TY - CONF
AB - We evaluate the usefulness of persistent homology in the analysis of heart rate variability. In our approach we extract several topological descriptors characterising datasets of RR-intervals, which are later used in classical machine learning algorithms. By this method we are able to differentiate the group of patients with the history of transient ischemic attack and the group of hypertensive patients.
AU - Graff, Grzegorz
AU - Graff, Beata
AU - Jablonski, Grzegorz
AU - Narkiewicz, Krzysztof
ID - 8580
SN - 9781728157511
T2 - 11th Conference of the European Study Group on Cardiovascular Oscillations: Computation and Modelling in Physiology: New Challenges and Opportunities,
TI - The application of persistent homology in the analysis of heart rate variability
ER -
TY - CONF
AB - Even though Delaunay originally introduced his famous triangulations in the case of infinite point sets with translational periodicity, a software that computes such triangulations in the general case is not yet available, to the best of our knowledge. Combining and generalizing previous work, we present a practical algorithm for computing such triangulations. The algorithm has been implemented and experiments show that its performance is as good as the one of the CGAL package, which is restricted to cubic periodicity.
AU - Osang, Georg F
AU - Rouxel-Labbé, Mael
AU - Teillaud, Monique
ID - 8703
SN - 18688969
T2 - 28th Annual European Symposium on Algorithms
TI - Generalizing CGAL periodic Delaunay triangulations
VL - 173
ER -
TY - CHAP
AB - We study the Gromov waist in the sense of t-neighborhoods for measures in the Euclidean space, motivated by the famous theorem of Gromov about the waist of radially symmetric Gaussian measures. In particular, it turns our possible to extend Gromov’s original result to the case of not necessarily radially symmetric Gaussian measure. We also provide examples of measures having no t-neighborhood waist property, including a rather wide class
of compactly supported radially symmetric measures and their maps into the Euclidean space of dimension at least 2.
We use a simpler form of Gromov’s pancake argument to produce some estimates of t-neighborhoods of (weighted) volume-critical submanifolds in the spirit of the waist theorems, including neighborhoods of algebraic manifolds in the complex projective space. In the appendix of this paper we provide for reader’s convenience a more detailed explanation of the Caffarelli theorem that we use to handle not necessarily radially symmetric Gaussian
measures.
AU - Akopyan, Arseniy
AU - Karasev, Roman
ED - Klartag, Bo'az
ED - Milman, Emanuel
ID - 74
SN - 00758434
T2 - Geometric Aspects of Functional Analysis
TI - Gromov's waist of non-radial Gaussian measures and radial non-Gaussian measures
VL - 2256
ER -
TY - THES
AB - Many methods for the reconstruction of shapes from sets of points produce ordered simplicial complexes, which are collections of vertices, edges, triangles, and their higher-dimensional analogues, called simplices, in which every simplex gets assigned a real value measuring its size. This thesis studies ordered simplicial complexes, with a focus on their topology, which reflects the connectedness of the represented shapes and the presence of holes. We are interested both in understanding better the structure of these complexes, as well as in developing algorithms for applications.
For the Delaunay triangulation, the most popular measure for a simplex is the radius of the smallest empty circumsphere. Based on it, we revisit Alpha and Wrap complexes and experimentally determine their probabilistic properties for random data. Also, we prove the existence of tri-partitions, propose algorithms to open and close holes, and extend the concepts from Euclidean to Bregman geometries.
AU - Ölsböck, Katharina
ID - 7460
KW - shape reconstruction
KW - hole manipulation
KW - ordered complexes
KW - Alpha complex
KW - Wrap complex
KW - computational topology
KW - Bregman geometry
SN - 2663-337X
TI - The hole system of triangulated shapes
ER -
TY - JOUR
AB - Slicing a Voronoi tessellation in ${R}^n$ with a $k$-plane gives a $k$-dimensional weighted Voronoi tessellation, also known as a power diagram or Laguerre tessellation. Mapping every simplex of the dual weighted Delaunay mosaic to the radius of the smallest empty circumscribed sphere whose center lies in the $k$-plane gives a generalized discrete Morse function. Assuming the Voronoi tessellation is generated by a Poisson point process in ${R}^n$, we study the expected number of simplices in the $k$-dimensional weighted Delaunay mosaic as well as the expected number of intervals of the Morse function, both as functions of a radius threshold. As a by-product, we obtain a new proof for the expected number of connected components (clumps) in a line section of a circular Boolean model in ${R}^n$.
AU - Edelsbrunner, Herbert
AU - Nikitenko, Anton
ID - 7554
IS - 4
JF - Theory of Probability and its Applications
SN - 0040585X
TI - Weighted Poisson–Delaunay mosaics
VL - 64
ER -