论文标题
$ \ mathbb r^2 $的ra transform的新分辨率分析，用于粗糙边缘的功能
Novel resolution analysis for the Radon transform in $\mathbb R^2$ for functions with rough edges
论文作者
论文摘要
令$ f $为$ \ mathbb r^2 $中的函数，它在平滑曲线$ \ mathcal s $ ac and零曲率上具有跳跃。 We consider a family of functions $f_ε$ with jumps across a family of curves $\mathcal S_ε$.每个$ \ MATHCALS_ε$都是$ o（ε）$ - 大小的$ \ Mathcal S $的扰动，它像$ O（ε^{ - 1/2}）$沿$ \ Mathcal s $相似。令$f_ε^{\ text {rec}} $是从其离散ra transform数据中的$f_ε$重建，其中$ε$是数据采样率。一个简单的渐近线（AS $ε\ to0 $）公式，可在任何$ o（ε）$ - 大小$ \ Mathcal s $中近似于$f_ε^{\ text {rec}} $。数值实验表明，即使对于非滑动（即，仅H {Ö} lder连续）$ \ MATHCALS_ε$，该公式也非常准确。在本文中，我们提供了这个结果的完整证明，其中说$f_ε^{\ text {rec}} $之间的误差的幅度，其近似值为$ o（ε^{1/2} \ ln（1/ε））$。主要的假设是函数$ h_0（\ cdot，ε）$的级别集，该级别对扰动$ \ mathcal s \ to \ Mathcals_ε$进行参数不太密度。
Let $f$ be a function in $\mathbb R^2$, which has a jump across a smooth curve $\mathcal S$ with nonzero curvature. We consider a family of functions $f_ε$ with jumps across a family of curves $\mathcal S_ε$. Each $\mathcal S_ε$ is an $O(ε)$-size perturbation of $\mathcal S$, which scales like $O(ε^{-1/2})$ along $\mathcal S$. Let $f_ε^{\text{rec}}$ be the reconstruction of $f_ε$ from its discrete Radon transform data, where $ε$ is the data sampling rate. A simple asymptotic (as $ε\to0$) formula to approximate $f_ε^{\text{rec}}$ in any $O(ε)$-size neighborhood of $\mathcal S$ was derived heuristically in an earlier paper of the author. Numerical experiments revealed that the formula is highly accurate even for nonsmooth (i.e., only H{ö}lder continuous) $\mathcal S_ε$. In this paper we provide a full proof of this result, which says that the magnitude of the error between $f_ε^{\text{rec}}$ and its approximation is $O(ε^{1/2}\ln(1/ε))$. The main assumption is that the level sets of the function $H_0(\cdot,ε)$, which parametrizes the perturbation $\mathcal S\to\mathcal S_ε$, are not too dense.
