Near-best C2C^2 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 $C^2$ 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.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

Recent Results on Near-Best Spline Quasi-Interpolants

Roughly speaking, a near-best (abbr. NB) quasi-interpolant (abbr. QI) is an approximation operator of the form $Q_af=\sum_{\alpha\in A} \Lambda_\alpha (f) B_\alpha$ where the $B_\alpha$'s are B-splines and the $\Lambda_\alpha (f)$'s are linear discrete or integral forms acting on the given function $f$. These forms depend on a finite number of coefficients which are the components of vectors $a_\alpha$ for $\alpha\in A$. The index $a$ refers to this sequence of vectors. In order that $Q_a p=p$ for all polynomials $p$ belonging to some subspace included in the space of splines generated by the $B_\alpha$'s, each vector $a_\alpha$ must lie in an affine subspace $V_\alpha$, i.e. satisfy some linear constraints. However there remain some degrees of freedom which are used to minimize $\Vert a_\alpha \Vert_1$ for each $\alpha\in A$. It is easy to prove that $\max \{\Vert a_\alpha \Vert_1 ; \alpha\in A\}$ is an upper bound of $\Vert Q_a \Vert_{\infty}$: thus, instead of minimizing the infinite norm of $Q_a$, which is a difficult problem, we minimize an upper bound of this norm, which is much easier to do. Moreover, the latter problem has always at least one solution, which is associated with a NB QI. In the first part of the paper, we give a survey on NB univariate or bivariate spline QIs defined on uniform or non-uniform partitions and already studied by the author and coworkers. In the second part, we give some new results, mainly on univariate and bivariate integral QIs on {\sl non-uniform} partitions: in that case, NB QIs are more difficult to characterize and the optimal properties strongly depend on the geometry of the partition. Therefore we have restricted our study to QIs having interesting shape properties and/or infinite norms uniformly bounded independently of the partition

Quasi-interpolation by C1 quartic splines on type-1 triangulations

This work was initiated during the visiting on 2017, March of the first and third authors to the Department of Mathematics of the University of Torino, and partially realized during the visiting of the fourth author to the Department of Applied Mathematics of the University of Granada on 2017, November. They thank the financial support of both institutions and the Gruppo Nazionale per il Calcolo Scientifico (GNCS) - INdAM.In this paper we construct two new families of C1 quartic quasi-interpolating splines on type-1 triangulations approximating regularly distributed data. The splines are directly determined by setting their Bernstein–Bézier coefficients to appropriate combinations of the given data values instead of defining the approximating splines as linear combinations of compactly supported bivariate spanning functions and do not use prescribed derivatives at any point of the domain. The quasi-interpolation operators provided by the proposed schemes interpolate the data values at the vertices of the triangulation, reproduce cubic polynomials and yield approximation order four for smooth functions. We also propose some numerical tests that confirm the theoretical results

Near-best bivariate spline quasi-interpolants on a four-directional mesh of the plane

Spline quasi-interpolants (QIs) are practical and effective approximation operators. In this paper, we construct QIs with optimal approximation orders and small infinity norms called near-best discrete and integral quasi-interpolants which are based on $\Omega$-~splines, i.e. B-splines with regular lozenge supports on the uniform four directional mesh of the plane. These quasi-interpolants are obtained so as to be exact on some space of polynomials and to minimize an upper bound of their infinity norms which depend on a finite number of free parameters. We show that this problem has always a solution, which is not unique in general. Concrete examples of these types of quasi-interpolants are given in the last section

Recursive subdivision algorithms for curve and surface design

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.In this thesis, the author studies recursIve subdivision algorithms for curves and surfaces. Several subdivision algorithms are constructed and investigated. Some graphic examples are also presented. Inspired by the Chaikin's algorithm and the Catmull-Clark's algorithm, some non-uniform schemes, the non-uniform corner cutting scheme and the recursive subdivision algorithm for non-uniform B-spline curves, are constructed and analysed. The adapted parametrization is introduced to analyse these non-uniform algorithms. In order to solve the surface interpolation problem, the Dyn-Gregory-Levin's 4-point interpolatory scheme is generalized to surfaces and the 10-point interpolatory subdivision scheme for surfaces is formulated. The so-called Butterfly Scheme, which was firstly introduced by Dyn, Gregory Levin in 1988, is just a special case of the scheme. By studying the Cross-Differences of Directional Divided Differences, a matrix approach for analysing uniform subdivision algorithms for surfaces is established and the convergence of the 10-point scheme over both uniform and non-uniform triangular networks is studied. Another algorithm, the subdivision algorithm for uniform bi-quartic B-spline surfaces over arbitrary topology is introduced and investigated. This algorithm is a generalization of Doo-Sabin's and Catmull-Clark's algorithms. It produces uniform Bi-quartic B-spline patches over uniform data. By studying the local subdivision matrix, which is a circulant, the tangent plane and curvature properties of the limit surfaces at the so-called Extraordinary Points are studied in detail.The Chinese Educational Commission and The British Council (SBFSS/1987