Flooding edge or node weighted graphs
Résumé
Reconstruction closings have all properties of a physical flooding of a topographic surface. They are precious for simplifying gradient images or, filling unwanted catchment basins, on which a subsequent watershed transform extracts the targeted objects. Flooding a topographic surface may be modeled as flooding a node weighted graph (TG), with unweighted edges, the node weights representing the ground level. The progression of a flooding may also be modeled on the region adjacency graph (RAG) of a topographic surface. On a RAG each node represents a catchment basin and edges connect neighboring nodes. The edges are weighted by the altitude of the pass point between both adjacent regions. The graph is flooded from sources placed at the marker positions and each node is assigned to the source by which it has been flooded. The level of the flood is represented on the nodes on each type of graphs. The same flooding may thus be modeled on a TG or on a RAG. We characterize all valid floodings on both types of graphs, as they should verify the laws of hydrostatics. We then show that each flooding of a node weighted graph also is a flooding of an edge weighted graph with appropriate edge weights. The highest flooding under a ceiling function may be interpreted as the shortest distance to the root for the ultrametric flooding distance in an augmented graph. The ultrametric distance between two nodes is the minimal altitude of a flooding for which both nodes are flooded. This remark permits to flood edge or node weighted graphs by using shortest path algorithms. It appears that the collection of all lakes of a RAG has the structure of a dendrogram, on which the highest flooding under a ceiling function may be rapidly found.