Injectivity of the Heisenberg X-ray transform
Injectivity of the Heisenberg X-ray transform
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DOI：
10.1016/j.jfa.2020.108886
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发表时间：
2020-04
影响因子：
1.7
通讯作者：
Steven Flynn
中科院分区：
文献类型：
--
作者：
Steven Flynn
我们在次黎曼流形上开始了X射线层析成像的研究，其中海森堡群是最简单的非平凡例子。用群傅里叶变换的语言，我们证明了傅里叶切片定理的算子值化身，并应用这个新工具来证明海森堡群上的一个充分正则函数是由它在次黎曼测地线上的线积分决定的。我们还考虑了驯服度量g的家庭近似的次黎曼度量，并证明了相关的X射线变换是内射的所有<$0。这一结果给出了一个具体的例子，内射X射线变换在一个几何丰富的共轭点。
We initiate the study of X-ray tomography on sub-Riemannian manifolds, for which the Heisenberg group exhibits the simplest nontrivial example. With the language of the group Fourier transform, we prove an operator-valued incarnation of the Fourier Slice Theorem, and apply this new tool to show that a sufficiently regular function on the Heisenberg group is determined by its line integrals over sub-Riemannian geodesics. We also consider the family of taming metrics g ϵ approximating the sub-Riemannian metric, and show that the associated X-ray transform is injective for all ϵ> 0. This result gives a concrete example of an injective X-ray transform in a geometry with an abundance of conjugate points.