In this communication we propose a new and automatic strategy for the multi-scale extraction of vessels. The objective is to obtain a good representation of the vessels. That is to say a precise characterization of their centerlines and diameters. The adopted solution requires the generation of an image scale-space in which the various levels of details allow to process arteries of any diameter. The proposed method is implemented using the Partial Differential Equations (PDE) and differential geometry formalisms. The differential geometry allows, by the computation of a new valley response, to characterize the centerlines of vessels as well as the bottom lines of the valleys of the image surface. The information given by the centerlines and valley response at different scales are used to obtain the 2D multi-scale centerlines of the arteries. To that purpose, we construct a multi-scale adjacency graph which permits to keep the K strongest detections. Then, the detection we obtain is coded as an attributed graph. The suggested algorithm is applied in the scope of two kinds of angiograms: coronaries and retinal angiograms.