A Lattice Approach to Image Segmentation

After a formal definition of segmentation as the largest partition of the space according to a criterion σ and a function f, the notion of a morphological connection is reminded. It is used as an input to a central theorem of the paper (Theorem 8), that identifies segmentation with the connections that are based on connective criteria. Just as connections, the segmentations can then be regrouped by suprema and infima. The generality of the theorem makes it valid for functions from any space to any other one. Two propositions make precise the AND and OR combinations of connective criteria. The soundness of the approach is demonstrated by listing a series of segmentation techniques. One considers first the cases when the segmentation under study does not involve initial seeds. Various modes of regularity are discussed, which all derive from Lipschitz functions. A second category of examples involves the presence of seeds around which the partition of the space is organized. An overall proposition shows that these examples are a matter for the central theorem. Watershed and jump connection based segmentations illustrate this type of situation. The third and last category of examples deals with cases when the segmentation occurs in an indirect space, such as an histogram, and is then projected back on the actual space under study. The relationships between filtering and segmentation are then investigated. A theoretical chapter introduces and studies the two notions of a pulse opening and of a connected operator. The conditions under which a family of pulse openings can yield a connected filter are clarified. The ability of segmentations to generate pyramids, or hierarchies, is analyzed. 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. The last section is based on one example, and goes back over the controversy about “lattice” versus “functional” optimization. The problem is now tackled via a case of colour segmentation, where the saturation serves as a cursor between luminance and hue. The emphasis is put on the difficulty of grouping the various necessary optimizations into a single one.