TY - CONF
AB - Topological data analysis offers a rich source of valuable information to study vision problems. Yet, so far we lack a theoretically sound connection to popular kernel-based learning techniques, such as kernel SVMs or kernel PCA. In this work, we establish such a connection by designing a multi-scale kernel for persistence diagrams, a stable summary representation of topological features in data. We show that this kernel is positive definite and prove its stability with respect to the 1-Wasserstein distance. Experiments on two benchmark datasets for 3D shape classification/retrieval and texture recognition show considerable performance gains of the proposed method compared to an alternative approach that is based on the recently introduced persistence landscapes.
AU - Reininghaus, Jan
AU - Huber, Stefan
AU - Bauer, Ulrich
AU - Kwitt, Roland
ID - 1483
TI - A stable multi-scale kernel for topological machine learning
ER -
TY - CONF
AB - Motivated by biological questions, we study configurations of equal-sized disks in the Euclidean plane that neither pack nor cover. Measuring the quality by the probability that a random point lies in exactly one disk, we show that the regular hexagonal grid gives the maximum among lattice configurations.
AU - Edelsbrunner, Herbert
AU - Iglesias Ham, Mabel
AU - Kurlin, Vitaliy
ID - 1495
T2 - Proceedings of the 27th Canadian Conference on Computational Geometry
TI - Relaxed disk packing
VL - 2015-August
ER -
TY - CONF
AB - The concept of well group in a special but important case captures homological properties of the zero set of a continuous map f from K to R^n on a compact space K that are invariant with respect to perturbations of f. The perturbations are arbitrary continuous maps within L_infty distance r from f for a given r > 0. The main drawback of the approach is that the computability of well groups was shown only when dim K = n or n = 1. Our contribution to the theory of well groups is twofold: on the one hand we improve on the computability issue, but on the other hand we present a range of examples where the well groups are incomplete invariants, that is, fail to capture certain important robust properties of the zero set. For the first part, we identify a computable subgroup of the well group that is obtained by cap product with the pullback of the orientation of R^n by f. In other words, well groups can be algorithmically approximated from below. When f is smooth and dim K < 2n-2, our approximation of the (dim K-n)th well group is exact. For the second part, we find examples of maps f, f' from K to R^n with all well groups isomorphic but whose perturbations have different zero sets. We discuss on a possible replacement of the well groups of vector valued maps by an invariant of a better descriptive power and computability status.
AU - Franek, Peter
AU - Krcál, Marek
ID - 1510
TI - On computability and triviality of well groups
VL - 34
ER -
TY - JOUR
AB - The Heat Kernel Signature (HKS) is a scalar quantity which is derived from the heat kernel of a given shape. Due to its robustness, isometry invariance, and multiscale nature, it has been successfully applied in many geometric applications. From a more general point of view, the HKS can be considered as a descriptor of the metric of a Riemannian manifold. Given a symmetric positive definite tensor field we may interpret it as the metric of some Riemannian manifold and thereby apply the HKS to visualize and analyze the given tensor data. In this paper, we propose a generalization of this approach that enables the treatment of indefinite tensor fields, like the stress tensor, by interpreting them as a generator of a positive definite tensor field. To investigate the usefulness of this approach we consider the stress tensor from the two-point-load model example and from a mechanical work piece.
AU - Zobel, Valentin
AU - Jan Reininghaus
AU - Hotz, Ingrid
ID - 1531
JF - Mathematics and Visualization
TI - Visualizing symmetric indefinite 2D tensor fields using The Heat Kernel Signature
VL - 40
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 -
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 - 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 - 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 - 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 - 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 - 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 - 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 -
TY - JOUR
AU - Edelsbrunner, Herbert
AU - Strelkova, Nataliya
ID - 2912
IS - 6
JF - Uspekhi Mat. Nauk
TI - Configuration space for shortest networks
VL - 67
ER -
TY - JOUR
AU - Dolbilin, Nikolai
AU - Edelsbrunner, Herbert
AU - Musin, Oleg
ID - 2941
IS - 4
JF - Russian Mathematical Surveys
TI - On the optimality of functionals over triangulations of Delaunay sets
VL - 67
ER -
TY - CONF
AB - We study the task of interactive semantic labeling of a segmentation hierarchy. To this end we propose a framework interleaving two components: an automatic labeling step, based on a Conditional Random Field whose dependencies are defined by the inclusion tree of the segmentation hierarchy, and an interaction step that integrates incremental input from a human user. Evaluated on two distinct datasets, the proposed interactive approach efficiently integrates human interventions and illustrates the advantages of structured prediction in an interactive framework.
