七维柠檬和苹果Radon变换的微局部性质及其在康普顿散射层析成像中的应用
Microlocal properties of seven-dimensional lemon and apple Radon transforms with applications in Compton scattering tomography
摘要：
我们提出了一个微局部分析的两个新的Radon变换的兴趣在康普顿散射层析成像（CST），映射compounds支持的$L^2$功能，他们的积分在七维集的苹果和柠檬表面。具体地说，我们证明了苹果和柠檬变换是椭圆傅里叶积分算子（FIO），满足Bolker条件。在分析了完整的七维情形后，我们将注意力集中在具有固定中心轴的苹果曲面和柠檬曲面的$n$ -D子集上，其中$n<7$。这种曲面积分的子集在机场行李和安全检查中有应用。当数据维数受限时，苹果变换违反了Bolker条件，并且在苹果-圆柱相交处出现伪影。证明了当函数的支集限制在条带$\{0<z<1\}$上时，柠檬变换满足Bolker条件.
英文摘要：
We present a microlocal analysis of two novel Radon transforms of interest in Compton Scattering Tomography (CST), which map compactly supported $L^2$ functions to their integrals over seven-dimensional sets of apple and lemon surfaces. Specifically, we show that the apple and lemon transforms are elliptic Fourier Integral Operators (FIO), which satisfy the Bolker condition. After an analysis of the full seven-dimensional case, we focus our attention on $n$-D subsets of apple and lemon surfaces with fixed central axis, where $n<7$. Such subsets of surface integrals have applications in airport baggage and security screening. When the data dimensionality is restricted, the apple transform is shown to violate the Bolker condition, and there are artifacts which occur on apple-cylinder intersections. The lemon transform is shown to satisfy the Bolker condition, when the support of the function is restricted to the strip $\{0<z<1\}$.
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