Rocio Gonzalez-Diaz

Integral Operators for Computing Homology Generators at Any Dimension

Starting from an nDgeometrical 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.

Towards Digital Cohomology

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 I via 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.

A combinatorial method for computing Steenrod squares

We present here a combinatorial method for computing Steenrod squares of a simplicial set X. This method is essentially based on the determination of explicit formulae for the component morphisms of a higher diagonal approximation (i.e., a family of morphisms measuring the lack of commutativity of the cup product on the cochain level) in terms of face operators of X. A generalization of this method to Steenrod reduced powers is sketched.

Human Gait Identification Using Persistent Homology

This 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.

3D Well-composed Polyhedral Complexes

A binary three-dimensional (3D) image I is well-composed if the boundary surface of its continuous analog is a 2D manifold. Since 3D images are not often well-composed, there are several voxel-based methods (“repairing” algorithms) for turning them into well-composed ones but these methods either do not guarantee the topological equivalence between the original image and its corresponding well-composed one or involve sub-sampling the whole image. In this paper, we present a method to locally “repair” the cubical complex Q(I) (embedded in R 3 ) associated to I to obtain a polyhedral complex P(I) homotopy equivalent to Q(I) such that the boundary of every connected component of P(I) is a 2D manifold. The reparation is performed via a new codification system for P(I) under the form of a 3D grayscale image that allows an efficient access to cells and their faces.

Cup products on polyhedral approximations of 3D digital images

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.

Incremental-decremental algorithm for computing AT-models and persistent homology

In 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.

A new topological entropy-based approach for measuring similarities among piecewise linear functions

In this paper we present a novel methodology based on a topological entropy, the so-called persistent entropy, for addressing the comparison between discrete piecewise linear functions. The comparison is certified by the stability theorem for persistent entropy that is presented here. The theorem is used in the implementation of a new algorithm. The algorithm transforms a discrete piecewise linear function into a filtered simplicial complex that is analyzed via persistent homology and persistent entropy. Persistent entropy is used as a discriminant feature for solving the supervised classification problem of real long-length noisy signals of DC electrical motors. The quality of classification is stated in terms of the area under receiver operating characteristic curve (AUC=93.87%). HighlightsDefinition of a new entropy from the persistent barcode.Proof of the stability theorem for the persistent entropy.Development of a new entropy-based methodology for studying piecewise linear function.Development of a new entropy-based algorithm for the classification of real signals

Cubical Cohomology Ring of 3D Photographs

Cohomology groups and the 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. This could facilitate efficient algorithms for the computation of topological invariants in the image context. In this article, we present a constructive process, made up by several algorithms, to compute the cohomology ring of 3D binary‐valued digital photographs represented by cubical complexes. Starting from a cubical complex Q that represents such a 3D picture whose foreground has one connected component, we first compute the homological information on the boundary of the object, ∂Q, by an incremental technique; using a face reduction algorithm, we then compute it on the whole object; finally, applying explicit formulas for cubical complexes (without making use of any additional triangulation), the cohomology ring is computed from such information.

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 R3.