Showing papers by "Herbert Edelsbrunner published in 2013"

3D phenotyping and quantitative trait locus mapping identify core regions of the rice genome controlling root architecture

Abstract: 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 × 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.

Persistent Homology: Theory and Practice

Abstract: 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. 2010 Mathematics Subject Classification. Primary 55N99; Secondary 68W30.

Genotypic recognition and spatial responses by rice roots

Abstract: 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 to which 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.

Homology and robustness of level and interlevel sets

Abstract: Given a continuous function $f\colon \mathbb{X} \to \mathbb{R}$ 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 $\mathbb{X} = \mathbb{R}^3$ has ramifications in the fields of medical imaging and scientific visualization.

The Morse theory of \v{C}ech and Delaunay filtrations

Abstract: Given a finite set of points in $\mathbb R^n$ and a positive radius, we consider the \v{C}ech, Delaunay-\v{C}ech, alpha, and wrap complexes as examples of a generalized discrete Morse theory. We prove that the latter three are simple-homotopy equivalent, and the same is true for their weighted versions. Our results have applications in high-dimensional data analysis and in the reconstruction of shapes from sampled data.

The Morse theory of \v{C}ech and Delaunay complexes

Abstract: Given a finite set of points in $\mathbb R^n$ and a radius parameter, we study the Cech, Delaunay-Cech, Delaunay (or Alpha), and Wrap complexes in the light of generalized discrete Morse theory. Establishing the Cech and Delaunay complexes as sublevel sets of generalized discrete Morse functions, we prove that the four complexes are simple-homotopy equivalent by a sequence of simplicial collapses, which are explicitly described by a single discrete gradient field.

Add Isotropic Gaussian Kernels at Own Risk: More and More Resilient Modes in Higher Dimensions

Abstract: The fact that a sum of isotropic Gaussian kernels can have more modes than kernels is surprising. Extra (ghost) modes do not exist in $${{\mathbb{R }}}^1$$R1 and are generally not well studied in higher dimensions. We study a configuration of $$n+1$$n+1 Gaussian kernels for which there are exactly $$n+2$$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 $${{\mathbb{R }}}^2$$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.

Persistent Homology in Image Processing

Abstract: Taking images is an efficient way to collect data about the physical world. It can be done fast and in exquisite detail. By definition, image processing is the field that concerns itself with the computation aimed at harnessing the information contained in images [10]. This talk is concerned with topological information. Our main thesis is that persistent homology [5] is a useful method to quantify and summarize topological information, building a bridge that connects algebraic topology with applications. We provide supporting evidence for this thesis by touching upon four technical developments in the overlap between persistent homology and image processing.

3D kinetic alpha complexes and their implementation

Abstract: 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.

Stable length estimates of tube-like shapes

Abstract: 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.

Analysis of Dynamic Message Passing Programs (A framework for the analysis of depth-bounded systems )

Abstract: Motivated by the analysis of highly dynamic message-passing systems, i.e. unbounded thread creation, mobility, etc. We present a framework for the analysis of depth-bounded systems. Depth-bounded systems are one of the most expressive known fragment of the π-calculus for which interesting verification problems are still decidable. Even though they are infinite state systems depth-bounded systems are well-structured, thus can be analyzed algorithmically. We give an interpretation of depth-bounded systems as graph-rewriting systems. This gives more flexibility and ease of use to apply depth-bounded systems to other type of systems like shared memory concurrency. First, we develop an adequate domain of limits for depth-bounded systems, a prerequisite for the effective representation of downward-closed sets. Downwardclosed sets are needed by forward saturation-based algorithms to represent potentially infinite sets of states. Then, we present an abstract interpretation framework to compute the covering set of well-structured transition systems Because, in general, the covering set is not computable, our abstraction overapproximates the actual covering set. Our abstraction captures the essence of acceleration based-algorithms while giving up enough precision to ensure convergence. We have implemented the analysis in the Picasso tool and show that it is accurate in practice. Finally, we build some further analyses like termination using the covering set as starting point.

Alpha, Betti and the Megaparsec Universe: on the Topology of the Cosmic Web

Abstract: We study the topology of the Megaparsec Cosmic Web in terms of the scale-dependent Betti numbers, which formalize the topological information content of the cosmic mass distribution. While the Betti numbers do not fully quantify topology, they extend the information beyond conventional cosmological studies of topology in terms of genus and Euler characteristic. The richer information content of Betti numbers goes along the availability of fast algorithms to compute them. For continuous density fields, we determine the scale-dependence of Betti numbers by invoking the cosmologically familiar filtration of sublevel or superlevel sets defined by density thresholds. For the discrete galaxy distribution, however, the analysis is based on the alpha shapes of the particles. These simplicial complexes constitute an ordered sequence of nested subsets of the Delaunay tessellation, a filtration defined by the scale parameter, $\alpha$. As they are homotopy equivalent to the sublevel sets of the distance field, they are an excellent tool for assessing the topological structure of a discrete point distribution. In order to develop an intuitive understanding for the behavior of Betti numbers as a function of $\alpha$, and their relation to the morphological patterns in the Cosmic Web, we first study them within the context of simple heuristic Voronoi clustering models. Subsequently, we address the topology of structures emerging in the standard LCDM scenario and in cosmological scenarios with alternative dark energy content. The evolution and scale-dependence of the Betti numbers is shown to reflect the hierarchical evolution of the Cosmic Web and yields a promising measure of cosmological parameters. We also discuss the expected Betti numbers as a function of the density threshold for superlevel sets of a Gaussian random field.