In their seminal work on homological sensor networks, de Silva and Ghrist showed the surprising fact that it's possible to certify the coverage of a coordinate-free sensor network even with very minimal knowledge of the space to be covered. Here, coverage means that every point in the domain (except possibly those very near the boundary) has a nearby sensor. More generally, their algorithm takes a pair of nested neighborhood graphs along with a labeling of vertices as either boundary or interior and computes the relative homology of a simplicial complex induced by the graphs. This approach, called the Topological Coverage Criterion (TCC), requires some assumptions about the underlying geometric domain as well as some assumptions about the relationship of the input graphs to the domain. The goal of this paper is to generalize these assumptions and show how the TCC can be applied to both much more general domains as well as very weak assumptions on the input. We give a new, simpler proof of the de Silva-Ghrist Topological Coverage Criterion that eliminates any assumptions about the smoothness of the boundary of the underlying space, allowing the results to be applied to much more general problems. The new proof factors the geometric, topological, and combinatorial aspects, allowing us to provide a coverage condition that supports thick boundaries, k-coverage, and weighted coverage, in which sensors have varying radii. From Sensor Coverage to Data CoverageProblems in Homological Sensor Networks (HSNs) are usually stated in the vocabulary of sensor networks. There are sensors and coverage regions, and an important problem is to determine if the sensing region of a collection of sensors covers a given domain, given only the neighborhood relationships between the sensors and some indication of which sensors are near the boundary. The locations (coordinates) of the points is not assumed, nor is the shape (topology) of the domain to be covered. Although phrased in the language of sensor networks, the problem may be understood more generally as one of data coverage; it answers when a data set * Supported by NSF grants CCF-1464379 and CCF-1525978.† University of Connecticut. ‡ University of Connecticut. § University of Connecticut.sufficiently covers a domain, and holes in coverage can be viewed as gaps in the data. A surprising result by de Silva & Ghrist is that there exists a computable, sufficient condition called the Topological Coverage Criterion (TCC) to certify coverage without knowing the locations of the sensors when the domain's boundary is smooth [4]. All that is required is that the sensors have unique identifiers, can detect nearby sensors, can differentiate whether neighboring sensors are "close" or "very close" (in a technical sense to be defined below), and can detect if the boundary of the domain is close. Developed over a series of papers [8,17,5,4], the most general version of the TCC can be understood as a purely geometric problem on point sets P in R d with unknown coordinates in an unknown d...
In this paper we consider adaptive sampling's local-feature size, used in surface reconstruction and geometric inference, with respect to an arbitrary landmark set rather than the medial axis and relate it to a path-based adaptive metric on Euclidean space. We prove a near-duality between adaptive samples in the Euclidean metric space and uniform samples in this alternate metric space which results in topological interleavings between the offsets generated by this metric and those generated by an linear approximation of it. After smoothing the distance function associated to the adaptive metric, we apply a result from the theory of critical points of distance functions to the interleaved spaces which yields a computable homology inference scheme assuming one has Hausdorff-close samples of the domain and the landmark set.
We generalize the local-feature size definition of adaptive sampling used in surface reconstruction to relate it to an alternative metric on Euclidean space. In the new metric, adaptive samples become uniform samples, making it simpler both to give adaptive sampling versions of homological inference results and to prove topological guarantees using the critical points theory of distance functions. This ultimately leads to an algorithm for homology inference from samples whose spacing depends on their distance to a discrete representation of the complement space.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.