7,638 research outputs found
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
Polynomial cubic splines with tension properties
In this paper we present a new class of spline functions with tension properties. These splines are composed by polynomial cubic pieces and therefore are conformal to the standard, NURBS based CAD/CAM systems
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
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
Scalar and Hermite subdivision schemes
AbstractA criterion of convergence for stationary nonuniform subdivision schemes is provided. For periodic subdivision schemes, this criterion is optimal and can be applied to Hermite subdivision schemes which are not necessarily interpolatory. For the Merrien family of Hermite subdivision schemes which involve two parameters, we are able to describe explicitly the values of the parameters for which the Hermite subdivision scheme is convergent
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