Everything about Offset filtration
The offset filtration is a growing sequence of metric balls used to detect the size and scale of topological features of a data set. The offset filtration commonly arises in persistent homology and the field of topological data analysis. Utilizing a union of balls to approximate the shape of geometric objects was first suggested by Frosini in 1992 in the context of submanifolds of Euclidean space. The construction was independently explored by Robins in 1998, and expanded to considering the collection of offsets indexed over a series of increasing scale parameters, in order to observe the stability of topological features with respect to attractors. Homological persistence as introduced in these papers by Frosini and Robins was subsequently formalized by Edelsbrunner et al. in their seminal 2002 paper Topological Persistence and Simplification. Since then, the offset filtration has become a primary example in the study of computational topology and data analysis. Wikipedia →
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- Read the overview: Offset filtrationA 5-minute orientation
- Read the landmark paper: Modeling the Entire Filtration Process from Depth Filtration to Surface Filtration with a Multilayer Construction and an Offset for Dust Cake GrowthCited 2 times
- Go sideways: Multicover bifiltrationA closely related topic
- Go sideways: Subdivision bifiltrationA closely related topic
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- Multicover bifiltration The multicover bifiltration is a two-parameter sequence of nested topological spaces derived from the covering of a finite set in a metric space by growing…
- Subdivision bifiltration In topological data analysis, a subdivision bifiltration is a collection of filtered simplicial complexes, typically built upon a set of data points in…
- Persistent homology In topological data analysis, persistent homology is a method for computing topological features of a space at different spatial resolutions. More persistent…
- Topological data analysis In applied mathematics, topological data analysis (TDA) is an approach to the analysis of datasets using techniques from topology. Extraction of information…
- Čech complex In algebraic topology and topological data analysis, the Čech complex is an abstract simplicial complex constructed from a point cloud in any metric space…
- Persistent Betti number In persistent homology, a persistent Betti number is a multiscale analog of a Betti number that tracks the number of topological features that persist…
- Vietoris–Rips filtration In topological data analysis, the Vietoris–Rips filtration (sometimes shortened to "Rips filtration") is the collection of nested Vietoris–Rips complexes…
- Persistence module A persistence module is a mathematical structure in persistent homology and topological data analysis that formally captures the persistence of topological…