42 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
Computing robustness and persistence for images
We are interested in 3-dimensional images given as arrays of voxels with intensity values. Extending these values to acontinuous function, we study the robustness of homology classes in its level and interlevel sets, that is, the amount of perturbationneeded to destroy these classes. The structure of the homology classes and their robustness, over all level and interlevel sets, can bevisualized by a triangular diagram of dots obtained by computing the extended persistence of the function. We give a fast hierarchicalalgorithm using the dual complexes of oct-tree approximations of the function. In addition, we show that for balanced oct-trees, thedual complexes are geometrically realized in and can thus be used to construct level and interlevel sets. We apply these tools tostudy 3-dimensional images of plant root systems
Hierarchies and Ranks for Persistence Pairs
We develop a novel hierarchy for zero-dimensional persistence pairs, i.e.,
connected components, which is capable of capturing more fine-grained spatial
relations between persistence pairs. Our work is motivated by a lack of spatial
relationships between features in persistence diagrams, leading to a limited
expressive power. We build upon a recently-introduced hierarchy of pairs in
persistence diagrams that augments the pairing stored in persistence diagrams
with information about which components merge. Our proposed hierarchy captures
differences in branching structure. Moreover, we show how to use our hierarchy
to measure the spatial stability of a pairing and we define a rank function for
persistence pairs and demonstrate different applications.Comment: Topology-based Methods in Visualization 201
Universality of the Bottleneck Distance for Extended Persistence Diagrams
The extended persistence diagram is an invariant of piecewise linear
functions, introduced by Cohen-Steiner, Edelsbrunner, and Harer. The bottleneck
distance has been introduced by the same authors as an extended pseudometric on
the set of extended persistence diagrams, which is stable under perturbations
of the function. We address the question whether the bottleneck distance is the
largest possible stable distance, providing an affirmative answer.Comment: 20 pages + 12 pages appendix, 18 figures, LaTeX; removal of appendix
on "stable functors on M" which has moved to arXiv:2108.09298, added and
improved figures, added a note of caution regarding variants of the
bottleneck distance, rewrote the proof of lemma 4.6 (formerly lemma 4.4),
added appendix B including the connection to the original definition of
extended persistence, several minor edit