Most background noise in medical Positron Emission Tomography (PET) imaging originates from photons which Compton scatter before they leave the body of the patient. In only 4% of total events do both photons leave the body un-scattered and are therefore useful for current PET imaging. In an additional 8% of the events one of the photons scatters only once. This class of events can be analyzed correctly with Compton reconstruction. A detector with sufficiently good energy, position, and multiple hit resolution can recognize these events and process them properly. By virtue of recent technological progress, Liquid Xenon Time Projection Chambers (LXeTPCs) match these requirements. A ring of small LXeTPCs can measure the energy with better than 4% FWHM and the position with less than 1 mm RMS. It also has the capability to reconstruct all interaction points. These data are required for Compton imaging. With an improved energy resolution over existing crystal scintillators (LSO/LYSO,BGO and GSO), more of the remaining background noise is rejected, instead of being erroneously assumed as full energy peak events.