【正文】 shape plexity of the given set. In the literature the term \thinning is not used 2 in a unique interpretation besides that it always denotes a connectivity preserving reduction operation applied to digital images, involving iterations of transformations of speci_ed contour points into background points. A subset Q _ I of object points is reduced by a de_ned set D in one iteration, and the result Q0 = Q n D bees Q for the next iteration. Topologypreserving skeletonization is a special case of thinning resulting in a connected set of digital arcs or curves. A digital curve is a path p =p0。 p1。 :::。 I(p)) of an image are pixels (2D case) or voxels (3D case). The range of a (scalar) image is f0。 1g. We only use binary images I in this report. Let hIi be the set of all pixel locations with value 1, . hIi = I?1(1). The image carrier is de_ned on an orthogonal grid in 2D or 3D 3 space. There are two options: using the grid cell model a 2D pixel location p is a closed square (2cell) in the Euclidean plane and a 3D pixel location is a closed cube (3cell) in the Euclidean space, where edges are of length 1 and parallel to the coordinate axes, and centers have integer coordinates. As a second option, using the grid point model a 2D or 3D pixel location is a grid point. Two pixel locations p and q in the grid cell model are called 0adjacent i_ p 6= q and they share at least one vertex (which is a 0cell). Note that this speci_es 8adjacency in 2D or 26adjacency in 3D if the grid point model is used. Two pixel locations p and q in the grid cell model are called 1 adjacent i_ p 6= q and they share at least one edge (which is a 1cell). Note that this speci_es 4adjacency in 2D or 18adjacency in 3D if the grid point model is used. Finally, two 3D pixel locations p and q in the grid cell model are called 2adjacent i_ p 6= q and they share at least one face (which is a 2cell). Note that this speci_es 6adjacency if the grid point model is used. Any of these adjacency relations A_, _ 2 f0。 2。 6。 26g, is irreexive and symmetric on an image carrier C. The _neighborhood N_(p) of a pixel location p includes p and its _adjacent pixel locations. Coordinates of 2D grid points are denoted by (i。 i。m are the numbers of rows and columns of C. In 3Dwe use integer coordinates (i。 k). Based on neighborhood relations we de_ne connectedness as usual: two points p。 p1。 :::。 q) from point p to point q, p 6= q, is the smallest positive integer n such that there exists a sequence of distinct grid points p = p0,p1。 :::。 q) has all properties of a metric. Given a binary digital image. We transform this image into a new one which represents at each point p 2 hIi the d4distance to pixels having value zero. The transformation includes two steps. We apply functions f1 to the image I in standard scan order, producing I_(i。 j。 j)), and f2 in reverse standard scan order, producing T(i。 j。 j)), as follows: f1(i。 I(i。 j) = 0 minfI_(i ? 1。 I_(i。 j) = 1 and i 6= 1 or j 6= 1 5 m+ n otherwise f2(i。 I_(i。 j)。 j)+ 1。 j + 1) + 1g The resulting image T is the distance transform image of I. Note that T is a set f[(i。 T(i。 j)。 j)] 2 T_ i_ none of the four points in A4((i。 j)+1. For all remaining points (i。 j) = 0. This image T_ is called distance skeleton. Now we apply functions g1 to the distance skeleton T_ in standard scan order, producing T__(i。 j。 j)), and g2 to the result of g1 in reverse standard scan order, producing T___(i。 j。 j)), as follows: g1(i。 T_(i。 j)。 j)? 1。 j ? 1) ? 1g g2(i。 T__(i。 j)。 j)? 1。 j + 1) ? 1g The result