The Core Experiment CE-Shape-1 for shape descriptors performed for the MPEG-7 standard gave a unique opportunity to compare various shape descriptors for non-rigid shapes with a single closed contour. There are two main differences with respect to other comparison results reported in the literature: (1) For each shape descriptor the experiments were carried out by an institute that is in favor of this descriptor. This implies that the parameters for each system were optimally determined and the implementations were thoroughly rested. (2) It was possible to compare the performance of shape descriptors based on totally different mathematical approaches. A more theoretical comparison of these descriptors seems to be extremely hard. In this paper we report on the MPEG-7 Core Experiment CE-Shape.

One of the most widely used methods for preprocessing binary images is thinning. The popularity of this method rests on the fact that considerable data reduction is achieved while retaining “essential” properties of the original image. Moreover, topological features, which cannot be verified by a genuinely parallel method (by Minsky and Papert22) are more easily treated in thinned images. For these reasons, many articles on this topic were published in the literature. Most of them are concerned with modifications of existing methods in order to yield “nicer” results out of the thinning process. Also, many results of numerical experiments are available in different publications. The aim of this paper is to show that the quite natural requirement of invariance of the results obtained by thinning leads nearly automatically to a method proposed by the authors in different publications. Moreover, this method is genuinely parallel and well-defined so that it is possible to investigate it theoretically. The practical feasibility of the method is also discussed.

Demonstrates that a specific method for thinning binary pictures which was proposed by the authors can be formulated as a labeling process thus leading to an invertible result. This can be interpreted that the 'form' of an object is retained under application of the method.>

A general parametrization for lines which is based on homogeneous coordinates is presented. It is shown that various parametrizations published in the literature can be obtained from this general scheme. By suitable parametrization of the Hough transform, favorable properties can be achieved. They include insights into the geometry of the parameter (or Hough) space, its boundedness, and consistent discretization of both image and Hough space.>

There exist very complex sets that contain interior points and that are irreducible under thinning. The authors investigate the structure of such irreducible sets. It is shown that irreducible sets containing interior points have essentially the structure of a counterexample given by C. Arcelli (1981). Finally the question is raised of under which conditions irreducible sets that have a large kernel of interior points are not possible. It turns out that a very simple condition will suffice.< >

Weierstrass' ~-function and its derivative can be approximated in the complex plane by a rational expression as described in [3,4].The following programs evaluate these approximations for the case of a unit period parallelogram.They are written according to the rules given in [2].Programs PEQ and PEQ1 evaluate Weierstrass' ~-function in the equianharmonic case for a unit period parallelogram defined by the complex numberssee [1, Sec.18.13] for notation).Because of the periodicity of Weierstrass' functions it is sufficient to calculate them only in the fundamental rectangle of all z = x +/y with 1 1 ,/5 J5 --<x_~ -__<y