276 research outputs found
Shape preserving interpolatory subdivision schemes
Stationary interpolatory subdivision schemes which preserve shape properties such as convexity or monotonicity are constructed. The schemes are rational in the data and generate limit functions that are at least . The emphasis is on a class of six-point convexity preserving subdivision schemes that generate limit functions. In addition, a class of six-point monotonicity preserving schemes that also leads to limit functions is introduced. As the algebra is far too complicated for an analytical proof of smoothness, validation has been performed by a simple numerical methodology
Exponential Splines and Pseudo-Splines: Generation versus reproduction of exponential polynomials
Subdivision schemes are iterative methods for the design of smooth curves and
surfaces. Any linear subdivision scheme can be identified by a sequence of
Laurent polynomials, also called subdivision symbols, which describe the linear
rules determining successive refinements of coarse initial meshes. One
important property of subdivision schemes is their capability of exactly
reproducing in the limit specific types of functions from which the data is
sampled. Indeed, this property is linked to the approximation order of the
scheme and to its regularity. When the capability of reproducing polynomials is
required, it is possible to define a family of subdivision schemes that allows
to meet various demands for balancing approximation order, regularity and
support size. The members of this family are known in the literature with the
name of pseudo-splines. In case reproduction of exponential polynomials instead
of polynomials is requested, the resulting family turns out to be the
non-stationary counterpart of the one of pseudo-splines, that we here call the
family of exponential pseudo-splines. The goal of this work is to derive the
explicit expressions of the subdivision symbols of exponential pseudo-splines
and to study their symmetry properties as well as their convergence and
regularity.Comment: 25 page
Ellipse-preserving Hermite interpolation and subdivision
We introduce a family of piecewise-exponential functions that have the
Hermite interpolation property. Our design is motivated by the search for an
effective scheme for the joint interpolation of points and associated tangents
on a curve with the ability to perfectly reproduce ellipses. We prove that the
proposed Hermite functions form a Riesz basis and that they reproduce
prescribed exponential polynomials. We present a method based on Green's
functions to unravel their multi-resolution and approximation-theoretic
properties. Finally, we derive the corresponding vector and scalar subdivision
schemes, which lend themselves to a fast implementation. The proposed vector
scheme is interpolatory and level-dependent, but its asymptotic behaviour is
the same as the classical cubic Hermite spline algorithm. The same convergence
properties---i.e., fourth order of approximation---are hence ensured
A global approach to the refinement of manifold data
A refinement of manifold data is a computational process, which produces a
denser set of discrete data from a given one. Such refinements are closely
related to multiresolution representations of manifold data by pyramid
transforms, and approximation of manifold-valued functions by repeated
refinements schemes. Most refinement methods compute each refined element
separately, independently of the computations of the other elements. Here we
propose a global method which computes all the refined elements simultaneously,
using geodesic averages. We analyse repeated refinements schemes based on this
global approach, and derive conditions guaranteeing strong convergence.Comment: arXiv admin note: text overlap with arXiv:1407.836
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