Separating Topological Noise from Features Using Persistent Entropy

Topology is the branch of mathematics that studies shapes and maps among them. From the algebraic definition of topology a new set of algorithms have been derived. These algorithms are identified with “computational topology” or often pointed out as Topological Data Analysis (TDA) and are used for investigating high-dimensional data in a quantitative manner. Persistent homology appears as a fundamental tool in Topological Data Analysis. It studies the evolution of k−dimensional holes along a sequence of simplicial complexes (i.e. a filtration). The set of intervals representing birth and death times of k−dimensional holes along such sequence is called the persistence barcode. k−dimensional holes with short lifetimes are informally considered to be topological noise, and those with a long lifetime are considered to be topological feature associated to the given data (i.e. the filtration). In this paper, we derive a simple method for separating topological noise from topological features using a novel measure for comparing persistence barcodes called persistent entropy.Ministerio de Economía y Competitividad MTM2015-67072-

Towards Emotion Recognition: A Persistent Entropy Application

Emotion recognition and classification is a very active area of research. In this paper, we present a first approach to emotion classification using persistent entropy and support vector machines. A topology-based model is applied to obtain a single real number from each raw signal. These data are used as input of a support vector machine to classify signals into 8 different emotions (calm, happy, sad, angry, fearful, disgust and surprised)

Encoding Specific 3D Polyhedral Complexes Using 3D Binary Images

We build upon the work developed in [4] in which we presented a method to “locally repair” the cubical complex Q(I) associated to a 3D binary image I, to obtain a “well-composed” 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 Andalucía FQM-369Ministerio de Economía y Competitividad MTM2012-32706Ministerio de Economía 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 0–barcode encoding lifetime of connected components on the image sequence over time. This information may not coincide with the one provided by the 0–barcode encoding the 0–persistent 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 ∈ K, a spatiotemporal path from v to the “oldest” spatiotemporally-connected vertex is computed and the corresponding spatiotemporal 0–bar is added to the spatiotemporal 0–barcode.Junta de Andalucía FQM-369Ministerio de Economía y Competitividad MTM2012-3270

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 183:59–77, 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-

On the stability of persistent entropy and new summary functions for Topological Data Analysis

Persistent entropy of persistence barcodes, which is based on the Shannon entropy, has been recently defined and successfully applied to different scenarios: characterization of the idiotypic immune network, detection of the transition between the preictal and ictal states in EEG signals, or the classification problem of real long-length noisy signals of DC electrical motors, to name a few. In this paper, we study properties of persistent entropy and prove its stability under small perturbations in the given input data. From this concept, we define three summary functions and show how to use them to detect patterns and topological features

Computation of cohomology operations of finite simplicial complexes

We propose a method for calculating cohomology operations on finite simplicial complexes. Of course, there exist well–known methods for computing (co)homology groups, for example, the “reduction algorithm” consisting in reducing the matrices corresponding to the differential in each dimension to the Smith normal form, from which one can read off the (co)homology groups of the complex [Mun84], or the “incremental algorithm” for computing Betti numbers [DE93]. Nevertheless, little is known about general methods for computing cohomology operations. For any finite simplicial complex K, we give a procedure including the computation of some primary and secondary cohomology operations. This method is based on the transcription of the reduction algorithm mentioned above, in terms of a special type of algebraic homotopy equivalences, called contractions [McL75], of the (co)chain complex of K to a “minimal” (co)chain complex M(K). More concretely, whenever the ground ring is a field or the (co)homology of K is free, then M(K) is isomorphic to the (co)homology of K. Combining this contraction with the combinatorial formulae for Steenrod reduced pth powers at cochain level developed in [GR99] and [Gon00], these operations at cohomology level can be computed. Finally, a method for calculating Adem secondary cohomology operations Φq : Ker(Sq2Hq (K)) → Hq+3(K)/Sq2Hq (K) is showed

Persistence Partial Matchings Induced by Morphisms between Persistence Modules

The notion of persistence partial matching, as a generalization of partial matchings between persistence modules, is introduced. We study how to obtain a persistence partial matching Gf , and a partial matching Mf , induced by a morphism f between persistence modules, both being linear with respect to direct sums of morphisms. Some of their properties are also provided, including their stability after a perturbation of the morphism f, and their relationship with other induced partial matchings already de ned in TDA.Ministerio de Ciencia e Innovación PID2019-107339GB-I0

An example in combinatorial cohomology

Steenrod cohomology operations are algebraic tools for distinguishing non–homeomorphic topological spaces. In this paper, starting off from the general method developed in [4] for Steenrod squares and Steenrod reduced powers, we present an explicit combinatorial formulation for the Steenrod reduced power Pp 1 : Hq(X;Fp) ! Hqp−1(X;Fp), at cocycle level, where p is an odd prime, q a non–negative integer, X a simplicial set and Fp the finite field with p elements. We design an algorithm for computing Pp 1 on the cohomology of the classifying space of Zp and we generalize this process to any simplicial set at cohomology level