Computing large deformation metric mappings via geodesic flows of diffeomorphisms
Computing large deformation metric mappings via geodesic flows of diffeomorphisms
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DOI：
10.1023/b:visi.0000043755.93987.aa
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发表时间：
2005-02-01
影响因子：
19.5
通讯作者：
Younes, L
中科院分区：
文献类型：
--
作者：
Beg, MF;Miller, MI;Younes, L
本文研究了Dupuis等（1998）和Trouve（1995）研究的大变形双纯度量映射问题的Euler-Lagrange方程，其中两个图像10，1通过坐标的双纯变换I-0 o φ（-1）= I连接，其中p = 01是曲线φ（t）在t = 1时的端点，t是[0，1]中满足（phi）over dot（t）= v（t）（phi（t））的元素，t是[0，1]中具有phi（0）= id的元素。变分问题采取[GRAPHICS]的形式，其中平行于（V）的平行于（t）是速度场v（t）（.）上的适当Sobolev范数，第二项是图像的匹配，平行于，平行于（L2）表示平方误差范数，本文导出了描述极小化向量场的Euler-Lagrange方程，其中t是[0，1]的元素，假设范数足够光滑，以保证解的存在性。我们描述了使用半拉格朗日方法计算粒子流的欧拉方程的实现，并显示了各种例子的解决方案。同样，我们计算度量距离的几个解剖结构上测量的积分（0）（1）平行于（v）dt的测地线最短路径上的tov（t）平行。
This paper examine the Euler-Lagrange equations for the solution of the large deformation diffeomorphic metric mapping problem studied in Dupuis et al. (1998) and Trouve (1995) in which two images 10, 1, are given and connected via the diffeomorphic change of coordinates I-0 o phi(-1) = I, where p = 01 is the end point at t = 1 of curve phi(t), t is an element of [0, 1] satisfying (phi)over dot(t) = v(t)(phi(t)), t is an element of [0, 1] with phi(0) = id. The variational problem takes the form[GRAPHICS]where parallel tov(t)parallel to(V) is an appropriate Sobolev norm on the velocity field v(t)(.), and the second term enforces matching of the images with parallel to.parallel to(L2) representing the squared-error norm.In this paper we derive the Euler-Lagrange equations characterizing the minimizing vector fields vt(,) t is an element of [0, 1] assuming sufficient smoothness of the norm to guarantee existence of solutions in the space of diffeomorphisms. We describe the implementation of the Euler equations using semi-lagrangian method of computing particle flows and show the solutions for various examples. As well, we compute the metric distance on several anatomical configurations as measured by integral(0)(1) parallel tov(t)parallel to(v)dt on the geodesic shortest paths.