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Elongated black objects in black-and-white pictures can be ``thinned'' to arcs and curves, without changing their connectedness, by (repeatedly) deleting black border points whose deletion does not locally disconnect the black points in their neighborhoods. This technique generalizes to gray-scale pictures if we use a weighted definition of connectedness: two points are ``connected'' if there is a path joining them on which no point is lighter than either of them. We can then ``thin'' dark objects by changing each point's gray level to the minimum of its neighbors' gray levels, provided this does not disconnect any pair of points in its neighborhood. Examples illustrating the performance of this technique are given.