Ordinary and directed combinatorial homotopy, applied to image analysis and concurrency. (English) Zbl 1027.68140

Summary: Combinatorial homotopical tools developed in previous works, and consisting essentially of intrinsic homotopy theories for simplicial complexes and directed simplicial complexes, can be applied to explore mathematical models representing images, or directed images, or concurrent processes.
An image, represented by a metric space \(X\), can be explored at a variable resolution \(\varepsilon>0\), by equipping it with a structure \(t_\varepsilon X\) of simplicial complex depending on \(\varepsilon\); this complex can be further analysed by homotopy groups \(\pi^\varepsilon_n(X) =\pi_n(t,X)\) and homology groups \(H_n^\varepsilon(X)= H_n(t_\varepsilon X)\). Loosely speaking, these objects detect singularities which can be captured by an \(n\)-dimensional grid, with edges bound by \(\varepsilon\); this works equally well for continuous or discrete regions of euclidean spaces.
Similarly, a directed image, represented by an “asymmetric metric space”, produces a family of directed simplicial complexes \(s_\varepsilon X\) and can be explored by the fundamental \(n\)-category \(\uparrow\Pi^\varepsilon_n (X)\) of the latter. The same directed tools can be applied to combinatorial models of concurrent automata, like Chu-spaces.


68U10 Computing methodologies for image processing
68Q85 Models and methods for concurrent and distributed computing (process algebras, bisimulation, transition nets, etc.)
54E35 Metric spaces, metrizability
54E15 Uniform structures and generalizations
55U10 Simplicial sets and complexes in algebraic topology
05C38 Paths and cycles
55Q05 Homotopy groups, general; sets of homotopy classes
55N99 Homology and cohomology theories in algebraic topology
54G99 Peculiar topological spaces