Morphological operators on graph based on geodesic distance map

In this article, we are interested in the graph-based mathematical morphology operators (dilations, erosions, openings, closings, alternated filters) defined in [1] [2]. These operators depend on a size parameter and, as often in mathematical morphology; they are obtained by iterative successions of elementary dilations/erosions. The number of iterations of the elementary operators depends directly of the parameter size. Thus, this leads to running times that increase with respect to the parameter size. In order to optimize this computation time, we propose another algorithmic variant that is based on the computation of geodesic distance maps in graphs. The computed distance map allows us to determine (by thresholding), for any value of the parameter size, dilations and erosions that map a set of vertices to a set of edges and a set of edges to a set of vertices. The proposed algorithm allows the operators to be computed with a single (linear-time) iteration. Therefore, the processing time is improved compared to the time of the multi-iterations original method and does not depend of the parameter size anymore.