Search results for " Geometry"
showing 10 items of 2294 documents
Compactness of a conformal boundary of the Euclidean unit ball
2011
We study conformal metrics d‰ on the Euclidean unit ball B n : We assume that either the density ‰ associated with the metric d‰ satisfies a logarithmic volume growth condition for small balls or that ‰ satisfies a Harnack inequality and a suitable sub-Euclidean volume growth condition. We prove that the ‰-boundary @‰ B n is homeomorphic to S ni1 if and only if @‰ B n is compact. In the planar case, the compactness of @‰ B 2 is further equivalent to local connectivity of the ‰-boundary together with the boundedness of (B 2 ;d‰):
A comparison theorem for the mean exit time from a domain in a K�hler manifold
1992
Let M be a Kahler manifold with Ricci and antiholomorphic Ricci curvature bounded from below. Let ω be a domain in M with some bounds on the mean and JN-mean curvatures of its boundary ∂ω. The main result of this paper is a comparison theorem between the Mean Exit Time function defined on ω and the Mean Exit Time from a geodesic ball of the complex projective space ℂℙ n (λ) which involves a characterization of the geodesic balls among the domain ω. In order to achieve this, we prove a comparison theorem for the mean curvatures of hypersurfaces parallel to the boundary of ω, using the Index Lemma for Submanifolds.
Nonlinear Nonhomogeneous Robin Problems with Almost Critical and Partially Concave Reaction
2020
We consider a nonlinear Robin problem driven by a nonhomogeneous differential operator, with reaction which exhibits the competition of two Caratheodory terms. One is parametric, $$(p-1)$$-sublinear with a partially concave nonlinearity near zero. The other is $$(p-1)$$-superlinear and has almost critical growth. Exploiting the special geometry of the problem, we prove a bifurcation-type result, describing the changes in the set of positive solutions as the parameter $$\lambda >0$$ varies.
Approximated overlap error for the evaluation of feature descriptors on 3D scenes
2013
This paper presents a new framework to evaluate feature descriptors on 3D datasets. The proposed method employs the approximated overlap error in order to conform with the reference planar evaluation case of the Oxford dataset based on the overlap error. The method takes into account not only the keypoint centre but also the feature shape and it does not require complex data setups, depth maps or an accurate camera calibration. Only a ground-truth fundamental matrix should be computed, so that the dataset can be freely extended by adding further images. The proposed approach is robust to false positives occurring in the evaluation process, which do not introduce any relevant changes in the …
Generic attribute deviation metric for assessing mesh simplification algorithm quality
2002
International audience; This paper describes an efficient method to compare two triangular meshes. Meshes considered here contain geometric features as well as other surface attributes such as material colors, texture, temperature, radiation, etc. Two deviation measurements are presented to assess the differences between two meshes. The first measurement, called geometric deviation, returns geometric differences. The second measurement , called attribute deviation, returns attribute differences regardless of the attribute type. In this paper we present an application of this method to the Mesh Simplification Algorithm (MSA) quality assessment according to the appearance attributes. This ass…
Approximation of functions over manifolds : A Moving Least-Squares approach
2021
We present an algorithm for approximating a function defined over a $d$-dimensional manifold utilizing only noisy function values at locations sampled from the manifold with noise. To produce the approximation we do not require any knowledge regarding the manifold other than its dimension $d$. We use the Manifold Moving Least-Squares approach of (Sober and Levin 2016) to reconstruct the atlas of charts and the approximation is built on-top of those charts. The resulting approximant is shown to be a function defined over a neighborhood of a manifold, approximating the originally sampled manifold. In other words, given a new point, located near the manifold, the approximation can be evaluated…
Fully Automatic Trunk Packing with Free Placements
2010
We present a new algorithm to compute the volume of a trunk according to the SAE J1100 standard. Our new algorithm uses state-of-the-art methods from computational geometry and from combinatorial optimization. It finds better solutions than previous approaches for small trunks.
Topology-based goodness-of-fit tests for sliced spatial data
2023
In materials science and many other application domains, 3D information can often only be extrapolated by taking 2D slices. In topological data analysis, persistence vineyards have emerged as a powerful tool to take into account topological features stretching over several slices. In the present paper, we illustrate how persistence vineyards can be used to design rigorous statistical hypothesis tests for 3D microstructure models based on data from 2D slices. More precisely, by establishing the asymptotic normality of suitable longitudinal and cross-sectional summary statistics, we devise goodness-of-fit tests that become asymptotically exact in large sampling windows. We illustrate the test…
On the reducibility of geometric constraint graphs
2018
Geometric modeling by constraints, whose applications are of interest to communities from various fields such as mechanical engineering, computer aided design, symbolic computation or molecular chemistry, is now integrated into standard modeling tools. In this discipline, a geometric form is specified by the relations that the components of this form must verify instead of explicitly specifying these components. The purpose of the resolution is to deduce the form satisfying all these constraints. Various methods have been proposed to solve this problem. We will focus on the so-called graph-based or graph-based methods with application to the two-dimensional space.
Optimal rates of convergence for persistence diagrams in Topological Data Analysis
2013
Computational topology has recently known an important development toward data analysis, giving birth to the field of topological data analysis. Topological persistence, or persistent homology, appears as a fundamental tool in this field. In this paper, we study topological persistence in general metric spaces, with a statistical approach. We show that the use of persistent homology can be naturally considered in general statistical frameworks and persistence diagrams can be used as statistics with interesting convergence properties. Some numerical experiments are performed in various contexts to illustrate our results.