886 research outputs found
Detection of Small Holes by the Scale-Invariant Robust Density-Aware Distance (RDAD) Filtration
A novel topological-data-analytical (TDA) method is proposed to distinguish,
from noise, small holes surrounded by high-density regions of a probability
density function whose mass is concentrated near a manifold (or more generally,
a CW complex) embedded in a high-dimensional Euclidean space. The proposed
method is robust against additive noise and outliers. In particular, sample
points are allowed to be perturbed away from the manifold. Traditional TDA
tools, like those based on the distance filtration, often struggle to
distinguish small features from noise, because of their short persistence. An
alternative filtration, called Robust Density-Aware Distance (RDAD) filtration,
is proposed to prolong the persistence of small holes surrounded by
high-density regions. This is achieved by weighting the distance function by
the density in the sense of Bell et al. Distance-to-measure is incorporated to
enhance stability and mitigate noise due to the density estimation. The utility
of the proposed filtration in identifying small holes, as well as its
robustness against noise, are illustrated through an analytical example and
extensive numerical experiments. Basic mathematical properties of the proposed
filtration are proven.Comment: 47 pages, 60 figures, GitHub repo: https://github.com/c-siu/RDA
Structure in the 3D Galaxy Distribution: I. Methods and Example Results
Three methods for detecting and characterizing structure in point data, such
as that generated by redshift surveys, are described: classification using
self-organizing maps, segmentation using Bayesian blocks, and density
estimation using adaptive kernels. The first two methods are new, and allow
detection and characterization of structures of arbitrary shape and at a wide
range of spatial scales. These methods should elucidate not only clusters, but
also the more distributed, wide-ranging filaments and sheets, and further allow
the possibility of detecting and characterizing an even broader class of
shapes. The methods are demonstrated and compared in application to three data
sets: a carefully selected volume-limited sample from the Sloan Digital Sky
Survey redshift data, a similarly selected sample from the Millennium
Simulation, and a set of points independently drawn from a uniform probability
distribution -- a so-called Poisson distribution. We demonstrate a few of the
many ways in which these methods elucidate large scale structure in the
distribution of galaxies in the nearby Universe.Comment: Re-posted after referee corrections along with partially re-written
introduction. 80 pages, 31 figures, ApJ in Press. For full sized figures
please download from: http://astrophysics.arc.nasa.gov/~mway/lss1.pd
The Topology of Wireless Communication
In this paper we study the topological properties of wireless communication
maps and their usability in algorithmic design. We consider the SINR model,
which compares the received power of a signal at a receiver against the sum of
strengths of other interfering signals plus background noise. To describe the
behavior of a multi-station network, we use the convenient representation of a
\emph{reception map}. In the SINR model, the resulting \emph{SINR diagram}
partitions the plane into reception zones, one per station, and the
complementary region of the plane where no station can be heard. We consider
the general case where transmission energies are arbitrary (or non-uniform).
Under that setting, the reception zones are not necessarily convex or even
connected. This poses the algorithmic challenge of designing efficient point
location techniques as well as the theoretical challenge of understanding the
geometry of SINR diagrams. We achieve several results in both directions. We
establish a form of weaker convexity in the case where stations are aligned on
a line. In addition, one of our key results concerns the behavior of a
-dimensional map. Specifically, although the -dimensional map might
be highly fractured, drawing the map in one dimension higher "heals" the zones,
which become connected. In addition, as a step toward establishing a weaker
form of convexity for the -dimensional map, we study the interference
function and show that it satisfies the maximum principle. Finally, we turn to
consider algorithmic applications, and propose a new variant of approximate
point location.Comment: 64 pages, appeared in STOC'1
Contours in Visualization
This thesis studies the visualization of set collections either via or defines as the relations among contours.
In the first part, dynamic Euler diagrams are used to communicate and improve semimanually the result of clustering methods which allow clusters to overlap arbitrarily. The contours of the Euler diagram are rendered as implicit surfaces called blobs in computer graphics. The interaction metaphor is the moving of items into or out of these blobs. The utility of the method is demonstrated on data arising from the analysis of gene expressions. The method works well for small datasets of up to one hundred items and few clusters.
In the second part, these limitations are mitigated employing a GPU-based rendering of Euler diagrams and mixing textures and colors to resolve overlapping regions better. The GPU-based approach subdivides the screen into triangles on which it performs a contour interpolation, i.e. a fragment shader determines for each pixel which zones of an Euler diagram it belongs to. The rendering speed is thus increased to allow multiple hundred items. The method is applied to an example comparing different document clustering results.
The contour tree compactly describes scalar field topology. From the viewpoint of graph drawing, it is a tree with attributes at vertices and optionally on edges. Standard tree drawing algorithms emphasize structural properties of the tree and neglect the attributes. Adapting popular graph drawing approaches to the problem of contour tree drawing it is found that they are unable to convey this information. Five aesthetic criteria for drawing contour trees are proposed and a novel algorithm for drawing contour trees in the plane that satisfies four of these criteria is presented. The implementation is fast and effective for contour tree sizes usually used in interactive systems and also produces readable pictures for larger trees.
