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Tables of Contents for Advances in Digital and Computational Geometry
Chapter/Section Title
Page #
Page Count
Preface
v
1 Digital Geometry
1
54
1.1 Introduction
1
2
1.2 Digitization
3
1
1.3 Topology
4
2
1.4 Distance and Size
6
1
1.5 Convexity and Elongatedness
6
1
1.6 Three Dimensions
7
1
1.7 Other Grids
7
1
1.8 Gray Levels
8
1
1.9 Concluding Remarks
8
1
1.10 Bibliography
9
46
2 Topological Projection of Planar Discrete Patterns
55
32
2.1 Introduction
56
2
2.2 Patterns and Connectivity
58
7
2.2.1 Patterns and Operations
58
3
2.2.2 Connectivity and Neighborhood
61
4
2.3 Topological Equivalence of Binary Patterns
65
7
2.3.1 Topology and Connectivity
65
3
2.3.2 Topology of Gray-Scale Patterns
68
3
2.3.3 Topological Equivalence of Planar Patterns
71
1
2.4 Enumeration of the Number of Holes
72
7
2.4.1 Topological Projection of Planar Patterns
72
4
2.4.2 Convex Closure and Holes
76
3
2.5 Convex Closure and Genuses of Spatial Patterns
79
4
2.6 Conclusions
83
1
2.7 Bibliography
84
3
3 Discrete Integral Geometry and Stochastic Images
87
26
3.1 Discrete Objects
87
9
3.1.1 Incidence Structures
87
2
3.1.2 Homogeneous Incidence Structures
89
1
3.1.3 Z(n) as Incidence Structure
90
1
3.1.4 Objects in Z(n)
91
3
3.1.5 Similarity of Objects
94
2
3.2 Discrete Integral Geometry
96
9
3.2.1 Motions of Objects in Z(n)
96
1
3.2.2 Count Measures and Intersections of Objects
97
2
3.2.3 Applications of the Intersection Formula
99
3
3.2.4 Count Formulas
102
3
3.3 Stochastic Images
105
5
3.3.1 Expectation Values
105
5
3.4 Bibliography
110
3
4 On Approximation of Planar One-Dimensional Continua
113
48
4.1 Introduction
113
4
4.2 The Jordan Content and Grid Topology
117
2
4.3 Intrinsic Geometry: Basic Notions
119
9
4.4 Method of Shrinking
128
5
4.5 Method of Expanding
133
8
4.6 Implicit Forms and Characteristic Sets
141
3
4.7 Computational Geometry: Algorithms
144
8
4.8 Examples
152
2
4.9 Conclusion
154
4
4.10 Bibliography
158
3
5 Approximation and Representation of 3D Objects
161
34
5.1 Introduction
161
7
5.1.1 Soundness Properties
162
2
5.1.2 Jordan Faces
164
3
5.1.3 Jordan Surfaces
167
1
5.2 Object Digitization
168
7
5.2.1 Orthogonal Grids at Different Resolutions
168
2
5.2.2 Digitization Schemes
170
5
5.3 Convergence Analysis
175
15
5.3.1 Volume and Surface Area Calculation
176
3
5.3.2 Approximation of Planes
179
4
5.3.3 Integration of Jordan Face Gradients
183
3
5.3.4 Solution of a Cauchy Problem
186
4
5.4 Conclusions
190
1
5.5 Bibliography
191
4
6 Digitization Models to Discrete Shape Constraints
195
32
6.1 Introduction
195
2
6.2 Digital Representations
197
1
6.3 Relation between Continuous and Digital Shape Features
198
2
6.4 Continuous Representations of Real Objects
200
5
6.5 Digitization and Segmentation
205
2
6.6 Topology Preservation
207
3
6.7 Digitizations Produce Well-Composed Images
210
3
6.8 Spatial Reasoning with Digital Representations
213
3
6.9 An Optimal Threshold for Gray-Level Document Images
216
8
6.9.1 Well-Composedness of Document Images
218
1
6.9.2 Weak Connectivity
219
5
6.10 Bibliography
224
3
7 Performance Analysis of Voronoi Algorithms
227
28
7.1 Introduction
227
1
7.2 The Voronoi Tessellation
228
9
7.2.1 Basic Notions
228
1
7.2.2 A Divide and Conquer Algorithm
229
1
7.2.3 Distance Functions
229
3
7.2.4 Approximations of the Euclidean Distance
232
1
7.2.5 Implementation of the Region Growing Method
