second order method to diffeomorphic matching of anterior leaflet and posterior leaflet snapshots. We obtain a quadratic convergence for data sets consisting of hundreds of points. To further enhance the computational efficiency for large data sets, we introduce new representations of shapes and develop a multi-scale method. Finally, we incorporate a stretching fraction in the cost function to explore the elastic model and provide a computationally feasible algorithm including the elasticity energy. The performance of the algorithm is illustrated by numerical results for examples from medical 3D-imaging of the mitral valve to reduce excessive contraction and stretching.