The Geometry of Nonparametric Filament Estimation

Genovese, Christopher R.; Perone-Pacifico, Marco; Verdinelli, Isabella; Wasserman, Larry

doi:10.1080/01621459.2012.682527

Cited by 39 publications

(30 citation statements)

References 43 publications

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“…From a more general perspective all these methods are attempting to find structure in multivariate data with geometric and topological ideas entering the definition of the methodology explicitly (cf. Genovese et al 2012a).…”

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confidence: 99%

Theoretical analysis of nonparametric filament estimation

Qiao¹,

Polonik²

2016

Ann. Statist.

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This paper provides a rigorous study of the nonparametric estimation of filaments or ridge lines of a probability density f . Points on the filament are considered as local extrema of the density when traversing the support of f along the integral curve driven by the vector field of second eigenvectors of the Hessian of f . We 'parametrize' points on the filaments by such integral curves, and thus both the estimation of integral curves and of filaments will be considered via a plug-in method using kernel density estimation. We establish rates of convergence and asymptotic distribution results for the estimation of both the integral curves and the filaments. The main theoretical result establishes the asymptotic distribution of the uniform deviation of the estimated filament from its theoretical counterpart. This result utilizes the extreme value behavior of non-stationary Gaussian processes indexed by manifolds M h , h ∈ (0, 1] as h → 0.

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confidence: 99%

Theoretical analysis of nonparametric filament estimation

Qiao¹,

Polonik²

2016

Ann. Statist.

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“…In statistics, several approaches have been proposed to address the problem of detection and extraction of filamentary structures in point cloud data. For example, Arial-Castro et al [4] use multiscale anisotropic strips to detect linear structure, while Genovese et al [25,27] and more recently [26] base their approach upon density gradient descents or medial axis techniques. These methods apply to data corrupted by outliers embedded in Euclidean spaces and focus on the inference of individual filaments without focussing on the global geometric structure of the filaments network.…”

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confidence: 99%

Gromov–Hausdorff Approximation of Filamentary Structures Using Reeb-Type Graphs

Chazal

Huang

Sun

2015

Discrete Comput Geom

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In many real-world applications, data appear to be sampled around 1-dimensional filamentary structures that can be seen as topological metric graphs. In this paper, we address the metric reconstruction problem of such filamentary structures from data sampled around them. We prove that they can be approximated with respect to the Gromov-Hausdorff distance by well-chosen Reeb graphs (and some of their variants) and provide an efficient and easy-to-implement algorithm to compute such approximations in almost linear time. We illustrate the performance of our algorithm on a few datasets.

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“…In statistics, several approaches have been proposed to address the problem of detection and extraction of filamentary structures in point cloud data. For example Arial-Castro et al [4] use multiscale anisotropic strips to detect linear structure while [25,27] and more recently [26] base their approach upon density gradient descents or medial axis techniques. These methods apply to data corrupted by outliers embedded in Euclidean spaces and focus on the inference of individual filaments without focus on the global geometric structure of the filaments network.…”

Section: Introductionmentioning

confidence: 99%

Gromov-Hausdorff Approximation of Filament Structure Using Reeb-type Graph

Chazal

Sun

2014

Proceedings of the Thirtieth Annual Symposium on Computational Geometry

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In many real-world applications data appear to be sampled around 1-dimensional filamentary structures that can be seen as topological metric graphs. In this paper we address the metric reconstruction problem of such filamentary structures from data sampled around them. We prove that they can be approximated, with respect to the Gromov-Hausdorff distance by well-chosen Reeb graphs (and some of their variants) and we provide an efficient and easy to implement algorithm to compute such approximations in almost linear time. We illustrate the performances of our algorithm on a few data sets.

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The Geometry of Nonparametric Filament Estimation

Cited by 39 publications

References 43 publications

Theoretical analysis of nonparametric filament estimation

Theoretical analysis of nonparametric filament estimation

Gromov–Hausdorff Approximation of Filamentary Structures Using Reeb-Type Graphs

Gromov-Hausdorff Approximation of Filament Structure Using Reeb-type Graph

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