AU - Zankl, Georg
AU - Haxhimusa, Yll
AU - Ion, Adrian
ID - 2971
TI - Interactive labeling of image segmentation hierarchies
VL - 7476
ER -
TY - JOUR
AB - 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.
AU - Berberich, Eric
AU - Halperin, Dan
AU - Kerber, Michael
AU - Pogalnikova, Roza
ID - 3115
IS - 4
JF - Discrete & Computational Geometry
TI - Deconstructing approximate offsets
VL - 48
ER -
TY - JOUR
AB - We introduce a strategy based on Kustin-Miller unprojection that allows us to construct many hundreds of Gorenstein codimension 4 ideals with 9 × 16 resolutions (that is, nine equations and sixteen first syzygies). Our two basic games are called Tom and Jerry; the main application is the biregular construction of most of the anticanonically polarised Mori Fano 3-folds of Altinok's thesis. There are 115 cases whose numerical data (in effect, the Hilbert series) allow a Type I projection. In every case, at least one Tom and one Jerry construction works, providing at least two deformation families of quasismooth Fano 3-folds having the same numerics but different topology. © 2012 Copyright Foundation Compositio Mathematica.
AU - Brown, Gavin
AU - Kerber, Michael
AU - Reid, Miles
ID - 3120
IS - 4
JF - Compositio Mathematica
TI - Fano 3 folds in codimension 4 Tom and Jerry Part I
VL - 148
ER -
TY - CONF
AB - When searching for characteristic subpatterns in potentially noisy graph data, it appears self-evident that having multiple observations would be better than having just one. However, it turns out that the inconsistencies introduced when different graph instances have different edge sets pose a serious challenge. In this work we address this challenge for the problem of finding maximum weighted cliques.
We introduce the concept of most persistent soft-clique. This is subset of vertices, that 1) is almost fully or at least densely connected, 2) occurs in all or almost all graph instances, and 3) has the maximum weight. We present a measure of clique-ness, that essentially counts the number of edge missing to make a subset of vertices into a clique. With this measure, we show that the problem of finding the most persistent soft-clique problem can be cast either as: a) a max-min two person game optimization problem, or b) a min-min soft margin optimization problem. Both formulations lead to the same solution when using a partial Lagrangian method to solve the optimization problems. By experiments on synthetic data and on real social network data, we show that the proposed method is able to reliably find soft cliques in graph data, even if that is distorted by random noise or unreliable observations.
AU - Quadrianto, Novi
AU - Lampert, Christoph
AU - Chen, Chao
ID - 3127
T2 - Proceedings of the 29th International Conference on Machine Learning
TI - The most persistent soft-clique in a set of sampled graphs
ER -
TY - CONF
AB - Let K be a simplicial complex and g the rank of its p-th homology group Hp(K) defined with ℤ2 coefficients. We show that we can compute a basis H of Hp(K) and annotate each p-simplex of K with a binary vector of length g with the following property: the annotations, summed over all p-simplices in any p-cycle z, provide the coordinate vector of the homology class [z] in the basis H. The basis and the annotations for all simplices can be computed in O(n ω ) time, where n is the size of K and ω < 2.376 is a quantity so that two n×n matrices can be multiplied in O(n ω ) time. The precomputed annotations permit answering queries about the independence or the triviality of p-cycles efficiently.
