Geometry of Digital Spaces

"There are only a few books that address the processing of digital spaces...in a mathematically rigorous manner: this book is one of them... A valuable contribution to formally study and present modern digital data processing methods." -Mathematical Reviews

• 1 Cloning Flies on Sugar Cubes.- 1.1 What Is Our Game?.- 1.2 A Methodology for Extracting Object Boundaries.- 1.3 Flies in Flatland.- 1.4 Components Determined by Binary Relations.- 1.5 So, What Does a Flat Fly Do?.- 1.6 Back to the Cuberille.- 1.7 Algorithms for Fat Flies.- 1.8 Digraphs.- 1.9 So, What Can a Fat Fly Do?.- 1.10 Algorithms for Cloning Flies.- 1.11 An Efficient Implementation.- 1.12 Exercises.- 2 Enhancing the Cube.- 2.1 Why Study Noncubic Grids?.- 2.2 Other Spaces.- 2.3 Exercises.- 3 Digital Spaces.- 3.1 The Basic Definitions.- 3.2 Interiors and Exteriors.- 3.3 Connectedness in Digital Spaces.- 3.4 Isomorphisms between Digital Spaces.- 3.5 Exercises.- 4 Topological Digital Spaces.- 4.1 What Is a Topology?.- 4.2 Some Topological Digital Spaces.- 4.3 Many Digital Spaces Are Not Topological.- 4.4 Connectedness of Topological Interiors.- 4.5 Exercises.- 5 Binary Pictures.- 5.1 Digital Pictures.- 5.2 Fuzzy Segmentation.- 5.3 Boundaries in Binary Pictures.- 5.4 Jordan Pairs of Spel-Adjacencies.- 5.5 New Jordan Pairs from Old Ones.- 5.6 Exercises.- 6 Simply Connected Digital Spaces.- 6.1 N-Simply Connected Digital Spaces.- 6.2 Locally-Jordan Surfaces.- 6.3 Applications to Finding Jordan Pairs.- 6.4 1-Simply Connected Digital Spaces.- 6.5 Exercises.- 7 Jordan Graphs.- 7.1 The Theory of (Strong) Jordan Graphs.- 7.2 Jordan Surfaces.- 7.3 Spel-Manifolds.- 7.4 Exercises.- 8 Boundary Tracking.- 8.1 Tracking in Finitary 1-Simply Connected Spaces.- 8.2 Efficient Tracking of Boundary Elements.- 8.3 Boundary Tracking on Hypercubes.- 8.4 Proofs of the Boundary-Tracking Claims.- 8.5 Boundary Tracking in the FCC Grid.- 8.6 Pointers to Further Reading.- 8.7 Exercises.- Appendix List of Symbols.- References.