Abstract
In this paper, we present an algorithm to compute the filtered generalized Čech complex for a finite collection of disks in the plane, which do not necessarily have the same radius. The key step behind the algorithm is to calculate the minimum scale factor needed to ensure rescaled disks have a nonempty intersection, through a numerical approach, whose convergence is guaranteed by a generalization of the well-known Vietoris–Rips Lemma, which we also prove in an alternative way, using elementary geometric arguments. We give an algorithm for computing the 2-dimensional filtered generalized Čech complex of a finite collection of d-dimensional disks in R d , and we show the performance of our algorithm.
Highlights
In the study of data point clouds from a topological approach, the need to develop algorithms to construct different simplicial structures has arisen, such as the Vietoris–Rips complex, the Čech complex, the piecewise linear lower star complex, etc..Of particular interest to us is the generalized Čech complex structure, whereas the standardČech complex is induced by the intersection of a collection of disks with fixed radius, the generalized version admits different radii; when radii are rescaled, using the same scale factor each time, the corresponding simplicial complexes forms the filtered generalized Čech complex.There exist efficient algorithms to calculate the standard Čech complex, and software currently available to obtain the associated filtration; in [12] the authors propose an algorithm to approximate the Čech filtration
We must emphasize that our main algorithm (Algorithm 3) is only generalizable to higher-dimensional disk systems to obtain the 2-dimensional filtered generalized Čech structure, as we show as an application
We introduce the fundamental notions of Vietoris–Rips scale and Čech scale for a disk system, as the infimum over all rescaling factors such that the disk system becomes a Vietoris–Rips system or a
Summary
In the study of data point clouds from a topological approach (cf. [1,2,3,4,5]), the need to develop algorithms to construct different simplicial structures has arisen, such as the Vietoris–Rips complex, the Čech complex, the piecewise linear lower star complex, etc. (cf. [6,7]). We explain how their respective filtrations are induced by weight functions, and we propose an algorithm to obtain the Čech-weight function of a given disk system, associating to each Čech simplex its corresponding Čech scale.
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