Topic
Delta set
About: Delta set is a research topic. Over the lifetime, 361 publications have been published within this topic receiving 6999 citations.
Papers published on a yearly basis
Papers
2 Jun 2004
TL;DR: This paper tackles the problem of computing topological invariants of geometric objects in a robust manner, using only point cloud data sampled from the object, and produces a nested family of simplicial complexes, which represent the data at different feature scales, suitable for calculating persistent homology.
Abstract: This paper tackles the problem of computing topological invariants of geometric objects in a robust manner, using only point cloud data sampled from the object. It is now widely recognised that this kind of topological analysis can give qualitative information about data sets which is not readily available by other means. In particular, it can be an aid to visualisation of high dimensional data. Standard simplicial complexes for approximating the topological type of the underlying space (such as Cech, Rips, or a-shape) produce simplicial complexes whose vertex set has the same size as the underlying set of point cloud data. Such constructions are sometimes still tractable, but are wasteful (of computing resources) since the homotopy types of the underlying objects are generally realisable on much smaller vertex sets. We obtain smaller complexes by choosing a set of 'landmark' points from our data set, and then constructing a "witness complex" on this set using ideas motivated by the usual Delaunay complex in Euclidean space. The key idea is that the remaining (non-landmark) data points are used as witnesses to the existence of edges or simplices spanned by combinations of landmark points.
Our construction generalises the topology-preserving graphs of Martinetz and Schulten [MS94] in two directions. First, it produces a simplicial complex rather than a graph. Secondly it actually produces a nested family of simplicial complexes, which represent the data at different feature scales, suitable for calculating persistent homology [ELZ00, ZC04]. We find that in addition to the complexes being smaller, they also provide (in a precise sense) a better picture of the homology, with less noise, than the full scale constructions using all the data points. We illustrate the use of these complexes in qualitatively analyzing a data set of 3 × 3 pixel patches studied by David Mumford et al [LPM03].
467 citations
Book•
1 Jan 2008
TL;DR: In particular, we may interpret such a complex as a fami as discussed by the authors, which is a family of simplicial simplicial complex on a finite graph with vertex set V and edge set E.
Abstract: Let G be a finite graph with vertex set V and edge set E A graph complex on G is an abstract simplicial complex consisting of subsets of E In particular, we may interpret such a complex as a fami
334 citations
TL;DR: A new complex Y* of finite element spaces is constructed on the barycentric refinement of the mesh which can be seen as a realization of the simplicial chain complex on the original (unrefined) mesh, such that the L 2 duality is non-degenerate on Y i × X 2-i.
Abstract: Given a two dimensional oriented surface equipped with a simplicial mesh, the standard lowest order finite element spaces provide a complex X* centered on Raviart-Thomas divergence conforming vector fields. It can be seen as a realization of the simplicial cochain complex. We construct a new complex Y* of finite element spaces on the barycentric refinement of the mesh which can be seen as a realization of the simplicial chain complex on the original (unrefined) mesh, such that the L 2 duality is non-degenerate on Y i × X 2-i for each i ∈ {0,1,2}. In particular Y 1 is a space of curl-conforming vector fields which is L 2 dual to Raviart-Thomas div-conforming elements. When interpreted in terms of differential forms, these two complexes provide a finite-dimensional analogue of Hodge duality.
330 citations
TL;DR: In this article, a simplicial complex is associated with a monomial ideal in the polynomial ring generated by its vertex set over a field, and the algebraic properties of this ideal via combinatorial properties of simplicial complexes are studied.
Abstract: To a simplicial complex, we associate a square-free monomial ideal in the polynomial ring generated by its vertex set over a field. We study algebraic properties of this ideal via combinatorial properties of the simplicial complex. By generalizing the notion of a tree from graphs to simplicial complexes, we show that ideals associated to trees satisfy sliding depth condition, and therefore have normal and Cohen-Macaulay Rees rings. We also discuss connections with the theory of Stanley-Reisner rings.
255 citations
10 Jun 1994
TL;DR: This paper shows that inline-equation and f are homotopy equivalent if all such sets are contractible and homeomorphic if the sets can be further subdivided in a certain way so they form a regular CW complex.
Abstract: Given a subspace 𝒳 ⊆ Rd and a finite set S⊆Rd, we introduce the Delaunay simplicial complex, D𝒳, restricted by 𝒳. Its simplices are spanned by subsets T⊆S for which the common intersection of Voronoi cells meets 𝒳 in a non-empty set. By the nerve theorem,⋃D𝒳 and 𝒳 are homotopy equivalent if all such sets are contractible. This paper shows that ⋃D𝒳 and 𝒳 are homeomorphic if the sets can be further subdivided in a certain way so they form a regular CW complex.
249 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2019 | 3 |
| 2018 | 3 |
| 2017 | 18 |
| 2016 | 19 |
| 2015 | 21 |
| 2014 | 21 |