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Digital Geometry

Book Description

Digital Geometry is the first book to detail the concepts, algorithms, and practices of the discipline. This comphrehensive text and reference provides an introduction to the mathematical foundations of digital geometry, some of which date back to ancient times, and also discusses the key processes involved, such as geometric algorithms as well as operations on pictures.

Table of Contents

  1. Cover image
  2. Title page
  3. Table of Contents
  4. Copyright
  5. Preface
  6. Dedication
  7. Inside Front Cover
  8. Structure of this Book
  9. Errata
  10. Chapter 1: Introduction
    1. 1.1 Pictures
    2. 1.2 Digital Geometry and Related Disciplines
    3. 1.3 Exercises
    4. 1.4 Commented Bibliography
  11. Chapter 2: Grids and Digitization
    1. 2.1 The Grid Pointand Grid Cell Modesls
    2. 2.2 Connected Components
    3. 2.3 Digitization Models
    4. 2.4 Property Estimation
    5. 2.5 Exercises
    6. 2.6 Commented Bibliography
  12. Chapter 3: Metrics
    1. 3.1 Basics About Metrics
    2. 3.2 Grid Point Metrics
    3. 3.3 Grid Cell Metrics
    4. 3.4 Metrics on Pictures
    5. 3.5 Exercises
    6. 3.6 Commented Bibliography
  13. Chapter 4: Adjacency Graphs
    1. 4.1 Graphs, Adjacency Structures, and Adjacency Graphs
    2. 4.2 Some Basics of Graph Theory
    3. 4.3 Oriented Adjacency Graphs
    4. 4.4 Combinatorial Maps
    5. 4.5 Exercises
    6. 4.6 Commented Bibliography
  14. Chapter 5: Incidence Pseudographs
    1. 5.1 Incidence Structures
    2. 5.2 Boundaries, Frontiers, and the Euler Characteristic
    3. 5.3 The Regular Case
    4. 5.4 Pictures on Incidence Grids
    5. 5.5 Exercises
    6. 5.6 Commented Bibliography
  15. Chapter 6: Topology
    1. 6.1 Topologic Spaces
    2. 6.2 Digital Topologies
    3. 6.3 Topologic Concepts
    4. 6.4 Combinatorial Topology
    5. 6.5 Exercises
    6. 6.6 Commented Bibliography
  16. Chapter 7: Curves and Surfaces: Topology
    1. 7.1 Curves in the Euclidean Topology
    2. 7.2 Curves in Incidence Grids
    3. 7.3 Curves in Adjacency Grids
    4. 7.4 Surfaces in the Euclidean Topology
    5. 7.5 Surfaces and Separations in 3D Grids
    6. 7.6 Exercises
    7. 7.7 Commented Bibliography
  17. Chapter 8: Curves and Surfaces: Geometry
    1. 8.1 Planar Curves and Arcs
    2. 8.2 Space Curves and Arcs
    3. 8.3 Surfaces and Solids
    4. 8.3.3 Example: an ellipsoid
    5. 8.3.4 Gauss ’ definition of surface curvature
    6. 8.3.5 Principal, Gaussian, and mean surface curvature
    7. 8.3.6 Volume
    8. 8.3.7 Isothetic polyhedra
    9. 8.4 Surface Tracing and Approximation
    10. 8.5 Exercises
    11. 8.6 Commented Bibliography
  18. Chapter 9: 2D Straightness
    1. 9.1 Basics
    2. 9.2 Supporting Lines
    3. 9.3 Self-Similarity
    4. 9.4 Periodicity
    5. 9.5 Number-Theoretic Properties
    6. 9.6 Algorithms
    7. 9.7 Exercises
    8. 9.8 Commented Bibliography
  19. Chapter 10: 2D Arc Length; Curvature and Corners
    1. 10.1 The Length of a Digital Curve
    2. 10.2 Definitions of 2D Arc Length Estimators
    3. 10.3 Evaluation of 2D Arc Length Estimators
    4. 10.4 The Curvature of a Planar Digital Curve
    5. 10.5 Exercises
    6. 10.6 Commented Bibliography
  20. Chapter 11: 3D Straightness and Planarity
    1. 11.1 3D Straightness
    2. 11.2 Digital Planes in 3D Adjacency Grids
    3. 11.3 Digital Planes in the 3D Incidence Grid
    4. 11.4 DPS Recognition and Generation
    5. 11.5 Exercises
    6. 11.6 Commented Bibliography
  21. Chapter 12: 3D Arc Length, Surface Area, and Curvature
    1. 12.1 3D Arcs
    2. 12.2 Surface Area Estimation
    3. 12.3 Surface Curvature Estimation
    4. 12.4 Exercises
    5. 12.5 Commented Bibliography
  22. Chapter 13: Hulls and Diagrams
    1. 13.1 Hulls
    2. 13.2 2D Digital Convexity
    3. 13.3 Diagrams
    4. 13.4 Exercises
    5. 13.5 Commented Bibliography
  23. Chapter 14: Transformations
    1. 14.1 Geometries
    2. 14.2 Axiomatic Digital Geometry
    3. 14.3 Transformation Groups and Symmetries
    4. 14.4 Neighborhood-Preserving Transformations
    5. 14.5 Applying Transformations to Pictures
    6. 14.6 Magnification and Demagnification
    7. 14.7 Digital Tomography
    8. 14.8 Exercises
    9. 14.9 Commented Bibliography
  24. Chapter 15: Morphologic Operations
    1. 15.1 Dilation
    2. 15.2 Erosion
    3. 15.3 Combining Dilations and Erosions
    4. 15.4 Simplification
    5. 15.5 Segmentation
    6. 15.6 Decomposition
    7. 15.7 Exercises
    8. 15.8 Commented Bibliography
  25. Chapter 16: Deformations
    1. 16.1 Topology-Preserving Deformations and Simple Pixels
    2. 16.2 Shrinking
    3. 16.3 Thinning
    4. 16.4 Deformations of Curves
    5. 16.5 Interchangeable Pairs of Pixels
    6. 16.6 Deformations of 3D Pictures
    7. 16.7 Deformations of Multivalued Pictures
    8. 16.8 Exercises
    9. 16.9 Commented Bibliography
  26. Chapter 17: Picture Properties and Spatial Relations
    1. 17.1 Properties
    2. 17.2 Moments
    3. 17.3 Experimental Evaluation of Moment Estimates
    4. 17.4 Operations on Pictures and Invariant Properties
    5. 17.5 Spatial Relations
    6. 17.6 Exercises
    7. 17.7 Commented Bibliography
  27. List of Algorithms
  28. List of Symbols
  29. List of Axioms and Properties
  30. Bibliography
  31. Index