🍩 Database of Original & NonTheoretical Uses of Topology
(found 6 matches in 0.001074s)


The Euler Characteristic: A General Topological Descriptor for Complex Data (2021)
Alexander Smith, Victor ZavalaAbstract
Datasets are mathematical objects (e.g., point clouds, matrices, graphs, images, fields/functions) that have shape. This shape encodes important knowledge about the system under study. Topology is an area of mathematics that provides diverse tools to characterize the shape of data objects. In this work, we study a specific tool known as the Euler characteristic (EC). The EC is a general, lowdimensional, and interpretable descriptor of topological spaces defined by data objects. We revise the mathematical foundations of the EC and highlight its connections with statistics, linear algebra, field theory, and graph theory. We discuss advantages offered by the use of the EC in the characterization of complex datasets; to do so, we illustrate its use in different applications of interest in chemical engineering such as process monitoring, flow cytometry, and microscopy. We show that the EC provides a descriptor that effectively reduces complex datasets and that this reduction facilitates tasks such as visualization, regression, classification, and clustering. 
Topological Data Analysis Quantifies Biological NanoStructure From Single Molecule Localization Microscopy (2020)
Jeremy A. Pike, Abdullah O. Khan, Chiara Pallini, Steven G. Thomas, Markus Mund, Jonas Ries, Natalie S. Poulter, Iain B. StylesAbstract
AbstractMotivation. Localization microscopy data is represented by a set of spatial coordinates, each corresponding to a single detection, that form a point cl 
Persistent Homology to Quantify the Quality of SurfaceSupported Covalent Networks (2019)
Abraham Gutierrez, Mickaël Buchet, Sylvain ClairAbstract
Covalent networks formed by onsurface synthesis usually suffer from the presence of a large number of defects. We report on a methodology to characterize such twodimensional networks from their experimental images obtained by scanning probe microscopy. The computation is based on a persistent homology approach and provides a quantitative score indicative of the network homogeneity. We compare our scoring method with results previously obtained using minimal spanning tree analyses and we apply it to some molecular systems appearing in the existing literature. 
Using Persistent Homology as a New Approach for SuperResolution Localization Microscopy Data Analysis and Classification of γH2AX Foci/Clusters (2018)
Andreas Hofmann, Matthias Krufczik, Dieter W. Heermann, Michael HausmannAbstract
DNA double strand breaks (DSB) are the most severe damages in chromatin induced by ionizing radiation. In response to such environmentally determined stress situations, cells have developed repair mechanisms. Although many investigations have contributed to a detailed understanding of repair processes, e.g., homologous recombination repair or nonhomologous endjoining, the question is not sufficiently answered, how a cell decides to apply a certain repair process at a certain damage site, since all different repair pathways could simultaneously occur in the same cell nucleus. One of the first processes after DSB induction is phosphorylation of the histone variant H2AX to γH2AX in the given surroundings of the damaged locus. Since the spatial organization of chromatin is not random, it may be conclusive that the spatial organization of γH2AX foci is also not random, and rather, contributes to accessibility of special repair proteins to the damaged site, and thus, to the following repair pathway at this given site. The aim of this article is to demonstrate a new approach to analyze repair foci by their topology in order to obtain a cell independent method of categorization. During the last decade, novel superresolution fluorescence light microscopic techniques have enabled new insights into genome structure and spatial organization on the nanoscale in the order of 10 nm. One of these techniques is single molecule localization microscopy (SMLM) with which the spatial coordinates of single fluorescence molecules can precisely be determined and density and distance distributions can be calculated. This method is an appropriate tool to quantify complex changes of chromatin and to describe repair foci on the single molecule level. Based on the pointillist information obtained by SMLM from specifically labeled heterochromatin and γH2AX foci reflecting the chromatin morphology and repair foci topology, we have developed a new analytical methodology of foci or foci cluster characterization, respectively, by means of persistence homology. This method allows, for the first time, a cell independent comparison of two point distributions (here the point distributions of two γH2AX clusters) with each other of a selected ensample and to give a mathematical measure of their similarity. In order to demonstrate the feasibility of this approach, cells were irradiated by low LET (linear energy transfer) radiation with different doses and the heterochromatin and γH2AX foci were fluorescently labeled by antibodies for SMLM. By means of our new analysis method, we were able to show that the topology of clusters of γH2AX foci can be categorized depending on the distance to heterochromatin. This method opens up new possibilities to categorize spatial organization of point patterns by parameterization of topological similarity. 
Persistent Topology for CryoEm Data Analysis (2015)
Kelin Xia, GuoWei WeiAbstract
SummaryIn this work, we introduce persistent homology for the analysis of cryoelectron microscopy (cryoEM) density maps. We identify the topological fingerprint or topological signature of noise, which is widespread in cryoEM data. For low signaltonoise ratio (SNR) volumetric data, intrinsic topological features of biomolecular structures are indistinguishable from noise. To remove noise, we employ geometric flows that are found to preserve the intrinsic topological fingerprints of cryoEM structures and diminish the topological signature of noise. In particular, persistent homology enables us to visualize the gradual separation of the topological fingerprints of cryoEM structures from those of noise during the denoising process, which gives rise to a practical procedure for prescribing a noise threshold to extract cryoEM structure information from noise contaminated data after certain iterations of the geometric flow equation. To further demonstrate the utility of persistent homology for cryoEM data analysis, we consider a microtubule intermediate structure Electron Microscopy Data (EMD 1129). Three helix models, an alphatubulin monomer model, an alphatubulin and betatubulin model, and an alphatubulin and betatubulin dimer model, are constructed to fit the cryoEM data. The least square fitting leads to similarly high correlation coefficients, which indicates that structure determination via optimization is an illposed inverse problem. However, these models have dramatically different topological fingerprints. Especially, linkages or connectivities that discriminate one model from another, play little role in the traditional density fitting or optimization but are very sensitive and crucial to topological fingerprints. The intrinsic topological features of the microtubule data are identified after topological denoising. By a comparison of the topological fingerprints of the original data and those of three models, we found that the third model is topologically favored. The present work offers persistent homology based new strategies for topological denoising and for resolving illposed inverse problems. Copyright © 2015 John Wiley & Sons, Ltd.