A graph-based mathematical morphology reader

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

A uniqueness property of perfect fusion grids

electronic version (6 pp.)International audienc

Raising in watershed lattices

International audienc

New characterizations of minimum spanning trees and of saliency maps based on quasi-flat zones

We study three representations of hierarchies of partitions: dendrograms (direct representations), saliency maps, and minimum spanning trees. We provide a new bijection between saliency maps and hierarchies based on quasi-flat zones as used in image processing and characterize saliency maps and minimum spanning trees as solutions to constrained minimization problems where the constraint is quasi-flat zones preservation. In practice, these results form a toolkit for new hierarchical methods where one can choose the most convenient representation. They also invite us to process non-image data with morphological hierarchies

Dimensional operators for mathematical morphology on simplicial complexes

International audienceIn 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 interpreted 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.

Weighted fusion graphs: merging properties and watersheds

International audienc

Parallelization Strategy for Elementary Morphological Operators on Graphs

International audienceThis article focuses on the graph-based mathematical morphology operators presented in [J. Cousty et al, Morphological ltering on graphs, CVIU 2013]. These operators depend on a size parameter that species the number of iterations of elementary dilations/erosions. Thus, the associated running times increase with the size parameter. In this article, we present distance maps that allow us to recover (by thresh-olding) all considered dilations and erosions. The algorithms based on distance maps allow the operators to be computed with a single linear-time iteration, without any dependence to the size parameter. Then, we investigate a parallelization strategy to compute these distance maps. The idea is to build iteratively the successive level-sets of the distance maps, each level set being traversed in parallel. Under some reasonable assumptions about the graph and sets to be dilated, our parallel algorithm runs in O(n/p + K log 2 p) where n, p, and K are the size of the graph, the number of available processors, and the number of distinct level-sets of the distance map, respectively

Join, select, and insert: efficient out-of-core algorithms for hierarchical segmentation trees

Binary Partition Hierarchies (BPH) and minimum spanning trees are fundamental data structures involved in hierarchical analysis such as quasi-flat zones or watershed. However, classical BPH construction algorithms require to have the whole data in memory, which prevent the processing of large images that cannot fit entirely in the main memory of the computer. To cope with this problem, an algebraic framework leading to a high level calculus was introduced allowing an out-of-core computation of BPHs. This calculus relies on three operations: select, join, and insert. In this article, we introduce three efficient algorithms to perform these operations providing pseudo-code and complexity analysis

Discrete region merging and watersheds

incollectionThis paper summarizes some results of the authors concerning watershed divides and their use in region merging schemes.The first aspect deals with properties of watershed divides that can be used in particular for hierarchical region merging schemes. We introduce the mosaic to retrieve the altitude of points along the divide set. A desirable property is that, when two minima are separated by a crest in the original image, they are still separated by a crest of the same altitude in the mosaic. Our main result states that this is the case if and only if the mosaic is obtained through a topological thinning.The second aspect is closely related to the thinness of watershed divides. We present fusion graphs, a class of graphs in which any region can be always merged without any problem. This class is equivalent to the one in which watershed divides are thin. Topological thinnings do not always produce thin divides, even on fusion graphs. We also present the class of perfect fusion graphs, in which any pair of neighbouring regions can be merged through their common neighborhood. An important theorem states that the divides of any ultimate topological thinning are thin on any perfect fusion graph