This survey paper aims at providing a literary anthology of mathematical morphology on graphs. It describes in the English language many ideas stemming from a large number of different papers, hence providing a unified view of an active and diverse field of research.
In this work we study the framework of mathematical morphology on simplicial complex spaces. Simplicial complexes are widely used to represent multidimensional data, such as meshes, that are two dimensional complexes, or graphs, that can be interpret
ed as one dimensional complexes. Mathematical morphology is one of the most powerful frameworks for image processing, including the processing of digital structures, and is heavily used for many applications. However, mathematical morphology operators on simplicial complex spaces is not a concept fully developed in the literature. Specifically, we explore properties of the dimensional operators, small, versatile operators that can be used to define new operators on simplicial complexes, while maintaining properties from mathematical morphology. These operators can also be used to recover many morphological operators from the literature. Matlab code and additional material, including the proofs of the original properties, are freely available at url{https://code.google.com/p/math-morpho-simplicial-complexes.}
Hierarchical image segmentation provides region-oriented scalespace, i.e., a set of image segmentations at different detail levels in which the segmentations at finer levels are nested with respect to those at coarser levels. Most image segmentation
algorithms, such as region merging algorithms, rely on a criterion for merging that does not lead to a hierarchy, and for which the tuning of the parameters can be difficult. In this work, we propose a hierarchical graph based image segmentation relying on a criterion popularized by Felzenzwalb and Huttenlocher. We illustrate with both real and synthetic images, showing efficiency, ease of use, and robustness of our method.
