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High precision source characterization of intermediate mass-ratio black hole coalescences with gravitational waves: The importance of higher order multipoles

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Physical Review D 2021年第8期104卷 084068-084068页

作者： Tousif Islam Scott E. Field Carl-Johan Haster Rory Smith Department of Physics University of Massachusetts Dartmouth Massachusetts 02747 USA Department of Mathematics University of Massachusetts Dartmouth Massachusetts 02747 USA Center for Scientific Computing and Visualization Research University of Massachusetts Dartmouth Massachusetts 02747 USA Kavli Institute for Theoretical Physics University of California Santa Barbara California 93106 USA LIGO Laboratory Massachusetts Institute of Technology 185 Albany Street Cambridge Massachusetts 02139 USA Department of Physics and Kavli Institute for Astrophysics and Space Research Massachusetts Institute of Technology 77 Massachusetts Avenue Cambridge Massachusetts 02139 USA School of Physics and Astronomy Monash University Melbourne VIC 3800 Australia OzGrav: The ARC Centre of Excellence for Gravitational Wave Discovery Clayton VIC 3800 Australia

Intermediate mass-ratio inspiral (IMRI) binaries—containing stellar-mass black holes coalescing into intermediate-mass black holes (M>100 M⊙)—are a highly anticipated source of gravitational waves (GWs) for Advanced LIGO/Virgo. Their detection and source characterization would provide a unique probe of strong-field gravity and stellar evolution. Because of the asymmetric component masses and the large primary, these systems generically excite subdominant modes while reducing the importance of the dominant quadrupole mode. Including higher order harmonics can also result in a 10%–25% increase in signal-to-noise ratio for IMRIs, which may help to detect these systems. We show that by including subdominant gravitational-wave modes into the analysis we can achieve a precise characterization of IMRI source properties. For example, we find that the source properties for IMRIs can be measured to within 2%–15% accuracy at a fiducial signal-to-noise ratio of 25 if subdominant modes are included. When subdominant modes are neglected, the accuracy degrades to 9%–44% and significant biases are seen in chirp mass, mass ratio, primary spin, and luminosity distances. We further demonstrate that including subdominant modes in the waveform model can enable an informative measurement of both individual spin components and improve the source localization by a factor of ∼10. We discuss some important astrophysical implications of high precision source characterization enabled by subdominant modes such as constraining the mass gap and probing formation channels.

关键词： Gravitation Gravitational waves

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High-precision source characterization of intermediate mass-ratio black hole coalescences with gravitational waves: The importance of higher-order multipoles

arXiv

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arXiv 2021年

作者： Islam, Tousif Field, Scott E. Haster, Carl-Johan Smith, Rory Department of Physics University of Massachusetts DartmouthMA02747 United States Department of Mathematics University of Massachusetts DartmouthMA02747 United States Center for Scientific Computing and Visualization Research University of Massachusetts DartmouthMA02747 United States Kavli Institute for Theoretical Physics University of California Santa BarbaraCA93106 United States LIGO Laboratory Massachusetts Institute of Technology 185 Albany St CambridgeMA02139 United States Department of Physics Kavli Institute for Astrophysics and Space Research Massachusetts Institute of Technology 77 Massachusetts Ave CambridgeMA02139 United States School of Physics and Astronomy Monash University Vic3800 Australia OzGrav: The ARC Centre of Excellence for Gravitational Wave Discovery ClaytonVIC3800 Australia

Intermediate mass ratio inspiral (IMRI) binaries—containing stellar-mass black holes coalescing into intermediate-mass black holes (M > 100Mo )—are a highly anticipated source of gravitational waves (GWs) for Advanced LIGO/Virgo. Their detection and source characterization would provide a unique probe of strong-field gravity and stellar evolution. Due to the asymmetric component masses and the large primary, these systems generically excite subdominant modes while reducing the importance of the dominant quadrupole mode. Including higher order harmonics can also result in a 10% − 25% increase in signal-to-noise ratio for IMRIs, which may help to detect these systems. We show that by including subdominant GW modes into the analysis we can achieve a precise characterization of IMRI source properties. For example, we find that the source properties for IMRIs can be measured to within 2% − 15% accuracy at a fiducial signal-to-noise ratio of 25 if subdominant modes are included. When subdominant modes are neglected, the accuracy degrades to 9% − 44% and significant biases are seen in chirp mass, mass ratio, primary spin and luminosity distances. We further demonstrate that including subdominant modes in the waveform model can enable an informative measurement of both individual spin components and improve the source localization by a factor of ∼10. We discuss some important astrophysical implications of high-precision source characterization enabled by subdominant modes such as constraining the mass gap and probing formation channels. Copyright © 2021, The Authors. All rights reserved.

