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 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 - TY - JOUR AB - Given topological spaces X,Y, a fundamental problem of algebraic topology is understanding the structure of all continuous maps X→ Y. We consider a computational version, where X,Y are given as finite simplicial complexes, and the goal is to compute [X,Y], that is, all homotopy classes of suchmaps.We solve this problem in the stable range, where for some d ≥ 2, we have dim X ≤ 2d-2 and Y is (d-1)-connected; in particular, Y can be the d-dimensional sphere Sd. The algorithm combines classical tools and ideas from homotopy theory (obstruction theory, Postnikov systems, and simplicial sets) with algorithmic tools from effective algebraic topology (locally effective simplicial sets and objects with effective homology). In contrast, [X,Y] is known to be uncomputable for general X,Y, since for X = S1 it includes a well known undecidable problem: testing triviality of the fundamental group of Y. In follow-up papers, the algorithm is shown to run in polynomial time for d fixed, and extended to other problems, such as the extension problem, where we are given a subspace A ⊂ X and a map A→ Y and ask whether it extends to a map X → Y, or computing the Z2-index-everything in the stable range. Outside the stable range, the extension problem is undecidable. AU - Čadek, Martin AU - Krcál, Marek AU - Matoušek, Jiří AU - Sergeraert, Francis AU - Vokřínek, Lukáš AU - Wagner, Uli ID - 2184 IS - 3 JF - Journal of the ACM TI - Computing all maps into a sphere VL - 61 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 - CONF AB - In this paper, we introduce planar matchings on directed pseudo-line arrangements, which yield a planar set of pseudo-line segments such that only matching-partners are adjacent. By translating the planar matching problem into a corresponding stable roommates problem we show that such matchings always exist. Using our new framework, we establish, for the first time, a complete, rigorous definition of weighted straight skeletons, which are based on a so-called wavefront propagation process. We present a generalized and unified approach to treat structural changes in the wavefront that focuses on the restoration of weak planarity by finding planar matchings. AU - Biedl, Therese AU - Huber, Stefan AU - Palfrader, Peter ID - 10892 SN - 0302-9743 T2 - 25th International Symposium, ISAAC 2014 TI - Planar matchings for weighted straight skeletons VL - 8889 ER - TY - BOOK AB - This monograph presents a short course in computational geometry and topology. In the first part the book covers Voronoi diagrams and Delaunay triangulations, then it presents the theory of alpha complexes which play a crucial role in biology. The central part of the book is the homology theory and their computation, including the theory of persistence which is indispensable for applications, e.g. shape reconstruction. The target audience comprises researchers and practitioners in mathematics, biology, neuroscience and computer science, but the book may also be beneficial to graduate students of these fields. AU - Edelsbrunner, Herbert ID - 6853 SN - 2191-530X TI - A Short Course in Computational Geometry and Topology ER - TY - CONF AB - We propose a method for visualizing two-dimensional symmetric positive definite tensor fields using the Heat Kernel Signature (HKS). The HKS is derived from the heat kernel and was originally introduced as an isometry invariant shape signature. Each positive definite tensor field defines a Riemannian manifold by considering the tensor field as a Riemannian metric. On this Riemmanian manifold we can apply the definition of the HKS. The resulting scalar quantity is used for the visualization of tensor fields. The HKS is closely related to the Gaussian curvature of the Riemannian manifold and the time parameter of the heat kernel allows a multiscale analysis in a natural way. In this way, the HKS represents field related scale space properties, enabling a level of detail analysis of tensor fields. This makes the HKS an interesting new scalar quantity for tensor fields, which differs significantly from usual tensor invariants like the trace or the determinant. A method for visualization and a numerical realization of the HKS for tensor fields is proposed in this chapter. To validate the approach we apply it to some illustrating simple examples as isolated critical points and to a medical diffusion tensor data set. AU - Zobel, Valentin AU - Reininghaus, Jan AU - Hotz, Ingrid ID - 10886 SN - 1612-3786 T2 - Topological Methods in Data Analysis and Visualization III TI - Visualization of two-dimensional symmetric positive definite tensor fields using the heat kernel signature ER - TY - CHAP AB - The Morse-Smale complex can be either explicitly or implicitly represented. Depending on the type of representation, the simplification of the Morse-Smale complex works differently. In the explicit representation, the Morse-Smale complex is directly simplified by explicitly reconnecting the critical points during the simplification. In the implicit representation, on the other hand, the Morse-Smale complex is given by a combinatorial gradient field. In this setting, the simplification changes the combinatorial flow, which yields an indirect simplification of the Morse-Smale complex. The topological complexity of the Morse-Smale complex is reduced in both representations. However, the simplifications generally yield different results. In this chapter, we emphasize properties of the two representations that cause these differences. We also provide a complexity analysis of the two schemes with respect to running time and memory consumption. AU - Günther, David AU - Reininghaus, Jan AU - Seidel, Hans-Peter