TY - JOUR
AB - We investigate weighted straight skeletons from a geometric, graph-theoretical, and combinatorial point of view. We start with a thorough definition and shed light on some ambiguity issues in the procedural definition. We investigate the geometry, combinatorics, and topology of faces and the roof model, and we discuss in which cases a weighted straight skeleton is connected. Finally, we show that the weighted straight skeleton of even a simple polygon may be non-planar and may contain cycles, and we discuss under which restrictions on the weights and/or the input polygon the weighted straight skeleton still behaves similar to its unweighted counterpart. In particular, we obtain a non-procedural description and a linear-time construction algorithm for the straight skeleton of strictly convex polygons with arbitrary weights.
AU - Biedl, Therese
AU - Held, Martin
AU - Huber, Stefan
AU - Kaaser, Dominik
AU - Palfrader, Peter
ID - 1582
IS - 2
JF - Computational Geometry: Theory and Applications
TI - Weighted straight skeletons in the plane
VL - 48
ER -
TY - JOUR
AB - We study the characteristics of straight skeletons of monotone polygonal chains and use them to devise an algorithm for computing positively weighted straight skeletons of monotone polygons. Our algorithm runs in O(nlogn) time and O(n) space, where n denotes the number of vertices of the polygon.
AU - Biedl, Therese
AU - Held, Martin
AU - Huber, Stefan
AU - Kaaser, Dominik
AU - Palfrader, Peter
ID - 1583
IS - 2
JF - Information Processing Letters
TI - A simple algorithm for computing positively weighted straight skeletons of monotone polygons
VL - 115
ER -
TY - JOUR
AB - We investigate weighted straight skeletons from a geometric, graph-theoretical, and combinatorial point of view. We start with a thorough definition and shed light on some ambiguity issues in the procedural definition. We investigate the geometry, combinatorics, and topology of faces and the roof model, and we discuss in which cases a weighted straight skeleton is connected. Finally, we show that the weighted straight skeleton of even a simple polygon may be non-planar and may contain cycles, and we discuss under which restrictions on the weights and/or the input polygon the weighted straight skeleton still behaves similar to its unweighted counterpart. In particular, we obtain a non-procedural description and a linear-time construction algorithm for the straight skeleton of strictly convex polygons with arbitrary weights.
AU - Biedl, Therese
AU - Held, Martin
AU - Huber, Stefan
AU - Kaaser, Dominik
AU - Palfrader, Peter
ID - 1584
IS - 5
JF - Computational Geometry: Theory and Applications
TI - Reprint of: Weighted straight skeletons in the plane
VL - 48
ER -
TY - CHAP
AB - The straight skeleton of a polygon is the geometric graph obtained by tracing the vertices during a mitered offsetting process. It is known that the straight skeleton of a simple polygon is a tree, and one can naturally derive directions on the edges of the tree from the propagation of the shrinking process. In this paper, we ask the reverse question: Given a tree with directed edges, can it be the straight skeleton of a polygon? And if so, can we find a suitable simple polygon? We answer these questions for all directed trees where the order of edges around each node is fixed.
AU - Aichholzer, Oswin
AU - Biedl, Therese
AU - Hackl, Thomas
AU - Held, Martin
AU - Huber, Stefan
AU - Palfrader, Peter
AU - Vogtenhuber, Birgit
ID - 1590
T2 - Graph Drawing and Network Visualization
TI - Representing directed trees as straight skeletons
VL - 9411
ER -
TY - JOUR
AB - We study the problem of robust satisfiability of systems of nonlinear equations, namely, whether for a given continuous function f:K→ ℝn on a finite simplicial complex K and α > 0, it holds that each function g: K → ℝn such that ||g - f || ∞ < α, has a root in K. Via a reduction to the extension problem of maps into a sphere, we particularly show that this problem is decidable in polynomial time for every fixed n, assuming dimK ≤ 2n - 3. This is a substantial extension of previous computational applications of topological degree and related concepts in numerical and interval analysis. Via a reverse reduction, we prove that the problem is undecidable when dim K > 2n - 2, where the threshold comes from the stable range in homotopy theory. For the lucidity of our exposition, we focus on the setting when f is simplexwise linear. Such functions can approximate general continuous functions, and thus we get approximation schemes and undecidability of the robust satisfiability in other possible settings.