关键词： Signal to noise ratio

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Bayesian Inference for Gravitational Waves from Binary Neutron Star Mergers in Third Generation Observatories

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Physical Review Letters 2021年第8期127卷 081102-081102页

作者： Rory Smith Ssohrab Borhanian Bangalore Sathyaprakash Francisco Hernandez Vivanco Scott E. Field Paul Lasky Ilya Mandel Soichiro Morisaki David Ottaway Bram J. J. Slagmolen Eric Thrane Daniel Töyrä Salvatore Vitale School of Physics and Astronomy Monash University Victoria 3800 Australia OzGrav: The ARC Centre of Excellence for Gravitational Wave Discovery Clayton Victoria 3800 Australia Institute for Gravitation and the Cosmos Department of Physics Pennsylvania State University University Park Pennsylvania 16802 USA Department of Astronomy and Astrophysics Pennsylvania State University University Park Pennsylvania 16802 USA School of Physics and Astronomy Cardiff University Cardiff CF24 3AA United Kingdom Department of Mathematics and Center for Scientific Computing and Visualization Research University of Massachusetts Dartmouth Massachusetts 02747 USA Department of Physics University of Wisconsin-Milwaukee Milwaukee Wisconsin 53201 USA OzGrav University of Adelaide Adelaide South Australia 5005 Australia OzGrav ANU Centre for Gravitational Astrophysics Research Schools of Physics and Astronomy and Astrophysics The Australian National University Australian Capital Territory 2601 Australia LIGO Laboratory Massachusetts Institute of Technology Cambridge Massachusetts 02139 USA Department of Physics and Kavli Institute for Astrophysics and Space Research Massachusetts Institute of Technology 77 Massachusetts Avenue Cambridge Massachusetts 02139 USA

Third generation (3G) gravitational-wave detectors will observe thousands of coalescing neutron star binaries with unprecedented fidelity. Extracting the highest precision science from these signals is expected to be challenging owing to both high signal-to-noise ratios and long-duration signals. We demonstrate that current Bayesian inference paradigms can be extended to the analysis of binary neutron star signals without breaking the computational bank. We construct reduced-order models for ∼90-min-long gravitational-wave signals covering the observing band (5–2048 Hz), speeding up inference by a factor of ∼1.3×104 compared to the calculation times without reduced-order models. The reduced-order models incorporate key physics including the effects of tidal deformability, amplitude modulation due to Earth’s rotation, and spin-induced orbital precession. We show how reduced-order modeling can accelerate inference on data containing multiple overlapping gravitational-wave signals, and determine the speedup as a function of the number of overlapping signals. Thus, we conclude that Bayesian inference is computationally tractable for the long-lived, overlapping, high signal-to-noise-ratio events present in 3G observatories.

关键词： Experimental studies of gravity Gravitational waves

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Bayesian inference for gravitational waves from binary neutron star mergers in third-generation observatories

arXiv

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arXiv 2021年

作者： Smith, Rory Borhanian, Ssohrab Sathyaprakash, Bangalore Vivanco, FranciscoHernandez Field, Scott E. Lasky, Paul Mandel, Ilya Morisaki, Soichiro Ottaway, David Slagmolen, Bram Thrane, Eric Töyrä, Daniel Vitale, Salvatore School of Physics and Astronomy Monash University VIC3800 Australia OzGrav The ARC Centre of Excellence for Gravitational Wave Discovery ClaytonVIC3800 Australia Institute for Gravitation and the Cosmos Department of Physics Pennsylvania State University University ParkPA16802 United States Department of Astronomy and Astrophysics Penn State University University ParkPA16802 United States School of Physics and Astronomy Cardiff University CardiffCF24 3AA United Kingdom Department of Mathematics Center for Scientific Computing and Visualization Research University of Massachusetts DartmouthMA02747 United States Department of Physics University of Wisconsin-Milwaukee MilwaukeeWI53201 United States OzGrav University of Adelaide AdelaideSA5005 Australia OzGrav ANU Centre for Gravitational Astrophysics Research Schools of Physics and Astronomy and Astrophysics The Australian National University ACT2601 Australia LIGO Laboratory Massachusetts Institute of Technology CambridgeMA02139 United States Department of Physics Kavli Institute for Astrophysics and Space Research Massachusetts Institute of Technology 77 Massachusetts Ave CambridgeMA02139 United States

