The offset filtration at six scale parameters on a point cloud sampled from two circles of different sizes.
The offset filtration (also called the "union-of-balls"[1] or "union-of-disks"[2]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.[3] 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 (i.e., a growing sequence of balls), in order to observe the stability of topological features with respect to attractors.[4] 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.[5] Since then, the offset filtration has become a primary example in the study of computational topology and data analysis.
^Adams, Henry; Moy, Michael (2021). "Topology Applied to Machine Learning: From Global to Local". Frontiers in Artificial Intelligence. 4: 2. doi:10.3389/frai.2021.668302. ISSN 2624-8212. PMC 8160457. PMID 34056580.
^Edelsbrunner, Herbert (2014). A short course in computational geometry and topology. Cham. p. 35. ISBN 978-3-319-05957-0. OCLC 879343648.{{cite book}}: CS1 maint: location missing publisher (link)
^Frosini, Patrizio (1992-02-01). Casasent, David P. (ed.). "Measuring shapes by size functions". Intelligent Robots and Computer Vision X: Algorithms and Techniques. 1607. Boston, MA: 122–133. Bibcode:1992SPIE.1607..122F. doi:10.1117/12.57059. S2CID 121295508.
The offsetfiltration (also called the "union-of-balls" or "union-of-disks" filtration) is a growing sequence of metric balls used to detect the size and...
this is via taking the sublevel filtration of the distance to a point cloud, or equivalently, the offsetfiltration on the point cloud and taking its...
extension of the offsetfiltration that captures density information about the underlying data set by filtering the points of the offsets at each index according...
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body, increases the perfusion of glomerulus, thus increasing glomerular filtration rate (GFR). At the same time, loop diuretics inhibit the tubuloglomerular...
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areas is also able to reduce air pollution via the trees' absorption and filtration of pollutants, including carbon monoxide, sulfur dioxide, and ozone, in...
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