Discrete Images, Objects, and Functions in Zn

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.