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 - 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 - 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 - 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 - The fact that a sum of isotropic Gaussian kernels can have more modes than kernels is surprising. Extra (ghost) modes do not exist in ℝ1 and are generally not well studied in higher dimensions. We study a configuration of n+1 Gaussian kernels for which there are exactly n+2 modes. We show that all modes lie on a finite set of lines, which we call axes, and study the restriction of the Gaussian mixture to these axes in order to discover that there are an exponential number of critical points in this configuration. Although the existence of ghost modes remained unknown due to the difficulty of finding examples in ℝ2, we show that the resilience of ghost modes grows like the square root of the dimension. In addition, we exhibit finite configurations of isotropic Gaussian kernels with superlinearly many modes.
AU - Edelsbrunner, Herbert
AU - Fasy, Brittany Terese
AU - Rote, Günter
ID - 2815
IS - 4
JF - Discrete & Computational Geometry
TI - Add isotropic Gaussian kernels at own risk: More and more resilient modes in higher dimensions
VL - 49
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 -
TY - CONF
AB - We introduce the M-modes problem for graphical models: predicting the M label configurations of highest probability that are at the same time local maxima of the probability landscape. M-modes have multiple possible applications: because they are intrinsically diverse, they provide a principled alternative to non-maximum suppression techniques for structured prediction, they can act as codebook vectors for quantizing the configuration space, or they can form component centers for mixture model approximation. We present two algorithms for solving the M-modes problem. The first algorithm solves the problem in polynomial time when the underlying graphical model is a simple chain. The second algorithm solves the problem for junction chains. In synthetic and real dataset, we demonstrate how M-modes can improve the performance of prediction. We also use the generated modes as a tool to understand the topography of the probability distribution of configurations, for example with relation to the training set size and amount of noise in the data.
AU - Chen, Chao
AU - Kolmogorov, Vladimir
AU - Yan, Zhu
AU - Metaxas, Dimitris
AU - Lampert, Christoph
ID - 2901
TI - Computing the M most probable modes of a graphical model
VL - 31
ER -
TY - CONF
AB - Motivated by an application in cell biology, we describe an extension of the kinetic data structures framework from Delaunay triangulations to fixed-radius alpha complexes. Our algorithm is implemented
using CGAL, following the exact geometric computation paradigm. We report on several
techniques to accelerate the computation that turn our implementation applicable to the underlying biological
problem.
AU - Kerber, Michael
AU - Edelsbrunner, Herbert
ID - 2906
T2 - 2013 Proceedings of the 15th Workshop on Algorithm Engineering and Experiments
TI - 3D kinetic alpha complexes and their implementation
ER -
TY - JOUR
AB - In this paper, we present the first output-sensitive algorithm to compute the persistence diagram of a filtered simplicial complex. For any Γ > 0, it returns only those homology classes with persistence at least Γ. Instead of the classical reduction via column operations, our algorithm performs rank computations on submatrices of the boundary matrix. For an arbitrary constant δ ∈ (0, 1), the running time is O (C (1 - δ) Γ R d (n) log n), where C (1 - δ) Γ is the number of homology classes with persistence at least (1 - δ) Γ, n is the total number of simplices in the complex, d its dimension, and R d (n) is the complexity of computing the rank of an n × n matrix with O (d n) nonzero entries. Depending on the choice of the rank algorithm, this yields a deterministic O (C (1 - δ) Γ n 2.376) algorithm, an O (C (1 - δ) Γ n 2.28) Las-Vegas algorithm, or an O (C (1 - δ) Γ n 2 + ε{lunate}) Monte-Carlo algorithm for an arbitrary ε{lunate} > 0. The space complexity of the Monte-Carlo version is bounded by O (d n) = O (n log n).
AU - Chen, Chao
AU - Kerber, Michael
ID - 2939
IS - 4
JF - Computational Geometry: Theory and Applications
TI - An output sensitive algorithm for persistent homology
VL - 46
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
AU - Edelsbrunner, Herbert
AU - Strelkova, Nataliya
ID - 2849
IS - 6
JF - Russian Mathematical Surveys
TI - On the configuration space of Steiner minimal trees
VL - 67
ER -
TY - JOUR
AB - We present an algorithm for simplifying linear cartographic objects and results obtained with a computer program implementing this algorithm.
AU - Edelsbrunner, Herbert
AU - Musin, Oleg
AU - Ukhalov, Alexey
AU - Yakimova, Olga
AU - Alexeev, Vladislav
AU - Bogaevskaya, Victoriya
AU - Gorohov, Andrey
AU - Preobrazhenskaya, Margarita
ID - 2902
IS - 6
JF - Modeling and Analysis of Information Systems
TI - Fractal and computational geometry for generalizing cartographic objects
VL - 19
ER -
TY - CONF
AB - In order to enjoy a digital version of the Jordan Curve Theorem, it is common to use the closed topology for the foreground and the open topology for the background of a 2-dimensional binary image. In this paper, we introduce a single topology that enjoys this theorem for all thresholds decomposing a real-valued image into foreground and background. This topology is easy to construct and it generalizes to n-dimensional images.
AU - Edelsbrunner, Herbert
AU - Symonova, Olga
ID - 2903
TI - The adaptive topology of a digital image
ER -
TY - JOUR
AB - Generalized van der Corput sequences are onedimensional, infinite sequences in the unit interval. They are generated from permutations in integer base b and are the building blocks of the multi-dimensional Halton sequences. Motivated by recent progress of Atanassov on the uniform distribution behavior of Halton sequences, we study, among others, permutations of the form P(i) = ai (mod b) for coprime integers a and b. We show that multipliers a that either divide b - 1 or b + 1 generate van der Corput sequences with weak distribution properties. We give explicit lower bounds for the asymptotic distribution behavior of these sequences and relate them to sequences generated from the identity permutation in smaller bases, which are, due to Faure, the weakest distributed generalized van der Corput sequences.
AU - Pausinger, Florian
ID - 2904
IS - 3
JF - Journal de Theorie des Nombres des Bordeaux
SN - 2118-8572
TI - Weak multipliers for generalized van der Corput sequences
VL - 24
ER -