Vladimir Kovalevsky

SRH Hochschule Berlin | SRH · Computer Science

The paper presents a new set of axioms of digital topology, which are easily understandable for application developers. They define a class of locally finite (LF) topological spaces. An important property of LF spaces satisfying the axioms is that the neighborhood relation is antisymmetric and transitive. Therefore any connected and non-trivial LF...

Some results on certain finite and locally finite spaces which may be used in image processing are reviewed, and algorithms related to these results are given and discussed.

The paper presents some algorithms in digital geometry based on the topology of cell complexes. The paper contains an axiomatic justification of the necessity of using cell complexes in digital geometry. Algorithms for solving the following problems are presented: tracing of curves and surfaces, recognition of digital straight line segments (DSS),...

This paper describes a new algorithm of computing the convex hull of a 3-dimensional object. The convex hull generated by this algorithm is an abstract polyhedron being described by a new data structure, the cell list, suggested by one of the authors. The correctness of the algorithm is proved and experimental results are presented. The convex hull...

The paper presents a new method of investigating topological properties of three-dimensional manifolds by means of computers. Manifolds are represented as block complexes. The paper contains definitions and a theorem necessary to transfer some basic knowledge of the classical topology to finite topological spaces. The method is based on subdividing...

The paper presents an analysis of sources of errors when estimating derivatives of numerical or noisy functions. A method of minimizing the errors is suggested. When being applied to the estimation of the curvature of digital curves, the analysis shows that under the conditions typical for digital image processing the curvature can rarely be estima...

The paper presents a new method of investigating topological properties of three-dimensional manifolds by means of computers. Manifolds are represented as finite cell complexes. The paper contains definitions and a theorem necessary to transfer some basic knowledge of the classical topology to finite topological spaces. The method is based on subdi...

The paper presents an introduction to computer topology with applications to image processing and computer graphics. Basic topological notions such as connectivity, frontier, manifolds, surfaces, combinatorial homeomorphism etc. are recalled and adapted for locally finite topological spaces. The paper describes data structures for explicitly repres...

The paper presents some theorems about interlaced spheres of different dimensions in multidimensional spaces. Two spheres S k and S m are called interlaced with each other if their intersection is empty, however, one of them crosses each topological ball whose boundary is the other sphere. We describe a method of simulating interlaced spheres in co...

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A new method of representing a surface in the 3D space as a single digitally continuous sequence of faces is described. The method is based on topological properties of quasi-manifolds. It is realized as tracing the boundary of a growing set of labeled faces. As the result the surface is encoded as a single sequence of mutually adjacent faces. Each...

A new classification of digital curves into boundary curves and visual curves of different thickness is suggested. A fast algorithm for recognizing digital straight line segments in boundary curves is presented. The algorithm is applied to encode the boundaries of homogeneous regions in digitized images. The code is economical and enables an exact...

The author suggests therefore an essentially digital approach: the surface to be reconstructed is considered as a polyhedron. The image contains then a finite number of plane polygonal regions each having a constant grey value. The problem consists in finding the heights of the vertices of the polyhedron. The heights must satisfy a system of quadra...

A concept for geometry in a locally finite topological space without use of infinitesimals is presented. The topological space under consideration is an abstract cell complex which is a particular case of a T0-space. The concept is in accordance with classical axioms and therefore free of contradictions an paradoxes. Coordinates are introduced inde...

The definition and recognition algorithm for a digital straight segment (DSS) is presented. The points of a DSS must have a limited distance from that edge. A recognition algorithm is given which uses only integer arithmetic and needs an average of about 10 such operations per point. The definition of a digital circular arc (DCA) which uses the not...

The notion of a cellular complex which is well known in the topology is applied to describe the structure of images. It is shown that the topology of cellular complexes is the only possible topology of finite sets. Under this topology no contradictions or paradoxes arise when defining connected subsets and their boundaries. Ways of encoding images...

The author asserts that the goal of the system is clear but the sense of the notions 'implicit' and 'explicit' must be defined more precisely. It is expedient to distinguish between systems designed to deal with a single class of scenes and imaging techniques and those applicable to many classes. Any vision system must be designed on the basis of a...

For all kinds of stationary plane grids only the 6-neighborhood is consistent. It admits the definition of a digital curve, in particular a contour, as a set of pixel pairs. Thus curves become objects of zero thickness and may be coded with one bit per grid step.

This paper discusses the ways of finding consistency between the well-known statistical statement that "guessing destroys information" and the practically obvious advantage of hierarchical decisions. Certain nonstatistical sources of recognition errors are indicated, the influence of these sources increasing with the size of the image parts on whic...