École doctorale de l ' école Polytechnique Thèse

This chapter is dedicated to the de nition of relevant sets of shapes and that of de ning a metric on them. Following a recent research monograph by Delfour and Zolesio [33], we consider the characteristic functions of the subsets of R2 and their distance functions. The L2 norm of the di erence of characteristic functions, the L∞ and the W 1,2 norms of the di erence of distance functions de ne interesting topologies, in particular the well-known Hausdor distance. Because of practical considerations arising from the fact that we deal with image shapes de ned on nite grids of pixels we restrict our attention to subsets of R2 of positive reach in the sense of Federer [45], with smooth boundaries of bounded curvature. For this particular set of shapes we show that the three previous topologies are equivalent.