Search

Persistent homology is a powerful tool in topological data analysis (TDA) to compute, study, and encode efficiently multi-scale topological features and is being increasingly used in digital image classification. The topological features represent a number of connected components, cycles, and voids that describe the shape of data. Persistent homology extracts the birth and death of these topological features through a filtration process. The lifespan of these features can be represented using persistent diagrams (topological signatures). Cubical homology is a more efficient method for extracting topological features from a 2D image and uses a collection of cubes to compute the homology, which fits the digital image structure of grids. In this research, we propose a cubical homology-based algorithm for extracting topological features from 2D images to generate their topological signatures. Additionally, we propose a novel score measure, which measures the significance of each of the sub-simplices in terms of persistence. In addition, gray-level co-occurrence matrix (GLCM) and contrast limited adapting histogram equalization (CLAHE) are used as supplementary methods for extracting features. Supervised machine learning models are trained on selected image datasets to study the efficacy of the extracted topological features. Among the eight tested models with six published image datasets of varying pixel sizes, classes, and distributions, our experiments demonstrate that cubical homology-based machine learning with the deep residual network (ResNet 1D) and Light Gradient Boosting Machine (lightGBM) shows promise with the extracted topological features.

ResumenLa matemática difusa generaliza los conceptos tradicionales de la matemática utilizando los llamados conjuntos difusos, lo que permite modelar y estudiar de manera más apropiada fenómenos caracterizados por su imprecisión. Estas generalizaciones incluyen conceptos de álgebra, análisis y topología, entre otros. Por otra parte, los complejos cúbicos tienen aplicaciones al procesamiento de imágenes digitales y al estudio de los sistemas dinámicos, pero en la literatura actual no hay una extensión de sus propiedades utilizando conjuntos difusos. En este documento se propone una generalización del concepto de complejo cúbico y de algunas de sus características, tales como realización poliédrica, conexidad, componente conexa y agujero, utilizando conjuntos difusos. Se definen los árboles superior e inferior de un complejo cúbico difuso, los cuales proporcionan información sobre la manera en la que están relacionados sus extremos regionales. También se definen los grupos de homología de estos complejos cúbicos difusos y se demuestra que el rango del 0-grupo de homología de un nivel dado es igual al número de máximos regionales de dicho nivel. Por último, se muestra cómo se le puede asociar un complejo cúbico difuso a una imagen digital bidimensional en tonos de gris para estudiar algunas de sus propiedades topológicas. AbstractFuzzy mathematics generalize concepts of traditional mathematics using fuzzy sets. This enables to study and model more properly phenomens characterized by imprecision. These generalizations includes concepts of algebra, analysis and topology. On the other side, cubical complexes have applications in digital image processing and in the study of dynamical systems, but in the actual literature there is not an extension of their properties using fuzzy sets. In this paper is proposed a generalization of the concept of cubical complex and of some of their properties, such as connectedness, polyhedral realization, connected component and holes, using fuzzy sets. The upper and lower trees of a fuzzy cubical complex are defined, which give information about the way in which its regional extrema are related. The homology groups of a fuzzy cubical complex are defined and it is shown that the rank of the 0-homology group of a given level is equal with the number of regional maxima of that level. Finally, it is shown how to associate a fuzzy cubical complex with a bidimensional digital grayscale image in order to study somo of its topological properties.

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 “repair” 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).

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 183:59–77, 2015), we presented a method to “locally repair” 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 “minimal” codification of P(I ), we finally present a method to compute the 2-cells in the boundary surface of P(I ). Ministerio de Economía y Competitividad MTM2015-67072-P

Let $\pi$ be a finitely presented group. If h is a non trivial homology class in Hn($\pi$; Z), a theorem of Gromov (see [Gro83], 6) asserts the existence of regular geometric cycles which represent h, whose relative systolic volume is as close as desired to the systolic volume of h, in which we can control the volume of balls of radius less than half of the cycle's relative systol. The aim of this note is to explain and provide a complete proof of this result., Comment: in French

The geometric models of higher dimensional automata (HDA) and Dijkstra's PV-model are cubically subdivided topological spaces with a local partial order. If a cubicalization of a topological space is free of immersed cubic Möbius bands, then there are consistent choices of direction in all cubes, such that any n-cube in the cubic subdivision is dihomeomorphic to [0,1]n with the induced partial order from Rn. After subdivision once, any cubicalized space has a cubical local partial order. In particular, all triangularized spaces have a cubical local partial order. This implies in particular that the underlying geometry of an HDA may be quite complicated.

We study finite foldable cubical complexes of nonpositive curvature (in the sense of A.D. Alexandrov). We show that such a complex X admits a graph of spaces decomposition. It is also shown that when dim X=3, X contains a closed rank one geodesic in the 1-skeleton unless the universal cover of X is isometric to the product of two CAT(0) cubical complexes., Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-28.abs.html

Any diagram group over a finite semigroup presentation acts properly, freely, and cellularly by isometrices on a proper CAT(0) cubical complex. The existence of a proper, cellular action by isometries on a CAT(0) cubical complex has powerful consequences for the acting group G. One gets, for example, a proof that G satisfies the Baum–Connes conjecture. Any diagram group over a finite presentation of a finite semigroup is of type F ∞ .