AU - Weinkauf, Tino ED - Bremer, Peer-Timo ED - Hotz, Ingrid ED - Pascucci, Valerio ED - Peikert, Ronald ID - 10817 SN - 1612-3786 T2 - Topological Methods in Data Analysis and Visualization III. TI - Notes on the simplification of the Morse-Smale complex ER - TY - JOUR AB - Motivated by applications in biology, we present an algorithm for estimating the length of tube-like shapes in 3-dimensional Euclidean space. In a first step, we combine the tube formula of Weyl with integral geometric methods to obtain an integral representation of the length, which we approximate using a variant of the Koksma-Hlawka Theorem. In a second step, we use tools from computational topology to decrease the dependence on small perturbations of the shape. We present computational experiments that shed light on the stability and the convergence rate of our algorithm. AU - Edelsbrunner, Herbert AU - Pausinger, Florian ID - 2255 IS - 1 JF - Journal of Mathematical Imaging and Vision SN - 09249907 TI - Stable length estimates of tube-like shapes VL - 50 ER - TY - CONF AB - PHAT is a C++ library for the computation of persistent homology by matrix reduction. We aim for a simple generic design that decouples algorithms from data structures without sacrificing efficiency or user-friendliness. This makes PHAT a versatile platform for experimenting with algorithmic ideas and comparing them to state of the art implementations. AU - Bauer, Ulrich AU - Kerber, Michael AU - Reininghaus, Jan AU - Wagner, Hubert ID - 10894 SN - 0302-9743 T2 - ICMS 2014: International Congress on Mathematical Software TI - PHAT – Persistent Homology Algorithms Toolbox VL - 8592 ER - TY - GEN 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 T2 - arXiv TI - Sphere packing with limited overlap ER - TY - CONF AB - A straight skeleton is a well-known geometric structure, and several algorithms exist to construct the straight skeleton for a given polygon or planar straight-line graph. In this paper, we ask the reverse question: Given the straight skeleton (in form of a planar straight-line graph, with some rays to infinity), can we reconstruct a planar straight-line graph for which this was the straight skeleton? We show how to reduce this problem to the problem of finding a line that intersects a set of convex polygons. We can find these convex polygons and all such lines in $O(nlog n)$ time in the Real RAM computer model, where $n$ denotes the number of edges of the input graph. We also explain how our approach can be used for recognizing Voronoi diagrams of points, thereby completing a partial solution provided by Ash and Bolker in 1985. AU - Biedl, Therese AU - Held, Martin AU - Huber, Stefan ID - 2209 TI - Recognizing straight skeletons and Voronoi diagrams and reconstructing their input ER - TY - CONF AB - A straight skeleton is a well-known geometric structure, and several algorithms exist to construct the straight skeleton for a given polygon. In this paper, we ask the reverse question: Given the straight skeleton (in form of a tree with a drawing in the plane, but with the exact position of the leaves unspecified), can we reconstruct the polygon? We show that in most cases there exists at most one polygon; in the remaining case there is an infinite number of polygons determined by one angle that can range in an interval. We can find this (set of) polygon(s) in linear time in the Real RAM computer model. AU - Biedl, Therese AU - Held, Martin AU - Huber, Stefan ID - 2210 T2 - 29th European Workshop on Computational Geometry TI - Reconstructing polygons from embedded straight skeletons ER - TY - JOUR AB - This extended abstract is concerned with the irregularities of distribution of one-dimensional permuted van der Corput sequences that are generated from linear permutations. We show how to obtain upper bounds for the discrepancy and diaphony of these sequences, by relating them to Kronecker sequences and applying earlier results of Faure and Niederreiter. AU - Pausinger, Florian ID - 2304 JF - Electronic Notes in Discrete Mathematics TI - Van der Corput sequences and linear permutations VL - 43 ER - TY - CONF AB - We consider several basic problems of algebraic topology, with connections to combinatorial and geometric questions, from the point of view of computational complexity. The extension problem asks, given topological spaces X; Y , a subspace A ⊆ X, and a (continuous) map f : A → Y , whether f can be extended to a map X → Y . For computational purposes, we assume that X and Y are represented as finite simplicial complexes, A is a subcomplex of X, and f is given as a simplicial map. In this generality the problem is undecidable, as follows from Novikov's result from the 1950s on uncomputability of the fundamental group π1(Y ). We thus study the problem under the assumption that, for some k ≥ 2, Y is (k - 1)-connected; informally, this means that Y has \no holes up to dimension k-1" (a basic example of such a Y is the sphere Sk). We prove that, on the one hand, this problem is still undecidable for dimX = 2k. On the other hand, for every fixed k ≥ 2, we obtain an algorithm that solves the extension problem in polynomial time assuming Y (k - 1)-connected and dimX ≤ 2k - 1. For dimX ≤ 2k - 2, the algorithm also provides a classification of all extensions up to homotopy (continuous deformation). This relies on results of our SODA 2012 paper, and the main new ingredient is a machinery of objects with polynomial-time homology, which is a polynomial-time analog of objects with effective homology developed earlier by Sergeraert et al. We also consider the computation of the higher homotopy groups πk(Y ), k ≥ 2, for a 1-connected Y . Their computability was established by Brown in 1957; we show that πk(Y ) can be computed in polynomial time for every