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 -