122 research outputs found
Removal and Contraction Operations in D Generalized Maps for Efficient Homology Computation
In this paper, we show that contraction operations preserve the homology of
D generalized maps, under some conditions. Removal and contraction
operations are used to propose an efficient algorithm that compute homology
generators of D generalized maps. Its principle consists in simplifying a
generalized map as much as possible by using removal and contraction
operations. We obtain a generalized map having the same homology than the
initial one, while the number of cells decreased significantly.
Keywords: D Generalized Maps; Cellular Homology; Homology Generators;
Contraction and Removal Operations.Comment: Research repor
3D Well-composed Polyhedral Complexes
A binary three-dimensional (3D) image is well-composed if the boundary
surface of its continuous analog is a 2D manifold. Since 3D images are not
often well-composed, there are several voxel-based methods ("repairing"
algorithms) for turning them into well-composed ones but these methods either
do not guarantee the topological equivalence between the original image and its
corresponding well-composed one or involve sub-sampling the whole image.
In this paper, we present a method to locally "repair" the cubical complex
(embedded in ) associated to to obtain a polyhedral
complex homotopy equivalent to such that the boundary of every
connected component of is a 2D manifold. The reparation is performed via
a new codification system for under the form of a 3D grayscale image
that allows an efficient access to cells and their faces
Topological signature for periodic motion recognition
In this paper, we present an algorithm that computes the topological
signature for a given periodic motion sequence. Such signature consists of a
vector obtained by persistent homology which captures the topological and
geometric changes of the object that models the motion. Two topological
signatures are compared simply by the angle between the corresponding vectors.
With respect to gait recognition, we have tested our method using only the
lowest fourth part of the body's silhouette. In this way, the impact of
variations in the upper part of the body, which are very frequent in real
scenarios, decreases considerably. We have also tested our method using other
periodic motions such as running or jumping. Finally, we formally prove that
our method is robust to small perturbations in the input data and does not
depend on the number of periods contained in the periodic motion sequence.Comment: arXiv admin note: substantial text overlap with arXiv:1707.0698
Geometric Objects and Cohomology Operations
Cohomology operations (including the cohomology ring) of a geometric object
are finer algebraic invariants than the homology of it.
In the literature, there exist various algorithms for computing the homology
groups of simplicial complexes but concerning the algorithmic treatment of
cohomology operations, very little is known. In this paper, we establish a
version of the incremental algorithm for computing homology which saves
algebraic information, allowing us the computation of the cup product and the
effective evaluation of the primary and secondary cohomology operations on the
cohomology of a finite simplicial complex. We study the computational
complexity of these processes and a program in Mathematica for cohomology
computations is presented.Comment: International Conference Computer Algebra and Scientific Computing
CASC'02. Big Yalta, Crimea (Ucrania). Septiembre 200
Simplification Techniques for Maps in Simplicial Topology
This paper offers an algorithmic solution to the problem of obtaining
"economical" formulae for some maps in Simplicial Topology, having, in
principle, a high computational cost in their evaluation. In particular, maps
of this kind are used for defining cohomology operations at the cochain level.
As an example, we obtain explicit combinatorial descriptions of Steenrod k-th
powers exclusively in terms of face operators
An algorithm to compute minimal Sullivan algebras
In this note, we give an algorithm that starting with a Sullivan algebra
gives us its minimal model. This algorithm is a kind of modified AT-model
algorithm used to compute in the past other kinds of topology information such
as (co)homology, cup products on cohomology and persistent homology. Taking as
input a (non-minimal) Sullivan algebra with an ordered finite set of
generators preserving the filtration defined on , we obtain as output a
minimal Sullivan algebra with the same rational cohomology as
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