Using annotations of edges in 2-complexes, we derive better algorithms for computing optimal basis and optimal homologous cycles in 1 - dimensional homology. Specifically, for computing an optimal basis of H1(K) , we improve the previously known time complexity from O(n 4) to O(n ω + n 2 g ω − 1). Here n denotes the size of the 2-skeleton of K and g the rank of H1(K) . Computing an optimal cycle homologous to a given 1-cycle is NP-hard even for surfaces and an algorithm taking 2 O(g) nlogn time is known for surfaces. We extend this algorithm to work with arbitrary 2-complexes in O(n ω ) + 2 O(g) n 2logn time using annotations.
AU - Busaryev, Oleksiy
AU - Cabello, Sergio
AU - Chen, Chao
AU - Dey, Tamal
AU - Wang, Yusu
ID - 3129
TI - Annotating simplices with a homology basis and its applications
VL - 7357
ER -
TY - CONF
AB - This note contributes to the point calculus of persistent homology by extending Alexander duality from spaces to real-valued functions. Given a perfect Morse function f: S n+1 →[0, 1 and a decomposition S n+1 = U ∪ V into two (n + 1)-manifolds with common boundary M, we prove elementary relationships between the persistence diagrams of f restricted to U, to V, and to M.
AU - Edelsbrunner, Herbert
AU - Kerber, Michael
ID - 3133
T2 - Proceedings of the twenty-eighth annual symposium on Computational geometry
TI - Alexander duality for functions: The persistent behavior of land and water and shore
ER -
TY - CONF
AB - It has been an open question whether the sum of finitely many isotropic Gaussian kernels in n ≥ 2 dimensions can have more modes than kernels, until in 2003 Carreira-Perpiñán and Williams exhibited n +1 isotropic Gaussian kernels in ℝ n with n + 2 modes. We give a detailed analysis of this example, showing that it has exponentially many critical points and that the resilience of the extra mode grows like √n. In addition, we exhibit finite configurations of isotropic Gaussian kernels with superlinearly many modes.
AU - Edelsbrunner, Herbert
AU - Fasy, Brittany
AU - Rote, Günter
ID - 3134
T2 - Proceedings of the twenty-eighth annual symposium on Computational geometry
TI - Add isotropic Gaussian kernels at own risk: More and more resilient modes in higher dimensions
ER -
TY - JOUR
AB - 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.
AU - Mileyko, Yuriy
AU - Edelsbrunner, Herbert
AU - Price, Charles
AU - Weitz, Joshua
ID - 3159
IS - 6
JF - PLoS One
TI - Hierarchical ordering of reticular networks
VL - 7
ER -
TY - JOUR
AB - First we note that the best polynomial approximation to vertical bar x vertical bar on the set, which consists of an interval on the positive half-axis and a point on the negative half-axis, can be given by means of the classical Chebyshev polynomials. Then we explore the cases when a solution of the related problem on two intervals can be given in elementary functions.
AU - Pausinger, Florian
ID - 6588
IS - 1
JF - Journal of Mathematical Physics, Analysis, Geometry
SN - 1812-9471
TI - Elementary solutions of the Bernstein problem on two intervals
VL - 8
ER -
TY - JOUR
AB - 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.
AU - Edelsbrunner, Herbert
AU - Kerber, Michael
ID - 3256
IS - 2
JF - Discrete & Computational Geometry
TI - Dual complexes of cubical subdivisions of ℝn
VL - 47
ER -
TY - CONF
AB - We propose a mid-level statistical model for image segmentation that composes multiple figure-ground hypotheses (FG) obtained by applying constraints at different locations and scales, into larger interpretations (tilings) of the entire image. Inference is cast as optimization over sets of maximal cliques sampled from a graph connecting all non-overlapping figure-ground segment hypotheses. Potential functions over cliques combine unary, Gestalt-based figure qualities, and pairwise compatibilities among spatially neighboring segments, constrained by T-junctions and the boundary interface statistics of real scenes. Learning the model parameters is based on maximum likelihood, alternating between sampling image tilings and optimizing their potential function parameters. State of the art results are reported on the Berkeley and Stanford segmentation datasets, as well as VOC2009, where a 28% improvement was achieved.
AU - Ion, Adrian
AU - Carreira, Joao
AU - Sminchisescu, Cristian
ID - 3265
TI - Image segmentation by figure-ground composition into maximal cliques
ER -