Third-generation (3G) gravitational-wave detectors will observe thousands of coalescing neutron star binaries with unprecedented fidelity. Extracting the highest precision science from these signals is expected to be challenging owing to both high signal-to-noise ratios and long-duration signals. We demonstrate that current Bayesian inference paradigms can be extended to the analysis of binary neutron star signals without breaking the computational bank. We construct reduced order models for ∼ 90 minute long gravitational-wave signals, covering the observing band (5−2048 Hz), speeding up inference by a factor of ∼ 1.3 × 104 compared to the calculation times without reduced order models. The reduced order models incorporate key physics including the effects of tidal deformability, amplitude modulation due to the Earth's rotation, and spin-induced orbital precession. We show how reduced order modeling can accelerate inference on data containing multiple, overlapping gravitational-wave signals, and determine the speedup as a function of the number of overlapping signals. Thus, we conclude that Bayesian inference is computationally tractable for the long-lived, overlapping, high signal-to-noise-ratio events present in 3G observatories. Copyright © 2021, The Authors. All rights reserved.

关键词： Gravitational effects

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Visual computing for archaeological artifacts with integral invariant filters in 3D

Visual computing for archaeological artifacts with integral ...

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2017 Eurographics Workshop on Graphics and Cultural Heritage, GCH 2017

作者： Mara, H. Krmker, S. Heidelberg University IWR Interdisciplinary Center for Scientific Computing Germany FCGL Forensic Computational Geometry Laboratory VNGG - Visualization and Numerical Geometry Group Mathematikon Neuenheimer Feld 205 Heidelberg69120 Germany

ISBN: (纸本)9783038680376

3D-artifacts from ancient civilizations contain many different kinds of information in form of forensic trace evidence, e.g., tool marks from styli or fingerprints on wax sealings. These very fine structures are increasingly captured by various 3D-acquisiton techniques and stored as irregular meshes. We introduce filter algorithms for the processing of these datasets to finally extract meaningful information at predefined scales. Therefore, Multiscale-Integral Invariants (MSII) are introduced as robust filter methods with their four different variants, using volume, patch, surface and line integrals for their specific sensitivity on mean curvature, Gaussian curvature or noise detection. Smoothing as known from 2D-raster image processing cannot be applied directly. It needs adaptation to the irregular structure of the triangular grids describing 2D-manifolds in *** introduce a fast 1-ring smoothing with a skillful weighting by distance and area of the neighboring points and triangles. Finally, we apply our technique to the various motivating examples for showing the results as false color images with isolines, indicating the respective field of function values, e.g., curvature in various norms or correlations in the feature space. Smooth isolines are indicators for the successful removal of noise. We finally compare the fully automated results with a manual graphic rendering of a faded handwriting found in the tomb of the empress Gisela of Swabia. © 2017 GCH 2017 - Eurographics Workshop on Graphics and Cultural Heritage. All rights reserved.

关键词： Image processing

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An environment for sustainable research software in Germany and beyond: Current state, open challenges, and call for action

