Computational Proximity Excursions in the Topology of Digital Images 1st edition by James Peters 3319302604 9783319302607

1 Computational Proximity

1.1 Computational Proximity Framework

1.2 Points, Regions, Connectedness and Point-Free Geometry

1.3 Choice of Probe

1.4 Proximities

1.5 Di Concilio Strong Contact

1.6 Strongly Far Proximity

1.7 Convexity Structures

1.8 Descriptive Proximity

1.9 Descriptive Strong Proximity

1.10 Nerves

1.11 Voronoï Diagrams and Mesh Nerves

1.12 Origins, Variations and Applications of Voronoï Diagrams

1.13 Mesh Nerves on a Digital Image

1.14 Singer Needle Tip Image Mesh: A Step Toward Object Recognition

1.15 Topological Spaces: Setting for Computational Proximity

1.16 Connectedness and Strongly Near Connectedness

1.17 Connected and Strongly Connected Mesh Nerves

1.18 Boundedness, Bornology and Bornological Nerves

1.19 Proximal Boundedness, Strong Proximal Boundednessƒ

1.20 Local Proximity Spaces and Local Strong Proximity Spaces

References

2 Proximities Revisited

2.1 Cech Proximity

2.2 Cech Closure of a Set

2.3 Near Edge Sets

2.4 Lodato Proximity

2.5 Descriptive Lodato Proximity

2.6 Delaunay Triangulation

2.7 Voronoï Diagrams Revisited

2.7.1 Sites

2.8 Some Results for Voronoï Regions

2.9 Dirichlet Tessellation Quality and Digital Image Quality

2.10 Tessellation Region Centroids

2.11 Centroid-Based Voronoï Mesh on an Image

References

3 Distance and Proximally Continuous

3.1 Metrics and Metric Topology

3.2 Distances: Euclidean and Taxicab Metrics

3.3 Metric Proximity

3.4 Closeness Metric

3.5 Some Recent History of Near Sets

3.6 Descriptive Similarity Distance

3.7 Descriptively Near Sets Can Be Spatially Disjoint Sets

3.8 Neighbourhoods of Points in Euclidean Space

3.9 2D Picture Descriptive Neighbourhood of a Point

3.9.1 Unbounded Descriptive Neighbourhood of a Picture Point

3.9.2 Bounded Descriptive Neighbourhood of a Picture Point

3.9.3 Bounded Indistinguishable Descriptive Neighbourhood of a Picture Point

3.9.4 Unbounded Indistinguishable Descriptive Neighbourhood of a Picture Point

References

4 Image Geometry and Nearness Expressions for Image and Scene Analysis

4.1 Image Geometry

4.1.1 Delaunay Triangulation

4.1.2 Voronoï Diagrams

4.2 Nearness Expressions: Basic Notions

4.3 Near and Strongly Near Sets Revisited

4.3.1 Lodato Proximity Revisited

4.3.2 Descriptive Lodato Proximity Revisited

4.3.3 Strong Proximity Revisited

4.3.4 Descriptive Strong Proximity Revisited

4.4 Di Concilio–Gerla Overlap Revisited

4.5 Wallman Proximity Revisited

4.6 Far and Strongly Far Sets Revisited

References

5 Homotopic Maps, Shapes and Borsuk–Ulam Theorem

5.1 Antipodal Points and Hyperspheres

5.2 Borsuk–Ulam Theorem

5.3 Homotopic Maps and Deformation Retracts

5.4 Object Description, Structured Sets and Shape Features

5.4.1 Euclidean Feature Space

5.4.2 Feature Vectors

5.5 Strongly Proximal Object and Feature Spaces

5.6 Strong Proximal Borsuk–Ulam Theorem

5.7 Strong Proximal Region-Based Borsuk–Ulam Theorem

5.8 Complexes, Nuclei and Applied Homotopy

5.9 Strong Proximal Homotopy

5.10 Borsuk–Ulam Theorem Extended to Hyperbolic Surfaces by A. Tozzi

References

6 Visibility, Hausdorffness, Algebra and Separation Spaces

6.1 Visibility

6.2 Separation Axioms

6.3 R0 Symmetric Space

6.4 Descriptive R0 Space

6.4.1 Near Sets in L-Proximity Spaces

6.4.2 Descriptive L-Proximity

6.4.3 Descriptive Proximity Space Revisited

6.5 T0 Anti-Symmetric Space

6.6 T1 Space

6.7 Hausdorff T2 Space

6.8 Hausdorffness and Visibility

6.9 Partial Descriptive Groupoids in Hausdorff Spaces

6.10 Set-Based Probes and Proximal Delaunay Groupoids

6.11 Voronoï Groupoids

6.12 T3 Space

6.13 T4 Space

6.14 Visual Set Patterns in Descriptive Separation Spaces

6.14.1 Visual Patterns in Descriptive T1 Spaces

6.14.2 Visual Patterns in Descriptive T2 Spaces

6.14.3 Set Pattern Generators

References

7 Strongly Near Sets and Overlapping Dirichlet Tessellation Regions

7.1 Known Mesh Seed or Generating Points

7.2 Julia Set Image Example

7.3 Strongly Near Regions and Mesh Cells

7.4 Back to Earth

References

8 Proximal Manifolds

8.1 Manifolds: Basic Notions

8.2 Charts and Atlases

8.3 Proximal Voronoï Manifolds, Atlases and Charts

8.4 Mean Shift Manifold

8.5 Digital Image Manifolds

8.6 Mean-Shift Image Manifolds

8.7 Image Saliency Manifolds

8.8 Selfie Voronoï Manifold

8.9 Manifolds with a Proximity

References

9 Watershed, Smirnov Measure, Fuzzy Proximity and Sorted Near Sets

9.1 See-Through MM Segments Via Dirichlet Tessellation on an Image

9.2 Strongly Near Voronoï Region Interiors on MM Segmentations

9.3 Extension of Smirnov Proximity Measure

9.4 Fuzzy Lodato Proximity and Fuzzy Lodato–Smirnov Membership Function

9.5 Descriptive Smirnov Proximity

9.6 Application: Sorting Near Sets Using Smirnov Proximity Measures

9.7 Fuzzy Descriptive Lodato Proximity Space

9.8 Image Dilation

9.9 MorphologicalBinarize Operation with Lower and Upper Bounds

9.10 Dilation Method

9.11 Dilation and Watershed Segmentation

9.12 Gradient Filtering Watershed Components

References

10 Strong Connectedness Revisited

10.1 Connectedness: Basic Concepts

10.2 Strongly Connected Subsets in a Voronoï Mesh

References

11 Helly’s Theorem and Strongly Proximal Helly Theorem

11.1 Helly’s Theorem

11.2 Strongly Near Version of Helly’s Theorem

11.3 Descriptive Helly’s Theorem

11.4 Polyforms

References

12 Nerves and Strongly Near Nerves

12.1 Polyform Nerves

References

13 Connnectedness Patterns

13.1 Connectedness in Voronoï Meshes

13.2 Nearness of Collections of Strongly Connected Sets

13.3 Picture Mesh Patterns

13.4 Pattern Axioms and Object Signature Axioms

13.5 Keypoints Mesh Sites

13.6 Keypoints Mesh Patterns

References

14 Nerve Patterns

14.1 Voronoï Mesh Nerves

14.2 Nerves with a Nucleus

14.3 Partial Ordering of Nuclei

14.4 Voronoï Mesh Nerve Patterns