Microlocal properties of seven-dimensional lemon and apple Radon transforms with applications in Compton scattering tomography

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 $\{0<z<1\}$.