Four-directional Mesh Research Articles

Optimal bivariate [formula omitted] cubic quasi-interpolation on a type-2 triangulation

In [A. Guessab, O. Nouisser, G. Schmeisser, Multivariate approximation by a combination of modified Taylor polynomials, J. Comput. Appl. Math. 196 (2006) 162–179], a general method is proposed to increase the approximation order of approximation operators. In this work, by using these enhancement techniques, we introduce and study new schemes based on a C 1 -spline quasi-interpolant on a type-2 triangulation. They are designed for approximating real-valued functions defined on R 2 . The proposed method is based on the following idea: from a discrete quasi-interpolant defined by the quadratic box-spline exact on P 2 and by judiciously choosing the first-order Taylor coefficients, we derive a cubic differential quasi-interpolant yielding optimal approximation order. In addition, when the derivatives are not available or are extremely expensive to compute, we approximate them by finite difference approximations having the desired accuracy to derive a new class of discrete quasi-interpolants. As an essential difference to some of the existing methods, we only use the given data values and, then, we do not modify the original triangulation. Finally, we present some numerical tests which confirm the efficiency of the newly quasi-interpolant and demonstrate good visual quality. In particular, we compare it with a differential quasi-interpolant done by Lai [M.-J. Lai, Approximation order from bivariate C 1 -cubics on a four-directional mesh is full, Comput. Aided Geom. Design 11 (2) (1994) 215–223] which is also exact on P 3 but uses third order partial derivatives.

A tension approach to controlling the shape of cubic spline surfaces on FVS triangulations

We propose a parametric tensioned version of the FVS macro-element to control the shape of the composite surface and remove artificial oscillations, bumps and other undesired behaviour. In particular, this approach is applied to C 1 cubic spline surfaces over a four-directional mesh produced by two-stage scattered data fitting methods.

On near-best discrete quasi-interpolation on a four-directional mesh

Spline quasi-interpolants are practical and effective approximation operators. In this paper, we construct QIs with optimal approximation orders and small infinity norms called near-best discrete quasi-interpolants which are based on Ω -splines, i.e. B-splines with octagonal supports on the uniform four-directional mesh of the plane. These quasi-interpolants are exact on some space of polynomials and they minimize an upper bound of their infinity norms depending on a finite number of free parameters. We show that this problem has always a solution, in general nonunique. Concrete examples of such quasi-interpolants are given in the last section.

Smoothed hat functions in subdivision

Following the general idea of smoothing through convolution, we discuss schemes for smoothing hat functions on the three-directional and the four-directional mesh. The smoothing masks–which in the four-directional case are matrix masks–have the same support as the refinement mask for the hat functions. Several smoothing schemes are presented, and in some cases the smoothness properties of the smoothed functions are analysed. In particular, the results for the four-directional case follow new and original lines of research.

Near-best operators based on a [formula omitted] quartic spline on the uniform four-directional mesh

We present some results about the construction of quasi-interpolant operators based on a special C 2 quartic B-spline. We show that these operators, called near-best quasi-interpolants, have the best approximation order and small infinity norms. They are obtained by solving a minimization problem that admits always a solution. We give an error bound of these quasi-interpolants and we illustrate our results by a numerical example.

Subdivision Scheme of Quartic Bivariate Splines on a Four-Directional Mesh

In this paper we give a new deflnition of minimally and quasi-minimally supported C 2 quartic bivariate B-splines associated with the four-directional mesh of the plane, introduced in (7,19), which is convenient to show that theses B-splines satisfy the reflnement equation and we determine the associated matrix mask, we prove that the family of these B-splines is stable and the associated subdivision scheme converges. These results can be extended to various cases in the spline space of class C 3k+2 and degree 4k + 4, but in these cases the supports of the masks are larger.

Refinable bivariate quartic and quintic [formula omitted]-splines for quadrilateral subdivisions

Refinable compactly supported bivariate C 2 quartic and quintic spline function vectors on the four-directional mesh are introduced in this paper to generate matrix-valued templates for approximation and Hermite interpolatory surface subdivision schemes, respectively, for both the 2 and 1-to-4 split quadrilateral topological rules. These splines have their full local polynomial preservation orders. In addition, we extend our study to parametric approach and use the symmetric properties of our refinable quintic spline components as a guideline to reduce the number of free parameters in constructing second order C 2 Hermite interpolatory quadrilateral subdivision schemes with precisely six components.

Dimension of [formula omitted]-splines on type-6 tetrahedral partitions

We consider a linear space of piecewise polynomials in three variables which are globally smooth, i.e. trivariate C 1 -splines of arbitrary polynomial degree. The splines are defined on type-6 tetrahedral partitions, which are natural generalizations of the four-directional mesh. By using Bernstein–Bézier techniques, we analyze the structure of the spaces and establish formulae for the dimension of the smooth splines on such uniform type partitions.

Scattered Data Fitting by Direct Extension of Local Polynomials to Bivariate Splines

We present a new scattered data fitting method, where local approximating polynomials are directly extended to smooth (C 1 or C 2) splines on a uniform triangulation (the four-directional mesh). The method is based on designing appropriate minimal determining sets consisting of whole triangles of domain points for a uniformly distributed subset of . This construction allows to use discrete polynomial least squares approximations to the local portions of the data directly as parts of the approximating spline. The remaining Bernstein-Bae#169;zier coefficients are efficiently computed by extension, i.e., using the smoothness conditions. To obtain high quality local polynomial approximations even for difficult point constellations (e.g., with voids, clusters, tracks), we adaptively choose the polynomial degrees by controlling the smallest singular value of the local collocation matrices. The computational complexity of the method grows linearly with the number of data points, which facilitates its application to large data sets. Numerical tests involving standard benchmarks as well as real world scattered data sets illustrate the approximation power of the method, its efficiency and ability to produce surfaces of high visual quality, to deal with noisy data, and to be used for surface compression.

