Discrete Images, Objects, and Functions in Zn

Voss' book which deals with theoretical problems of digital image processing is an indispensable source of information for all those who are involved in the design, implementation, and application of mathematically sound algorithms in image processing; it is written for engineers, mathematicians, and computer scientists. The author, who is a well-known expert in the field of digital image processing, presents important new research which generalizes the currently used two-dimensional theory to n dimensions.

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.