Inversion of the Attenuated Geodesic X-Ray Transform over Functions and Vector Fields on Simple Surfaces
Inversion of the Attenuated Geodesic X-Ray Transform over Functions and Vector Fields on Simple Surfaces
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简单曲面上函数和矢量场的衰减测地X射线变换的反演
DOI：
10.1137/15m1016412
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发表时间：
2015
期刊：
影响因子：
--
通讯作者：
F. Monard
中科院分区：
文献类型：
--
作者：
F. Monard
我们推导了函数上的衰减测地 X 射线变换的显式重构公式，并且在非零衰减的情况下，在一类具有边界的简单黎曼表面上，导出了矢量场。这些公式部分依赖于新的显式方法来构造反投影算子的连续右逆（以及全纯积分因子），而这在以前是无法以系统形式获得的。函数的重构以两种方式呈现，后者是由数值考虑推动的，并在本文的最后一节（附录之前）成功实现。构造上述右逆矩阵需要某些 Fredholm 方程，首先出现在 [L. Pestov 和 G. Uhlmann，Int。数学。资源。 Not.，2004 (2004)，第 4331--4347 页]，可颠倒。最后一个条件是否会降低整体方法对简单曲面的严格子集的适用性目前仍然悬而未决。
We derive explicit reconstruction formulas for the attenuated geodesic X-ray transform over functions and, in the case of nonvanishing attenuation, vector fields, on a class of simple Riemannian surfaces with boundary. These formulas partly rely on new explicit approaches to construct continuous right-inverses for backprojection operators (and, in turn, holomorphic integrating factors), which were previously unavailable in a systematic form. The reconstruction of functions is presented in two ways, the latter being motivated by numerical considerations and successfully implemented in the last section of the paper (before the appendix). Constructing the aforementioned right-inverses requires that certain Fredholm equations, first appearing in [L. Pestov and G. Uhlmann, Int. Math. Res. Not., 2004 (2004), pp. 4331--4347], be invertible. Whether this last condition reduces the applicability of the overall approach to a strict subset of simple surfaces remains open at present.