Limited Data Problems for the Generalized Radon Transform in ℝn
Limited Data Problems for the Generalized Radon Transform in ℝn
复制标题
ℝn 中广义 Radon 变换的有限数据问题
DOI：
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发表时间：
2015
影响因子：
2
通讯作者：
E. T. Quinto
中科院分区：
文献类型：
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作者：
J. Frikel;E. T. Quinto
我们考虑$mathbb{R}^n$中超平面上的广义Radon变换（定义为光滑权函数）。我们分析了一般的滤波反投影型重建方法的有限数据与一般的伪微分算子给出的过滤器。我们在$mathbb{R}^n$中提供可见和添加的奇点的微局部特征，并定义不生成添加伪影的重建算子的修改版本。我们计算了我们的一般重建算子作为伪微分算子的符号，并提供了过滤器的条件下，重建算子是椭圆形的可见奇点。如果根据这些条件选择过滤器，我们表明，几乎所有的可见奇点可以可靠地恢复。我们的工作推广了$mathbb{R}^2$中经典线变换和经典重构算子（使用特定滤波器）的结果。在我们的证明中，我们采用了一个一般的范例，是基于微积分的傅立叶积分算子。由于这种技术不依赖于重建算子的显式表达式，它使我们能够分析更一般的成像情况。
We consider the generalized Radon transform (defined in terms of smooth weight functions) on hyperplanes in $mathbb{R}^n$. We analyze general filtered backprojection type reconstruction methods for limited data with filters given by general pseudodifferential operators. We provide microlocal characterizations of visible and added singularities in $mathbb{R}^n$ and define modified versions of reconstruction operators that do not generate added artifacts. We calculate the symbol of our general reconstruction operators as pseudodifferential operators, and provide conditions for the filters under which the reconstruction operators are elliptic for the visible singularities. If the filters are chosen according to those conditions, we show that almost all visible singularities can be recovered reliably. Our work generalizes the results for the classical line transforms in $mathbb{R}^2$ and the classical reconstruction operators (that use specific filters). In our proofs, we employ a general paradigm that is based on the calculus of Fourier integral operators. Since this technique does not rely on explicit expressions of the reconstruction operators, it enables us to analyze more general imaging situations.