Molecular imaging by PET is a powerful tool in modern clinical practice for cancer diagnosis. Nevertheless, improvements are needed with respect to the spatial resolution and sensitivity of the technique for its application to specific human organs (breast, prostate, brain, etc.), and to small animals. Presently, commercial PET scanners do not detect the depth of interaction of photons in scintillators, which results in a not negligible parallax error. We describe here a novel concept of PET scanner design that provides full three-dimensional (3D) gamma reconstruction with high spatial resolution over the total detector volume, free of parallax errors. It uses matrices of long scintillators read at both ends by hybrid photon detectors. This so-called 3D axial concept also enhances the gamma detection efficiency since it allows one to reconstruct a significant fraction of Compton scattered events. In this note, we describe the concept, a possible design and the expected performance of this new PET device. We also report about first characterization measurements of 10 cm long YAP:Ce scintillation crystals.