Ulrich Eckhardt

Weber's problem and weiszfeld's algorithm in general spaces

For solving the Euclidean distance Weber problem Weiszfeld proposed an iterative method. This method can also be applied to generalized Weber problems in Banach spaces. Examples for generalized Weber problems are: minimal surfaces with obstacles, Fermats principle in geometrical optics and brachistochrones with obstacles.

Topologies for the digital spaces Z 2 and Z 3

We show that there are only two topologies in Z2 and five topologies in Z3 whose connected sets are connected in the intuitive sense. Both topologies for Z2 are well known (e.g., one is presented in D. Marcus, F. Wyse et al., Amer. Math. Monthly 77 (1979) 1119, and the second in E. Khalimsky et al., Topology and its Applications 36 (1990) 1) and found applications in computer graphics and computer vision (e.g., A. Rosenfeld, Amer. Math. Monthly 77 (1979) pp. 621, and T.Y. Kong et al., Amer. Math. Monthly 98 (1991) 901). Two of the five topologies for Z3 are products of the topologies known from Z1 and Z2. The remaining three topologies for Z3 are also generated from the two topologies for Z2 however, they are not product topologies.

Linear convergence of generalized Weiszfeld's method

Weiszfelds method is widely used for solving problems of optimal location. It is shown that a very general variant of this method converges linearly thus generalizing a result of I. N. Katz.ZusammenfassungZur Lösung von Standortoptimierungsproblemen wird häufig eine auf E. Weiszfeld zurückgehende Methode verwendet, von der I. N. Katz lokal lineare Konvergenz nachweisen konnte. Dieses Resultat wird in der vorliegenden Arbeit verallgemeinert.

Theorems on the dimension of convex sets

Abstract This paper gives theorems on the dimension of solution sets of semi-infinite systems of linear inequalities. As an application, some results of Clark and Williams are generalized.

The Euler Characteristics of Discrete Objects and Discrete Quasi-Objects

Assuming planar 4-connectivity and spatial 6-connectivity, we first introduce the curvature indices of the boundary of a discrete object, and, using these indices of points, we define the vertex angles of discrete surfaces as an extension of the chain codes of digital curves. Second, we prove the relation between the number of point indices and the numbers of holes, genus, and cavities of an object. This is the angular Euler characteristic of a discrete object. Third, we define quasi-objects as the connected simplexes. Geometric relations between discrete quasi-objects and discrete objects permit us to define the Euler characteristic for the planar 8-connected, and the spatial 18- and 26-connected objects using these for the planar 4-connected and the spatial 6-connected objects. Our results show that the planar 4-connectivity and the spatial 6-connectivity define the Euler characteristics of point sets in a discrete space. Finally, we develop an algorithm for the computation of these characteristics of discrete objects.

Digital lines and digital convexity

Euclidean geometry on a computer is concerned with the translation of geometric concepts into a discrete world in order to cope with the requirements of representation of abstract geometry on a computer. The basic constructs of digital geometry are digital lines, digital line segments and digitally convex sets. The aim of this paper is to review some approaches for such digital objects. It is shown that digital objects share much of the properties of their continuous counterparts. Finally, it is demonstrated by means of a theorem due to Tietze (1929) that there are fundamental differences between continuous and discrete concepts.

Root Images of Median Filters

Median filters are frequently used in signal analysis because of their smoothing properties and their insensitivity with respect to outliers in the data. Since median filters are nonlinear filters, the tools of linear theory are not applicable to them. One approach to deal with nonlinear filters consists in investigating their root images (fixed elements or signals transparent to the filter). Whereas for one-dimensional median filters the set of all root signals can be completely characterized, this is not true for higher dimensional filters.In 1989, Döhler stated a result on certain root images for two-dimensional median filters. Although Döhlers results are true for a wide class of median filters, his arguments were not correct and his assertions do not hold universally. In this paper we give a rigorous proof of Döhlers results. Moreover, his approach is generalized to the d-dimensional case.

THE STEWART-HAMILTON EQUATIONS AND THE INDICATOR DILUTION METHOD*

The so-called Stewart–Hamilton equations are generally accepted as a theoretical basis of the indicator dilution method in heart diagnosis. In this paper we find the exact physical and mathematical conditions under which these equations are indeed valid for a very simple model situation. At the same time the nature of the transfer function relating the processes of indicator injection and sampling is studied. The paper ends by demonstrating that there are indeed practical experimental designs where the Stewart–Hamilton equations are fulfilled in the simple model given.

Polygonal Representations of Digital Sets

Abstract In the context of discrete curve evolution the following problem is of relevance: decompose the boundary of a plane digital object into convex and concave parts. Such a decomposition is very useful for describing the form of an object, e.g. for shape databases. Although the problem is relatively trivial in ordinary plane geometry, in digital geometry its statement becomes a very difficult task due to the fact that in digital geometry there is no simple set-complement duality. The paper is based on results given by Hübler et al. The main new contribution of the paper is the generalization of the concepts introduced by these authors to nonconvex sets. The digital geometric “low level” segmentation of the boundary of a digital object can be used as a starting basis for further reduction of the boundary by means of discrete evolution.

Continuity of the discrete curve evolution

Recently Latecki and Lakamper (Computer Vision and Image Understanding 73:3, March 1999) reported a novel process for a discrete curve evolution. This process has various application possibilities, in particular, for noise removal and shape simplification of boundary curves in digital images. In this paper we prove that the process of the discrete curve evolution is continuous: if polygon Q is close to polygon P, then the polygons obtained by their evolution remain close. This result follows directly from the fact that the evolution of Q corresponds to the evolution of P if Q approximates P. This intuitively means that first all vertices of Q are deleted that are not close to any vertex of P, and then, whenever a vertex of P is deleted, then a vertex of Q that is close to it is deleted in the corresponding evolution step of Q.