@article{3159,
abstract = {The structure of hierarchical networks in biological and physical systems has long been characterized using the Horton-Strahler ordering scheme. The scheme assigns an integer order to each edge in the network based on the topology of branching such that the order increases from distal parts of the network (e.g., mountain streams or capillaries) to the "root" of the network (e.g., the river outlet or the aorta). However, Horton-Strahler ordering cannot be applied to networks with loops because they they create a contradiction in the edge ordering in terms of which edge precedes another in the hierarchy. Here, we present a generalization of the Horton-Strahler order to weighted planar reticular networks, where weights are assumed to correlate with the importance of network edges, e.g., weights estimated from edge widths may correlate to flow capacity. Our method assigns hierarchical levels not only to edges of the network, but also to its loops, and classifies the edges into reticular edges, which are responsible for loop formation, and tree edges. In addition, we perform a detailed and rigorous theoretical analysis of the sensitivity of the hierarchical levels to weight perturbations. In doing so, we show that the ordering of the reticular edges is more robust to noise in weight estimation than is the ordering of the tree edges. We discuss applications of this generalized Horton-Strahler ordering to the study of leaf venation and other biological networks.},
author = {Mileyko, Yuriy and Edelsbrunner, Herbert and Price, Charles and Weitz, Joshua},
journal = {PLoS One},
number = {6},
publisher = {Public Library of Science},
title = {{Hierarchical ordering of reticular networks}},
doi = {10.1371/journal.pone.0036715},
volume = {7},
year = {2012},
}
@article{3310,
abstract = {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.},
author = {Bendich, Paul and Cabello, Sergio and Edelsbrunner, Herbert},
journal = {Pattern Recognition Letters},
number = {11},
pages = {1436 -- 1444},
publisher = {Elsevier},
title = {{A point calculus for interlevel set homology}},
doi = {10.1016/j.patrec.2011.10.007},
volume = {33},
year = {2012},
}
@article{3256,
abstract = {We use a distortion to define the dual complex of a cubical subdivision of ℝ n as an n-dimensional subcomplex of the nerve of the set of n-cubes. Motivated by the topological analysis of high-dimensional digital image data, we consider such subdivisions defined by generalizations of quad- and oct-trees to n dimensions. Assuming the subdivision is balanced, we show that mapping each vertex to the center of the corresponding n-cube gives a geometric realization of the dual complex in ℝ n.},
author = {Edelsbrunner, Herbert and Kerber, Michael},
journal = {Discrete & Computational Geometry},
number = {2},
pages = {393 -- 414},
publisher = {Springer},
title = {{Dual complexes of cubical subdivisions of ℝn}},
doi = {10.1007/s00454-011-9382-4},
volume = {47},
year = {2012},
}
@article{3331,
abstract = {Computing the topology of an algebraic plane curve C means computing a combinatorial graph that is isotopic to C and thus represents its topology in R2. We prove that, for a polynomial of degree n with integer coefficients bounded by 2ρ, the topology of the induced curve can be computed with bit operations ( indicates that we omit logarithmic factors). Our analysis improves the previous best known complexity bounds by a factor of n2. The improvement is based on new techniques to compute and refine isolating intervals for the real roots of polynomials, and on the consequent amortized analysis of the critical fibers of the algebraic curve.},
author = {Kerber, Michael and Sagraloff, Michael},
journal = { Journal of Symbolic Computation},
number = {3},
pages = {239 -- 258},
publisher = {Elsevier},
title = {{A worst case bound for topology computation of algebraic curves}},
doi = {10.1016/j.jsc.2011.11.001},
volume = {47},
year = {2012},
}
@article{3115,
abstract = {We consider the offset-deconstruction problem: 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 P with a disk of fixed radius? If it does, we also seek a preferably simple-looking solution P; then, P's offset constitutes an accurate, vertex-reduced, and smoothened approximation of Q. We give an O(nlogn)-time exact decision algorithm that handles any polygonal shape, assuming the real-RAM model of computation. A variant of the algorithm, which we have implemented using the cgal library, is based on rational arithmetic and answers the same deconstruction problem up to an uncertainty parameter δ its running time additionally depends on δ. If the input shape is found to be approximable, this algorithm also computes an approximate solution for the problem. It also allows us to solve parameter-optimization problems induced by the offset-deconstruction problem. For convex shapes, the complexity of the exact decision algorithm drops to O(n), which is also the time required to compute a solution P with at most one more vertex than a vertex-minimal one.},
author = {Berberich, Eric and Halperin, Dan and Kerber, Michael and Pogalnikova, Roza},
journal = {Discrete & Computational Geometry},
number = {4},
pages = {964 -- 989},
publisher = {Springer},
title = {{Deconstructing approximate offsets}},
doi = {10.1007/s00454-012-9441-5},
volume = {48},
year = {2012},
}