63 research outputs found

    Normal Cone Approximation and Offset Shape Isotopy

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    This work adresses the problem of the approximation of the normals of the offsets of general compact sets in euclidean spaces. It is proven that for general sampling conditions, it is possible to approximate the gradient vector field of the distance to general compact sets. These conditions involve the ÎĽ\mu-reach of the compact set, a recently introduced notion of feature size. As a consequence, we provide a sampling condition that is sufficient to ensure the correctness up to isotopy of a reconstruction given by an offset of the sampling. We also provide a notion of normal cone to general compact sets which is stable under perturbation

    Stability of Curvature Measures

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    We address the problem of curvature estimation from sampled compact sets. The main contribution is a stability result: we show that the gaussian, mean or anisotropic curvature measures of the offset of a compact set K with positive ÎĽ\mu-reach can be estimated by the same curvature measures of the offset of a compact set K' close to K in the Hausdorff sense. We show how these curvature measures can be computed for finite unions of balls. The curvature measures of the offset of a compact set with positive ÎĽ\mu-reach can thus be approximated by the curvature measures of the offset of a point-cloud sample. These results can also be interpreted as a framework for an effective and robust notion of curvature

    Geometric Inference on Kernel Density Estimates

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    We show that geometric inference of a point cloud can be calculated by examining its kernel density estimate with a Gaussian kernel. This allows one to consider kernel density estimates, which are robust to spatial noise, subsampling, and approximate computation in comparison to raw point sets. This is achieved by examining the sublevel sets of the kernel distance, which isomorphically map to superlevel sets of the kernel density estimate. We prove new properties about the kernel distance, demonstrating stability results and allowing it to inherit reconstruction results from recent advances in distance-based topological reconstruction. Moreover, we provide an algorithm to estimate its topology using weighted Vietoris-Rips complexes.Comment: To appear in SoCG 2015. 36 pages, 5 figure

    Dehn paternity bounds and hyperbolicity tests

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    Thesis advisor: George R. MeyerhoffRecent advances in normal surface algorithms enable the determination by computer of the hyperbolicity of compact orientable 3-manifolds with zero Euler characteristic and nonempty boundary. Recent advances in hyperbolic geometry enable the determination by computer of the Dehn paternity relation between two orientable compact hyperbolic 3-manifolds. Presented here is an exposition of these developments, along with prototype implementations of one of these determinations in software. These have applications to two questions about Mom technology.Thesis (PhD) — Boston College, 2015.Submitted to: Boston College. Graduate School of Arts and Sciences.Discipline: Mathematics

    Properties of Gauss digitized sets and digital surface integration

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    International audienceThis paper presents new topological and geometrical properties of Gauss digitizations of Euclidean shapes, most of them holding in arbitrary dimension dd. We focus on rr-regular shapes sampled by Gauss digitization at gridstep hh. The digitized boundary is shown to be close to the Euclidean boundary in the Hausdorff sense, the minimum distance d2h\frac{\sqrt{d}}{2}h being achieved by the projection map ξ\xi induced by the Euclidean distance. Although it is known that Gauss digitized boundaries may not be manifold when d≥3d \ge 3, we show that non-manifoldness may only occur in places where the normal vector is almost aligned with some digitization axis, and the limit angle decreases with hh. We then have a closer look at the projection of the digitized boundary onto the continuous boundary by ξ\xi. We show that the size of its non-injective part tends to zero with hh. This leads us to study the classical digital surface integration scheme, which allocates a measure to each surface element that is proportional to the cosine of the angle between an estimated normal vector and the trivial surface element normal vector. We show that digital integration is convergent whenever the normal estimator is multigrid convergent, and we explicit the convergence speed. Since convergent estimators are now available in the litterature, digital integration provides a convergent measure for digitized objects

    Mathematical Imaging and Surface Processing

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    Within the last decade image and geometry processing have become increasingly rigorous with solid foundations in mathematics. Both areas are research fields at the intersection of different mathematical disciplines, ranging from geometry and calculus of variations to PDE analysis and numerical analysis. The workshop brought together scientists from all these areas and a fruitful interplay took place. There was a lively exchange of ideas between geometry and image processing applications areas, characterized in a number of ways in this workshop. For example, optimal transport, first applied in computer vision is now used to define a distance measure between 3d shapes, spectral analysis as a tool in image processing can be applied in surface classification and matching, and so on. We have also seen the use of Riemannian geometry as a powerful tool to improve the analysis of multivalued images. This volume collects the abstracts for all the presentations covering this wide spectrum of tools and application domains

    Shape Smoothing using Double Offsets

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    It has been observed for a long time that the operation consisting of offseting a solid by a quantity rr and then offseting its complement by $

    A Statistical Approach to Topological Data Analysis

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    Until very recently, topological data analysis and topological inference methods mostlyrelied on deterministic approaches. The major part of this habilitation thesis presents astatistical approach to such topological methods. We first develop model selection toolsfor selecting simplicial complexes in a given filtration. Next, we study the estimationof persistent homology on metric spaces. We also study a robust version of topologicaldata analysis. Related to this last topic, we also investigate the problem of Wassersteindeconvolution. The second part of the habilitation thesis gathers our contributions inother fields of statistics, including a model selection method for Gaussian mixtures, animplementation of the slope heuristic for calibrating penalties, and a study of Breiman’spermutation importance measure in the context of random forests
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