AU - Franek, Peter
AU - Krcál, Marek
ID - 1682
IS - 4
JF - Journal of the ACM
TI - Robust satisfiability of systems of equations
VL - 62
ER -
TY - JOUR
AB - We present a software platform for reconstructing and analyzing the growth of a plant root system from a time-series of 3D voxelized shapes. It aligns the shapes with each other, constructs a geometric graph representation together with the function that records the time of growth, and organizes the branches into a hierarchy that reflects the order of creation. The software includes the automatic computation of structural and dynamic traits for each root in the system enabling the quantification of growth on fine-scale. These are important advances in plant phenotyping with applications to the study of genetic and environmental influences on growth.
AU - Symonova, Olga
AU - Topp, Christopher
AU - Edelsbrunner, Herbert
ID - 1793
IS - 6
JF - PLoS One
TI - DynamicRoots: A software platform for the reconstruction and analysis of growing plant roots
VL - 10
ER -
TY - GEN
AU - Symonova, Olga
AU - Topp, Christopher
AU - Edelsbrunner, Herbert
ID - 9737
TI - Root traits computed by DynamicRoots for the maize root shown in fig 2
ER -
TY - CONF
AB - Persistent homology is a recent grandchild of homology that has found use in
science and engineering as well as in mathematics. This paper surveys the method as well
as the applications, neglecting completeness in favor of highlighting ideas and directions.
AU - Edelsbrunner, Herbert
AU - Morozovy, Dmitriy
ID - 2905
TI - Persistent homology: Theory and practice
ER -
TY - JOUR
AB - Watermarking techniques for vector graphics dislocate vertices in order to embed imperceptible, yet detectable, statistical features into the input data. The embedding process may result in a change of the topology of the input data, e.g., by introducing self-intersections, which is undesirable or even disastrous for many applications. In this paper we present a watermarking framework for two-dimensional vector graphics that employs conventional watermarking techniques but still provides the guarantee that the topology of the input data is preserved. The geometric part of this framework computes so-called maximum perturbation regions (MPR) of vertices. We propose two efficient algorithms to compute MPRs based on Voronoi diagrams and constrained triangulations. Furthermore, we present two algorithms to conditionally correct the watermarked data in order to increase the watermark embedding capacity and still guarantee topological correctness. While we focus on the watermarking of input formed by straight-line segments, one of our approaches can also be extended to circular arcs. We conclude the paper by demonstrating and analyzing the applicability of our framework in conjunction with two well-known watermarking techniques.
AU - Huber, Stefan
AU - Held, Martin
AU - Meerwald, Peter
AU - Kwitt, Roland
ID - 1816
IS - 1
JF - International Journal of Computational Geometry and Applications
TI - Topology-preserving watermarking of vector graphics
VL - 24
ER -
TY - JOUR
AB - We prove polynomial upper bounds of geometric Ramsey numbers of pathwidth-2 outerplanar triangulations in both convex and general cases. We also prove that the geometric Ramsey numbers of the ladder graph on 2n vertices are bounded by O(n3) and O(n10), in the convex and general case, respectively. We then apply similar methods to prove an (Formula presented.) upper bound on the Ramsey number of a path with n ordered vertices.
AU - Cibulka, Josef
AU - Gao, Pu
AU - Krcál, Marek
AU - Valla, Tomáš
AU - Valtr, Pavel
ID - 1842
IS - 1
JF - Discrete & Computational Geometry
TI - On the geometric ramsey number of outerplanar graphs
VL - 53
ER -
TY - JOUR
AB - We study densities of functionals over uniformly bounded triangulations of a Delaunay set of vertices, and prove that the minimum is attained for the Delaunay triangulation if this is the case for finite sets.