233
2
7.2.6 The 3-Dimensional Voronoi Tessellation
235
2
7.3 Performance Analysis
237
12
7.3.1 Statistical Analysis
238
3
7.3.2 Cumulative Distribution Functions in the 2-D Case
241
4
7.3.3 Cumulative Distribution Functions in the 3-D Case
245
4
7.4 Parallel Implementation
249
3
7.4.1 Parallel 2-D Algorithm
249
2
7.4.2 Performance Measurements
251
1
7.4.3 Parallelization of the 3-D Voronoi Tessellation Algorithm
252
1
7.5 Conclusions
252
1
7.6 Bibliography
253
2
8 On Digital Convex Polygons
255
30
8.1 Introduction
255
1
8.2 Basic Notions
256
4
8.3 On Optimal Polygons P(n)
260
5
8.3.1 Limit Properties of m(n)
263
1
8.3.2 Related Results and Comments
264
1
8.4 On Optimal Polygons J(n)
265
4
8.4.1 Limit Properties of p(n)
267
1
8.4.2 Related Results and Comments
268
1
8.5 Limit Shapes of P(n) and J(n)
269
9
8.5.1 Limit Shape of P(n)
270
4
8.5.2 Limit Shape of J(n)
274
4
8.6 Relations between P(n) and J(n)
278
3
8.7 Conclusion
281
1
8.8 Bibliography
282
3
9 Deformation of Discrete Surfaces
285
32
9.1 Introduction
285
2
9.2 Discrete Combinatorial Topology
287
7
9.2.1 Unit Discrete Polyhedra
288
3
9.2.2 Polyhedral Discrete Complex
291
2
9.2.3 Simplicial Discrete Complex
293
1
9.3 Combinatorial Representation of Discrete Objects
294
1
9.3.1 Polyhedral Representation
294
1
9.3.2 Simplicial Representation
295
1
9.4 Construction of Discrete Object Surfaces
295
5
9.4.1 First Stage: Polyhedral Decomposition of Lattice Points
296
3
9.4.2 Second Stage: Object Separation by Connectivity
299
1
9.4.3 Third Stage: Surface Extraction from Discrete Objects
300
1
9.5 Algorithm of Discrete Closed Surface Construction
300
4
9.6 Topology of Points
304
1
9.7 Deformation and Topology
305
9
9.7.1 Polyhedral Deformation
306
1
9.7.2 Simplicial Deformation
306
2
9.7.3 Elementary Deformation Operations
308
1
9.7.4 Topology Preservation and Deformation
309
1
9.7.5 Deformation Algorithm
310
4
9.8 Conclusions
314
1
9.9 Bibliography
315
2
10 On Symmetry in Digital Geometry
317
24
10.1 Introduction
317
2
10.2 Transformations in the Continuous Plane
319
3
10.2.1 Symmetries
320
2
10.3 The Classical Symmetry Groups for Digital Geometry
322
4
10.4 A Digital Approach to Reflection
326
1
10.5 Arithmetic Representation of Lines
326
1
10.6 Construction of Arithmetic Reflections
327
4
10.7 Properties of Arithmetic Reflections
331
2
10.8 Examples
333
2
10.9 Alternative Constructions
335
1
10.10 Digital Symmetries
336
2
10.10.1 Digital Symmetry Groups
337
1
10.11 Conclusions
338
1
10.12 Bibliography
338
3
11 Computational Geometry
341
11.1 About this Bibliography
341
1
11.2 Concluding Remark
342
1
11.3 Bibliography
342
11.3.1 General References
342
1
11.3.2 Matching
343
1
11.3.3 Range Searching
343
1
11.3.4 Point Location
344
1
11.3.5 Enclosure and Separation
344
1
11.3.6 Intersection
345
1
11.3.7 Rays and Lines
346
1
11.3.8 Polygons
347
1
11.3.9 Packing and Covering
348
1
11.3.10 Arrangements and Partitions
348
1
11.3.11 Triangulations and Subdivisions
349
2
11.3.12 Distance and Proximity
351
2
11.3.13 Voronoi Diagrams, Delaunay Triangulations
353
1
11.3.14 Centers
354
1
11.3.15 Convexity and Hulls
354
1
11.3.16 Visibility
355
2
11.3.17 Paths and Networks
357
2
11.3.18 Moving Objects and Motion Planning
359
2
11.3.19 Reconstruction and Shape Approximation
361
1
11.3.20 Robust Computation
361
1
11.3.21 Randomization and Sampling
362
1
11.3.22 Software
362
1
11.3.23 Miscellaneous
363