48 research outputs found
Homology groups of cubical sets
The paper is devoted to homology groups of cubical sets with coefficients in
contravariant systems of Abelian groups. The study is based on the proof of the
assertion that the homology groups of the category of cubes with coefficients
in the diagram of Abelian groups are isomorphic to the homology groups of
normalized complex of the cubical Abelian group corresponding to this diagram.
The main result shows that the homology groups of a cubical set with
coefficients in a contravariant system of Abelian groups are isomorphic to the
values of left derived functors of the colimit functor on this contravariant
system. This is used to obtain the isomorphism criterion for homology groups of
cubical sets with coefficients in contravariant systems, and also to construct
spectral sequences for the covering of a cubical set and for a morphism between
cubical sets.Comment: 24 page
Cubical Cohomology Ring of 3D Photographs
Cohomology and cohomology ring of three-dimensional (3D) objects are
topological invariants that characterize holes and their relations. Cohomology
ring has been traditionally computed on simplicial complexes. Nevertheless,
cubical complexes deal directly with the voxels in 3D images, no additional
triangulation is necessary, facilitating efficient algorithms for the
computation of topological invariants in the image context. In this paper, we
present formulas to directly compute the cohomology ring of 3D cubical
complexes without making use of any additional triangulation. Starting from a
cubical complex that represents a 3D binary-valued digital picture whose
foreground has one connected component, we compute first the cohomological
information on the boundary of the object, by an incremental
technique; then, using a face reduction algorithm, we compute it on the whole
object; finally, applying the mentioned formulas, the cohomology ring is
computed from such information
Euler characteristic surfaces
We study the use of the Euler characteristic for multiparameter topological
data analysis. Euler characteristic is a classical, well-understood topological
invariant that has appeared in numerous applications, including in the context
of random fields. The goal of this paper is to present the extension of using
the Euler characteristic in higher-dimensional parameter spaces. While
topological data analysis of higher-dimensional parameter spaces using stronger
invariants such as homology continues to be the subject of intense research,
Euler characteristic is more manageable theoretically and computationally, and
this analysis can be seen as an important intermediary step in multi-parameter
topological data analysis. We show the usefulness of the techniques using
artificially generated examples, and a real-world application of detecting
diabetic retinopathy in retinal images
A deep learning approach to 3D segmentation of brain vasculature
The segmentation of blood-vessels is an important preprocessing step for the quantitative analysis of brain vasculature. We approach the segmentation task for two-photon brain angiograms using a fully convolutional 3D deep neural network.Published versio
Homological methods in feature extraction of multidimensional images
We show that the newly developed homology algorithms are helpful in imaging problems on the example of an algorithm extracting one dimensional features from a noisy image. We indicate that in some situations the global nature of this algorithm may become advantageous when compared with the standard algorithms based on skeletonization and pruning. The algorithm works in every dimension. ©2009 IEEE.published_or_final_versionThe 2nd International Congress on Image and Signal Processing (CISP'09), Tianjin, China, 17-19 October 2009. In Proceedings of 2nd CISP, 2009, p. 1-
Computing Persistent Homology within Coq/SSReflect
Persistent homology is one of the most active branches of Computational
Algebraic Topology with applications in several contexts such as optical
character recognition or analysis of point cloud data. In this paper, we report
on the formal development of certified programs to compute persistent Betti
numbers, an instrumental tool of persistent homology, using the Coq proof
assistant together with the SSReflect extension. To this aim it has been
necessary to formalize the underlying mathematical theory of these algorithms.
This is another example showing that interactive theorem provers have reached a
point where they are mature enough to tackle the formalization of nontrivial
mathematical theories
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