Thread-like structures are pervasive across scales, from polymeric proteins to root systems to galaxy filaments, and their characteristics can be readily investigated in the network formalism. Yet, network links usually represent only parts of filaments, which, when neglected, may lead to erroneous conclusions from network-based analyses. The existing alternatives to detect filaments in network representations require tuning of parameters over a large range of values and treat all filaments equally, thus, precluding automated analysis of diverse filamentous systems. Here, we propose a fully automated and robust optimisation-based approach to detect filaments of consistent intensities and angles in a given network. We test and demonstrate the accuracy of our solution with contrived, biological, and cosmic filamentous structures. In particular, we show that the proposed approach provides powerful automated means to study properties of individual actin filaments in their network context. Our solution is made publicly available as an open-source tool, "DeFiNe", facilitating decomposition of any given network into individual filaments.