A generalized map is a topological model that allows to represent implicitly differenttypes of cells (vertices, edges, volumes, . . . ) and their relationship by using a set of dartsand some involutions. Generalized maps are used to model 3D meshes and images.Anyway there exists only few tools to compare theses generalized maps. Our main goalis to define some tools tolerant to error to compare them.We define a similarity measure based on the size of the common part of two generalizedmaps, called maximum common submap. Then we define two types of submaps,partial and induced, the induced submap needs to preserve all the involutions whereasthe partial one can allow some involutions to be removed. Then we define a set of operationsto modify a generalized map into another and the associated edit distance. Theedit distance is equal to the minimal cost of all the sequences of operations that modifya generalized map into the other. This edit distance can use labels to consider additionalinformation, with the operation called ’substitution’. Labels are set on darts. Wenext showa relation between our edit distance and the distance based on the maximumcommon submap.Computing theses distance are aNP-hard problem.We propose a greedy algorithmcomputing an approximation of it. We also propose a set of heuristics based on thedescription of the neighborhoob of the darts to help the greedy algorithm.We try thesesheuristics on a set of generalized maps randomly generated where a lower bound of thedistance is known. We also propose some applications of our similarity measures inthe image analysis domain. We compare our edit distance on generalized maps withthe edit distance on graphs. We also define a set of labels specific on images and 3Dmeshes. And we show that the matching computed by our algorithm construct a linkbetween images’s areas.
