The paper studies local convexity properties of parts of dig- ital boundaries. An online and linear-time algorithm is introduced for the decomposition of a digital boundary into convex and concave parts. In addition, other data are computed in the same time without any extra cost: the hull of each convex or concave part as well as the Bezout points of each edge of those hulls. The proposed algorithm involves well- understood algorithms: adding a point to the front or removing a point from the back of a digital straight segment and computing the set of maximal segments. The output of the algorithm is useful either for a polygonal representation of digital boundaries or for a segmentation into circular arcs.