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

TL;DR

This paper analyzes the microlocal properties of seven-dimensional apple and lemon Radon transforms used in Compton Scattering Tomography, revealing their ellipticity, Bolker condition satisfaction, and implications for imaging artifacts.

Contribution

It provides a microlocal analysis showing that the apple and lemon Radon transforms are elliptic FIOs satisfying the Bolker condition under certain restrictions, with implications for imaging applications.

Findings

Apple transform violates Bolker condition on restricted data

Lemon transform satisfies Bolker condition in specific support

Artifacts occur on apple-cylinder intersections

Abstract

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