Near-best $C^2$ quartic spline quasi-interpolants on type-6 tetrahedral partitions of bounded domains
Provides a practical quasi-interpolation scheme for 3D volume data reconstruction, but the approach is incremental, extending existing box spline methods to bounded domains.
The paper develops near-best $C^2$ quartic spline quasi-interpolants on type-6 tetrahedral partitions for reconstructing gridded volume data, achieving approximation order four with minimized infinity norm. Numerical tests demonstrate competitive approximation properties compared to other spline methods.
In this paper, we present new quasi-interpolating spline schemes defined on 3D bounded domains, based on trivariate $C^2$ 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.