Dynamical models that explain the formation of spatial structures of RNA molecules have reached a complexity that requires novel visualization methods to analyze these model\''s validity. The fourth part of the thesis focuses on the visualization of so-called folding landscapes of a growing RNA molecule. Folding landscapes describe the energy of a molecule as a function of its spatial configuration; they are huge and high dimensional. Their most salient features are described by their so-called barrier tree -- a contour tree for discrete observation spaces. The changing folding landscapes of a growing RNA chain are visualized as an animation of the corresponding barrier tree sequence. The animation is created as an adaption of the foresight layout with tolerance algorithm for dynamic graph layout. The adaptation requires changes to the concept of supergraph and it layout.
The thesis finishes with some thoughts on how these approaches can be combined and how the task the application should support can help inform the choice of visualization modality
Medial Axis Transform using Ridge Following
The intent of this investigation has been to find a robust algorithm for generation of the medial axis transform (MAT). The MAT is an invertible, object centered, shape representation defined as the collection of the centers of disks contained in the shape but not in any other such disk. Its uses include feature extraction, shape smoothing, and data compression. MAT generating algorithms include brushfire, Voronoi diagrams, and ridge following. An improved implementation of the ridge following algorithm is given. Orders of the MAT generating algorithms are compared. The effects of the number of edges in the polygonal approximation, shape area, number of holes, and number/distribution of concave vertices are shown from test results. Finally, a set of useful extensions to the ridge following algorithm are discussed
The influence of grain size distribution on strain hardening behavior for dual phase steels using statistica ly informed artificial microstructure model and crystal plasticity
Dual phase steels are well suited to the automotive application. Their microstructures comprise constituents of strong distinction in mechanical properties. As a result, dual phase steels exhibit remarkably high-energy absorption as well as an excellent combination of strength and ductility. Various deformation mechanisms can be observed on the microscale owing to their heterogeneous composition. A reliable microstructure-based simulation approach for describing these deformations is hence needed. Therefore, the approach to generate artificial dual phase microstructure models based on the quantitative results of metallographic microstructure analysis
and their statistical representation is developed. This method captures several microstructural features such as microstructure morphology and thus enables a simulation-based analysis of the influence of these features on the meso- and macroscopic material behavior. The algorithm input contains representative information about individual phase grain size and orientation distributions. The statistical parameters to represent the grain size distribution function are then input into a multiplicatively weighted Voronoi tessellation based algorithm to generate artificial microstructure geometry models that are applicable to bimodal distribution and with which microstructure deformation (finite element) simulations can be performed. By implementation of the phenomenological based crystal plasticity model to the generated artificial microstructure model, the influence of grain size distribution on the strain hardening behavior
can be investigated
The Multiscale Morphology Filter: Identifying and Extracting Spatial Patterns in the Galaxy Distribution
We present here a new method, MMF, for automatically segmenting cosmic
structure into its basic components: clusters, filaments, and walls.
Importantly, the segmentation is scale independent, so all structures are
identified without prejudice as to their size or shape. The method is ideally
suited for extracting catalogues of clusters, walls, and filaments from samples
of galaxies in redshift surveys or from particles in cosmological N-body
simulations: it makes no prior assumptions about the scale or shape of the
structures.}Comment: Replacement with higher resolution figures. 28 pages, 17 figures. For
Full Resolution Version see:
http://www.astro.rug.nl/~weygaert/tim1publication/miguelmmf.pd
The persistent cosmic web and its filamentary structure II: Illustrations
The recently introduced discrete persistent structure extractor (DisPerSE,
Soubie 2010, paper I) is implemented on realistic 3D cosmological simulations
and observed redshift catalogues (SDSS); it is found that DisPerSE traces
equally well the observed filaments, walls, and voids in both cases. In either
setting, filaments are shown to connect onto halos, outskirt walls, which
circumvent voids. Indeed this algorithm operates directly on the particles
without assuming anything about the distribution, and yields a natural
(topologically motivated) self-consistent criterion for selecting the
significance level of the identified structures. It is shown that this
extraction is possible even for very sparsely sampled point processes, as a
function of the persistence ratio. Hence astrophysicists should be in a
position to trace and measure precisely the filaments, walls and voids from
such samples and assess the confidence of the post-processed sets as a function
of this threshold, which can be expressed relative to the expected amplitude of
shot noise. In a cosmic framework, this criterion is comparable to friend of
friend for the identifications of peaks, while it also identifies the connected
filaments and walls, and quantitatively recovers the full set of topological
invariants (Betti numbers) {\sl directly from the particles} as a function of
the persistence threshold. This criterion is found to be sufficient even if one
particle out of two is noise, when the persistence ratio is set to 3-sigma or
more. The algorithm is also implemented on the SDSS catalogue and used to locat
interesting configurations of the filamentary structure. In this context we
carried the identification of an ``optically faint'' cluster at the
intersection of filaments through the recent observation of its X-ray
counterpart by SUZAKU. The corresponding filament catalogue will be made
available online.Comment: A higher resolution version is available at
http://www.iap.fr/users/sousbie together with complementary material (movie
and data). Submitted to MNRA
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