Digital homotopy relations and digital homology theories

[EN] In this paper we prove results relating to two homotopy relations and four homology theories developed in the topology of digital images.We introduce a new type of homotopy relation for digitally continuous functions which we call ``strong homotopy.'' Both digital homotopy and strong homotopy are natural digitizations of classical topological homotopy: the difference between them is analogous to the difference between digital 4-adjacency and 8-adjacency in the plane.We also consider four different digital homology theories: a simplicial homology theory by Arslan et al which is the homology of the clique complex, a singular simplicial homology theory by D. W. Lee, a cubical homology theory by Jamil and Ali, and a new kind of cubical homology for digital images with $c_1$-adjacency which is easily computed, and generalizes a construction by Karaca \& Ege. We show that the two simplicial homology theories are isomorphic to each other, but distinct from the two cubical theories.We also show that homotopic maps have the same induced homomorphisms in the cubical homology theory, and strong homotopic maps additionally have the same induced homomorphisms in the simplicial theory.Staecker, PC. (2021). Digital homotopy relations and digital homology theories. Applied General Topology. 22(2):223-250. https://doi.org/10.4995/agt.2021.13154OJS223250222H. Arslan, I. Karaca and A. Öztel, Homology groups of n-dimensional digital images, in: Turkish National Mathematics Symposium XXI (2008), 1-13.L. Boxer, A classical construction for the digital fundamental group, J. Math. Imaging Vision 10, no. 1 (1999), 51-62. https://doi.org/10.1023/A:1008370600456L. Boxer, Generalized normal product adjacency in digital topology, Appl. Gen. Topol. 18, no. 2 (2017), 401-427. https://doi.org/10.4995/agt.2017.7798L. Boxer, I. Karaca and A. Öztel, Topological invariants in digital images, J. Math. Sci. Adv. Appl. 11, no. 2 (2011), 109-140.L. Boxer and P. C. Staecker, Remarks on fixed point assertions in digital topology, Appl. Gen. Topol. 20, no. 1 (2019), 135-153. https://doi.org/10.4995/agt.2019.10474O. Ege and I. Karaca, Fundamental properties of digital simplicial homology groups, American Journal of Computer Technology and Application 1 (2013), 25-41.S.-E. Han, Non-product property of the digital fundamental group, Inform. Sci. 171, no. 1-3 (2005), 73-91. https://doi.org/10.1016/j.ins.2004.03.018A. Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002.S. S. Jamil and D. Ali, Digital Hurewicz theorem and digital homology theory, arxiv eprint 1902.02274v3.T. Kaczynski, K. Mischaikow and M. Mrozek, Computing homology. Algebraic topological methods in computer science (Stanford, CA, 2001), Homology Homotopy Appl. 5, no. 2 (2003), 233-256. https://doi.org/10.4310/HHA.2003.v5.n2.a8I. Karaca and O. Ege, Cubical homology in digital images, International Journal of Information and Computer Science, 1 (2012), 178-187.D. W. Lee, Digital singular homology groups of digital images, Far East Journal of Mathematics 88 (2014), 39-63.G. Lupton, J. Oprea and N. Scoville, A fundamental group for digital images, preprint.W. S. Massey, A Basic Course in Algebraic Topology,Graduate Texts in Mathematics, 127. Springer-Verlag, New York, 1991. https://doi.org/10.1007/978-1-4939-9063-4A. Rosenfeld, 'Continuous' functions on digital pictures, Pattern Recognition Letters 4 (1986), 177-184. https://doi.org/10.1016/0167-8655(86)90017-

Generating Second Order (Co)homological Information within AT-Model Context

In this paper we design a new family of relations between (co)homology classes, working with coefficients in a field and starting from an AT-model (Algebraic Topological Model) AT(C) of a finite cell complex C These relations are induced by elementary relations of type “to be in the (co)boundary of” between cells. This high-order connectivity information is embedded into a graph-based representation model, called Second Order AT-Region-Incidence Graph (or AT-RIG) of C. This graph, having as nodes the different homology classes of C, is in turn, computed from two generalized abstract cell complexes, called primal and dual AT-segmentations of C. The respective cells of these two complexes are connected regions (set of cells) of the original cell complex C, which are specified by the integral operator of AT(C). In this work in progress, we successfully use this model (a) in experiments for discriminating topologically different 3D digital objects, having the same Euler characteristic and (b) in designing a parallel algorithm for computing potentially significant (co)homological information of 3D digital objects.Ministerio de Economía y Competitividad MTM2016-81030-PMinisterio de Economía y Competitividad TEC2012-37868-C04-0

Chain Homotopies for Object Topological Representations

This paper presents a set of tools to compute topological information of simplicial complexes, tools that are applicable to extract topological information from digital pictures. A simplicial complex is encoded in a (non-unique) algebraic-topological format called AM-model. An AM-model for a given object K is determined by a concrete chain homotopy and it provides, in particular, integer (co)homology generators of K and representative (co)cycles of these generators. An algorithm for computing an AM-model and the cohomological invariant HB1 (derived from the rank of the cohomology ring) with integer coefficients for a finite simplicial complex in any dimension is designed here. A concept of generators which are "nicely" representative cycles is also presented. Moreover, we extend the definition of AM-models to 3D binary digital images and we design algorithms to update the AM-model information after voxel set operations (union, intersection, difference and inverse)

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 $\mathbb{R}^3$) 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