Firearm identification is crucial in many investigative scenario. The crime scene often contains traces left by firearms in terms of bullets and cartridges. Traces analysis is a fundamental step in the Forensics Ballistics Analysis Process to identify which firearm fired a specific cartridge. In this paper we present a fully automated technique to compare cartridges represented as a set of 3D point-clouds. The overall approach is based on Siamese Neural Network learning paradigm that we use to build a suitable embedding space where the 3D point-cloud of the cartridges are compared. The proposed approach has been assessed by considering the NBTRD dataset. Obtained results support the exploitation of the proposed technique in ballistic analysis.