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.
By definition, transverse intersections are stable under in-finitesimal perturbations. Using persistent homology, we extend 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… (More)
Given a 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. After characterizing these groups, we show… (More)
The (apparent) contour of a smooth mapping from a 2-manifold to the plane, f : M → R 2 , 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.
We define the robustness of a level set homology class of a function f : X → R 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 = R 3 has ramifications in medical imaging and… (More)
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… (More)
Stress urinary incontinence (SUI) affects 10-20% of women in the general population. Surgery for stress incontinence has been performed on women for over a century, but with the advent of new urogynaecological sling procedures for its management, urological surgeons are having to deal with an increasing number of patients presenting with associated… (More)
There are many problems for which the best known algorithms take time much greater than the worst known lower bound. This suggests the existence of more efficient algorithms. In computational geometry, one such barrier is the Θ(n 2) barrier. There exists problems for which the best algorithm runs in worst case time Θ(n 2). However, their lower bounds are… (More)
Imagine two binary trees T 1 and T 2 with n nodes. The rotation distance between T 1 and T 2 is the minimum number of rotations needed to convert one tree into another. What is the maximum rotation distance between any two binary trees with n nodes? For n ≥ 11, 2n − 6 rotations are sufficient and this bound is tight [STT98]. The bound itself isn't as… (More)