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In this paper we develop a new computational technique called boundary scale-space theory. This tech- nique is based on the topol1 ogical paradigm consisting of representing a geometric subdivided object K using a one-parameter family of geometric objects { Ki }i ≥ 1 all of them having the same number of closed pieces than K. Each piece of Ki ( ∀i ≥ 1) presents the same interior part than the corresponding one in K, and a different boundary part depending on the scale i. Working with coefficients in a field, a scale is installed for the algebraic boundary of each piece and a new invariant for cell complex isomorphisms is given in terms of the Betti numbers of the generated boundary-scale-space cell complexes. Moreover, the so called homology boundary scale-space model of K ( hbss -model for short) is introduced here. Thismodel consists of a hierarchical graph whose nodes are the homology generators of the different bound- ary scale levels and whose edges are specified by homology generators of consecutive boundary scaleindices linked by ( hbss -transition maps) preserving homology classes. Various codes for each connectedsubgraph of an hbss -model are defined, which besides being fast and efficient similarity measures for cel- lular structures, they are as well relevant interpretive tools for the hbss -model. Finally, experimentations mainly aimed at clarifying and understanding the notion of hbss -model, as well as conjecturing about new graph isomorphism invariants (seeing graphs as a 1-dimensional cell complexes), are performed. Ministerio de Economía y Competitividad MTM2016-81030-P Ministerio de Economía y Competitividad TEC2016-77785-P

In [18], a membrane parallel theoretical framework for computing (co)homology information of foreground or background of binary digital images is developed. Starting from this work, we progress here in two senses: (a) providing advanced topological information, such as (co)homology torsion and efficiently answering to any decision or classification problem for sum of k-xels related to be a (co)cycle or a (co)boundary; (b) optimizing the previous framework to be implemented in using GPGPU computing. Discrete Morse theory, Effective Homology Theory and parallel computing techniques are suitably combined for obtaining a homological encoding, called algebraic minimal model, of a Region-Of-Interest (seen as cubical complex) of a presegmented k-D digital image.

We present here a combinatorial method for computing cup-$i$ products and 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. This description can be considered as a translation of the most ancient definition of Steenrod squares to the general setting of the Simplicial Topology., 25 pages