Involved people : Gilles Bertrand, Michel Couprie, Jean-Christophe Everat.
We extended the topological concepts of the 2D binary images to the 2D graylevel images, i.e. to functions whose field is Z 2 (this work can be easily extended to the functions whose field is Z 3 ). For that we consider the different "cross-sections" of a function : let F be a 2D graylevel image, i.e. a function of Z 2 in Z , we call cross-section of F at the level K the subset of Z 2 : F K = { X in Z 2 , f(x) > = K }. We will say that a transformation "preserves the topology" of F if this transformation "preserves the topology" (in the binary sense) of all the cross-sections F K , for any K in Z . The concepts of destructible point and constructible point generalize, within this framework, the usual concept of simple point. More generally, we showed that any point of a graylevel image can be characterized locally by its "topological type" : there are eleven categories of points in this classification.
One obtains a homotopic kernel of an image, by selecting a destructible point, by lowering the value of this point down to a value where it is not destructible any more, and by repeating these operations until stability (fig. 9). We proposed several concepts of homotopic kernel, and defined filtering operators on such kernels. This enabled us to segment graylevel images without using any parameter (fig. 10). The first results of this new approach of the segmentation seem thus very promising [ BEC97 , Eve97 ].
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