An efficient divide-and-conquer algorithm for morphological filters

Morphological filters, evolved from the traditional envelope filter, are function oriented filtration techniques. A recent research on the implementation of morphological filters was based on the theoretical link between morphological operations and the alpha shape. However the Delaunay triangulation on which the alpha shape method depends is costly for large areal data. This paper proposes a divide-and-conquer method as an optimization to the alpha shape method aiming to speed up its performance. The large areal surface is divided into small sub-surfaces so that the alpha shape method is executed on the partitioned surfaces in a fast manner. The contact points are searched on each sub-surface and merged into a super set on which the alpha shape method is applied again to archive the updated result. The recursion process is repeated until the contact points of the whole surface are obtained. The morphological envelope could be computed recursively without the 3D Delaunay triangulation to the whole surface data. Meanwhile this method retains almost all the merits of the alpha shape method. The experiment shows that the result obtained by the divide-and-conquer algorithm is consistent with the one generated by applying the alpha shape method directly. The performance evaluation reveals that the divide-and-conquer algorithm achieved superior performances over the original alpha shape method.