• Publications
  • Influence
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.
Computing Persistent Homology
TLDR
The homology of a filtered d-dimensional simplicial complex K is studied as a single algebraic entity and a correspondence is established that provides a simple description over fields that enables a natural algorithm for computing persistent homology over an arbitrary field in any dimension.
The Theory of Multidimensional Persistence
TLDR
This paper proposes the rank invariant, a discrete invariant for the robust estimation of Betti numbers in a multifiltration, and proves its completeness in one dimension.
Computing Persistent Homology
Abstract We show that the persistent homology of a filtered d-dimensional simplicial complex is simply the standard homology of a particular graded module over a polynomial ring. Our analysis
Computational topology
TLDR
The aim of this chapter is not to be comprehensive, but to describe the fundamental concepts, methods, and structures that permeate the field of computational topology.
On the Local Behavior of Spaces of Natural Images
TLDR
A theoretical model for the high-density 2-dimensional submanifold of ℳ showing that it has the topology of the Klein bottle and a polynomial representation is used to give coordinatization to various subspaces ofℳ.
Hierarchical morse complexes for piecewise linear 2-manifolds
TLDR
While Morse complexes are defined only in the smooth category, the construction is extended to the piecewise linear category by ensuring structural integrity and simulating differentiability and then simplified by cancelling pairs of critical points in order of increasing persistence.
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.
Fast construction of the Vietoris-Rips complex
  • A. Zomorodian
  • Computer Science, Mathematics
    Comput. Graph.
  • 1 June 2010
...
...