Geometry of
Locally Finite Spaces Abstract The book presents a
selfcontained theory of locally finite spaces including a set of new axioms,
numerous definitions and theorems concerning the properties of that spaces.
It also presents a way of defining digital geometry in locally finite spaces
independently of Euclidean geometry. A large portion of the book is devoted
to applications to computer imagery. Locally finite spaces give the possibility to
overcome the existing discrepancy between theory and applications: The
traditional way of research consists in making theory in Euclidean space with
real coordinates while applications deal only with finite discrete sets and
rational numbers. The reason is that even the smallest part of the Euclidean
space cannon be explicitly represented in a computer and computations with
irrational numbers are impossible since there exists no arithmetic of
irrational numbers. Locally finite spaces are on the one hand
theoretically consistent and conform with classical topology and on the other
hand explicitly representable in a computer. Coordinates are rational or
integer numbers. New data structures
and numerous geometric and topological algorithms are presented. Most
algorithms are accompanied by a pseudocode based on the C++ programming
language. 

Recommendations “It is a real pleasure to read the book. I will recommend it to several
people and libraries, because it continues to tell to young researchers that
topology of cell complexes is not only the best way for the theory,
especially for multicolor images, but also the only practical topology for
programming large scale software.”. Jean
Françon, Professor em., University Louis Pasteur, France "It is certainly an impressive work. Much of what it covers
cannot be found in any other book that I know of. The book will be an
important reference for me and, I believe, for anyone else who has a serious
interest in digital geometry. I expect to make much use of it in the future. T. Yung Kong, Professor of Computer Science, City University of New
York." “The author has developed a
new theory and proved numerous theorems concerning the properties of locally
finite spaces. The book, being the first one it its own way, is addressed to
specialists in digital topology and digital geometry. It is important for
them”. S. V. Matveev, Professor of Chelyabinsk State University,
Corresponding member of Russian Academy of Sciences 

Contents 1 Introduction 1.1 About
the Development of the Locally Finite
Topology 1.2 Contents of the Monograph 1.3 The Aims of the
Monograph 2 Axiomatic Approach to Digital Topology 2.1 Why Do We Introduce a New Set of Axioms 2.2 Axioms of Digital Topology 2.3 Relation
between the Suggested and Classical Axioms 2.4 Deducing
the Properties of ALF Spaces from Axioms 2.5 Previous Work 3
Theory of Abstract Cell Complexes 3.1
Topology of Complexes 3.2 Cartesian Complexes and
Combinatorial Coordinates 3.3 AC Complexes Compared
with other Locally Finite Spaces 3.4
Combinatorial Homeomorphism, Balls and Spheres 3.5
Justification of the above Definitions 3.6
Definition of the Combinatorial Homeomorphism 3.7
Generalized Boundary and Boundary of a Space 3.8
Orientation of AC Complexes 3.9
Combinatorial Manifolds 3.10 Block
Complexes 3.11
Consistency of (m, n)Adjacencies 3.12
Completely Connected Spaces 3.13
Problems to Be Solved 
4 Mappings among Locally Finite Spaces 4.1 ConnectednessPreserving Correspondences
(CPM) 4.2 CPMs and Combinatorial Homeomorphism 4.3 Some Properties of Manifolds
and of Block Complexes 4.4 Problems to Be Solved 5 Interlaced Spheres in Locally Finite Spaces 5.1 Preface 5.2 Interlaced Multidimensional Spheres 5.3 Examples of Interlaced Spheres 6 Digital Geometry based on Topology of Complexes 6.1 Preface 6.2 Classification of Digital Curves 6.3 Introduction to Digital Analytical
Geometry 7 Linear Inequalities in Locally Finite Spaces 7.1 Digital Collinearity, HalfSpaces and
Convex Sets 7.2 Digital Straight Segments (DSS) 7.3 Theory of Digital Plane Patches 8 Surfaces in a ThreeDimensional Space 8.1 Introduction 8.2 Frontiers of Sets of Voxels 8.3 QuasiManifolds and Their Properties 8.4 Adjacency of Principal Cells of a
QuasiManifold 9 Digital Arcs and Their Recognition 9.1 Some Definitions 9.2 Recognition of Digital Arcs 10 Data Structures 10.1 The Standard Grid 10.2 The Combinatorial Grid 

10.3 Data Structures Using Lists of Space
Elements 10.4 The TwoDimensional Cell List 10.5 The ThreeDimensional Cell List 11 Applications and Algorithms 11.1 Recommendations for Applications 11.2 Tracing Boundaries in 2D Images and 2D
Subspaces 11.3 Encoding of DSSs with Additional
Parameters 11.4 Reconstruction of Multidimensional
Images from Boundaries 11.5 Labeling Connected Component in an
nD Space 11.6 Skeletons of Subsets in a TwoDimensional Space 11.7 Algorithms for Topological Investigations 11.8 Conclusion of Section 11 and Problems to
Be Solved 12 Convex Hulls in a ThreeDimensional Space 12.1 Introduction 12.2 The Algorithm 12.3 Proof of Correctness 12.4 Results of Computer Experiments 13 Algorithms for Tracing and Encoding of Surfaces 13.1 Introduction 13.2 The DepthFirst Method 13.3 Euler Circuits 13.4 Spiral Tracing 13.5 The Economical Hoop Code 13.6 Problems to Be Solved 14 Topics for Discussion 14.1 Real Numbers and Derivatives References;
Index 
About the author Vladimir A. Kovalevsky
received his diploma in physics from the Kharkov University (Ukraine) in
1950, the first doctoral degree in technical sciences from the Central Institute
of Metrology (Leningrad) in 1957, and the second doctoral degree in computer
science from the Institute of Cybernetics of the Academy of Sciences of the
Ukraine (Kiev) in 1968. From 1961 to 1983 he served as Head of Department of
Pattern Recognition at that Institute. He has been living in Germany since 1983. From 1983 to 1989
he was researcher at the Central Institute of Cybernetics of the Academy of
Sciences of the GDR, Berlin. From 1989 to 2004 he was professor of computer
science at the University of Applied Sciences Berlin with an interruption for
three years (19982001). In that time he was scientific collaborator at the
University of Rostock. He worked as visiting researcher at the University of
Pennsylvania (1990), at the Manukau Institute of Technology, New Zeeland
(2005) and as lecturer at the Chonbuk National University, South Korea (2009). He has been plenary
speaker at conferences in Europe, America and New Zeeland. His research
interests include digital geometry, digital topology, computer vision, image
processing and pattern recognition. He has published four monographs and more
than 180 journal and conference papers in image analysis, digital geometry
and digital topology. 

Vladimir A. Kovalevsky Geometry of Locally Finite Spaces
A
selfcontained theory with axioms, definitions
and theorems as well as numerous algorithms for computer imagery 330 pages, 120 figures (12 in color), 85 literature references Soft cover: € 27.50; (sold out). Foil cover: € 32.30; ISBN 9783981225204 The author: Vladimir
A. Kovalevsky emal:
kovalev@beuthhochschule.de www.kovalevsky.de © 2008 Publishing House Dr. Baerbel Kovalevski Heisterbachstrasse
23 B, 12559 Berlin, Germany To ordered the book send your name and mailing address to fax
+493065908707 or click to: Email
to the Publishing House email:
Baerbel.Kovalev@online.de You will obtain
the book with an invoice for remitting the price + postage 