arXiv

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arXiv 2020年

作者： Anzt, Hartwig Bach, Felix Druskat, Stephan Löffler, Frank Loewe, Axel Renard, Bernhard Y. Seemann, Gunnar Struck, Alexander Achhammer, Elke Aggarwal, Piush Appel, Franziska Bader, Michael Brusch, Lutz Busse, Christian Chourdakis, Gerasimos Dabrowski, Piotr W. Ebert, Peter Flemisch, Bernd Friedl, Sven Fritzsch, Bernadette Funk, Maximilian D. Gast, Volker Goth, Florian Grad, Jean-Noël Hermann, Sibylle Hohmann, Florian Janosch, Stephan Kutra, Dominik Linxweiler, Jan Muth, Thilo Peters-Kottig, Wolfgang Rack, Fabian Raters, Fabian H.C. Rave, Stephan Reina, Guido Reißig, Malte Ropinski, Timo Schaarschmidt, Joerg Seibold, Heidi Thiele, Jan P. Uekerman, Benjamin Unger, Stefan Weeber, Rudolf Germany Innovative Computing Lab University of Tennessee KnoxvilleTN United States Department of English Studies Friedrich Schiller University Jena Germany Germany Department of Computer Science Humboldt-Universität zu Berlin Germany Heinz Nixdorf Chair for Distributed Information Systems Friedrich Schiller University Jena Germany Michael Stifel Center Jena Germany Center for Computation and Technology Louisiana State University Baton Rouge United States Germany Robert Koch Institute Berlin Germany Hasso Plattner Institute Faculty of Digital Engineering University of Potsdam Germany Institute for Experimental Cardiovascular Medicine University Heart Centre Freiburg Bad Krozingen Germany Faculty of Medicine University of Freiburg Freiburg Germany Matters of Activity. Image Space Material. Cluster of Excellence at Humboldt-Universität zu Berlin Germany Technische Universität München Germany Language Technology Lab Universität Duisburg-Essen Germany Germany Department of Informatics Technical University of Munich Germany Technische Universität Dresden Germany Deutsches Krebsforschungszentrum Heidelberg Germany Scientific Computing in Computer Science Technical University Munich Germany School of Computing Communication and Business Hochschule für Technik und Wirtschaft Berlin Germany Center for Bioinformatics Saar Saarland Informatics Campus Germany Institute for Modelling Hydraulic and Environmental Systems University of Stuttgart Germany Berlin Institute of Health Germany Scientific Computing Alfred Wegener Institute Helmholtz Center for Polar and Marine Research Bremerhaven Germany Max-Planck-Gesellschaft e.V. Institut für Theoretische Physik und Astrophysik Universität Würzburg Germany Institut für Computerphysik University of Stuttgart Germany University Library University of Stuttgart Germany Universität Bremen Germany Max Planck Institute of Molecular Cell Biology and Genetics Dresden Germany European Molecular Biology Laboratory Heidelb

Research software has become a central asset in academic research. It optimizes existing and enables new research methods, implements and embeds research knowledge, and constitutes an essential research product in itself. Research software must be sustainable in order to understand, replicate, reproduce, and build upon existing research or conduct new research effectively. In other words, software must be available, discoverable, usable, and adaptable to new needs, both now and in the future. Research software therefore requires an environment that supports sustainability. Hence, a change is needed in the way research software development and maintenance are currently motivated, incentivized, funded, structurally and infrastructurally supported, and legally treated. Failing to do so will threaten the quality and validity of research. In this paper, we identify challenges for research software sustainability in Germany and beyond, in terms of motivation, selection, research software engineering personnel, funding, infrastructure, and legal aspects. Besides researchers, we specifically address political and academic decision-makers to increase awareness of the importance and needs of sustainable research software practices. In particular, we recommend strategies and measures to create an environment for sustainable research software, with the ultimate goal to ensure that software-driven research is valid, reproducible and sustainable, and that software is recognized as a first class citizen in research. This paper is the outcome of two workshops run in Germany in 2019, at deRSE19 - the first International Conference of Research Software Engineers in Germany - and a dedicated DFG-supported follow-up workshop in Berlin. Copyright © 2020, The Authors. All rights reserved.

关键词： Software design

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A large-scale ENIGMA multisite replication study of brain age in depression

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Neuroimage: Reports 2022年第4期2卷 100149-100149页

