189 research outputs found
Efficiently Storing Well-Composed Polyhedral Complexes Computed Over 3D Binary Images
A 3D binary image I can be naturally represented
by a combinatorial-algebraic structure called cubical complex
and denoted by Q(I ), whose basic building blocks are
vertices, edges, square faces and cubes. In Gonzalez-Diaz
et al. (Discret Appl Math 183:59â77, 2015), we presented a
method to âlocally repairâ Q(I ) to obtain a polyhedral complex
P(I ) (whose basic building blocks are vertices, edges,
specific polygons and polyhedra), homotopy equivalent to
Q(I ), satisfying that its boundary surface is a 2D manifold.
P(I ) is called a well-composed polyhedral complex over the
picture I . Besides, we developed a new codification system
for P(I ), encoding geometric information of the cells
of P(I ) under the form of a 3D grayscale image, and the
boundary face relations of the cells of P(I ) under the form
of a set of structuring elements. In this paper, we build upon
(Gonzalez-Diaz et al. 2015) and prove that, to retrieve topological
and geometric information of P(I ), it is enough to
store just one 3D point per polyhedron and hence neither
grayscale image nor set of structuring elements are needed.
From this âminimalâ codification of P(I ), we finally present
a method to compute the 2-cells in the boundary surface of
P(I ).Ministerio de EconomĂa y Competitividad MTM2015-67072-
Encoding Specific 3D Polyhedral Complexes Using 3D Binary Images
We build upon the work developed in [4] in which we presented
a method to âlocally repairâ the cubical complex Q(I) associated
to a 3D binary image I, to obtain a âwell-composedâ polyhedral complex
P(I), homotopy equivalent to Q(I). There, we developed a new codification
system for P(I), called ExtendedCubeMap (ECM) representation,
that encodes: (1) the (geometric) information of the cells of P(I) (i.e.,
which cells are presented and where), under the form of a 3D grayscale
image gP ; (2) the boundary face relations between the cells of P(I),
under the form of a set BP of structuring elements.
In this paper, we simplify ECM representations, proving that geometric
and topological information of cells can be encoded using just a 3D
binary image, without the need of using colors or sets of structuring
elements. We also outline a possible application in which well-composed
polyhedral complexes can be useful.Junta de AndalucĂa FQM-369Ministerio de EconomĂa y Competitividad MTM2012-32706Ministerio de EconomĂa y Competitividad MTM2015-67072-
Spatiotemporal Barcodes for Image Sequence Analysis
Taking as input a time-varying sequence of two-dimensional
(2D) binary images, we develop an algorithm for computing a spatiotemporal
0âbarcode encoding lifetime of connected components on the image
sequence over time. This information may not coincide with the one provided
by the 0âbarcode encoding the 0âpersistent homology, since the
latter does not respect the principle that it is not possible to move backwards
in time. A cell complex K is computed from the given sequence,
being the cells of K classified as spatial or temporal depending on whether
they connect two consecutive frames or not. A spatiotemporal path is
defined as a sequence of edges of K forming a path such that two edges
of the path cannot connect the same two consecutive frames. In our
algorithm, for each vertex v â K, a spatiotemporal path from v to the
âoldestâ spatiotemporally-connected vertex is computed and the corresponding
spatiotemporal 0âbar is added to the spatiotemporal 0âbarcode.Junta de AndalucĂa FQM-369Ministerio de EconomĂa y Competitividad MTM2012-3270
Well-Composed Cell Complexes
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 â3
3D well-composed polyhedral complexes
A binary three-dimensional (3D) image II is well-composed if the boundary surface of its continuous analog is a 2D manifold. In this paper, we present a method to locally ârepairâ the cubical complex Q(I)Q(I) (embedded in R3R3) associated to II to obtain a polyhedral complex P(I)P(I) homotopy equivalent to Q(I)Q(I) such that the boundary surface of P(I)P(I) is a 2D manifold (and, hence, P(I)P(I) is a well-composed polyhedral complex). For this aim, we develop a new codification system for a complex KK, called ExtendedCubeMap (ECM) representation of KK, that codifies: (1) the information of the cells of KK (including geometric information), under the form of a 3D grayscale image gPgP; and (2) the boundary face relations between the cells of KK, under the form of a set BPBP of structuring elements that can be stored as indexes in a look-up table. We describe a procedure to locally modify the ECM representation EQEQ of the cubical complex Q(I)Q(I) to obtain an ECM representation of a well-composed polyhedral complex P(I)P(I) that is homotopy equivalent to Q(I)Q(I). The construction of the polyhedral complex P(I)P(I) is accomplished for proving the results though it is not necessary to be done in practice, since the image gPgP (obtained by the repairing process on EQEQ) together with the set BPBP codify all the geometric and topological information of P(I)P(I)
