TY - JOUR AB - Given a continuous function f:X-R 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 f. In addition, we quantify the robustness of the homology classes under perturbations of f using well groups, and we show how to read the ranks of these groups from the same extended persistence diagram. The special case X=R3 has ramifications in the fields of medical imaging and scientific visualization. AU - Bendich, Paul AU - Edelsbrunner, Herbert AU - Morozov, Dmitriy AU - Patel, Amit ID - 2859 IS - 1 JF - Homology, Homotopy and Applications TI - Homology and robustness of level and interlevel sets VL - 15 ER - TY - JOUR AB - By definition, transverse intersections are stable under in- finitesimal perturbations. Using persistent homology, we ex- tend this notion to sizeable perturbations. Specifically, we assign to each homology class of the intersection its robust- ness, the magnitude of a perturbation necessary to kill it, and prove that robustness is stable. Among the applications of this result is a stable notion of robustness for fixed points of continuous mappings and a statement of stability for con- tours of smooth mappings. AU - Edelsbrunner, Herbert AU - Morozov, Dmitriy AU - Patel, Amit ID - 3377 IS - 3 JF - Foundations of Computational Mathematics TI - Quantifying transversality by measuring the robustness of intersections VL - 11 ER - TY - CHAP AB - The (apparent) contour of a smooth mapping from a 2-manifold to the plane, f: M → R2 , is the set of critical values, that is, the image of the points at which the gradients of the two component functions are linearly dependent. Assuming M is compact and orientable and measuring difference with the erosion distance, we prove that the contour is stable. AU - Edelsbrunner, Herbert AU - Morozov, Dmitriy AU - Patel, Amit ID - 3795 T2 - Topological Data Analysis and Visualization: Theory, Algorithms and Applications TI - The stability of the apparent contour of an orientable 2-manifold ER - TY - CONF AB - We define the robustness of a level set homology class of a function f:XR as the magnitude of a perturbation necessary to kill the class. Casting this notion into a group theoretic framework, we compute the robustness for each class, using a connection to extended persistent homology. The special case X=R3 has ramifications in medical imaging and scientific visualization. AU - Bendich, Paul AU - Edelsbrunner, Herbert AU - Morozov, Dmitriy AU - Patel, Amit ID - 3848 TI - The robustness of level sets VL - 6346 ER - TY - CONF AB - Using ideas from persistent homology, the robustness of a level set of a real-valued function is defined in terms of the magnitude of the perturbation necessary to kill the classes. Prior work has shown that the homology and robustness information can be read off the extended persistence diagram of the function. This paper extends these results to a non-uniform error model in which perturbations vary in their magnitude across the domain. AU - Bendich, Paul AU - Edelsbrunner, Herbert AU - Kerber, Michael AU - Patel, Amit ID - 3849 TI - Persistent homology under non-uniform error VL - 6281 ER - TY - CONF AB - Generalizing the concept of a Reeb graph, the Reeb space of a multivariate continuous mapping identifies points of the domain that belong to a common component of the preimage of a point in the range. We study the local and global structure of this space for generic, piecewise linear mappings on a combinatorial manifold. AU - Herbert Edelsbrunner AU - Harer, John AU - Amit Patel ID - 3974 TI - Reeb spaces of piecewise linear mappings ER -