A Topological Characterization of Thinning

Abstract A large number of skeletonization algorithms for binary images use the method of thinning: successive layers of pixels are deleted from the figure until it becomes one pixel thick. In this paper we analyze the topological properties of the set D of pixels to be deleted from a figure F in order to get a skeleton. We characterize them by the concept of strong k -deletability ( k = 4 or 8). For individual pixels, strong k -deletability is equivalent to a more general property that we call k -deletability, which is well-known connectivity requirement assumed—at least implicity—in all existing thinning algorithms. We show then that a strongly k -deletable subset D of a figure F can be deleted by a succession of deletions of individual pixels p 1 ,..., p t , where each p i is k -deletable from F\{ p j ¶ j i }. This justifies our definition of strong deletability and shows that any topologically valid skeleton can be obtained by some thinning process.