We propose a method for computing the cohomology ring of three-dimensional (3D) digital binaryvalued pictures. We obtain the cohomology ring of a 3D digital binary-valued picture I, via a simplicial complex K(I ) topologically representing (up to isomorphisms of pictures) the picture I. The usefulness of a simplicial description of the "digital" cohomology ring of 3D digital binary-valued pictures is tested by means of a small program visualizing the different steps of the method. Some examples concerning topological thinning, the visualization of representative (co)cycles of (co)homology generators and the computation of the cup product on the cohomology of simple pictures are showed.

a b s t r a c tIn persistent homology, the persistence barcode encodes pairs of simplices meaning birth and death of homology classes. Persistence barcodes depend on the ordering of the simplices (called a filter) of the given simplicial complex. In this paper, we define the notion of "minimal" barcodes in terms of entropy. Starting from a given filtration of a simplicial complex K, an algorithm for computing a "proper" filter (a total ordering of the simplices preserving the partial ordering imposed by the filtration as well as achieving a persistence barcode with small entropy) is detailed, by way of computation, and subsequent modification, of maximum matchings on subgraphs of the Hasse diagram associated to K. Examples demonstrating the utility of computing such a proper ordering on the simplices are given.

Persistent homology and persistent entropy have recently become useful tools for patter recognition. In this paper, we find requirements under which persistent entropy is stable to small perturbations in the input data and scale invariant. In addition, we describe two new stable summary functions combining persistent entropy and the Betti curve. Finally, we use the previously defined summary functions in a material classification task to show their usefulness in machine learning and pattern recognition.

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

a b s t r a c tA binary three-dimensional (3D) image I 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) (embedded in R 3 ) associated to I to obtain a polyhedral complex P(I) homotopy equivalent to Q (I) such that the boundary surface of P(I) is a 2D manifold (and, hence, P(I) is a well-composed polyhedral complex). For this aim, we develop a new codification system for a complex K , called ExtendedCubeMap (ECM) representation of K , that codifies: (1) the information of the cells of K (including geometric information), under the form of a 3D grayscale image g P ; and (2) the boundary face relations between the cells of K , under the form of a set B P of structuring elements that can be stored as indexes in a look-up table. We describe a procedure to locally modify the ECM representation E Q of the cubical complex Q (I) to obtain an ECM representation of a well-composed polyhedral complex P(I) that is homotopy equivalent to Q (I). The construction of the polyhedral complex P(I) is accomplished for proving the results though it is not necessary to be done in practice, since the image g P (obtained by the repairing process on E Q ) together with the set B P codify all the geometric and topological information of P(I).