21 research outputs found
Near-best quartic spline quasi-interpolants on type-6 tetrahedral partitions of bounded domains
In this paper, we present new quasi-interpolating spline schemes defined on
3D bounded domains, based on trivariate quartic box splines on type-6
tetrahedral partitions and with approximation order four. Such methods can be
used for the reconstruction of gridded volume data. More precisely, we propose
near-best quasi-interpolants, i.e. with coefficient functionals obtained by
imposing the exactness of the quasi-interpolants on the space of polynomials of
total degree three and minimizing an upper bound for their infinity norm. In
case of bounded domains the main problem consists in the construction of the
coefficient functionals associated with boundary generators (i.e. generators
with supports not completely inside the domain), so that the functionals
involve data points inside or on the boundary of the domain.
We give norm and error estimates and we present some numerical tests,
illustrating the approximation properties of the proposed quasi-interpolants,
and comparisons with other known spline methods. Some applications with real
world volume data are also provided.Comment: In the new version of the paper, we have done some minor revisions
with respect to the previous version, CALCOLO, Published online: 10 October
201
A Practical Box Spline Compendium
Box splines provide smooth spline spaces as shifts of a single generating
function on a lattice and so generalize tensor-product splines. Their elegant
theory is laid out in classical papers and a summarizing book. This compendium
aims to succinctly but exhaustively survey symmetric low-degree box splines
with special focus on two and three variables. Tables contrast the lattices,
supports, analytic and reconstruction properties, and list available
implementations and code.Comment: 15 pages, 10 figures, 8 table
Characterization of bivariate hierarchical quartic box splines on a three-directional grid
International audienceWe consider the adaptive refinement of bivariate quartic C 2-smooth box spline spaces on the three-directional (type-I) grid G. The polynomial segments of these box splines belong to a certain subspace of the space of quar-tic polynomials, which will be called the space of special quartics. Given a bounded domain Ω ⊂ R 2 and finite sequence (G ℓ) ℓ=0,...,N of dyadically refined grids, we obtain a hierarchical grid by selecting mutually disjoint cells from all levels such that their union covers the entire domain. Using a suitable selection procedure allows to define a basis spanning the hierarchical box spline space. The paper derives a characterization of this space. Under certain mild assumptions on the hierarchical grid, the hierarchical spline space is shown to contain all C 2-smooth functions whose restrictions to the cells of the hierarchical grid are special quartic polynomials. Thus, in this case we can give an affirmative answer to the completeness questions for the hierarchical box spline basis