fixed k ≥ 2. On the other hand, Anick proved in 1989 that computing πk(Y ) is #P-hard if k is a part of input, where Y is a cell complex with certain rather compact encoding. We strengthen his result to #P-hardness for Y given as a simplicial complex. AU - Čadek, Martin AU - Krcál, Marek AU - Matoušek, Jiří AU - Vokřínek, Lukáš AU - Wagner, Uli ID - 2807 T2 - 45th Annual ACM Symposium on theory of computing TI - Extending continuous maps: Polynomiality and undecidability ER - TY - CONF AB - We consider the problem of deciding whether the persistent homology group of a simplicial pair (K, L) can be realized as the homology H* (X) of some complex X with L ⊂ X ⊂ K. We show that this problem is NP-complete even if K is embedded in ℝ3. As a consequence, we show that it is NP-hard to simplify level and sublevel sets of scalar functions on S3 within a given tolerance constraint. This problem has relevance to the visualization of medical images by isosurfaces. We also show an implication to the theory of well groups of scalar functions: not every well group can be realized by some level set, and deciding whether a well group can be realized is NP-hard. AU - Attali, Dominique AU - Bauer, Ulrich AU - Devillers, Olivier AU - Glisse, Marc AU - Lieutier, André ID - 2812 T2 - Proceedings of the 29th annual symposium on Computational Geometry TI - Homological reconstruction and simplification in R3 ER - TY - JOUR AB - Identification of genes that control root system architecture in crop plants requires innovations that enable high-throughput and accurate measurements of root system architecture through time. We demonstrate the ability of a semiautomated 3D in vivo imaging and digital phenotyping pipeline to interrogate the quantitative genetic basis of root system growth in a rice biparental mapping population, Bala x Azucena. We phenotyped >1,400 3D root models and >57,000 2D images for a suite of 25 traits that quantified the distribution, shape, extent of exploration, and the intrinsic size of root networks at days 12, 14, and 16 of growth in a gellan gum medium. From these data we identified 89 quantitative trait loci, some of which correspond to those found previously in soil-grown plants, and provide evidence for genetic tradeoffs in root growth allocations, such as between the extent and thoroughness of exploration. We also developed a multivariate method for generating and mapping central root architecture phenotypes and used it to identify five major quantitative trait loci (r2 = 24-37%), two of which were not identified by our univariate analysis. Our imaging and analytical platform provides a means to identify genes with high potential for improving root traits and agronomic qualities of crops. AU - Topp, Christopher AU - Iyer Pascuzzi, Anjali AU - Anderson, Jill AU - Lee, Cheng AU - Zurek, Paul AU - Symonova, Olga AU - Zheng, Ying AU - Bucksch, Alexander AU - Mileyko, Yuriy AU - Galkovskyi, Taras AU - Moore, Brad AU - Harer, John AU - Edelsbrunner, Herbert AU - Mitchell Olds, Thomas AU - Weitz, Joshua AU - Benfey, Philip ID - 2822 IS - 18 JF - PNAS TI - 3D phenotyping and quantitative trait locus mapping identify core regions of the rice genome controlling root architecture VL - 110 ER - TY - CONF AB - Mathematical objects can be measured unambiguously, but not so objects from our physical world. Even the total length of tubelike shapes has its difficulties. We introduce a combination of geometric, probabilistic, and topological methods to design a stable length estimate for tube-like shapes; that is: one that is insensitive to small shape changes. AU - Edelsbrunner, Herbert AU - Pausinger, Florian ID - 2843 T2 - 17th IAPR International Conference on Discrete Geometry for Computer Imagery TI - Stable length estimates of tube-like shapes VL - 7749 ER - 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 - Root system growth and development is highly plastic and is influenced by the surrounding environment. Roots frequently grow in heterogeneous environments that include interactions from neighboring plants and physical impediments in the rhizosphere. To investigate how planting density and physical objects affect root system growth, we grew rice in a transparent gel system in close proximity with another plant or a physical object. Root systems were imaged and reconstructed in three dimensions. Root-root interaction strength was calculated using quantitative metrics that characterize the extent towhich the reconstructed root systems overlap each other. Surprisingly, we found the overlap of root systems of the same genotype was significantly higher than that of root systems of different genotypes. Root systems of the same genotype tended to grow toward each other but those of different genotypes appeared to avoid each other. Shoot separation experiments excluded the possibility of aerial interactions, suggesting root communication. Staggered plantings indicated that interactions likely occur at root tips in close proximity. Recognition of obstacles also occurred through root tips, but through physical contact in a size-dependent manner. These results indicate that root systems use two different forms of communication to recognize objects and alter root architecture: root-root recognition, possibly mediated through root exudates, and root-object recognition mediated by physical contact at the root tips. This finding suggests that root tips act as local sensors that integrate rhizosphere information into global root architectural changes. AU - Fang, Suqin AU - Clark, Randy AU - Zheng, Ying AU - Iyer Pascuzzi, Anjali AU - Weitz, Joshua AU - Kochian, Leon AU - Edelsbrunner, Herbert AU - Liao, Hong AU - Benfey, Philip ID - 2887 IS - 7 JF - PNAS TI - Genotypic recognition and spatial responses by rice roots VL - 110 ER -