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Register a new userStability of Curvature Measures
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Added by
Frederic Chazal
Publication date
2008
fields
Informatics Engineering
and research's language is
English
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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 $mu$-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 $mu$-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.
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For $pin (1,2]$ and a bounded, convex, nonempty, open set $Omegasubsetmathbb R^2$ let $mu_p(bar{Omega},cdot)$ be the $p$-capacitary curvature measure (generated by the closure $bar{Omega}$ of $Omega$) on the unit circle $mathbb S^1$. This paper shows that such a problem of prescribing $mu_p$ on a planar convex domain: Given a finite, nonnegative, Borel measure $mu$ on $mathbb S^1$, find a bounded, convex, nonempty, open set $Omegasubsetmathbb R^2$ such that $dmu_p(bar{Omega},cdot)=dmu(cdot)$ is solvable if and only if $mu$ has centroid at the origin and its support $mathrm{supp}(mu)$ does not comprise any pair of antipodal points. And, the solution is unique up to translation. Moreover, if $dmu_p(bar{Omega},cdot)=psi(cdot),dell(cdot)$ with $psiin C^{k,alpha}$ and $dell$ being the standard arc-length element on $mathbb S^1$, then $partialOmega$ is of $C^{k+2,alpha}$.
We consider the geodesic X-ray transform acting on solenoidal tensor fields on a compact simply connected manifold with strictly convex boundary and non-positive curvature. We establish a stability estimate of the form $L^2mapsto H^{1/2}_{T}$, where the $H^{1/2}_{T}$-space is defined using the natural parametrization of geodesics as initial boundary points and incoming directions (fan-beam geometry); only tangential derivatives at the boundary are used. The proof is based on the Pestov identity with boundary term localized in frequency.
In topological data analysis, persistent homology is used to study the shape of data. Persistent homology computations are completely characterized by a set of intervals called a bar code. It is often said that the long intervals represent the topological signal and the short intervals represent noise. We give evidence to dispute this thesis, showing that the short intervals encode geometric information. Specifically, we prove that persistent homology detects the curvature of disks from which points have been sampled. We describe a general computational framework for solving inverse problems using the average persistence landscape, a continuous mapping from metric spaces with a probability measure to a Hilbert space. In the present application, the average persistence landscapes of points sampled from disks of constant curvature results in a path in this Hilbert space which may be learned using standard tools from statistical and machine learning.
We introduce a parametrized notion of genericity for Delaunay triangulations which, in particular, implies that the Delaunay simplices of $delta$-generic point sets are thick. Equipped with this notion, we study the stability of Delaunay triangulations under perturbations of the metric and of the vertex positions. We quantify the magnitude of the perturbations under which the Delaunay triangulation remains unchanged.
We describe a general family of curved-crease folding tessellations consisting of a repeating lens motif formed by two convex curved arcs. The third author invented the first such design in 1992, when he made both a sketch of the crease pattern and a vinyl model (pictured below). Curve fitting suggests that this initial design used circular arcs. We show that in fact the curve can be chosen to be any smooth convex curve without inflection point. We identify the ruling configuration through qualitative properties that a curved folding satisfies, and prove that the folded form exists with no additional creases, through the use of differential geometry.
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