Digital Topology

The aim of this paper is to give an introduction into the field of digital topology. This topic of research arose in connection with image processing. It is important also in all applications of artificial intelligence dealing with spatial structures. In this article, a simple but representative model of the Euclidean plane, called the digital plane, is studied. The problem is to introduce a satisfactory ‘topology’ for this essentially discrete structure. It turns out, that all known approaches, which come from different directions of applications and theory, converge to virtually one concept of 4/8– or 8/4– connectedness. A very natural approach to problems of discrete topology is the concept of semi–topological spaces.

[1]  Werner Nagel.  Image Analysis and Mathematical Morphology. Volume 2: Theoretical Advances. Edited by Jean Serra , 1870 .

[2]  W. Blaschke.  Vorlesungen über Integralgeometrie , 1937 .

[3]  F. Thorne,et al.  Geometry of Numbers , 2017, Algebraic Number Theory.

[4]  Peter E. Hart,et al.  GRAPHICAL-DATA-PROCESSING RESEARCH STUDY AND EXPERIMENTAL INVESTIGATION , 1964 .

[5]  Frank Harary,et al.  Graph Theory , 2016 .

[6]  Marvin Minsky,et al.  Perceptrons: An Introduction to Computational Geometry , 1969 .

[7]  Dana S. Scott,et al.  Outline of a Mathematical Theory of Computation , 1970 .

[8]  M. V. Novikov,et al.  Academy of Sciences of the Ukrainian SSR , 1970 .

[9]  G. Matheron.  Random Sets and Integral Geometry , 1976 .

[10]  Dana S. Scott,et al.  Data Types as Lattices , 1976, SIAM J. Comput..

[11]  Azriel Rosenfeld,et al.  Fuzzy Digital Topology , 1979, Inf. Control..

[12]  Jean-Marc Chassery,et al.  Connectivity and consecutivity in digital pictures , 1979 .

[13]  Azriel Rosenfeld,et al.  Three-Dimensional Digital Topology , 1981, Inf. Control..

[14]  FUZZY PRETOPOLOGICAL STRUCTURES AND FORMATION OF COALITIONS , 1982 .

[15]  T. Pavlidis.  Algorithms for Graphics and Image Processing , 1981, Springer Berlin Heidelberg.

[16]  Jean Serra,et al.  Image Analysis and Mathematical Morphology , 1983 .

[17]  C. B. Wilson,et al.  The mathematical description of shape and form , 1984 .

[18]  Henk Barendregt,et al.  The Lambda Calculus: Its Syntax and Semantics , 1985 .

[19]  A. W. Roscoe,et al.  Continuous analogs of axiomatized digital surfaces , 1984, Comput. Vis. Graph. Image Process..

[20]  Azriel Rosenfeld,et al.  'Continuous' functions on digital pictures , 1986, Pattern Recognit. Lett..

[21]  Mark A. Roth.  Theoretical advances in *** , 1986, CSC '86.

[22]  Efim Khalimsky,et al.  Topological structures in computer science , 1987 .

[23]  P. P. Das,et al.  Knight's distance in digital geometry , 1988, Pattern Recognit. Lett..

[24]  Azriel Rosenfeld,et al.  Digital topology: Introduction and survey , 1989, Comput. Vis. Graph. Image Process..

[25]  T. Yung Kong,et al.  A digital fundamental group , 1989, Comput. Graph..

[26]  Vladimir A. Kovalevsky,et al.  Finite topology as applied to image analysis , 1989, Comput. Vis. Graph. Image Process..

[27]  D. Stoyan,et al.  Stochastic Geometry and Its Applications , 1989 .

[28]  P. R. Meyer,et al.  Computer graphics and connected topologies on finite ordered sets , 1990 .

[29]  L. L. Miller,et al.  Topological Approach for Testing Equivalence in Heterogeneous Relational Databases , 1990, Comput. J..

[30]  Azriel Rosenfeld,et al.  If we use 4- or 8-connectedness for both the objects and the background, the Euler characteristics is not locally computable , 1990, Pattern Recognition Letters.

[31]  Gabor T. Herman,et al.  On topology as applied to image analysis , 1990, Comput. Vis. Graph. Image Process..

[32]  Azriel Rosenfeld,et al.  Winding and Euler numbers for 2D and 3D digital images , 1991, CVGIP Graph. Model. Image Process..

[33]  Ralph Kopperman,et al.  A Jordan surface theorem for three-dimensional digital spaces , 1991, Discret. Comput. Geom..

[34]  E. H. Kronheimer.  The topology of digital images , 1992 .

[35]  A. W. Roscoe,et al.  Concepts of digital topology , 1992 .

[36]  Gabor T. Herman,et al.  Discrete multidimensional Jordan surfaces , 1992, CVGIP Graph. Model. Image Process..

[37]  Norman Levitt,et al.  The euler characteristic is the unique locally determined numerical homotopy invariant of finite complexes , 1992, Discret. Comput. Geom..

[38]  Longin Jan Latecki.  Digitale und Allgemeine Topologie in der bildhaften Wissensrepräsentation , 1992, DISKI.

[39]  Azriel Rosenfeld,et al.  Holes and Genus of 2D and 3D Digital Images , 1993, CVGIP Graph. Model. Image Process..

[40]  Longin Jan Latecki,et al.  Topological connectedness and 8-connectedness in digital pictures , 1993 .

[41]  ULRICH ECKHARDT,et al.  Invariant Thinning , 1993, Int. J. Pattern Recognit. Artif. Intell..

[42]  Longin Jan Latecki,et al.  Digitizations preserving topological and differential geometric properties , 1995, Other Conferences.