CT scans without X-rays: Parallel-beam imaging from nonlinear current flows

Parallel-beam X-ray computed tomography (CT) and electrical impedance tomography (EIT) are two imaging modalities that stem from completely different underlying physics, and for decades have been thought to have little in common either practically or mathematically. CT is only mildly ill-posed and uses straight X-rays as measurement energy, which admits simple linear mathematics. However, CT relies on exposing targets to ionizing radiation and requires cumbersome setups with expensive equipment. In contrast, EIT uses harmless electrical currents as measurement energy and can be implemented using simple low-cost portable setups. But EIT is burdened by nonlinearity stemming from the curved paths of electrical currents, as well as extreme ill-posedness that causes characteristic low spatial resolution. In practical EIT reconstruction methods, nonlinearity and ill-posedness have been considered intertwined in a complicated fashion. In this work, we demonstrate a surprising connection between CT and EIT, first announced in the theoretical work by Greenleaf et al., 2018, which partly unravels the main problems of EIT and leads directly to a proposed reconstruction technique that we call virtual hybrid parallel-beam tomography (VHPT). We show that hidden deep within EIT data is information that possesses the same linear geometry as parallel-beam CT data. This admits a fundamental restructuring of EIT, separating ill-posedness and nonlinearity into simple modular sub-problems, and yields 'virtual radiographs' and CT-like images that reveal previously concealed information. Furthermore, as proof of concept, we present VHPT images of simulated and experimentally collected data.