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2 with the natural Euclidian metric d of the space IR in which X is embedded. Obviously, d < d . X Bx(x,A) A = 1,2,... B (x, A ) A=1,2,... Figure 4-1 : Disks with géodésie metric 4-2 : Disks with Euclidian metric In order to have a better understanding of fig. 4-1, imagine that the particles are a string of ponds, and that a stone is thrown into one of them. A front of ripples appears, and we observe them at successive moments. From the metric d , we can define the géodésie distance between X a point x of X and a subset Y of X. d (x,Y) is the smallest géodésie X distance between x and any point y of Y : dx(x,Y) = inf dx(x,y) y C Y The main interest in the géodésie distance function lies in that it is perfectly suited to deal with connectivity problems. An illustration of this is provided by the following example. Consider two biological images X and Y. X is a population of cells with parts of broken cells, artefacts, etc. Y is the population of the nuclei. X and Y are obtained using a double staining technique (see fig. 5). The only cells that must be studied are the complété cells containing a nucleus. The other ones are just artefacts and must be disregarded. How can we detect cells of X having a nucleus ?