59 research outputs found
Generalized quadratic B-splines generated by Merrien subdivision algorithm and some applications
A new global basis of B-splines is defined in the space of generalized
quadratic splines (GQS) generated by Merrien subdivision algorithm. Then,
refinement equations for these B-splines and the associated corner-cutting
algorithm are given. Afterwards, several applications are presented. First a
global construction of monotonic and/or convex generalized splines
interpolating monotonic and/or convex data. Second, convergence of sequences of
control polygons to the graph of a GQS. Finally, a Lagrange interpolant and a
quasi-interpolant which are exact on the space of affine polynomials and whose
infinite norms are uniformly bounded independently of the partition.Comment: 2004-1
Hermite Subdivision with Shape Constraints on a Rectangular Mesh
International audienceWe study a two parameter version of the Hermite subdivision scheme introduced in [7], wish gives interpolants on rectangular meshes. We prove -convergence for a range of the two parameters. By introducing a control grid we can choose the parameters in the scheme so that the interpolant inherits positivity and/or directional monotonicity from the initial data. Several examples are given showing that a desired shape can be achieved even if we use only very crude estimates for the initial slopes
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
Interpolatory Subdivision with Shape Constraints for Curves
International audienceWe derive two reformulations of the Hermite subdivision scheme introduced in [12]. One where we separate computation of values and derivatives and one based of refinement of a control polygon. We show that the latter leads to a subdivision matrix which is totally positive. Based on this we give algorithms for constructing subdivision curves that preserve positivity, monotonicity, and convexity
Hermite Subdivision Schemes and Taylor Polynomials
International audienceWe propose a general study of the convergence of a Hermite subdivision scheme of degree in dimension 1. This is done by linking Hermite subdivision schemes and Taylor polynomials and by associating a so-called Taylor subdivision (vector) scheme . The main point of investigation is a spectral condition. If the subdivision scheme of the finite differences of is contractive, then is and is . We apply this result to two families of Hermite subdivision schemes, the first one is interpolatory, the second one is a kind of corner cutting, both of them use Obreshkov interpolation polynomial
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
- …