Parallelization Strategy for Elementary Morphological Operators on Graphs: Distance-Based Algorithms and Implementation on Multicore Shared-Memory Architecture

This article focuses on the graph-based mathematical morphology operators presented in [J. Cousty et al., "Morphological filtering on graphs", CVIU 2013]. These operators depend on a size parameter that specifies 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 thresholding 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 ini¾?$$On/p + K \log _2 p$$ wherei¾?n,i¾?p, andi¾?K are the size of the graph, the number of available processors, and the number of distinct level-sets of the distance map, respectively.