16,341 research outputs found
Discrete schemes for Gaussian curvature and their convergence
In this paper, several discrete schemes for Gaussian curvature are surveyed.
The convergence property of a modified discrete scheme for the Gaussian
curvature is considered. Furthermore, a new discrete scheme for Gaussian
curvature is resented. We prove that the new scheme converges at the regular
vertex with valence not less than 5. By constructing a counterexample, we also
show that it is impossible for building a discrete scheme for Gaussian
curvature which converges over the regular vertex with valence 4. Finally,
asymptotic errors of several discrete scheme for Gaussian curvature are
compared
Discrete curvature approximations and segmentation of polyhedral surfaces
The segmentation of digitized data to divide a free form surface into patches is one of the key steps required to perform a reverse engineering process of an object. To this end, discrete curvature approximations are introduced as the basis of a segmentation process that lead to a decomposition of digitized data into areas that will help the construction of parametric surface patches. The approach proposed relies on the use of a polyhedral representation of the object built from the digitized data input. Then, it is shown how noise reduction, edge swapping techniques and adapted remeshing schemes can participate to different preparation phases to provide a geometry that highlights useful characteristics for the segmentation process. The segmentation process is performed with various approximations of discrete curvatures evaluated on the polyhedron produced during the preparation phases. The segmentation process proposed involves two phases: the identification of characteristic polygonal lines and the identification of polyhedral areas useful for a patch construction process. Discrete curvature criteria are adapted to each phase and the concept of invariant evaluation of curvatures is introduced to generate criteria that are constant over equivalent meshes. A description of the segmentation procedure is provided together with examples of results for free form object surfaces
Discrete exterior calculus (DEC) for the surface Navier-Stokes equation
We consider a numerical approach for the incompressible surface Navier-Stokes
equation. The approach is based on the covariant form and uses discrete
exterior calculus (DEC) in space and a semi-implicit discretization in time.
The discretization is described in detail and related to finite difference
schemes on staggered grids in flat space for which we demonstrate second order
convergence. We compare computational results with a vorticity-stream function
approach for surfaces with genus 0 and demonstrate the interplay between
topology, geometry and flow properties. Our discretization also allows to
handle harmonic vector fields, which we demonstrate on a torus.Comment: 21 pages, 9 figure
Chernoff's Theorem and Discrete Time Approximations of Brownian Motion on Manifolds
Let (S(t)) be a one-parameter family S = (S(t)) of positive integral
operators on a locally compact space L. For a possibly non-uniform partition of
[0,1] define a measure on the path space C([0,1],L) by using a) S(dt) for the
transition between cosecutive partition times of distance dt, and b) a suitable
continuous interpolation scheme (e.g. Brownian bridges or geodesics). If
necessary normalize to get a probability measure. We prove a version of
Chernoff's theorem of semigroup theory and tighness results which together
yield convergence in law of such measures as the partition gets finer. In
particular let L be a closed smooth submanifold of a Riemannian manifold M. We
prove convergence of Brownian motion on M, conditioned to visit L at all
partition times, to a process on L whose law has a Radon-Nikodym density with
repect to Brownian motion on L which contains scalar, mean and sectional
curvature terms. Various approximation schemes for Brownian motion are also
given. These results substantially extend earlier work by the authors and by
Andersson and Driver.Comment: 35 pages, revised version for publication, more detailed expositio
ADI splitting schemes for a fourth-order nonlinear partial differential equation from image processing
We present directional operator splitting schemes for the numerical solution of a fourth-order, nonlinear partial differential evolution equation which arises in image processing. This equation constitutes the H−1-gradient flow of the total variation and represents a prototype of higher-order equations of similar type which are popular in imaging for denoising, deblurring and inpainting problems. The efficient numerical solution of this equation is very challenging due to the stiffness of most numerical schemes. We show that the combination of directional splitting schemes with implicit time-stepping provides a stable and computationally cheap numerical realisation of the equation
Variational image regularization with Euler's elastica using a discrete gradient scheme
This paper concerns an optimization algorithm for unconstrained non-convex
problems where the objective function has sparse connections between the
unknowns. The algorithm is based on applying a dissipation preserving numerical
integrator, the Itoh--Abe discrete gradient scheme, to the gradient flow of an
objective function, guaranteeing energy decrease regardless of step size. We
introduce the algorithm, prove a convergence rate estimate for non-convex
problems with Lipschitz continuous gradients, and show an improved convergence
rate if the objective function has sparse connections between unknowns. The
algorithm is presented in serial and parallel versions. Numerical tests show
its use in Euler's elastica regularized imaging problems and its convergence
rate and compare the execution time of the method to that of the iPiano
algorithm and the gradient descent and Heavy-ball algorithms
- …