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Homological computation using spanning trees


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Author: Molina Abril, Helena
Real Jurado, Pedro
Department: Universidad de Sevilla. Departamento de Matemática Aplicada I (ETSII)
Date: 2009
Published in: Progress in Pattern Recognition, Image Analysis, Computer Vision, and Applications, Lecture Notes in Computer Science, Vol. 5856 p. 272-278
Document type: Chapter of Book
Abstract: We introduce here a new F2 homology computation algorithm based on a generalization of the spanning tree technique on a finite 3-dimensional cell complex K embedded in ℝ3. We demonstrate that the complexity of this algorithm is linear in the number of cells. In fact, this process computes an algebraic map φ over K, called homology gradient vector field (HGVF), from which it is possible to infer in a straightforward manner homological information like Euler characteristic, relative homology groups, representative cycles for homology generators, topological skeletons, Reeb graphs, cohomology algebra, higher (co)homology operations, etc. This process can be generalized to others coefficients, including the integers, and to higher dimension.
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