Offset filtration

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.

  1. ^ 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.
  2. ^ 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)
  3. ^ 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.
  4. ^ Robins, Vanessa (1999-01-01). "Towards computing homology from approximations" (PDF). Topology Proceedings. 24: 503–532.
  5. ^ Edelsbrunner; Letscher; Zomorodian (2002). "Topological Persistence and Simplification". Discrete & Computational Geometry. 28 (4): 511–533. doi:10.1007/s00454-002-2885-2. ISSN 0179-5376.