250 research outputs found
H\"older Regularity of Geometric Subdivision Schemes
We present a framework for analyzing non-linear -valued
subdivision schemes which are geometric in the sense that they commute with
similarities in . It admits to establish
-regularity for arbitrary schemes of this type, and
-regularity for an important subset thereof, which includes all
real-valued schemes. Our results are constructive in the sense that they can be
verified explicitly for any scheme and any given set of initial data by a
universal procedure. This procedure can be executed automatically and
rigorously by a computer when using interval arithmetics.Comment: 31 pages, 1 figur
Piecewise polynomial monotonic interpolation of 2D gridded data
International audienceA method for interpolating monotone increasing 2D scalar data with a monotone piecewise cubic C-continuous surface is presented. Monotonicity is a sufficient condition for a function to be free of critical points inside its domain. The standard axial monotonicity for tensor-product surfaces is however too restrictive. We therefore introduce a more relaxed monotonicity constraint. We derive sufficient conditions on the partial derivatives of the interpolating function to ensure its monotonicity. We then develop two algorithms to effectively construct a monotone C surface composed of cubic triangular BĂ©zier surfaces interpolating a monotone gridded data set. Our method enables to interpolate given topological data such as minima, maxima and saddle points at the corners of a rectangular domain without adding spurious extrema inside the function domain. Numerical examples are given to illustrate the performance of the algorithm
Long-time behavior of a finite volume discretization for a fourth order diffusion equation
We consider a non-standard finite-volume discretization of a strongly
non-linear fourth order diffusion equation on the -dimensional cube, for
arbitrary . The scheme preserves two important structural properties
of the equation: the first is the interpretation as a gradient flow in a mass
transportation metric, and the second is an intimate relation to a linear
Fokker-Planck equation. Thanks to these structural properties, the scheme
possesses two discrete Lyapunov functionals. These functionals approximate the
entropy and the Fisher information, respectively, and their dissipation rates
converge to the optimal ones in the discrete-to-continuous limit. Using the
dissipation, we derive estimates on the long-time asymptotics of the discrete
solutions. Finally, we present results from numerical experiments which
indicate that our discretization is able to capture significant features of the
complex original dynamics, even with a rather coarse spatial resolution.Comment: 27 pages, minor change
Non-linear subdivision using local spherical coordinates
In this paper, we present an original non-linear subdivision scheme suitable for univariate data, plane curves and discrete triangulated surfaces, while keeping the complexity acceptable. The proposed technique is compared to linear subdivision methods having an identical support. Numerical criteria are proposed to verify basic properties, such as convergence of the scheme and the regularity of the limit function
Shape criteria of Bernstein-BĂ©zier polynomials over simplexes
AbstractThis paper discusses the criteria of convexity, monotonicity, and positivity of Bernstein-BĂ©zier polynomials over simplexes
Error bounded approximate reparametrization of NURBS curves
Journal ArticleThis paper reports research on solutions to the following reparametrization problem: approximate c(r(t)) by a NURBS where c is a NURBS curve and r may, or may not, be a NURBS function. There are many practical applications of this problem including establishing and exploring correspondence in geometry, creating related speed profiles along motion curves for animation, specifying speeds along tool paths, and identifying geometrically equivalent, or nearly equivalent, curve mappings. A framework for the approximation problem is described using two related algorithmic schemes. One constrains the shape of the approximation to be identical to the original curve c. The other relaxes this constraint. New algorithms for important cases of curve reparametrization are developed from within this framework. They produce results with bounded error and address approximate arc length parametrizations of curves, approximate inverses of NURBS functions, and reparametrizations that establish user specified tolerances as bounds on the Frechet distance between parametric curves
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