Related papers: A varifold approach to surface approximation
In computer vision and medical imaging, the problem of matching structures finds numerous applications from automatic annotation to data reconstruction. The data however, while corresponding to the same anatomy, are often very different in…
We introduce variational approximations for curve evolutions in two-dimensional Riemannian manifolds that are conformally flat, i.e.\ conformally equivalent to the Euclidean space. Examples include the hyperbolic plane, the hyperbolic disk,…
This paper presents a method for computing two-dimensional constant mean curvature surfaces. The method in question uses the variational aspect of the problem to implement an efficient algorithm. In principle it is a flow like method in…
This paper establishes the consistency of spectral approaches to data clustering. We consider clustering of point clouds obtained as samples of a ground-truth measure. A graph representing the point cloud is obtained by assigning weights to…
In this article a relation between curvature functionals for surfaces in the Euclidean space and area functionals in relative differential geometry will be given. Relative differential geometry can be described as the geometry of surfaces…
In this paper, we propose an efficient method to estimate the Weingarten map for point cloud data sampled from manifold embedded in Euclidean space. A statistical model is established to analyze the asymptotic property of the estimator. In…
We provide a general framework to construct finite dimensional approximations of the space of convex functions, which also applies to the space of c-convex functions and to the space of support functions of convex bodies. We give estimates…
While there is extensive literature on approximation of convex bodies by inscribed or circumscribed polytopes, much less is known in the case of generally positioned polytopes. Here we give upper and lower bounds for approximation of convex…
Approximating PDEs on surfaces by the diffuse interface approach allows us to use standard numerical tools to solve these problems. This makes it an attractive numerical approach. We extend this approach to vector-valued surface PDEs and…
In this paper, we study a natural discretization of the smooth Gaussian curvature on surfaces, which is defined as the quotient of the angle defect and the area of a geodesic disk at a vertex of a polyhedral surface. It is proved that each…
We propose a discrete surface theory in $\mathbb R^3$ that unites the most prevalent versions of discrete special parametrizations. This theory encapsulates a large class of discrete surfaces given by a Lax representation and, in…
We show that a realization of a closed connected PL-manifold of dimension n-1 in n-dimensional Euclidean space (n>2) is the boundary of a convex polyhedron (finite or infinite) if and only if the interior of each (n-3)-face has a point,…
A transformation based on mean curvature is introduced which morphs triangulated surfaces into round spheres.
We describe how to approximate the Riemann curvature tensor as well as sectional curvatures on possibly infinite-dimensional shape spaces that can be thought of as Riemannian manifolds. To this end, we extend the variational time…
We consider in this paper an area functional defined on submanifolds of fixed degree immersed into a graded manifold equipped with a Riemannian metric. Since the expression of this area depends on the degree, not all variations are…
We propose a computation of curvature of arbitrary two-dimensional surfaces of three-dimensional objects, which is a contribution to discrete gravity with potential applications in network geometry. We begin by linking each point of the…
We study hypersurfaces with fractional mean curvature in N-dimensional Euclidean space. These hypersurfaces are critical points of the fractional perimeter under a volume constraint. We use local inversion arguments to prove existence of…
We approximate boundaries of convex polytopes by smooth hypersurfaces $Y=Y_\varepsilon$ with {\it positive mean curvatures} and, by using basic geometric relations between the scalar curvatures of Riemannin manifolds and the mean curvatures…
We introduce a regularization method for mean curvature flow of a submanifold of arbitrary codimension in the Euclidean space, through higher order equations. We prove that the regularized problems converge to the mean curvature flow for…
In recent years new application areas have emerged in which one aims to capture the geometry of objects by means of three-dimensional point clouds. Often the obtained data consist of a dense sampling of the object's surface, containing many…