The geodesic X-ray transform with matrix weights
The geodesic X-ray transform with matrix weights
复制标题
具有矩阵权重的测地 X 射线变换
DOI：
10.1353/ajm.2019.0045
复制
发表时间：
2016
影响因子：
1.7
通讯作者：
Hanming Zhou
中科院分区：
文献类型：
--
作者：
G. Paternain;M. Salo;G. Uhlmann;Hanming Zhou
摘要：考虑具有严格凸边界的Dimension $ \ geq 3 $的紧凑型Riemannian歧管，以便歧管允许严格的凸功能。我们表明，在任意连接的存在下，衰减的射线变换是射层模量的自然障碍物的功能和一种形式的障碍物。我们还表明，连接和希格斯字段是由散射关系模量仪表转换唯一决定的。证据涉及将局部结果减少到局部结果，表明可以在边界处的严格凸度点附近将具有矩阵重量的大地X射线变换在本地倒置，并且基于严格凸出的填充功能对图层剥离参数进行了详细分析。作为一种有些惊人的推论，我们表明这些积分几何问题可以在尺寸$ \ geq 3 $具有非负截面曲率的严格凸流歧管上解决（较早以负截面曲率为单位）。我们还采用我们的方法来解决量子状态层析成像和极化层析成像中的一些反问题。
Abstract:Consider a compact Riemannian manifold of dimension $\geq 3$ with strictly convex boundary, such that the manifold admits a strictly convex function. We show that the attenuated ray transform in the presence of an arbitrary connection and Higgs field is injective modulo the natural obstruction for functions and one-forms. We also show that the connection and the Higgs field are uniquely determined by the scattering relation modulo gauge transformations. The proofs involve a reduction to a local result showing that the geodesic X-ray transform with a matrix weight can be inverted locally near a point of strict convexity at the boundary, and a detailed analysis of layer stripping arguments based on strictly convex exhaustion functions. As a somewhat striking corollary, we show that these integral geometry problems can be solved on strictly convex manifolds of dimension $\geq 3$ having non-negative sectional curvature (similar results were known earlier in negative sectional curvature). We also apply our methods to solve some inverse problems in quantum state tomography and polarization tomography.
黎曼流形中最小子流形的等周不等式：高维余维中的反例
DOI：
10.1007/s00526-011-0466-z
发表时间：
2012
影响因子：
2.1
作者：
Bangert;R öttgen
通讯作者：
R öttgen