Pattern Spectra from Partition Pyramids and Hierarchies

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Constrained connectivity relations partition the image definition domain into connected sets of maximal extent. The homogeneity and maximality of the resulting cells is subject to non-connective criteria that associate to logical predicates. The latter are based on attribute metrics, and are used to counter the leakage effect of the single-linkage clustering rule. Linkage is controlled by some dissimilarity measure and if unconstrained, may be used to generate regular connectivity classes. In this paper we introduce a hierarchical partition representation structure to map the evolution of components along the dissimilarity range in the absence of constraints. By contrast to earlier approaches, constraints may be put in place on the actual structure in the form of filters. This allows for custom and interactive segmentation of the image. Moreover, given an instance of the dissimilarity measure, one can retrieve all connected sets making up the corresponding connectivity class, directly from the hierarchy. The evolution of linkage relations with respect to the attributes on which the predicates are based on is used to compute a new type of pattern spectrum that is demonstrated on two real applications.