AU - Dolbilin, Nikolai
AU - Edelsbrunner, Herbert
AU - Glazyrin, Alexey
AU - Musin, Oleg
ID - 1876
IS - 3
JF - Moscow Mathematical Journal
TI - Functionals on triangulations of delaunay sets
VL - 14
ER -
TY - JOUR
AB - We propose an algorithm for the generalization of cartographic objects that can be used to represent maps on different scales.
AU - Alexeev, V V
AU - Bogaevskaya, V G
AU - Preobrazhenskaya, M M
AU - Ukhalov, A Y
AU - Edelsbrunner, Herbert
AU - Yakimova, Olga
ID - 1929
IS - 6
JF - Journal of Mathematical Sciences (United States)
TI - An algorithm for cartographic generalization that preserves global topology
VL - 203
ER -
TY - JOUR
AB - (Figure Presented) Data acquisition, numerical inaccuracies, and sampling often introduce noise in measurements and simulations. Removing this noise is often necessary for efficient analysis and visualization of this data, yet many denoising techniques change the minima and maxima of a scalar field. For example, the extrema can appear or disappear, spatially move, and change their value. This can lead to wrong interpretations of the data, e.g., when the maximum temperature over an area is falsely reported being a few degrees cooler because the denoising method is unaware of these features. Recently, a topological denoising technique based on a global energy optimization was proposed, which allows the topology-controlled denoising of 2D scalar fields. While this method preserves the minima and maxima, it is constrained by the size of the data. We extend this work to large 2D data and medium-sized 3D data by introducing a novel domain decomposition approach. It allows processing small patches of the domain independently while still avoiding the introduction of new critical points. Furthermore, we propose an iterative refinement of the solution, which decreases the optimization energy compared to the previous approach and therefore gives smoother results that are closer to the input. We illustrate our technique on synthetic and real-world 2D and 3D data sets that highlight potential applications.
AU - Günther, David
AU - Jacobson, Alec
AU - Reininghaus, Jan
AU - Seidel, Hans
AU - Sorkine Hornung, Olga
AU - Weinkauf, Tino
ID - 1930
IS - 12
JF - IEEE Transactions on Visualization and Computer Graphics
TI - Fast and memory-efficient topological denoising of 2D and 3D scalar fields
VL - 20
ER -
TY - CONF
AB - The classical sphere packing problem asks for the best (infinite) arrangement of non-overlapping unit balls which cover as much space as possible. We define a generalized version of the problem, where we allow each ball a limited amount of overlap with other balls. We study two natural choices of overlap measures and obtain the optimal lattice packings in a parameterized family of lattices which contains the FCC, BCC, and integer lattice.
AU - Iglesias Ham, Mabel
AU - Kerber, Michael
AU - Uhler, Caroline
ID - 2012
TI - Sphere packing with limited overlap
ER -
TY - CONF
AB - Persistent homology is a popular and powerful tool for capturing topological features of data. Advances in algorithms for computing persistent homology have reduced the computation time drastically – as long as the algorithm does not exhaust the available memory. Following up on a recently presented parallel method for persistence computation on shared memory systems [1], we demonstrate that a simple adaption of the standard reduction algorithm leads to a variant for distributed systems. Our algorithmic design ensures that the data is distributed over the nodes without redundancy; this permits the computation of much larger instances than on a single machine. Moreover, we observe that the parallelism at least compensates for the overhead caused by communication between nodes, and often even speeds up the computation compared to sequential and even parallel shared memory algorithms. In our experiments, we were able to compute the persistent homology of filtrations with more than a billion (109) elements within seconds on a cluster with 32 nodes using less than 6GB of memory per node.
