论文标题
分析一系列分布的离散数据中断层型重建的分辨率
Analysis of resolution of tomographic-type reconstruction from discrete data for a class of distributions
论文作者
论文摘要
令$ f（x）$，$ x \ in \ mathbb r^2 $，成为一个分段平滑功能，在光滑的表面$ \ mathcal s $上不连续性不连续。令$ f_ {λε} $表示$ f $的lambda断层扫描（LT）从其离散rad data $ \ hat f（α_k，p_j）$重建。沿每个变量的采样率为$ \simε$。首先，我们计算限制$ f_0（\ check x）= \ lim_ {ε\ to0}εf_{λε}（x_0+ε\ check x）$ for generic $ x_0 \ in \ in \ mathcal s $。一旦限制函数$ f_0（\ check x）$已知（我们称为离散过渡行为，或简短的DTB），可以很容易地找到重建的分辨率。接下来，我们表明$ \ mathcal s $的直段导致$ f_ {λε} $的非本地文物，并且这些文物与$ f_ {λε} $的有用奇异性具有相同的强度。我们还表明，$ f_ {λε}（x）$不会收敛到其连续模拟$f_λ=（ - δ）^{1/2} f $ as $ε\ to0 $，即使$ x \ not \ in \ in \ Mathcal s $也是如此。本文提出的数值实验结果证实了这些结论。我们还考虑了一类傅立叶积分运算符$ \ MATHCAL {B} $，具有与经典rad transfort相同的规范关系，以及一类分布$ g \ in \ Mathcal {e}'（z_n）$，$ z_n：$ z_n：= s^{n-1} {n-1} {n-1} {n-1} {n-1} \ times \ mathbb r $，以及ers wers wers wers wers wers wers wers wers wers wers wers wers n时何时使用。 $ \ MATHCAL {B} G $是从离散数据$ g（α_ {\ vec k}，p_j）$计算的。精确和LT重建是这种更通用理论的特殊情况。
Let $f(x)$, $x\in\mathbb R^2$, be a piecewise smooth function with a jump discontinuity across a smooth surface $\mathcal S$. Let $f_{Λε}$ denote the Lambda tomography (LT) reconstruction of $f$ from its discrete Radon data $\hat f(α_k,p_j)$. The sampling rate along each variable is $\simε$. First, we compute the limit $f_0(\check x)=\lim_{ε\to0}εf_{Λε}(x_0+ε\check x)$ for a generic $x_0\in\mathcal S$. Once the limiting function $f_0(\check x)$ 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 $\mathcal S$ lead to non-local artifacts in $f_{Λε}$, and that these artifacts are of the same strength as the useful singularities of $f_{Λε}$. We also show that $f_{Λε}(x)$ does not converge to its continuous analogue $f_Λ=(-Δ)^{1/2}f$ as $ε\to0$ even if $x\not\in\mathcal S$. Results of numerical experiments presented in the paper confirm these conclusions. We also consider a class of Fourier integral operators $\mathcal{B}$ with the same canonical relation as the classical Radon transform adjoint, and a class of distributions $g\in\mathcal{E}'(Z_n)$, $Z_n:=S^{n-1}\times\mathbb R$, and obtain easy to use formulas for the DTB when $\mathcal{B} g$ is computed from discrete data $g(α_{\vec k},p_j)$. Exact and LT reconstructions are particlular cases of this more general theory.
