论文标题
从散射断层扫描中的广义ra转换的微局部分析
Microlocal analysis of generalized Radon transforms from scattering tomography
论文作者
论文摘要
在这里，我们提出了对广义ra变换的新型微局部分析，描述了$ l^2 $紧凑型支持对$ c^{\ infty} $ curves $ q $的积分的积分。我们表明，ra变换是椭圆傅里叶积分算子（FIO），并提供了左图$π_l$的分析。我们的主要定理表明，$π_l$在且仅当$ g = q'/q $是沉浸式时就满足半全球Bolker的假设。提出了对可见奇异性的分析，之后我们得出了rafio的新型Sobolev平滑度估计。我们的理论在康普顿散射断层扫描（CST）和布拉格散射断层扫描（BST）中具有特定的兴趣应用。我们表明，CST和BST集成曲线满足了Bolker假设，并提供了CST和BST数据的模拟重建。此外，我们提供了不满足Bolker并提供图像伪像的模拟的“正弦”整合曲线的示例。重建中观察到的伪影被证明与我们的预测完全一致。
Here we present a novel microlocal analysis of generalized Radon transforms which describe the integrals of $L^2$ functions of compact support over surfaces of revolution of $C^{\infty}$ curves $q$. We show that the Radon transforms are elliptic Fourier Integral Operators (FIO) and provide an analysis of the left projections $Π_L$. Our main theorem shows that $Π_L$ satisfies the semi-global Bolker assumption if and only if $g=q'/q$ is an immersion. An analysis of the visible singularities is presented, after which we derive novel Sobolev smoothness estimates for the Radon FIO. Our theory has specific applications of interest in Compton Scattering Tomography (CST) and Bragg Scattering Tomography (BST). We show that the CST and BST integration curves satisfy the Bolker assumption and provide simulated reconstructions from CST and BST data. Additionally we give example "sinusoidal" integration curves which do not satisfy Bolker and provide simulations of the image artefacts. The observed artefacts in reconstruction are shown to align exactly with our predictions.