作者： Han, Laura K.M. Dinga, Richard Leenings, Ramona Hahn, Tim Cole, James H. Aftanas, Lyubomir I. Amod, Alyssa R. Besteher, Bianca Colle, Romain Corruble, Emmanuelle Couvy-Duchesne, Baptiste Danilenko, Konstantin V. Fuentes-Claramonte, Paola Gonul, Ali Saffet Gotlib, Ian H. Goya-Maldonado, Roberto Groenewold, Nynke A. Hamilton, Paul Ichikawa, Naho Ipser, Jonathan C. Itai, Eri Koopowitz, Sheri-Michelle Li, Meng Okada, Go Okamoto, Yasumasa Churikova, Olga S. Osipov, Evgeny A. Penninx, Brenda W.J.H. Pomarol-Clotet, Edith Rodríguez-Cano, Elena Sacchet, Matthew D. Shinzato, Hotaka Sim, Kang Stein, Dan J. Uyar-Demir, Aslihan Veltman, Dick J. Schmaal, Lianne Centre for Youth Mental Health The University of Melbourne Parkville VIC Australia Orygen Parkville VIC Australia Department of Cognitive Science & Artificial Intelligence Tilburg University Tilburg Netherlands Institute for Translational Psychiatry University of Münster Münster Germany Faculty of Mathematics and Computer Science University of Münster Münster Germany Centre for Medical Image Computing Department of Computer Science University College London United Kingdom Dementia Research Centre Queen Square Institute of Neurology University College London United Kingdom Scientific Research Institute of Neuroscience and Medicine Novosibirsk Russian Federation Novosibirsk State University Novosibirsk Russian Federation Department of Psychiatry & Mental Health Neuroscience Institute University of Cape Town South Africa Department of Psychiatry and Psychotherapy Jena University Hospital Jena Germany Service Hospitalo-Universitaire de Psychiatrie de Bicêtre Mood Center Paris Saclay Assistance Publique-Hôpitaux de Paris Hôpitaux Universitaires Paris-Saclay Hôpital de Bicêtre Le Kremlin Bicêtre F-94275 France MOODS Team INSERM 1018 CESP (Centre de Recherche en Epidémiologie et Santé des Populations) Université Paris-Saclay Faculté de Médecine Paris-Saclay Le Kremlin Bicêtre F-94275 France Sorbonne Université Institut du Cerveau - Paris Brain Institute - ICM CNRS Inria Inserm AP-HP Hôpital de La Pitié Salpêtrière Paris F-75013 France Institute for Molecular Bioscience The University of Queensland St Lucia 4072 QLD Australia FIDMAG Hermanas Hospitalarias Research Foundation Barcelona Spain CIBERSAM Barcelona Spain SoCAT Lab Department of Psychiatry School of Medicine Ege University Izmir Turkey Department of Psychology Stanford University Stanford CA United States Laboratory of Systems Neuroscience and Imaging in Psychiatry (SNIP-Lab) University Medical Center Göttingen Göttingen Germany Center for Social and Affective Neuroscience Department o

Background: Several studies have evaluated whether depressed persons have older appearing brains than their nondepressed peers. However, the estimated neuroimaging-derived “brain age gap” has varied from study to study, likely driven by differences in training and testing sample (size), age range, and used modality/features. To validate our previously developed ENIGMA brain age model and the identified brain age gap, we aim to replicate the presence and effect size estimate previously found in the largest study in depression to date (N = 2126 controls & N = 2675 cases;+1.08 years [SE 0.22], Cohen's d = 0.14, 95% CI: 0.08–0.20), in independent cohorts that were not part of the original study. Methods: A previously trained brain age model (***/enigma_brainage) based on 77 FreeSurfer brain regions of interest was used to obtain unbiased brain age predictions in 751 controls and 766 persons with depression (18–75 years) from 13 new cohorts collected from 20 different scanners. Meta-regressions were used to examine potential moderating effects of basic cohort characteristics (e.g., clinical and scan technical) on the brain age gap. Results: Our ENIGMA MDD brain age model generalized reasonably well to controls from the new cohorts (predicted age vs. age: r = 0.73, R² = 0.47, MAE = 7.50 years), although the performance varied from cohort to cohort. In these new cohorts, on average, depressed persons showed a significantly higher brain age gap of +1 year (SE 0.35) (Cohen's d = 0.15, 95% CI: 0.05–0.25) compared with controls, highly similar to our previous finding. Significant moderating effects of FreeSurfer version 6.0 (d = 0.41, p = 0.007) and Philips scanner vendor (d = 0.50, p < 0.0001) were found, leading to more positive effect size estimates. Conclusions: This study further validates our previously developed ENIGMA brain age algorithm. Importantly, we replicated the brain age gap in depression with a comparable effect size. Thus, two large-s

关键词： Biological aging Brain age Depression ENIGMA consortium Replication study

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Low-cost automatic inpainting for artifact suppression in facial images

Low-cost automatic inpainting for artifact suppression in fa...

