Multigrid-convergence of digital curvature estimators

Jacques-Olivier Lachaud

Displaying similar documents to “Multigrid-convergence of digital curvature estimators”

On the total curvature of surfaces in $E^{4}$

Peter Wintgen (1978)

Colloquium Mathematicae

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Curvature measures and fractals

Steffen Winter

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Curvature measures are an important tool in geometric measure theory and other fields of mathematics for describing the geometry of sets in Euclidean space. But the ’classical’ concepts of curvature are not directly applicable to fractal sets. We try to bridge this gap between geometric measure theory and fractal geometry by introducing a notion of curvature for fractals. For compact sets $F \subseteq ℝ^{d}$ (e.g. fractals), for which classical geometric characteristics such as curvatures or Euler characteristic...

On the motion of a curve by its binormal curvature

Jerrard, Robert L., Didier Smets (2015)

Journal of the European Mathematical Society

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We propose a weak formulation for the binormal curvature flow of curves in $ℝ^{3}$ . This formulation is sufficiently broad to consider integral currents as initial data, and sufficiently strong for the weak-strong uniqueness property to hold, as long as self-intersections do not occur. We also prove a global existence theorem in that framework.

Tangency properties of sets with finite geometric curvature energies

Sebastian Scholtes (2012)

Fundamenta Mathematicae

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We investigate tangential regularity properties of sets of fractal dimension, whose inverse thickness or integral Menger curvature energies are bounded. For the most prominent of these energies, the integral Menger curvature $ℳ_{p}^{α} (X) : = \int_{X} \int_{X} \int_{X} κ^{p} (x, y, z) d_{X}^{α} (x) d_{X}^{α} (y) d_{X}^{α} (z)$ , where κ(x,y,z) is the inverse circumradius of the triangle defined by x,y and z, we find that $ℳ_{p}^{α} (X) < \infty$ for p ≥ 3α implies the existence of a weak approximate α-tangent at every point of the set, if some mild density properties hold. This includes the scale invariant...

Curvature bounds for neighborhoods of self-similar sets

Steffen Winter (2011)

Commentationes Mathematicae Universitatis Carolinae

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In some recent work, fractal curvatures $C_{k}^{f} (F)$ and fractal curvature measures $C_{k}^{f} (F, \cdot)$ , $k = 0, ..., d$ , have been determined for all self-similar sets $F$ in $ℝ^{d}$ , for which the parallel neighborhoods satisfy a certain regularity condition and a certain rather technical curvature bound. The regularity condition is conjectured to be always satisfied, while the curvature bound has recently been shown to fail in some concrete examples. As a step towards a better understanding of its meaning, we discuss several equivalent...

Counterexample to the convexity of level sets of solutions to the mean curvature equation

Xu-Jia Wang (2014)

Journal of the European Mathematical Society

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The convexity of level sets of solutions to the mean curvature equation is a long standing open problem. In this paper we give a counterexample to it.

A pointwise inequality in submanifold theory

P. J. De Smet, F. Dillen, Leopold C. A. Verstraelen, L. Vrancken (1999)

Archivum Mathematicum

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We obtain a pointwise inequality valid for all submanifolds $M^{n}$ of all real space forms $N^{n + 2} (c)$ with $n \geq 2$ and with codimension two, relating its main scalar invariants, namely, its scalar curvature from the intrinsic geometry of $M^{n}$ , and its squared mean curvature and its scalar normal curvature from the extrinsic geometry of $M^{n}$ in $N^{m} (c)$ .

Curvature and Flow in Digital Space

Atsushi Imiya (2013)

Actes des rencontres du CIRM

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We first define the curvature indices of vertices of digital objects. Second, using these indices, we define the principal normal vectors of digital curves and surfaces. These definitions allow us to derive the Gauss-Bonnet theorem for digital objects. Third, we introduce curvature flow for isothetic polytopes defined in a digital space.

Existence, Consistency and computer simulation for selected variants of minimum distance estimators

Václav Kůs, Domingo Morales, Jitka Hrabáková, Iva Frýdlová (2018)

Kybernetika

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The paper deals with sufficient conditions for the existence of general approximate minimum distance estimator (AMDE) of a probability density function $f_{0}$ on the real line. It shows that the AMDE always exists when the bounded $φ$ -divergence, Kolmogorov, Lévy, Cramér, or discrepancy distance is used. Consequently, $n^{- 1 / 2}$ consistency rate in any bounded $φ$ -divergence is established for Kolmogorov, Lévy, and discrepancy estimators under the condition that the degree of variations of the corresponding...

Boundaries of prescribed mean curvature

Eduardo H. A. Gonzales, Umberto Massari, Italo Tamanini (1993)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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The existence of a singular curve in $R^{2}$ is proven, whose curvature can be extended to an $L^{2}$ function. The curve is the boundary of a two dimensional set, minimizing the length plus the integral over the set of the extension of the curvature. The existence of such a curve was conjectured by E. De Giorgi, during a conference held in Trento in July 1992.