AU - Bauer, Ulrich
AU - Kerber, Michael
AU - Reininghaus, Jan
ED - McGeoch, Catherine
ED - Meyer, Ulrich
ID - 2043
T2 - Proceedings of the Workshop on Algorithm Engineering and Experiments
TI - Distributed computation of persistent homology
ER -
TY - CHAP
AB - We present a parallel algorithm for computing the persistent homology of a filtered chain complex. Our approach differs from the commonly used reduction algorithm by first computing persistence pairs within local chunks, then simplifying the unpaired columns, and finally applying standard reduction on the simplified matrix. The approach generalizes a technique by Günther et al., which uses discrete Morse Theory to compute persistence; we derive the same worst-case complexity bound in a more general context. The algorithm employs several practical optimization techniques, which are of independent interest. Our sequential implementation of the algorithm is competitive with state-of-the-art methods, and we further improve the performance through parallel computation.
AU - Bauer, Ulrich
AU - Kerber, Michael
AU - Reininghaus, Jan
ED - Bremer, Peer-Timo
ED - Hotz, Ingrid
ED - Pascucci, Valerio
ED - Peikert, Ronald
ID - 2044
T2 - Topological Methods in Data Analysis and Visualization III
TI - Clear and Compress: Computing Persistent Homology in Chunks
ER -
TY - CONF
AB - We define a simple, explicit map sending a morphism f : M → N of pointwise finite dimensional persistence modules to a matching between the barcodes of M and N. Our main result is that, in a precise sense, the quality of this matching is tightly controlled by the lengths of the longest intervals in the barcodes of ker f and coker f . As an immediate corollary, we obtain a new proof of the algebraic stability theorem for persistence barcodes [5, 9], a fundamental result in the theory of persistent homology. In contrast to previous proofs, ours shows explicitly how a δ-interleaving morphism between two persistence modules induces a δ-matching between the barcodes of the two modules. Our main result also specializes to a structure theorem for submodules and quotients of persistence modules. Copyright is held by the owner/author(s).
AU - Bauer, Ulrich
AU - Lesnick, Michael
ID - 2153
T2 - Proceedings of the Annual Symposium on Computational Geometry
TI - Induced matchings of barcodes and the algebraic stability of persistence
ER -
TY - CONF
AB - Given a finite set of points in Rn and a positive radius, we study the Čech, Delaunay-Čech, alpha, and wrap complexes as instances of a generalized discrete Morse theory. We prove that the latter three complexes are simple-homotopy equivalent. Our results have applications in topological data analysis and in the reconstruction of shapes from sampled data. Copyright is held by the owner/author(s).
AU - Bauer, Ulrich
AU - Edelsbrunner, Herbert
ID - 2155
T2 - Proceedings of the Annual Symposium on Computational Geometry
TI - The morse theory of Čech and Delaunay filtrations
ER -
TY - CONF
AB - We propose a metric for Reeb graphs, called the functional distortion distance. Under this distance, the Reeb graph is stable against small changes of input functions. At the same time, it remains discriminative at differentiating input functions. In particular, the main result is that the functional distortion distance between two Reeb graphs is bounded from below by the bottleneck distance between both the ordinary and extended persistence diagrams for appropriate dimensions. As an application of our results, we analyze a natural simplification scheme for Reeb graphs, and show that persistent features in Reeb graph remains persistent under simplification. Understanding the stability of important features of the Reeb graph under simplification is an interesting problem on its own right, and critical to the practical usage of Reeb graphs. Copyright is held by the owner/author(s).
AU - Bauer, Ulrich
AU - Ge, Xiaoyin
AU - Wang, Yusu
ID - 2156
T2 - Proceedings of the Annual Symposium on Computational Geometry
TI - Measuring distance between Reeb graphs
ER -
TY - CONF
AB - We give evidence for the difficulty of computing Betti numbers of simplicial complexes over a finite field. We do this by reducing the rank computation for sparse matrices with to non-zero entries to computing Betti numbers of simplicial complexes consisting of at most a constant times to simplices. Together with the known reduction in the other direction, this implies that the two problems have the same computational complexity.
AU - Edelsbrunner, Herbert
AU - Parsa, Salman
ID - 2177
T2 - Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms
TI - On the computational complexity of betti numbers reductions from matrix rank
ER -