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8th International Conference on Computer Vision Theory and Applications, VISAPP 2013

作者： Sobiecki, André Telea, Alexandru Giraldi, Gilson Neves, Luiz Antonio Pereira Thomaz, Carlos Eduardo Scientific Visualization and Computer Graphics University of Groningen Nijenborgh 9 Groningen Netherlands Department of Computing National Laboratory of Scientific Computation Petŕopolis Brazil Department of Computing Parańa Federal University Curitiba Brazil Department of Artificial Intelligence Applied on Automation University Center of FEI Sao Bernardo do Campo Brazil

ISBN: (纸本)9789898565471

Facial images are often used in applications that need to recognize or identify persons. Many existing facial recognition tools have limitations with respect to facial image quality attributes such as resolution, face position, and artifacts present in the image. In this paper we describe a new low-cost framework for preprocessing low-quality facial images in order to render them suitable for automatic recognition. For this, we first detect artifacts based on the statistical difference between the target image and a set of pre-processed images in the database. Next, we eliminate artifacts by an inpainting method which combines information from the target image and similar images in our database. Our method has low computational cost and is simple to implement, which makes it attractive for usage in low-budget environments. We illustrate our method on several images taken from public surveillance databases, and compare our results with existing inpainting techniques.

关键词： Database systems

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A Maya use case: Adaptable scientific workflows with ADIOS for general relativistic astrophysics 13

A Maya use case: Adaptable scientific workflows with ADIOS f...

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Conference on Extreme Science and Engineering Discovery Environment, XSEDE 2013

作者： Slawinska, Magdalena Bode, Tanja Logan, Jeremy Clark, Michael Zou, Hongbo Kinsey, Matthew Wolf, Matthew Laguna, Pablo Klasky, Scott Georgia Institute of Technology College of Computing Atlanta GA United States Georgia Institute of Technology School of Physics Atlanta GA United States Remote Data Analysis and Visualization Center University of Tennessee Oak Ridge TN United States Oak Ridge National Laboratory Scientific Data Group Oak Ridge TN United States

ISBN: (纸本)9781450321709

There are many challenges in analyzing and visualizing data from current cutting-edge general relativistic astrophysics simulations. Many of the associated tasks are time-consuming with large performance degradation due to the magnitude and complexity of the data. The Adaptable I/O System (ADIOS) is a componentization of the I/O layer that has demonstrated remarkable I/O performance improvements on applications running on leadership class machines while also offering new in-memory "staging" operations for transforming data in-situ. We have been incorporating ADIOS staging technologies into our Maya numerical relativity code based on Cactus infrastructure and Carpet mesh refinement. Incorporating ADIOS into the Maya code is the first step toward enabling a more adaptable Maya workflow. We provide descriptions how we intend to leverage FlexPath (an ADIOS transport method) to provide a richer user experience in real-time visualization and interactive steering. © 2013 by the Association for computing Machinery, Inc.

关键词： visualization

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Topology exploration with hierarchical landscapes

Topology exploration with hierarchical landscapes

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Workshop at SIGGRAPH Asia 2012, WASA 2012

作者： Demir, Dogan Beketayev, Kenes Weber, Gunther H. Bremer, Peer-Timo Pascucci, Valerio Hamann, Bernd Scientific Computing and Imaging Institute School of Computing University of Utah United States Computational Research Divisition Lawrence Berkeley National Laboratory United States Center for Applied Scientific Computing Lawrence Livermore National Laboratory United States Institute for Data Analysis and Visualization Computer Science Department University of California Davis CA United States

ISBN: (纸本)9781450318358

Topological landscapes have been proposed as a visual metaphor for contour trees that does not require an understanding of the theory involved in defining contour trees. The idea is to create a representative terrain with the same topological structure as a given contour tree. This representation exploits the natural human ability to interpret topography and results in an intuitive visualization of otherwise abstract information. However, topological landscapes still suffer from some of the same limitations as traditional contour tree visualization. Most notable is the fact that for complex functions landscapes can quickly become highly complex, exceeding the limits of human comprehension as well as the available computing resources. To address this challenge, we propose a new framework for the dynamic creation and visualization of hierarchical topological landscapes. Our system provides an interactive visualization of complex functions by utilizing a hierarchical decomposition of the contour tree as well as focus+context type interactions. For three dimensional data, we link the landscape display to flexible isosurface extraction in order to correlate terrain features with their corresponding three dimensional counterparts. We demonstrate the utility and versatility of our approach on a variety of both low and high dimensional data sets. © 2012 ACM.

关键词： visualization

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