Uniqueness of the Perfect Fusion Grid on Zd

Authors:
Jean Cousty;Gilles Bertrand
Affiliations:
INRIA, Asclepios Research Project, Sophia Antipolis cedex, France and Université Paris-Est, Laboratoire d'Informatique Gaspard-Monge, équipe A3SI, ESIEE, Noisy-le-Grand, France;Université Paris-Est, Laboratoire d'Informatique Gaspard-Monge, équipe A3SI, ESIEE, Noisy-le-Grand, France
Venue:
Journal of Mathematical Imaging and Vision
Year:
2009

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Abstract

Region merging methods consist of improving an initial segmentation by merging some pairs of neighboring regions. In a graph, merging two regions, separated by a set of vertices, is not straightforward. The perfect fusion graphs defined in J. Cousty et al. (J. Math. Imaging Vis. 30:(1):87---104, 2008) verify all the basic properties required by region merging algorithms as used in image segmentation. Unfortunately, the graphs which are the most frequently used in image analysis (namely, those induced by the direct and the indirect adjacency relations) are not perfect fusion graphs. The perfect fusion grid, introduced in the above mentioned reference, is an adjacency relation on Zd which can be used in image analysis, which indeed induces perfect fusion graphs and which is "between" the graphs induced by the direct and the indirect adjacencies. One of the main results of this paper is that the perfect fusion grid is the only such graph whatever the dimension d.