FilDReaMS 1. Presentation of a new method for Filament Detection and Reconstruction at Multiple Scales

15 pages, 11 figures, 5 tablesContext. Filamentary structures appear to be ubiquitous in the interstellar medium. Being able to detect and characterize them is the first step toward understanding their origin, their evolution, and their role in the Galactic cycle of matter. Aims. We present a new method, called FilDReaMS, to detect and analyze filaments in a given image. This method is meant to be fast, user-friendly, multi-scale, and suited for statistical studies. Methods. The input image is scanned with a rectangular model bar, which makes it possible to uncover structures that can be locally approximated by this bar and to derive their orientations. The bar width can be varied over a broad range of values to probe filaments of different widths. Results. We performed several series of tests to validate the method and to assess its sensitivity to the level of noise, the filament aspect ratios, and the dynamic range of filament intensities. We found that the method exhibits very good performance at recovering the orientation of the filamentary structures, with an accuracy of 0.5{\deg} in nominal conditions, up to 3{\deg} in the worst case scenario with high level of noise. The filaments width is recovered with uncertainties better than 0.5 px (pixels) in most of the cases, which could extend up to 3 px in case of low signal-to-noise ratios. Some attempt to build a correspondence between Plummer-type filament profiles and outcomes of the method is proposed, but remains sensitive to the local environment. Conclusions. Our method is found to be robust and adapted to the identification and the reconstruction of filamentary structures in various environments, from diffuse to dense medium. It allows us to explore the hierarchical scales of these filamentary structures with a high reliability, especially when dealing with their orientation