Geometry of Digital Spaces

"La narraci6n literaria es la evocaci6n de las nostalgias. " ("Literary narration is the evocation of nostalgia. ") G. G. Marquez, interview in Puerta del Sol, VII, 4, 1996. A Personal Prehistory In 1972 I started cooperating with members of the Biodynamics Research Unit at the Mayo Clinic in Rochester, Minnesota, which was under the direction of Earl H. Wood. At that time, their ambitious (and eventually realized) dream was to build the Dynamic Spatial Reconstructor (DSR), a device capable of collecting data regarding the attenuation of X-rays through the human body fast enough for stop-action imaging the full extent of the beating heart inside the thorax. Such a device can be applied to study the dynamic processes of cardiopulmonary physiology, in a manner similar to the application of an ordinary cr (computerized tomography) scanner to observing stationary anatomy. The standard method of displaying the information produced by a cr scanner consists of showing two-dimensional images, corresponding to maps of the X-ray attenuation coefficient in slices through the body. (Since different tissue types attenuate X-rays differently, such maps provide a good visualization of what is in the body in those slices; bone - which attenuates X-rays a lot - appears white, air appears black, tumors typically appear less dark than the surrounding healthy tissue, etc. ) However, it seemed to me that this display mode would not be appropriate for the DSR.

Contenu

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.