Pairs of B-Splines with Small Support on the Four-Directional Mesh Generating a Partition of Unity

Let τ be the four-directional mesh of the plane and let Σ1 (respectively Λ1) be the unit square (respectively the lozenge) formed by four (respectively eight) triangles of τ. We study spaces of piecewise polynomial functions in Ck(R2) with supports Σ1 or Λ1 having sufficiently high degree n, which are invariant with respect to the group of symmetries of Σ1 or Λ1 and whose integer translates form a partition of unity. Such splines are called complete Σ1 and Λ1-splines. We first give a general study of spaces of linearly independent complete Σ1 and Λ1-splines of class Ck and degree n. Then, for any fixed k≥0, we prove the existence of complete Σ1 and Λ1-splines of class Ck and minimal degree, but they are not unique. Finally, we describe algorithms allowing to compute the Bernstein–Bezier coefficients of these splines.

Cubic Spline Prewavelets on the Four-Directional Mesh

Dedicated to Professor M. J. D. Powell on the occasion of his sixty-fifth birthday and his retirement. In this paper, we design differentiable, two-dimensional, piecewise polynomial cubic prewavelets of particularly small compact support. They are given in closed form, and provide stable, orthogonal decompositions of L 2 (R 2 ) . In particular, the splines we use in our prewavelet constructions give rise to stable bases of spline spaces that contain all cubic polynomials, whereas the more familiar box spline constructions cannot reproduce all cubic polynomials, unless resorting to a box spline of higher polynomial degree.

On the approximation power of bivariate quadratic C1 splines

In this paper we investigate the approximation power of local bivariate quadratic C 1 quasi-interpolating (q-i) spline operators with a four-directional mesh. In particular, we show that they can approximate a real function and its partial derivatives up to an optimal order and we derive local and global upper bounds both for the errors and for the spline partial derivatives, in the case the spline is more differentiable than the function. Then such general results are applied to prove new properties of two interesting q-i spline operators, proposed and partially studied in Chui and Wang (Sci. Sinica XXVII (1984) 1129–1142).

Quasi 4–8 subdivision

This paper presents a new scheme for subdivision surfaces based on four-directional meshes. It combines geometry-sensitive refinement with convolution smoothing. The scheme has a simple, efficient implementation and generates smooth well-shaped meshes.

B-splines on 3-D tetrahedron partition in four-directional mesh

It is more difficult to construct 3-D splines than in 2-D case. Some results in the three directional meshes of bivariate case have been extended to 3-D case and corresponding tetrahedron partition has been constructed. The support of related B-splines and their recurrent formulas on integration and differentiation-difference are obtained. The results of this paper can be extended into higher dimension spaces, and can be also used in wavelet analysis, because of the relationship between spline and wavelets.

Existence and Construction of Simple B-Splines of Class C k on a Four-Directional Mesh of the Plane.

In this paper we present a study of spaces of splines in Ck(R2) with supports the square Σ1 and the lozenge Λ1 formed respectively by four and eight triangles of the uniform four directional mesh of the plane. Such splines are called Σ1 and Λ1-splines. We first compute the dimension of the space of Σ1-splines. Then we prove the existence of a unique Σ1-spline of minimal degree for any fixed k≥0. By using this last result, we also prove the existence of a unique Σ1-spline of minimal degree. Finally, we describe algorithms allowing to compute the Bernstein–Bezier coefficients of Σ1-spline and Λ1-spline of minimal degree.

A new subdivision method for bivariate splines on the four-directional mesh

We present a new bivariate subdivision scheme based on two generators of a four-directional spline space. In particular, we are dealing with piecewise linear, continuous splines, and with piecewise cubic, continuously differentiable splines. The subdivision schemes rely on a matrix mask of small support. In addition, we present some results on related quasi-interpolating operators, and on approximation order of the underlying shift-invariant spaces.

Bivariate interpolatory rational splines

Smoothing conditions in terms of Bezier coefficients of piecewise rational functions on an arbitrary triangulation are derived. This facilitates the solution of the problem of bivariate rational spline interpolation, with or without convexity constraints, particularly on the three and four-directional meshes. For such a triangulation, we also derive the conformality condition that a bivariate rationale spline function must satisfy, and we demonstrate the interpolation scheme with a low-degree example.

Approximation order from bivariate C1-cubics on a four-directional mesh is full

We show that the space of bivariate C 1 piecewise cubic polynomial functions on a four-directional mesh of size h has the full approximation order, i.e., O( h 4).

Interpolation by bivariate quadratic splines on a four-directional mesh

We give sufficient conditions which guarantee that a set of points admits unique Lagrange interpolation by quadratic splines on a four-directional mesh. The poisedness of these sets will be proved by reducing one bivariate problem to a finite sequence of univariate problems.

Fortran subroutines for B-nets of box splines on three- and four-directional meshes

We describe an algorithm to compute the B-nets of bivariate box splines on a three-or four-directional mesh. Two pseudo Fortran programs for those B-nets are given.