On the local geometry of the pet reconstruction problem

Intrinsic uncertainties related to the ill-posedness of the PET image reconstruction inverse problem are investigated in this work. These uncertainties could lead to instabilities in the reconstructed images in particular when Deep Learning approaches are employed. We propose a framework, based on a Bayesian hypothesis testing, enabling to define various distinguishability measures between PET images and propose a local analysis of such a measure in a neighborhood of a reference image. We test numerically this approach in a synthetic experiment of a Biograph TruePoint TrueV acquisition using a 3D brain PET phantom derived from a [18F]-FDG exam with 100 anatomo-functional regions extracted. Our analysis allows us to highlight the key factors impacting the detectability of variations in a region of interest and to exhibit concrete examples of directions along which variations may not be detectable, and instabilities could appear in PET image reconstruction. In addition, it allows us to conduct a quantitative analysis of the role played by the injected radiotracer dose, and to define thresholds below which clinically significant variations cannot be detected.