Combinatorial optimization of cycles and bases

Erickson, Jeff

doi:10.1090/psapm/070/591

Cited by 13 publications

(11 citation statements)

References 108 publications

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“…The homology classes that correspond to each of these representative cycles are highlighted in Figure 10C . We remark that instead of the representative cycles produced by Dionysus one may want to use (approximate) shortest cycle representatives ( Jeff Erickson, 2012 ; Dey et al, 2018 ; Obayashi, 2018 ; Day et al, 2019 ).…”

Section: Resultsmentioning

confidence: 99%

Topological Data Analysis of C. elegans Locomotion and Behavior

Thomas¹,

Bates²,

Elchesen³

et al. 2021

Front. Artif. Intell.

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We apply topological data analysis to the behavior of C. elegans, a widely studied model organism in biology. In particular, we use topology to produce a quantitative summary of complex behavior which may be applied to high-throughput data. Our methods allow us to distinguish and classify videos from various environmental conditions and we analyze the trade-off between accuracy and interpretability. Furthermore, we present a novel technique for visualizing the outputs of our analysis in terms of the input. Specifically, we use representative cycles of persistent homology to produce synthetic videos of stereotypical behaviors.

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Section: Resultsmentioning

confidence: 99%

Topological Data Analysis of C. elegans Locomotion and Behavior

Thomas¹,

Bates²,

Elchesen³

et al. 2021

Front. Artif. Intell.

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“…[CSvdP10] show how to store and compute various geometrical properties for the contours of a contour tree. Erickson [Eri12] provides a detailed outline of multiple low‐dimensional localization techniques for simplicial chains. Dey et al .…”

Section: Related Workmentioning

confidence: 99%

Structural Analysis of Multivariate Point Clouds Using Simplicial Chains

Rieck

Leitte

2014

Computer Graphics Forum

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Topological and geometrical methods constitute common tools for the analysis of high-dimensional scientific data sets. Geometrical methods such as projection algorithms focus on preserving distances in the data set. Topological methods such as contour trees, by contrast, focus on preserving structural and connectivity information. By combining both types of methods, we want to benefit from their individual advantages. To this end, we describe an algorithm that uses persistent homology to analyse the topology of a data set. Persistent homology identifies high-dimensional holes in data sets, describing them as simplicial chains. We localize these chains using geometrical information of the data set, which we obtain from geodesic distances on a neighbourhood graph. The localized chains describe the structure of point clouds. We represent them using an interactive graph, in which each node describes a single chain and its geometrical properties. This graph yields a more intuitive understanding of multivariate point clouds and simplifies comparisons of time-varying data. Our method focuses on detecting and analysing inhomogeneous regions, i.e. holes, in a data set because these regions characterize data in a different manner, thereby leading to new insights. We demonstrate the potential of our method on data sets from particle physics, political science and meteorology.

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“…Over the last decades, several algorithms for detecting non trivial loops on general graphs have been proposed in computational geometry, see e.g. [Erickson 2012] and the references therein, but most of these results remain theoretical and have not turned into efficient implementations. The topic of extracting such topological features is not only relevant as a theoretical pursuit but has fundamental applications in geometry processing, covering tasks such as mesh parametrization, mesh repair, and feature recognition, and spills beyond to fields such as biotechnology and bioinformatics, see e.g.…”

Section: Introductionmentioning

confidence: 99%