Let f(x), x ∈ R2, be a piecewise smooth function with a jump discontinuity across a smooth surface S. Let fΛ denote the Lambda tomography (LT) reconstruction of f from its discrete Radon data f̂(αk, pj). The sampling rate along each variable is ∼ . First, we compute the limit f0(x̌) = lim →0 fΛ (x0 + x̌) for a generic x0 ∈ S. Once the limiting function f0(x̌) is known (which we call the discrete transition behavior, or DTB for short), the resolution of reconstruction can be easily found. Next, we show that straight segments of S lead to non-local artifacts in fΛ , and that these artifacts are of the same strength as the useful singularities of fΛ . We also show that fΛ (x) does not converge to its continuous analogue fΛ = (−∆)1/2f as → 0 even if x 6∈ S. Results of numerical experiments presented in the paper confirm these conclusions. We also consider a class of Fourier integral operators B with the same canonical relation as the classical Radon transform adjoint, and conormal distributions g ∈ E ′(Zn), Zn := Sn−1 × R, and obtain easy to use formulas for the DTB when Bg is computed from discrete data g(α~k, pj). Exact and LT reconstructions are particlular cases of this more general theory.

中文翻译：

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一类分布的离散数据的断层扫描型重建分辨率分析

令 f(x), x ∈ R2 是一个分段平滑函数，在平滑表面 S 上具有跳跃不连续性。令 fΛ 表示 f 从其离散氡数据 f̂(αk, pj) 的 Lambda 断层扫描 (LT) 重建。每个变量的采样率为 ∼ 。首先，我们计算泛型 x0 ∈ S 的极限 f0(x̌) = lim →0​​ fΛ (x0 + x̌)。一旦已知极限函数 f0(x̌)（我们称之为离散转移行为，或简称 DTB） )，可以很容易地找到重建的分辨率。接下来，我们表明 S 的直线段导致 fΛ 中的非局部伪影，并且这些伪影与 fΛ 的有用奇点具有相同的强度。我们还表明，即使 x 6∈ S，fΛ (x) 也不会收敛到它的连续模拟 fΛ = (-∆)1/2f as → 0。论文中给出的数值实验结果证实了这些结论。我们还考虑了一类与经典 Radon 变换伴随具有相同规范关系的傅立叶积分算子 B，以及共正态分布 g ∈ E ′(Zn), Zn := Sn−1 × R，并获得了易于使用的公式当 Bg 由离散数据 g(α~k, pj) 计算时的 DTB。精确和 LT 重建是这种更一般理论的特殊情况。