The notion of a connection is a non topological generalisation of connectivity. Its axiomatics lies on that the union of connected components that intersect is still connected. Such an approach yields powerful filters (levelings), and provides the more classical ones with additional properties (openings). Above all, connection opens a new way to formulate optimisation, by acting directly on partition lattices rather than on functionals.
Segmenting the space E on which a function f spreads out is defined as the largest partition of this space into homogeneous regions, according some criterion. It is proved that this problem admits a unique largest solution when the underlying criterion is connective, i.e. when it generates connections over all the subsets where it is satisfied, and uniquely in that case. The family of all possible segmentations of f forms a complete lattice S. This feature allows us to combine the optimal segmentations by means of infima, suprema, and iterations (partial segmentations). The generality of the theorem makes it valid for all functions from any space into any other one. Two propositions make precise the AND and OR combinations of connective criteria.
The soundness of the partition approach is illustrated by listing a comprehensive series of segmentation techniques. They are classified in three categories
Hierarchies of connected filters are approached within this framework. They turn out to be chains of nested partitions in lattice S. Then specific optimisations can be performed for each level of the hierarchy, and result in an optimal segmentation. A distinction is made between weak hierarchies where the partitions increase when going up in the pyramid, and the strong hierarchies where the various levels are structured as semi-groups, and particularly as granulometric semi-groups.