关于有限数据球面Radon变换中的伪影：平坦观测面
On Artifacts in Limited Data Spherical Radon Transform: Flat Observation Surfaces
摘要：
在这篇文章中，我们描述了强度的重建奇点和伪影的重建公式有限的数据球面Radon变换。也就是说，我们假设数据仅在$\mathbb{R}^n$（$n=2,3$）中的超平面的闭子集$Γ$上可用。我们认为，在一些以前的作品中研究的重建公式，假设数据只平滑到有穷级$k$附近的边界。对于二维空间中的问题，并且$Γ$是一条线段，伪影是通过沿着以$Γ$的端点为中心的圆旋转边界奇异性来产生的。我们表明，文物的$k$订单比原来的奇异性。对于三维空间中的问题，并且$Γ$是一个矩形，我们描述了伪影是通过旋转围绕$Γ$的顶点或边的边界奇异性产生的。通过围绕顶点旋转获得的伪影比原始奇异性平滑$2k$阶。同时，通过围绕边缘旋转获得的伪影比原始奇异性平滑$k$阶。对于二维和三维的问题，可见的奇点重建与正确的顺序。因此，我们成功地量化了J. Frikel和T.昆托
英文摘要：
In this article, we characterize the strength of the reconstructed singularities and artifacts in a reconstruction formula for limited data spherical Radon transform. Namely, we assume that the data is only available on a closed subset $Γ$ of a hyperplane in $\mathbb{R}^n$ ($n=2,3$). We consider a reconstruction formula studied in some previous works, under the assumption that the data is only smoothened out to a finite order $k$ near the boundary. For the problem in the two dimensional space and $Γ$ is a line segment, the artifacts are generated by rotating a boundary singularity along a circle centered at an end point of $Γ$. We show that the artifacts are $k$ orders smoother than the original singularity. For the problem in the three dimensional space and $Γ$ is a rectangle, we describe that the artifacts are generated by rotating a boundary singularity around either a vertex or an edge of $Γ$. The artifacts obtained by a rotation around a vertex are $2k$ orders smoother than the original singularity. Meanwhile, the artifacts obtained by a rotation around an edge are $k$ orders smoother than the original singularity. For both two and three dimensional problems, the visible singularities are reconstructed with the correct order. We, therefore, successfully quantify the geometric results obtained recently by J. Frikel and T. Quinto.
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