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Rocio conference papers/ponencias |
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Conference papers/Ponencias Guillaume Damiand, Rocío González-Díaz, Samuel Peltier: Removal Operations in nD
Generalized Maps for Efficient Homology computation. CTIC 2012 LNCS: 20-29 In this paper, we
present an efficient way for computing homology generators of nD generalized maps. The algorithm proceeds in two
steps: (1) cell removals reduces the number of cells while preserving
homology; (2) homology generator computation is performed on the reduced
object by reducing incidence matrices into their Smith-Agoston
normal form. In this paper, we provide a definition of cells that can be
removed while preserving homology. Some results on 2D and 3D homology
generators computation are presented. Doi: 10.1007/978-3-642-30238-1_3 Javier Lamar-León, Edel B. García Reyes, Rocío González-Díaz: Human Gait Identification Using Persistent Homology. CIARP 2012 LNCS: 244-251This paper shows
an image/video application using topological invariants for human gait
recognition. Using a background subtraction approach, a stack of silhouettes
is extracted from a subsequence and glued through their gravity centers,
forming a 3D digital image I. From this 3D representation, the
border simplicial complex ∂ K(I) is obtained. We order the triangles of
∂ K(I) obtaining a sequence of subcomplexes of ∂ K(I). The corresponding filtration F captures
relations among the parts of the human body when walking. Finally, a
topological gait signature is extracted from the persistence barcode
according to F. In this work we obtain 98.5% correct
classification rates on CASIA-B database. Doi: 10.1007/978-3-642-33275-3_30 Rocío González-Díaz, Adrian Ion, María José Jiménez , Regina Poyatos: Incremental-Decremental Algorithm for Computing AT-Models and Persistent Homology. CAIP (1) 2011 LNCS: 286-293In this paper, we
establish a correspondence between the incremental algorithm for computing
AT-models [8,9] and the one for computing persistent homology [6,14,15]. We
also present a decremental algorithm for computing
AT-models that allows to extend the persistence computation to a wider
setting. Finally, we show how to combine incremental and decremental
techniques for persistent homology computation. Doi: 10.1007/978-3-642-23672-3_35 Rocío
González-Díaz, María
José Jiménez, Belén
Medrano: Well-Composed Cell Complexes. DGCI
2011 LNCS: 153-162 Well-composed 3D
digital images, which are 3D binary digital images whose boundary surface is
made up by 2D manifolds, enjoy important topological and geometric properties
that turn out to be advantageous for some applications. In this paper, we
present a method to transform the cubical complex associated to a 3D binary
digital image (which is not generally a well-composed image) into a cell
complex that is homotopy equivalent to the first
one and whose boundary surface is composed by 2D manifolds. This way, the new
representation of the digital image can benefit from the application of
algorithms that are developed over surfaces embedded in R3. Doi: 10.1007/978-3-642-19867-0_13 Rocío González-Díaz, Javier Lamar, Ronald Umble: Cup Products on Polyhedral Approximations of 3D
Digital Images. IWCIA 2011 LNCS: 107-119 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. Doi: 10.1007/978-3-642-21073-0_12 Rocío González-Díaz, Adrian Ion, Mabel Iglesias Ham, Walter G. Kropatsch: Irregular Graph Pyramids and Representative Cocycles of Cohomology
Generators. GbRPR 2009 LNCS: 263-272 Structural
pattern recognition describes and classifies data based on the relationships
of features and parts. Topological invariants, like the Euler number,
characterize the structure of objects of any dimension. Cohomology
can provide more refined algebraic invariants to a topological space than
does homology. It assigns `quantities' to the chains used in homology to
characterize holes of any dimension. Graph pyramids can be used to describe
subdivisions of the same object at multiple levels of detail. This paper
presents cohomology in the context of structural
pattern recognition and introduces an algorithm to efficiently compute
representative cocycles (the basic elements of cohomology) in 2D using a graph pyramid. Extension to nD and application in the context of pattern recognition
are discussed. Doi: 10.1007/978-3-642-02124-4_27 Rocío
González-Díaz, María
José Jiménez, Belén
Medrano, Helena
Molina-Abril, Pedro
Real: Integral Operators for
Computing Homology Generators
at Any Dimension. CIARP
2008 LNCS: 356-363 Starting from an nD geometrical object, a cellular subdivision
of such an object provides an algebraic counterpart from which homology
information can be computed. In this paper, we develop a process to
drastically reduce the amount of data that represent the original object,
with the purpose of a subsequent homology computation. The technique applied
is based on the construction of a sequence of elementary chain homotopies (integral operators) which
algebraically connect the initial object with a simplified one with the same
homological information than the former. Doi: 10.1007/978-3-540-85920-8_44 Rocío González-Díaz, María José Jiménez, Belén Medrano, Pedro Real: A Graph-with-Loop Structure for a Topological
Representation of 3D Objects. CAIP
2007: 506-513 Given a cell complex
