Man wird dem einzelnen nicht gerecht, wenn man es gesondert ins Auge jaftt, ohne seinen Zusammenhang mit dem Ganzen zu beachten und dem Beziehungssystem Rechnung zu tragen, in dem es steht. Thomas Mann Science in general, as well as in each of its individual fields, is a part of human culture. In that sense, this book aims to contribute to uncovering a small part of the connections and relationships which bind image processing, categorized in informatics and technology, with the knowledge accumulated over the years on discrete structures. How does one consider problems, models, mathematical methods and prac­ tical applications? How does the search for ideas and the endeavour for know­ ledge in the original work of scientists find expression? Is there something to be learnt from science to date for future developments? Such questions have shaped the content and style of this book. Substantial impetus to the discrete theory of image processing was afforded by the work of Rosenfeld and colleagues. Other fruitful sources of ideas con­ sidered here are number theoretical problems (GauB, Minkowski) and integral geometric investigations (Blaschke, Santalo). Since the beginning of the 1980s I have strived to build upon these ideas a unified mathematical representation of discrete image processing working together with R.K1ette and P.Hufnagl.

Content.- 1 Neighborhood Structures.- 1.1 Finite Graphs.- 1.1.1 Historical Remarks.- 1.1.2 Elementary Theory of Sets and Relations.- 1.1.3 Elementary Graph Theory.- 1.2 Neighborhood Graphs.- 1.2.1 Graph Theory and Image Processing.- 1.2.2 Points, Edges, Paths, and Regions.- 1.2.3 Matrices of Adjacency.- 1.2.4 Graph Distances.- 1.3 Components in Neighborhood Structures.- 1.3.1 Search in Graphs and Labyrinths.- 1.3.2 Neighborhood Search.- 1.3.3 Graph Search in Images.- 1.3.4 Neighbored Sets and Separated Sets.- 1.3.5 Component Labeling.- 1.4 Dilatation and Erosion.- 1.4.1 Metric Spaces.- 1.4.2 Boundaries and Cores in Neighborhood Structures.- 1.4.3 Set Operations and Set Operators.- 1.4.4 Dilatation and Erosion.- 1.4.5 Opening and Closing.- 2 Incidence Structures.- 2.1 Homogeneous Incidence Structures.- 2.1.1 Topological Problems.- 2.1.2 Cellular Complexes.- 2.1.3 Incidence Structures.- 2.1.4 Homogeneous Incidence Structures.- 2.1.5 Zn as Incidence Structure.- 2.2 Oriented Neighborhood Structures.- 2.2.1 Orientation of a Neighborhood Structure.- 2.2.2 Euler Characteristic of a Neighborhood Structure.- 2.2.3 Border Meshes and Separation Theorem.- 2.2.4 Search in Oriented Neighborhood Structures.- 2.2.5 Coloring in Oriented Neighborhood Structures.- 2.3 Homogeneous Oriented Neighborhood Structures.- 2.3.1 Homogeneity in Neighborhood Structures.- 2.3.2 Toroidal Nets.- 2.3.3 Curvature of Border Meshes in Toroidal Nets.- 2.3.4 Planar Semi-Homogeneous Graphs.- 2.4 Objects in N-Dimensional Incidence Structures.- 2.4.1 Three-Dimensional Homogeneous Incidence Structures.- 2.4.2 Objects in Zn.- 2.4.3 Similarity of Objects.- 2.4.4 General Surface Formulas.- 2.4.5 Interpretation of Object Characteristics.- 3 Topological Laws and Properties.- 3.1 Objects and Surfaces.- 3.1.1 Surfaces in Discrete Spaces.- 3.1.2 Contur Following as Two-Dimensional Boundary Detection.- 3.1.3 Three-Dimensional Surface Detection.- 3.1.4 Curvature of Conturs and Surfaces.- 3.2 Motions and Intersections.- 3.2.1 Motions of Objects in Zn.- 3.2.2 Count Measures and Intersections of Objects.- 3.2.3 Applications of Intersection Formula.- 3.2.4 Count Formulas.- 3.2.5 Stochastic Images.- 3.3 Topology Preserving Operations.- 3.3.1 Topological Equivalence.- 3.3.2 Simple Points.- 3.3.3 Thinning.- 4 Geometrical Laws and Properties.- 4.1 Discrete Geometry.- 4.1.1 Geometry and Number Theory.- 4.1.2 Minkowski Geometry.- 4.1.3 Translative Neighborhood Structures.- 4.1.4 Digitalization Effects.- 4.2 Straight Lines.- 4.2.1 Rational Geometry.- 4.2.2 Digital Straight Lines in Z2.- 4.2.3 Continued Fractions.- 4.2.4 Straight Lines in Zn.- 4.3 Convexity.- 4.3.1 Convexity in Discrete Geometry.- 4.3.2 Maximal Convex Objects.- 4.3.3 Determination of Convex Hull.- 4.3.4 Convexity in Zn.- 4.4 Approximative Motions.- 4.4.1 Pythagorean Rotations.- 4.4.2 Shear Transformations.- 4.3.3 General Affine Transformations.- 5 Discrete Functions.- 5.1 One-Dimensional Periodical Discrete Functions.- 5.1.1 Functions.- 5.1.2 Space of Periodical Discrete Function.- 5.1.3 LSI-Operators and Convolutions.- 5.1.4 Products of Linear Operators.- 5.2 Algebraic Theory of Discrete Functions.- 5.2.1 Domain of Definition and Range of Values.- 5.2.2 Algebraical Structures.- 5.2.3 Convolution of Functions.- 5.2.4 Convolution Orthogonality.- 5.3 Orthogonal Convolution Bases.- 5.3.1 General Properties in OCB's.- 5.3.2 Fourier Transform.- 5.3.3 Number Theoretical Transforms.- 5.3.4 Two-Dimensional NTT.- 5.4 Inversion of Convolutions.- 5.4.1 Conditions for Inverse Elements.- 5.4.2 Deconvolutions and Texture Synthesis.- 5.4.3 Approximative Computation of Inverse Elements.- 5.4.4 Theory of Approximative Inversion.- 5.4.5 Examples of Inverse Filters.- 5.5 Differences and Sums of Functions.- 5.5.1 Differences of One-Dimensional Discrete Functions.- 5.5.2 Difference Equations and Z-Transform.- 5.5.3 Sums of Functions.- 5.5.4 Bernoulli's Polynomials.- 5.5.5 Determination of Moments.- 5.5.6 Final Comments.- 6 Summary and Symbols.- 7 References.- 8 Index.