# 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…

## 40 Citations

Optimal Topological Simplification of Discrete Functions on Surfaces

- MathematicsDiscret. Comput. Geom.
- 2012

The number of critical points of the resulting simplified function fδ achieves the lower bound dictated by the stability theorem of persistent homology and it is shown that the simplified function can be computed in linear time after persistence pairs have been computed.

Homological illusions of persistence and stability

- Mathematics
- 2008

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…

Generalized Topological Simplification of Scalar Fields on Surfaces

- Computer ScienceIEEE Transactions on Visualization and Computer Graphics
- 2012

A combinatorial algorithm for the general topological simplification of scalar fields on surfaces that enables the robust pruning of topological noise as selected by the user and is simple to implement, fast in practice and more general than previous techniques.

Proximity of persistence modules and their diagrams

- Mathematics, Computer ScienceSCG '09
- 2009

This paper presents new stability results that do not suffer from the restrictions of existing stability results, and makes it possible to compare the persistence diagrams of functions defined over different spaces, thus enabling a variety of new applications of the concept of persistence.

Move Schedules: Fast persistence computations in sparse dynamic settings

- Computer ScienceArXiv
- 2021

Results are presented showing that the decrease in operations to compute diagrams across a family of filtrations is proportional to the difference between the expected quadratic number of states, and the proposed sublinear coarsening.

Edge contraction in persistence-generated discrete Morse vector fields

- MathematicsComput. Graph.
- 2018

Topological Optimization with Big Steps

- Computer ScienceArXiv
- 2022

It is shown how the cycles and chains used in the persistence calculation can be used to prescribe gradients to larger subsets of the domain, and the number of steps required for the optimization is reduced by an order of magnitude.

Constructive Mayer-Vietoris Algorithm: Computing the Homology of Unions of Simplicial Complexes

- Mathematics
- 2010

In this research report, we present an efficient method for computing the homology of a large simplicial complex from the homologies of its sub-complexes. The method uses a constructive version of…

Data Analysis using Computational Topology and Geometric Statistics Mar 8 – Mar 13 , 2009 MEALS

- Mathematics
- 2009

(in alphabetic order by speaker surname) Speaker: Dominique Attali (CNRS, Grenoble) Title: Persistence-sensitive simplification of functions on surfaces in linear time. Abstract: Let f be a…

Topological Function Optimization for Continuous Shape Matching

- Computer ScienceComput. Graph. Forum
- 2018

The method is based on using the previously‐proposed persistence diagrams associated with real‐valued functions, and on the analysis of the derivatives of these diagrams with respect to changes in the function values allows for continuous optimization techniques to modify a given function, while optimizing an energy based purely on the values in the persistence diagrams.

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