Table of Contents

Cover image

Title page

Copyright

Preface

Dedication

Inside Front Cover

Structure of this Book

Errata

Chapter 1: Introduction

1.1 Pictures

1.2 Digital Geometry and Related Disciplines

1.3 Exercises

1.4 Commented Bibliography

Chapter 2: Grids and Digitization

2.1 The Grid Pointand Grid Cell Modesls

2.2 Connected Components

2.3 Digitization Models

2.4 Property Estimation

2.5 Exercises

2.6 Commented Bibliography

Chapter 3: Metrics

3.1 Basics About Metrics

3.2 Grid Point Metrics

3.3 Grid Cell Metrics

3.4 Metrics on Pictures

3.5 Exercises

3.6 Commented Bibliography

Chapter 4: Adjacency Graphs

4.1 Graphs, Adjacency Structures, and Adjacency Graphs

4.2 Some Basics of Graph Theory

4.3 Oriented Adjacency Graphs

4.4 Combinatorial Maps

4.5 Exercises

4.6 Commented Bibliography

Chapter 5: Incidence Pseudographs

5.1 Incidence Structures

5.2 Boundaries, Frontiers, and the Euler Characteristic

5.3 The Regular Case

5.4 Pictures on Incidence Grids

5.5 Exercises

5.6 Commented Bibliography

Chapter 6: Topology

6.1 Topologic Spaces

6.2 Digital Topologies

6.3 Topologic Concepts

6.4 Combinatorial Topology

6.5 Exercises

6.6 Commented Bibliography

Chapter 7: Curves and Surfaces: Topology

7.1 Curves in the Euclidean Topology

7.2 Curves in Incidence Grids

7.3 Curves in Adjacency Grids

7.4 Surfaces in the Euclidean Topology

7.5 Surfaces and Separations in 3D Grids

7.6 Exercises

7.7 Commented Bibliography

Chapter 8: Curves and Surfaces: Geometry

8.1 Planar Curves and Arcs

8.2 Space Curves and Arcs

8.3 Surfaces and Solids

8.3.3 Example: an ellipsoid

8.3.4 Gauss ’ definition of surface curvature

8.3.5 Principal, Gaussian, and mean surface curvature

8.3.6 Volume

8.3.7 Isothetic polyhedra

8.4 Surface Tracing and Approximation

8.5 Exercises

8.6 Commented Bibliography

Chapter 9: 2D Straightness

9.1 Basics

9.2 Supporting Lines

9.3 Self-Similarity

9.4 Periodicity

9.5 Number-Theoretic Properties

9.6 Algorithms

9.7 Exercises

9.8 Commented Bibliography

Chapter 10: 2D Arc Length; Curvature and Corners

10.1 The Length of a Digital Curve

10.2 Definitions of 2D Arc Length Estimators

10.3 Evaluation of 2D Arc Length Estimators

10.4 The Curvature of a Planar Digital Curve

10.5 Exercises

10.6 Commented Bibliography

Chapter 11: 3D Straightness and Planarity

11.1 3D Straightness

11.2 Digital Planes in 3D Adjacency Grids

11.3 Digital Planes in the 3D Incidence Grid

11.4 DPS Recognition and Generation

11.5 Exercises

11.6 Commented Bibliography

Chapter 12: 3D Arc Length, Surface Area, and Curvature

12.1 3D Arcs

12.2 Surface Area Estimation

12.3 Surface Curvature Estimation

12.4 Exercises

12.5 Commented Bibliography

Chapter 13: Hulls and Diagrams

13.1 Hulls

13.2 2D Digital Convexity

13.3 Diagrams

13.4 Exercises

13.5 Commented Bibliography

Chapter 14: Transformations

14.1 Geometries

14.2 Axiomatic Digital Geometry

14.3 Transformation Groups and Symmetries

14.4 Neighborhood-Preserving Transformations

14.5 Applying Transformations to Pictures

14.6 Magnification and Demagnification

14.7 Digital Tomography

14.8 Exercises

14.9 Commented Bibliography

Chapter 15: Morphologic Operations

15.1 Dilation

15.2 Erosion

15.3 Combining Dilations and Erosions

15.4 Simplification

15.5 Segmentation

15.6 Decomposition

15.7 Exercises

15.8 Commented Bibliography

Chapter 16: Deformations

16.1 Topology-Preserving Deformations and Simple Pixels

16.2 Shrinking

16.3 Thinning

16.4 Deformations of Curves

16.5 Interchangeable Pairs of Pixels

16.6 Deformations of 3D Pictures

16.7 Deformations of Multivalued Pictures

16.8 Exercises

16.9 Commented Bibliography

Chapter 17: Picture Properties and Spatial Relations

17.1 Properties

17.2 Moments

17.3 Experimental Evaluation of Moment Estimates

17.4 Operations on Pictures and Invariant Properties

17.5 Spatial Relations

17.6 Exercises

17.7 Commented Bibliography

List of Algorithms

List of Symbols

List of Axioms and Properties

Bibliography

Index