Topological tracking of connected components in image sequences
Persistent homology provides information about the lifetime of homology
classes along a filtration of cell complexes. Persistence barcode is a graphi-
cal representation of such information. A filtration might be determined by
time in a set of spatiotemporal data, but classical methods for computing
persistent homology do not respect the fact that we can not move back-
wards in time. In this paper, taking as input a time-varying sequence of
two-dimensional (2D) binary digital images, we develop an algorithm for en-
coding, in the so-called spatiotemporal barcode, lifetime of connected compo-
nents (of either the foreground or background) that are moving in the image
sequence over time (this information may not coincide with the one provided
by the persistence barcode). This way, given a connected component at a
specific time in the sequence, we can track the component backwards in time
until the moment it was born, by what we call a spatiotemporal path. The
main contribution of this paper with respect to our previous works lies in a
new algorithm that computes spatiotemporal paths directly, valid for both
foreground and background and developed in a general context, setting the
ground for a future extension for tracking higher dimensional topological
features in nD binary digital image sequences.Ministerio de EconomĂa y Competitividad MTM2015-67072-
Extending the notion of AT-Model for integer homology computation
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
A Graph-with-Loop Structure for a Topological Representation of 3D Objects
Given a cell complex K whose geometric realization |K| is embedded in R 3 and a continuous function h: |K|âR (called the height function), we construct a graph G h (K) which is an extension of the Reeb graph R h (|K|). More concretely, the graph G h (K) without loops is a subdivision of R h (|K|). The most important difference between the graphs G h (K) and R h (|K|) is that G h (K) preserves not only the number of connected components but also the number of âtunnelsâ (the homology generators of dimension 1) of K. The latter is not true in general for R h (|K|). Moreover, we construct a map Ï: G h (K)âK identifying representative cycles of the tunnels in K with the ones in G h (K) in the way that if e is a loop in G h (K), then Ï(e) is a cycle in K such that all the points in |Ï(e)| belong to the same level set in |K|
Integral operators for computing homology generators at any dimension
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
Essential Oils from Fruit and Vegetables, Aromatic Herbs, and Spices: Composition, Antioxidant, and Antimicrobial Activities
The use of essential oils (EOs) in the food industry is a popular research topic, as
they have antioxidant and antimicrobial activity and could be used as ingredients directly in food or
as bioactive component in food coating and food packaging. Thus, the study of their antioxidant
and antimicrobial activity is a crucial step to evaluate their use in food packaging/coating. In this
work, we evaluate the antioxidant and antimicrobial activities of 13 EOs from herbs, spices, fruits,
and vegetables. Briefly, the EOs from aromatic herbs and spices showed the highest antioxidant
and antimicrobial activity. Fennel essential oil reported the lowest antioxidant activity, however it
showed very good antimicrobial activity against Botrytis cinerea, one of the post-harvest pathogen
microorganisms in fruits and vegetables.In the field of food preservation, encapsulated Essential Oils (EOs) could be the best
non-toxic and eco-friendly tool for food preservative applications substituting the chemicals ones
that have several disadvantages for the environment and health. Thirteen commercial EOs from
plants, fruits, and vegetables were characterized by GC-MS. The antioxidant activity was measured
by DPPH and ABTS techniques. Antimicrobial activity was assessed by agar well-diffusion method
and the Minimum Inhibitory Concentration (MIC) by agar dilution method against six bacteria,
Candida albicans, and Botrytis cinerea. All the EOs tested have demonstrated antioxidant activity in the
range of IC50 0.01â105.32 mg/mL. Between them, cinnamon EOs were the best, followed by oregano
and thyme EOs. Fennel EO showed the lowest radical scavenging. MIC values ranged from 0.14 to
9 mg/mL. C. cassia, thyme, and oregano EOs were the most effective against the bacterial species
tested, and the yeast C. albicans. On the contrary, citric fruit EOs showed low or no inhibition against
most bacterial strains. The percentages of inhibition of mycelia growth of B. cinerea ranged from 3.4
to 98.5%. Thyme, oregano, mint, and fennel EOs showed the highest inhibition.European
Unionâs Horizon 2020 -No 81793
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