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 - The theory of persistent homology opens up the possibility to reason about topological features of a space or a function quantitatively and in combinatorial terms. We refer to this new angle at a classical subject within algebraic topology as a point calculus, which we present for the family of interlevel sets of a real-valued function. Our account of the subject is expository, devoid of proofs, and written for non-experts in algebraic topology. AU - Bendich, Paul AU - Cabello, Sergio AU - Edelsbrunner, Herbert ID - 3310 IS - 11 JF - Pattern Recognition Letters TI - A point calculus for interlevel set homology VL - 33 ER - TY - JOUR AB - The theory of intersection homology was developed to study the singularities of a topologically stratified space. This paper in- corporates this theory into the already developed framework of persistent homology. We demonstrate that persistent intersec- tion homology gives useful information about the relationship between an embedded stratified space and its singularities. We give, and prove the correctness of, an algorithm for the computa- tion of the persistent intersection homology groups of a filtered simplicial complex equipped with a stratification by subcom- plexes. We also derive, from Poincare ́ Duality, some structural results about persistent intersection homology. AU - Bendich, Paul AU - Harer, John ID - 3378 IS - 3 JF - Foundations of Computational Mathematics TI - Persistent intersection homology VL - 11 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 - JOUR AB - 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 $R^3$ and can thus be used to construct level and interlevel sets. We apply these tools tostudy 3-dimensional images of plant root systems. AU - Bendich, Paul AU - Edelsbrunner, Herbert AU - Kerber, Michael ID - 3901 IS - 6 JF - IEEE Transactions of Visualization and Computer Graphics TI - Computing robustness and persistence for images VL - 16 ER - TY - CONF AB - We study the reconstruction of a stratified space from a possibly noisy point sample. Specifically, we use the vineyard of the distance function restricted to a I-parameter family of neighborhoods of a point to assess the local homology of the stratified space at that point. We prove the correctness of this assessment under the assumption of a sufficiently dense sample. We also give an algorithm that constructs the vineyard and makes the local assessment in time at most cubic in the size of the Delaunay triangulation of the point sample. AU - Paul Bendich AU - Cohen-Steiner, David AU - Herbert Edelsbrunner AU - Harer, John AU - Morozov, Dmitriy ID - 3975 TI - Inferring local homology from sampled stratified spaces ER -