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… 

Figures from this paper

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
Persistence-sensitive simplication of functions on surfaces in linear time
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
TLDR
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
TLDR
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
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
TLDR
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
TLDR
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
TLDR
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
TLDR
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
TLDR
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.
...
...

References

SHOWING 1-10 OF 21 REFERENCES
Persistence barcodes for shapes
TLDR
This paper initiates a study of shape description and classification via the application of persistent homology to two tangential constructions on geometric objects, obtaining a shape descriptor, called a barcode, that is a finite union of intervals.
Topological estimation using witness complexes
TLDR
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.
CRITICAL POINTS AND CURVATURE FOR EMBEDDED POLYHEDRA
Recently a new insight into the Gauss-Bonnet Theorem and other problems in global differential geometry has come about through the connection between total curvature of embedded smooth manifolds and
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.
Natural Pseudo-Distance and Optimal Matching between Reduced Size Functions
TLDR
The matching distance is shown to be resistant to perturbations, implying that it is always smaller than the natural pseudo-distance, and it is proved that the lower bound so obtained is sharp and cannot be improved by any other distance between size functions.
Exploring Protein Folding Trajectories Using Geometric Spanners
TLDR
This work describes the 3-D structure of a protein using geometric spanners--geometric graphs with a sparse set of edges where paths approximate the n2 inter-atom distances, allowing one to easily detect formation of secondary and tertiary structures as the protein folds.
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
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.
Inequalities for the Curvature of Curves and Surfaces
TLDR
The difference between the total mean curvatures of two closed surfaces in R3 is bound in terms of their total absolute curvatures and the Fréchet distance between the volumes they enclose using a combination of methods from algebraic topology and integral geometry.
Extreme Elevation on a 2-Manifold
TLDR
An algorithm for finding points of locally maximum elevation is given, which is used to suggest mark cavities and protrusions and are useful in matching shapes as for example in protein docking.
...
...