Local spline approximation methods

Lyche, Tom; Schumaker, Larry L.

doi:10.1016/0021-9045(75)90091-x

Cited by 203 publications

(137 citation statements)

References 9 publications

Supporting

Mentioning

134

Contrasting

Unclassified

Order By: Relevance

“…Furthermore, according to classical results in approximation theory (see e.g. DeVore andLorentz 1993, Chap.5, Lyche andSchumaker 1975 for the univariate case and de Boor et al 1993, Chap.3, Dagnino andLamberti 2001;Foucher and Sablonnière 2008;Lyche and Schumaker 1975 for the bivariate one) and, in view of the exactness of P on P p [x, y] andP on Pp[z], we have that the rate of convergence is O(h p ) for the two-dimensional case and O(hp) for the univariate case, i.e.…”

Section: Construction Of Trivariate Blending Sums Of Quasi-interpolantsmentioning

confidence: 89%

“…Let {B α (x, y), α ∈ F }, F ⊂ Z 2 , be an appropriate set of bivariate B-spline functions spanning a space of bivariate splines of degree ρ defined on a uniform triangulation of Ω ′ of mesh size h > 0, and let {B α (z), α ∈F },F ⊂ Z, be an appropriate set of B-spline functions in one dimension, spanning a space of splines of degreē ρ defined on a uniform partition of Ω ′′ of step-length h. We consider bivariate and univariate quasi-interpolants (see e.g. de Boor 2001, Chap.12, Lyche andSchumaker 1975, for the univariate case and de Boor et al 1993, Chap.3, for the bivariate case), P f (x, y) andP f (z), of the form…”

Section: Construction Of Trivariate Blending Sums Of Quasi-interpolantsmentioning

confidence: 99%

See 1 more Smart Citation

On trivariate blending sums of univariate and bivariate quadratic spline quasi-interpolants on bounded domains

Remogna¹,

Sablonnière²

2011

Computer Aided Geometric Design

View full text Add to dashboard Cite

The aim of this paper is to investigate, in a bounded domain of R 3 , two blending sums of univariate and bivariate C 1 quadratic spline quasi-interpolants.The main problem consists in constructing the coefficient functionals associated with boundary generators, i.e. generators with supports not entirely inside the domain. In their definition, these functionals involve data points lying inside or on the boundary of the domain. Moreover, the weights of these functionals must be chosen so that the quasi-interpolants have the best approximation order and a reasonable infinite norm.We give their explicit constructions, infinite norms and error estimates. In order to illustrate the approximation properties of the proposed quasiinterpolants, some numerical examples are presented and compared with those obtained by some other trivariate quasi-interpolants given recently in the literature.

show abstract

Section: Construction Of Trivariate Blending Sums Of Quasi-interpolantsmentioning

confidence: 89%

Section: Construction Of Trivariate Blending Sums Of Quasi-interpolantsmentioning

confidence: 99%

On trivariate blending sums of univariate and bivariate quadratic spline quasi-interpolants on bounded domains

Remogna¹,

Sablonnière²

2011

Computer Aided Geometric Design

View full text Add to dashboard Cite

show abstract

“…Parametric NURBS surfaces are based on polynomial B-splines and are defined by a set of control points which allow local shape control [3,[5][6][7][8]. The main reason for using rational NURBS instead of (non-rational) polynomial Bsplines is that NURBS are able to exactly reproduce conic sections [9].…”

Section: Continuous Closed Surfacesmentioning

confidence: 99%

Smooth shapes with spherical topology: Beyond traditional modeling, efficient deformation, and interaction

2017

View full text Add to dashboard Cite

Existing shape models with spherical topology are typically designed either in the discrete domain using interpolating polygon meshes or in the continuous domain using smooth but non-interpolating schemes such as subdivision or NURBS. Both polygon models and subdivision methods require a large number of parameters to model smooth surfaces. NURBS need fewer parameters but have a complicated rational expression and non-uniform shifts in their formulation. We present a new method to construct deformable closed surfaces, which includes exact spheres, by combining the best of two worlds: a smooth, interpolating model with a continuously varying tangent plane and well-defined curvature at every point on the surface.Our formulation is considerably simpler than NURBS and requires fewer parameters than polygon meshes. We demonstrate the generality of our method with applications including intuitive user-interactive shape modeling, continuous surface deformation, shape morphing, reconstruction of shapes from parameterized point clouds, and fast iterative shape optimization for image segmentation. Comparisons with discrete methods and non-interpolating approaches highlight the advantages of our framework.

show abstract

“…In [4] two examples of sequences { f N } based on locally uniform partitions and satisfying (4)-(6) are provided for any positive integer p. These are the modified approximating splines and the modified optimal nodal splines, which are obtained by modifying the approximating splines [8] as well as the optimal nodal splines [1,2,3] in such a way that condition (5) is true for any positive integer p. In this paper, we consider sequences of approximating splines for which we can prove (4)-(6) without modifying their definition on [a, b]. In particular, we shall consider the Martensen spline operator, introduced in [9] and recently studied in [15,16].…”

Section: G(x + H) − G(x)| G ∈ C(j)mentioning

confidence: 99%

“…We are interested in evaluating numerically I( f ; λ; n − 1), defined in (3), by replacing f ∈ C n−1 ([−1, 1]) with its spline approximation M R ( f ) of degree n, defined in (8). We approximate I( f ; λ; n − 1) by the quadrature sum…”

Section: General Casementioning

confidence: 99%

Martensen splines and finite-part integrals

Demichelis

Sciarra

2014

Numer Algor

View full text Add to dashboard Cite

We state a uniform convergence theorem for finite-part integrals which are derivatives of weighted Cauchy principal value integrals. We prove that a sequence of Martensen splines, based on locally uniform meshes, satisfies the sufficient conditions required by the theorem. We construct the quadrature rules based on such splines and illustrate their behaviour by presenting some numerical results and comparisons with composite midpoint, Simpson and Newton-Cotes rules.

show abstract

Local spline approximation methods

Cited by 203 publications

References 9 publications

On trivariate blending sums of univariate and bivariate quadratic spline quasi-interpolants on bounded domains

On trivariate blending sums of univariate and bivariate quadratic spline quasi-interpolants on bounded domains

Smooth shapes with spherical topology: Beyond traditional modeling, efficient deformation, and interaction

Martensen splines and finite-part integrals

Contact Info

Product

Resources

About