Polynomial algorithms for subisomorphism of nD open combinatorial maps

Combinatorial maps describe the subdivision of objects in cells, and incidence and adjacency relations between cells, and they are widely used to model 2D and 3D images. However, there is no algorithm for comparing combinatorial maps, which is an important issue for image processing and analysis. In this paper, we address two basic comparison problems, i.e., map isomorphism, which involves deciding if two maps are equivalent, and submap isomorphism, which involves deciding if a copy of a pattern map may be found in a target map. We formally define these two problems for nD open combinatorial maps, we give polynomial time algorithms for solving them, and we illustrate their interest and feasibility for searching patterns in 2D and 3D images, as any child would aim to do when he searches Wally in Martin Handford’s books.

► Combinatorial maps are an extension of planar graphs in any dimension.
► We formally define map and submap isomorphism for nD open combinatorial maps.
► We give polynomial time algorithms for solving both problems.
► We illustrate the interest of these tools by searching 2D and 3D patterns.
► Comparisons with subgraph isomorphism demonstrate the interest of our approach.