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Cup products on polyhedral approximations of 3D digital images


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Opened Access Cup products on polyhedral approximations of 3D digital images

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Author: González Díaz, Rocío
Lamar León, Javier
Umble, Ronald
Department: Universidad de Sevilla. Departamento de Matemática Aplicada I
Date: 2011
Published in: Combinatorial Image Analysis, Lecture Notes in Computer Science, Vol. 6636 p. 107-119
Document type: Chapter of Book
Abstract: Let I be a 3D digital image, and let Q(I) be the associated cubical complex. In this paper we show how to simplify the combinatorial structure of Q(I) and obtain a homeomorphic cellular complex P(I) with fewer cells. We introduce formulas for a diagonal approximation on a general polygon and use it to compute cup products on the cohomology H *(P(I)). The cup product encodes important geometrical information not captured by the cohomology groups. Consequently, the ring structure of H *(P(I)) is a finer topological invariant. The algorithm proposed here can be applied to compute cup products on any polyhedral approximation of an object embedded in 3-space.
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