1,559 research outputs found
Generating Second Order (Co)homological Information within AT-Model Context
In this paper we design a new family of relations between
(co)homology classes, working with coefficients in a field and starting
from an AT-model (Algebraic Topological Model) AT(C) of a finite cell
complex C These relations are induced by elementary relations of type
“to be in the (co)boundary of” between cells. This high-order connectivity
information is embedded into a graph-based representation model,
called Second Order AT-Region-Incidence Graph (or AT-RIG) of C. This
graph, having as nodes the different homology classes of C, is in turn,
computed from two generalized abstract cell complexes, called primal
and dual AT-segmentations of C. The respective cells of these two complexes
are connected regions (set of cells) of the original cell complex C,
which are specified by the integral operator of AT(C). In this work in
progress, we successfully use this model (a) in experiments for discriminating
topologically different 3D digital objects, having the same Euler
characteristic and (b) in designing a parallel algorithm for computing
potentially significant (co)homological information of 3D digital objects.Ministerio de EconomĂa y Competitividad MTM2016-81030-PMinisterio de EconomĂa y Competitividad TEC2012-37868-C04-0
An entropy-based persistence barcode
In persistent homology, the persistence barcode encodes pairs of simplices meaning birth and death of homology classes. Persistence barcodes depend on the ordering of the simplices (called a filter) of the given simplicial complex. In this paper, we define the notion of “minimal” barcodes in terms of entropy. Starting from a given filtration of a simplicial complex K, an algorithm for computing a “proper” filter (a total ordering of the simplices preserving the partial ordering imposed by the filtration as well as achieving a persistence barcode with small entropy) is detailed, by way of computation, and subsequent modification, of maximum matchings on subgraphs of the Hasse diagram associated to K. Examples demonstrating the utility of computing such a proper ordering on the simplices are given
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