# Vines and vineyards by updating persistence in linear time

@inproceedings{CohenSteiner2006VinesAV, title={Vines and vineyards by updating persistence in linear time}, author={David Cohen-Steiner and Herbert Edelsbrunner and Dmitriy Morozov}, booktitle={SCG '06}, year={2006} }

Persistent homology is the mathematical core of recent work on shape, including reconstruction, recognition, and matching. Its pertinent information is encapsulated by a pairing of the critical values of a function, visualized by points forming a diagram in the plane. The original algorithm in [10] computes the pairs from an ordering of the simplices in a triangulation and takes worst-case time cubic in the number of simplices. The main result of this paper is an algorithm that maintains the…

## 225 Citations

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…

Persistence-sensitive simplication of functions on surfaces in linear time

- Mathematics
- 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…

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.

Extending Persistence Using Poincaré and Lefschetz Duality

- MathematicsFound. Comput. Math.
- 2009

An algebraic formulation is given that extends persistence to essential homology for any filtered space, an algorithm is presented to calculate it, and how it aids the ability to recognize shape features for codimension 1 submanifolds of Euclidean space is described.

Persistent Homology

- Mathematics
- 2017

The theory of persistent homology developed from 2000, motivated by practical problems related to approximation and reverse engineering [Rob99, ELZ00]. The main objective is to infer the topology of…

Tracking a Generator by Persistence

- Computer ScienceDiscret. Math. Algorithms Appl.
- 2010

This paper considers the problem of tracking generating cycles with temporal coherence, and builds upon the matrix framework proposed by Cohen-Steiner et al. to swap two consecutive simplices, so that the algorithm can process a reordering directly.

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.

Algorithmic Reconstruction of the Fiber of Persistent Homology on Cell Complexes

- MathematicsArXiv
- 2021

A depth first search algorithm is designed and implemented that recovers the polyhedra forming the fiber PH ́1pDq, providing a first insight into the statistical structure of these fibers, for general CW complexes.

Zigzag persistent homology and real-valued functions

- MathematicsSCG '09
- 2009

The algorithmic results provide a way to compute zigzag persistence for any sequence of homology groups, but combined with the structural results give a novel algorithm for computing extended persistence that is easily parallelizable and uses (asymptotically) less memory.

Defining and Computing Topological Persistence for 1-cocycles

- Mathematics, Computer ScienceArXiv
- 2010

It turns out that, instead of the standard persistence, one of its variants which the authors call level persistence can be leveraged for this purpose and it is worth mentioning that 1-cocyles appear in practice such as in data ranking or in discrete vector fields.

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