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Resolution Analysis of Inverting the Generalized Radon Transform from Discrete Data in $\mathbb{R}^3$
SIAM Journal on Mathematical Analysis ( IF 2 ) Pub Date : 2020-08-20 , DOI: 10.1137/19m1295039
Alexander Katsevich

SIAM Journal on Mathematical Analysis, Volume 52, Issue 4, Page 3990-4021, January 2020.
A number of practically important imaging problems involve inverting the generalized Radon transform (GRT) ${\mathcal R}$ of a function $f$ in $\mathbb{R}^3$. On the other hand, not much is known about the spatial resolution of the reconstruction from discretized data. In this paper we study how accurately and with what resolution the singularities of $f$ are reconstructed. The GRT integrates over a fairly general family of surfaces ${\mathcal S}_y$ in $\mathbb{R}^3$. Here $y$ is the parameter in the data space, which runs over an open set ${\mathcal V}\subset\mathbb{R}^3$. Assume that the data $g(y)=({\mathcal R} f)(y)$ are known on a regular grid $y_j$ with step-sizes $O(\epsilon)$ along each axis, and suppose ${\mathcal S}=\text{singsupp}(f)$ is a piecewise smooth surface. Let $f_\epsilon$ denote the result of reconstruction from the discrete data. We obtain explicitly the leading singular behavior of $f_\epsilon$ in an $O(\epsilon)$-neighborhood of a generic point $x_0\in{\mathcal S}$, where $f$ has a jump discontinuity. We also prove that under some generic conditions on ${\mathcal S}$ (which include, e.g., a restriction on the order of tangency of ${\mathcal S}_y$ and ${\mathcal S}$), the singularities of $f$ do not lead to nonlocal artifacts. For both computations, a connection with the uniform distribution theory turns out to be important. Finally, we present a numerical experiment, which demonstrates a good match between the theoretically predicted behavior and actual reconstruction.


中文翻译:

从$ \ mathbb {R} ^ 3 $中的离散数据反演广义Radon变换的分辨率分析

SIAM数学分析杂志,第52卷,第4期,第3990-4021页,2020年1月。
许多实际重要的成像问题涉及将$ \ mathbb {R} ^ 3 $中的函数$ f $的广义Radon变换(GRT)$ {\ mathcal R} $反转。另一方面,从离散数据重建的空间分辨率知之甚少。在本文中,我们研究了如何精确地重建$ f $的奇点并以何种分辨率进行重构。GRT集成了$ \ mathbb {R} ^ 3 $中相当普遍的曲面$ {\ mathcal S} _y $。$ y $是数据空间中的参数,它在开放集$ {\ mathcal V} \ subset \ mathbb {R} ^ 3 $上运行。假设数据$ g(y)=({\ mathcal R} f)(y)$在规则网格$ y_j $上已知,并且每个轴的步长为$ O(\ epsilon)$,并假定$ { \ mathcal S} = \ text {singsupp}(f)$是分段光滑的表面。令$ f_epsilon $表示从离散数据重建的结果。我们明确获得了在通用点$ x_0 \ in {\ mathcal S} $的$ O(\ epsilon)$邻域中$ f_ \ epsilon $的主导奇异行为,其中$ f $具有跳跃间断。我们还证明,在$ {\ mathcal S} $的某些通用条件下(例如,包括对$ {\ mathcal S} _y $和$ {\ mathcal S} $的相切顺序的限制),奇异性$ f $不会导致非本地工件。对于这两种计算,与均匀分布理论的联系非常重要。最后,我们提出了一个数值实验,证明了理论上预测的行为与实际重建之间的良好匹配。我们明确获得了在通用点$ x_0 \ in {\ mathcal S} $的$ O(\ epsilon)$邻域中$ f_ \ epsilon $的主导奇异行为,其中$ f $具有跳跃间断。我们还证明,在$ {\ mathcal S} $的某些通用条件下(例如,包括对$ {\ mathcal S} _y $和$ {\ mathcal S} $的相切顺序的限制),奇异性$ f $不会导致非本地工件。对于这两种计算,与均匀分布理论的联系非常重要。最后,我们提出了一个数值实验,证明了理论上预测的行为与实际重建之间的良好匹配。我们明确获得了在通用点$ x_0 \ in {\ mathcal S} $的$ O(\ epsilon)$邻域中$ f_ \ epsilon $的主导奇异行为,其中$ f $具有跳跃间断。我们还证明,在$ {\ mathcal S} $的某些通用条件下(例如,包括对$ {\ mathcal S} _y $和$ {\ mathcal S} $的相切顺序的限制),奇异性$ f $不会导致非本地工件。对于这两种计算,与均匀分布理论的联系非常重要。最后,我们提出了一个数值实验,证明了理论上预测的行为与实际重建之间的良好匹配。限制了$ {\ mathcal S} _y $和$ {\ mathcal S} $的相切顺序,$ f $的奇异性不会导致非局部伪像。对于这两种计算,与均匀分布理论的联系非常重要。最后,我们提出了一个数值实验,证明了理论上预测的行为与实际重建之间的良好匹配。限制了$ {\ mathcal S} _y $和$ {\ mathcal S} $的相切顺序,$ f $的奇异性不会导致非局部伪像。对于这两种计算,与均匀分布理论的联系非常重要。最后,我们提出了一个数值实验,证明了理论上预测的行为与实际重建之间的良好匹配。
更新日期:2020-08-20
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