K whose geometric realization |K | is embedded in R 3 and a continuous
function h: |K | → R (called the height function), we construct a graph Gh(K) which is an extension of the Reeb
graph Rh(|K|). More concretely, the graph Gh(K)
without loops is a subdivision of Rh(|K|). The most important difference
between the graphs Gh(K) and Rh(|K|) is that Gh(K) preserves not only the number of connected
components but also the number of “tunnels ” (the homology generators of
dimension 1) of K. The latter is not true in general for Rh(|K|). Moreover,
we construct a map ψ: Gh(K) → K
identifying representative cycles of the tunnels in K with the ones in Gh(K) in the way that if e is a loop in Gh(K), then ψ(e) is a cycle in K such that all the points in |ψ(e) | belong to the same
level set in |K|. 1 Reeb Graphs and Tunnels We are
interested in analyzing and visualizing intrinsic properties of geometric
models and scientific data. Specifically, Reeb graphs [13], which express the connectivity. Doi: 10.1007/978-3-540-74272-2_63 Rocío
González-Díaz, María
José Jiménez, Belén
Medrano, Pedro
Real: Extending the
Notion of AT-Model for Integer Homology
Computation. GbRPR 2007 LNCS: 330-339 When the ground
ring is a field, the notion of algebraic topological model (AT-model) is a
useful tool for computing (co)homology, representative (co)cycles of
(co)homology generators and the cup product on cohomology
of nD digital images as well as for controlling
topological information when the image suffers local changes [6,7,9]. In this
paper, we formalize the notion of λ-AT-model (λ being an integer) which extends the one of
AT-model and allows the computation of homological information in the integer
domain without computing the Smith Normal Form of the boundary matrices. We
present an algorithm for computing such a model, obtaining Betti numbers, the prime numbers p involved
in the invariant factors (corresponding to the torsion subgroup of the
homology), the amount of invariant factors that are a power of p and
a set of representative cycles of the generators of homology mod p,
for such p. Doi: 10.1007/978-3-540-72903-7_30 Rocío
González-Díaz, Belén
Medrano, Javier
Sánchez-Peláez, Pedro
Real: Simplicial Perturbation
Techniques and Effective Homology. CASC
2006 LNCS: 166-177 In this paper, we deal with the problem of the
computation of the homology of a finite simplicial
complex after an “elementary simplicial
perturbation” process such as the inclusion or elimination of a maximal
simplex or an edge contraction. To this aim we compute an algebraic
topological model that is a special chain homotopy
equivalence connecting the simplicial complex with
its homology (working with a field as the ground ring). Doi: 10.1007/11870814_14 Rocío
González-Díaz, Belén
Medrano, Javier
Sánchez-Peláez, Pedro
Real: Reusing Integer
Homology Information of Binary Digital Images. DGCI
2006 LNCS: 199-210 In this paper,
algorithms for computing integer (co)homology of a simplicial
complex of any dimension are designed, extending the work done in [1,2,3].
For doing this, the homology of the object is encoded in an
algebraic-topological format (that we call AM-model). Moreover, in the case
of 3D binary digital images, having as input AM-models for the images I and J,
we design fast algorithms for computing the integer homology of the
union, intersection and set diference of two sets I
and J. Doi: 10.1007/11907350_17 Rocío
González-Díaz, Belén
Medrano, Pedro
Real, Javier
Sánchez-Peláez: Algebraic Topological Analysis of Time-Sequence of Digital Images.
CASC
2005 LNCS: 208-219 This paper
introduces an algebraic framework for a topological analysis of time-varying
2D digital binary–valued images, each of them defined as 2D
arrays of pixels. Our answer is based on an algebraic-topological coding,
called AT–model, for a nD (n=2,3)
digital binary-valued image I consisting simply in taking I together
with an algebraic object depending on it. Considering AT–models for all the
2D digital images in a time sequence, it is possible to get an AT–model for
the 3D digital image consisting in concatenating the successive 2D digital
images in the sequence. If the frames are represented in a quadtree format, a similar positive result can be
derived. Doi: 10.1007/11555964_18 Rocío
González-Díaz, Pedro
Real: Towards Digital Cohomology. DGCI
2003 LNCS: 92-101 We propose a
method for computing the Z 2–cohomology
ring of a simplicial complex uniquely associated
with a three–dimensional digital binary–valued picture I. Binary digital
pictures are represented on the standard grid Z 3,
in which all grid points have integer coordinates. Considering a particular
14–neighbourhood system on this grid, we construct
a unique simplicial complex K(I)
topologically representing (up to isomorphisms of
pictures) the picture I. We then compute the cohomology
ring on Ivia the simplicial
complex K(I). The usefulness of a simplicial
description of the digital Z 2–cohomology ring of binary digital pictures is tested by
means of a small program visualizing the different steps of our method. Some
examples concerning topological thinning, the visualization of representative
generators of cohomology classes and the computation
of the cup product on the cohomology of simple 3D
digital pictures are showed. Doi: 10.1007/978-3-540-39966-7_8 |
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