27 research outputs found
Homology and Robustness of Level and Interlevel Sets
Given a function f: \Xspace \to \Rspace on a topological space, we consider
the preimages of intervals and their homology groups and show how to read the
ranks of these groups from the extended persistence diagram of . In
addition, we quantify the robustness of the homology classes under
perturbations of using well groups, and we show how to read the ranks of
these groups from the same extended persistence diagram. The special case
\Xspace = \Rspace^3 has ramifications in the fields of medical imaging and
scientific visualization
Variable sets over an algebra of lifetimes: a contribution of lattice theory to the study of computational topology
A topos theoretic generalisation of the category of sets allows for modelling
spaces which vary according to time intervals. Persistent homology, or more
generally, persistence is a central tool in topological data analysis, which
examines the structure of data through topology. The basic techniques have been
extended in several different directions, permuting the encoding of topological
features by so called barcodes or equivalently persistence diagrams. The set of
points of all such diagrams determines a complete Heyting algebra that can
explain aspects of the relations between persistent bars through the algebraic
properties of its underlying lattice structure. In this paper, we investigate
the topos of sheaves over such algebra, as well as discuss its construction and
potential for a generalised simplicial homology over it. In particular we are
interested in establishing a topos theoretic unifying theory for the various
flavours of persistent homology that have emerged so far, providing a global
perspective over the algebraic foundations of applied and computational
topology.Comment: 20 pages, 12 figures, AAA88 Conference proceedings at Demonstratio
Mathematica. The new version has restructured arguments, clearer intuition is
provided, and several typos correcte
Parametrized Homology via Zigzag Persistence
This paper develops the idea of homology for 1-parameter families of
topological spaces. We express parametrized homology as a collection of real
intervals with each corresponding to a homological feature supported over that
interval or, equivalently, as a persistence diagram. By defining persistence in
terms of finite rectangle measures, we classify barcode intervals into four
classes. Each of these conveys how the homological features perish at both ends
of the interval over which they are defined
Alexander Duality for Functions: the Persistent Behavior of Land and Water and Shore
This note contributes to the point calculus of persistent homology by
extending Alexander duality to real-valued functions. Given a perfect Morse
function and a decomposition such
that M = \U \cap V is an -manifold, we prove elementary relationships
between the persistence diagrams of restricted to , to , and to .Comment: Keywords: Algebraic topology, homology, Alexander duality,
Mayer-Vietoris sequences, persistent homology, point calculu
Bigraded Betti numbers and Generalized Persistence Diagrams
Commutative diagrams of vector spaces and linear maps over are
objects of interest in topological data analysis (TDA) where this type of
diagrams are called 2-parameter persistence modules. Given that quiver
representation theory tells us that such diagrams are of wild type, studying
informative invariants of a 2-parameter persistence module is of central
importance in TDA. One of such invariants is the generalized rank invariant,
recently introduced by Kim and M\'emoli. Via the M\"obius inversion of the
generalized rank invariant of , we obtain a collection of connected subsets
with signed multiplicities. This collection generalizes
the well known notion of persistence barcode of a persistence module over
from TDA. In this paper we show that the bigraded Betti numbers of
, a classical algebraic invariant of , are obtained by counting the
corner points of these subsets s. Along the way, we verify that an invariant
of 2-parameter persistence modules called the interval decomposable
approximation (introduced by Asashiba et al.) also encodes the bigraded Betti
numbers in a similar fashion.Comment: 26 pages, 7 figures; v3:improvements to expositio