A novel concept for a positron emission tomography (PET) camera module is proposed, which provides full 3D reconstruction with high resolution over the total detector volume, free of parallax errors. The key components are a matrix of long scintillator crystals and hybrid photon detectors (HPDs) with matched segmentation and integrated readout electronics. The HPDs read out the two ends of the scintillator package. Both excellent spatial (x, y, z) and energy resolution are obtained. The concept allows enhancing the detection efficiency by reconstructing a significant fraction of events which underwent Compton scattering in the crystals. The proof of concept will first be demonstrated with yttrium orthoaluminate perovskite (YAP):Ce crystals, but the final design will rely on other scintillators more adequate for PET applications (e.g. LSO:Ce or LaBr₃:Ce). A promising application of the proposed camera module, which is currently under development, is a high resolution 3D brain PET camera with an axial field-of-view of ~15 cm dedicated to brain research. The design philosophy and performance predictions based on analytical calculations and Monte Carlo simulations are presented. Image correction and reconstruction tools required to operate this transmissionless device in a research environment are also discussed. Better or similar performance parameters were obtained compared to other known designs at lower fabrication cost. The axial geometrical concept also seems to be promising for applications such as positron emission mammography.