@inproceedings{3850,
abstract = {Given a polygonal shape Q with n vertices, can it be expressed, up to a tolerance ε in Hausdorff distance, as the Minkowski sum of another polygonal shape with a disk of fixed radius? If it does, we also seek a preferably simple solution shape P;P’s offset constitutes an accurate, vertex-reduced, and smoothened approximation of Q. We give a decision algorithm for fixed radius in O(nlogn) time that handles any polygonal shape. For convex shapes, the complexity drops to O(n), which is also the time required to compute a solution shape P with at most one more vertex than a vertex-minimal one.},
author = {Berberich, Eric and Halperin, Dan and Kerber, Michael and Pogalnikova, Roza},
location = {Dortmund, Germany},
pages = {12 -- 23},
publisher = {TU Dortmund},
title = {{Polygonal reconstruction from approximate offsets}},
year = {2010},
}
@article{3901,
abstract = {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.},
author = {Bendich, Paul and Edelsbrunner, Herbert and Kerber, Michael},
journal = {IEEE Transactions of Visualization and Computer Graphics},
number = {6},
pages = {1251 -- 1260},
publisher = {IEEE},
title = {{Computing robustness and persistence for images}},
doi = {10.1109/TVCG.2010.139},
volume = {16},
year = {2010},
}
@inproceedings{3849,
abstract = {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.},
author = {Bendich, Paul and Edelsbrunner, Herbert and Kerber, Michael and Patel, Amit},
location = {Brno, Czech Republic},
pages = {12 -- 23},
publisher = {Springer},
title = {{Persistent homology under non-uniform error}},
doi = {10.1007/978-3-642-15155-2_2},
volume = {6281},
year = {2010},
}
@inproceedings{3968,
abstract = {We describe an algorithm for segmenting three-dimensional medical imaging data modeled as a continuous function on a 3-manifold. It is related to watershed algorithms developed in image processing but is closer to its mathematical roots, which are Morse theory and homological algebra. It allows for the implicit treatment of an underlying mesh, thus combining the structural integrity of its mathematical foundations with the computational efficiency of image processing.},
author = {Edelsbrunner, Herbert and Harer, John},
location = {Zermatt, Switzerland},
pages = {36 -- 50},
publisher = {Springer},
title = {{The persistent Morse complex segmentation of a 3-manifold}},
doi = {10.1007/978-3-642-10470-1_4},
volume = {5903},
year = {2009},
}