TY - CHAP AB - The Heat Kernel Signature (HKS) is a scalar quantity which is derived from the heat kernel of a given shape. Due to its robustness, isometry invariance, and multiscale nature, it has been successfully applied in many geometric applications. From a more general point of view, the HKS can be considered as a descriptor of the metric of a Riemannian manifold. Given a symmetric positive definite tensor field we may interpret it as the metric of some Riemannian manifold and thereby apply the HKS to visualize and analyze the given tensor data. In this paper, we propose a generalization of this approach that enables the treatment of indefinite tensor fields, like the stress tensor, by interpreting them as a generator of a positive definite tensor field. To investigate the usefulness of this approach we consider the stress tensor from the two-point-load model example and from a mechanical work piece. AU - Zobel, Valentin AU - Reininghaus, Jan AU - Hotz, Ingrid ED - Hotz, Ingrid ED - Schultz, Thomas ID - 1531 SN - 978-3-319-15089-5 T2 - Visualization and Processing of Higher Order Descriptors for Multi-Valued Data TI - Visualizing symmetric indefinite 2D tensor fields using The Heat Kernel Signature VL - 40 ER - TY - JOUR AB - We show that incorporating spatial dispersal of individuals into a simple vaccination epidemic model may give rise to a model that exhibits rich dynamical behavior. Using an SIVS (susceptible-infected-vaccinated-susceptible) model as a basis, we describe the spread of an infectious disease in a population split into two regions. In each subpopulation, both forward and backward bifurcations can occur. This implies that for disconnected regions the two-patch system may admit several steady states. We consider traveling between the regions and investigate the impact of spatial dispersal of individuals on the model dynamics. We establish conditions for the existence of multiple nontrivial steady states in the system, and we study the structure of the equilibria. The mathematical analysis reveals an unusually rich dynamical behavior, not normally found in the simple epidemic models. In addition to the disease-free equilibrium, eight endemic equilibria emerge from backward transcritical and saddle-node bifurcation points, forming an interesting bifurcation diagram. Stability of steady states, their bifurcations, and the global dynamics are investigated with analytical tools, numerical simulations, and rigorous set-oriented numerical computations. AU - Knipl, Diána AU - Pilarczyk, Pawel AU - Röst, Gergely ID - 1555 IS - 2 JF - SIAM Journal on Applied Dynamical Systems TI - Rich bifurcation structure in a two patch vaccination model VL - 14 ER - TY - CONF AB - Aiming at the automatic diagnosis of tumors from narrow band imaging (NBI) magnifying endoscopy (ME) images of the stomach, we combine methods from image processing, computational topology, and machine learning to classify patterns into normal, tubular, vessel. Training the algorithm on a small number of images of each type, we achieve a high rate of correct classifications. The analysis of the learning algorithm reveals that a handful of geometric and topological features are responsible for the overwhelming majority of decisions. AU - Dunaeva, Olga AU - Edelsbrunner, Herbert AU - Lukyanov, Anton AU - Machin, Michael AU - Malkova, Daria ID - 1568 T2 - Proceedings - 16th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing TI - The classification of endoscopy images with persistent homology ER - TY - CONF AB - My personal journey to the fascinating world of geometric forms started more than 30 years ago with the invention of alpha shapes in the plane. It took about 10 years before we generalized the concept to higher dimensions, we produced working software with a graphics interface for the three-dimensional case. At the same time, we added homology to the computations. Needless to say that this foreshadowed the inception of persistent homology, because it suggested the study of filtrations to capture the scale of a shape or data set. Importantly, this method has fast algorithms. The arguably most useful result on persistent homology is the stability of its diagrams under perturbations. AU - Edelsbrunner, Herbert ID - 1567 T2 - 23rd International Symposium TI - Shape, homology, persistence, and stability VL - 9411 ER - TY - JOUR AB - For a given self-map $f$ of $M$, a closed smooth connected and simply-connected manifold of dimension $m\geq 4$, we provide an algorithm for estimating the values of the topological invariant $D^m_r[f]$, which equals the minimal number of $r$-periodic points in the smooth homotopy class of $f$. Our results are based on the combinatorial scheme for computing $D^m_r[f]$ introduced by G. Graff and J. Jezierski [J. Fixed Point Theory Appl. 13 (2013), 63-84]. An open-source implementation of the algorithm programmed in C++ is publicly available at {\tt http://www.pawelpilarczyk.com/combtop/}. AU - Graff, Grzegorz AU - Pilarczyk, Pawel ID - 1563 IS - 1 JF - Topological Methods in Nonlinear Analysis TI - An algorithmic approach to estimating the minimal number of periodic points for smooth self-maps of simply-connected manifolds VL - 45 ER -