• Corpus ID: 4472239

Persistence-sensitive simplication of functions on surfaces in linear time

@inproceedings{Attali2009PersistencesensitiveSO,
  title={Persistence-sensitive simplication of functions on surfaces in linear time},
  author={Dominique Attali and Marc Glisse and Samuel Hornus and Francis Lazarus and Dmitriy Morozov},
  year={2009}
}
Persistence provides a way of grading the importance of homological features in the sublevel sets of a real-valued function. Following the definition given by Edelsbrunner, Morozov and Pascucci, an e-simplication of a function f is a function g in which the homological noise of persistence less than e has been removed. In this paper, we give an algorithm for constructing an e-simplication of a function defined on a triangulated surface in linear time. Our algorithm is very simple, easy to… 

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References

SHOWING 1-10 OF 27 REFERENCES
Vines and vineyards by updating persistence in linear time
TLDR
The main result of this paper is an algorithm that maintains the pairing in worst-case linear time per transposition in the ordering and uses the algorithm to compute 1-parameter families of diagrams which are applied to the study of protein folding trajectories.
Homological illusions of persistence and stability
In this thesis we explore and extend the theory of persistent homology, which captures topological features of a function by pairing its critical values. The result is represented by a collection of
Stability of Persistence Diagrams
The persistence diagram of a real-valued function on a topological space is a multiset of points in the extended plane. We prove that under mild assumptions on the function, the persistence diagram
Computing Persistent Homology
TLDR
The analysis establishes the existence of a simple description of persistent homology groups over arbitrary fields and derives an algorithm for computing individual persistent homological groups over an arbitrary principal ideal domain in any dimension.
A topological hierarchy for functions on triangulated surfaces
TLDR
This work combines topological and geometric methods to construct a multiresolution representation for a function over a two-dimensional domain and uses this data structure to extract topologically valid approximations that satisfy error bounds provided at runtime.
Persistence-sensitive simplification functions on 2-manifolds
TLDR
It is proved that for functions <i>f</i> on a 2-manifold such ε-simplification exists, and an algorithm to construct them in the piecewise linear case is given.
Topological Persistence and Simplification
TLDR
Fast algorithms for computing persistence and experimental evidence for their speed and utility are given for topological simplification within the framework of a filtration, which is the history of a growing complex.
Hierarchical Morse—Smale Complexes for Piecewise Linear 2-Manifolds
TLDR
Algorithm for constructing a hierarchy of increasingly coarse Morse—Smale complexes that decompose a piecewise linear 2-manifold by canceling pairs of critical points in order of increasing persistence is presented.
A linear-time algorithm for a special case of disjoint set union
TLDR
A linear-time algorithm for the special case of the disjoint set union problem in which the structure of the unions (defined by a “union tree”) is known in advance, which gives similar improvements in the efficiency of algorithms for solving a number of other problems.
Two linear time algorithms for MST on minor closed graph classes
This article presents two simple deterministic algorithms for finding the Minimum Spanning Tree in O(|V |+ |E|) time for any non-trivial class of graphs closed on graph minors. This applies in
...
...