Computational Topology
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Home > Art, Film & Photography > Computational Topology: Digital Topology, Simplicial Homology, Pixel Connectivity, Region Connection Calculus, Topological Data Analysis
Computational Topology: Digital Topology, Simplicial Homology, Pixel Connectivity, Region Connection Calculus, Topological Data Analysis

Computational Topology: Digital Topology, Simplicial Homology, Pixel Connectivity, Region Connection Calculus, Topological Data Analysis


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About the Book

Purchase includes free access to book updates online and a free trial membership in the publisher's book club where you can select from more than a million books without charge. Chapters: Digital Topology, Simplicial Homology, Pixel Connectivity, Region Connection Calculus, Topological Data Analysis, Spatial-Temporal Reasoning, Grid Cell Topology. Excerpt: In mathematics, in the area of algebraic topology, simplicial homology is a theory with a finitary definition, and is probably the most tangible variant of homology theory. Simplicial homology concerns topological spaces whose building blocks are n-simplexes, the n-dimensional analogs of triangles. By definition, such a space is homeomorphic to a simplicial complex (more precisely, the geometric realization of an abstract simplicial complex). Such a homeomorphism is referred to as a triangulation of the given space. Replacing n-simplexes by their continuous images in a given topological space gives singular homology. The simplicial homology of a simplicial complex is naturally isomorphic to the singular homology of its geometric realization. This implies, in particular, that the simplicial homology of a space does not depend on the triangulation chosen for the space. It has been shown that all manifolds up to 3 dimensions allow for a triangulation. This, together with the fact that it is now possible to resolve the simplicial homology of a simplicial complex automatically and efficiently, make this theory feasible for application to real life situations, such as image analysis, medical imaging, and data analysis in general. Let S be a simplicial complex. A simplicial k-chain is a formal sum of k-simplices, where is the i-th k-simplex.The group of k-chains on S, the free abelian group defined on the set of k-simplices in S, is denoted Ck. Consider a basis element of Ck, a k-simplex, The boundary operator is a homomorphism defined by: where the simplex is the i face of obta... More: http: //booksllc.net/?id=1060236


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Product Details
  • ISBN-13: 9781157807629
  • Publisher: Books LLC
  • Publisher Imprint: Books LLC
  • Height: 152 mm
  • Sub Title: Digital Topology, Simplicial Homology, Pixel Connectivity, Region Connection Calculus, Topological Data Analysis
  • Width: 229 mm
  • ISBN-10: 1157807623
  • Publisher Date: 09 Jun 2010
  • Binding: Paperback
  • Spine Width: 2 mm
  • Weight: 68 gr


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