【正文】 t p 6= q. A point of a digital arc which has exactly one neighbor is called an end point of this arc. Within this third class of operators (thinning algorithms) we may classify with respect to algorithmic strategies: individual pixels are either removed in a sequential order or in parallel. For example, the often cited algorithm by Hilditch [5] is an iterative process of testing and deleting contour pixels sequentially in standard raster scan order. Another sequential algorithm by Pavlidis [12] uses the de_nition of multiple points and proceeds by contour following. Examples of parallel algorithms in this third class are reduction operators which transform contour points into background points. Di_erences between these parallel algorithms are typically de_ned by tests implemented to ensure connectedness in a local neighborhood. The notion of a simple point is of basic importance for thinning and it will be shown in this report that di_erent de_nitions of simple points are actually equivalent. Several publications characterize properties of a set D of points (to be turned from object points to background points) to ensure that connectivity of object and background remain unchanged. The report discusses some of these properties in order to justify parallel thinning algorithms. Basics The used notation follows [17]. A digital image I is a function de_ned on a discrete set C , which is called the carrier of the image. The elements of C are grid points or grid cells, and the elements (p。 p2。 1 Digital Image Processing 1 Introduction Many operators have been proposed for presenting a connected ponent n a digital image by a reduced amount of data or simplied shape. In general we have to state that the development, choice and modi_cation of such algorithms in practical applications are domain and task dependent, and there is no \best method. However, it is interesting to note that there are several equivalences between published methods and notions, and characterizing such equivalences or di_erences should be useful to categorize the broad diversity of published methods for skeletonization. Discussing equivalences is a main intention of this report. Categories of Methods One class of shape reduction operators is based on distance transforms. A distance skeleton is a subset of points of a given ponent such that every point of this subset represents the center of a maximal disc (labeled with the radius of this disc) contained in the given ponent. As an example in this _rst class of operators, this report discusses one method for calculating a distance skeleton using the d4 distance function which is appropriate to digitized pictures. A second class of operators produces median or center lines of the digital object in a noniterative way. Normally such operators locate critical points _rst, and calculate a speci_ed path through the object by connecting these points. The third class of operators is characterized by iterative thinning. Historically, Listing [10] used already in 1862 the term linear skeleton for the result of a continuous deformation of the frontier of a connected subset of a Euclidean space without changing the connectivity of the original set, until only a set of lines and points remains. Many algorithms in image analysis are based on this general concept of thinning. The goal is a calculation of characteristic properties of digital objects which are not related to size or quantity. Methods should be independent from the position of a set in the plane or space, grid resolution (for digitizing this set) or the