Bernstein-Bézier polynomials on spheres and sphere-like surfaces

Alfeld, Peter; Neamtu, Marian; Schumaker, Larry L.

doi:10.1016/0167-8396(95)00030-5

Cited by 105 publications

(120 citation statements)

References 5 publications

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“…is called a homogeneous spherical Bernstein-Bézier (SBB-) polynomial of degree d. Due to this special representation, many properties of SBB-polynomials are analogous to those of classical planar Bernstein-Bézier polynomials [1]. However, evaluating integrals of spherical polynomials is considerably more difficult than in the planar setting.…”

Section: Preliminarymentioning

confidence: 99%

“…To ensure the C r continuity across each edge of ∆, we impose smoothness conditions which can be found in [1]. Let M denote the smoothness matrix such that (2).…”

Section: Computational Algorithmsmentioning

confidence: 99%

“…Spherical Bernstein-Bézier splines (introduced in [1] and studied in [2] and [7]) are well suited for scattered data interpolation/approximation problems. The spherical spline functions have many properties in common with classical polynomial splines over planar triangulations.…”

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confidence: 99%

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Spherical Splines for Data Interpolation and Fitting

Baramidze¹,

Lai²,

Shum³

2006

SIAM J. Sci. Comput.

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Abstract. We study minimal energy interpolation, discrete and penalized least squares approximation problems on the unit sphere using nonhomogeneous spherical splines. Several numerical experiments are conducted to compare approximating properties of homogeneous and nonhomogeneous splines. Our numerical experiments show that nonhomogeneous splines have certain advantages over homogeneous splines.Key words. spherical splines, data fitting AMS subject classifications. 65D05, 65D07, 65D171. Introduction. Contemporary research in atmospheric sciences, geodesy and geophysics requires the use of global data heterogeneously distributed in space around the Earth. For spherical data interpolation/approximation tensor products of univariate splines are not a good choice, since data locations are not usually spaced over a regular grid. Radial basis functions are not good candidates either, since the data values may have no rotational symmetry.Spherical Bernstein-Bézier splines (introduced in [1] and studied in [2] and [7]) are well suited for scattered data interpolation/approximation problems. The spherical spline functions have many properties in common with classical polynomial splines over planar triangulations. Moreover, many spline interpolation and approximation methods for planar scattered data problems have analogs in spherical setting [2].One of the disadvantages of using homogeneous spherical splines is that spline spaces of even and odd degrees have only zero function in common due to homogeneity of the basis. More explicitly, the spline space S

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Section: Preliminarymentioning

confidence: 99%

“…To ensure the C r continuity across each edge of ∆, we impose smoothness conditions which can be found in [1]. Let M denote the smoothness matrix such that (2).…”

Section: Computational Algorithmsmentioning

confidence: 99%

See 1 more Smart Citation

Spherical Splines for Data Interpolation and Fitting

Baramidze¹,

Lai²,

Shum³

2006

SIAM J. Sci. Comput.

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“…There is an extensive theory of splines defined on spherical triangulations which is remarkably similar to the theory of bivariate splines, see [1][2][3] and Chapters 13-14 of [13]. Such splines are piecewise spherical harmonics.…”

Section: Remarkmentioning

confidence: 99%

Computing bivariate splines in scattered data fitting and the finite-element method

Schumaker

2008

Numer Algor

Self Cite

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Abstract. A number of useful bivariate spline methods are global in nature, i.e., all of the coefficients of an approximating spline must be computed at one time. Typically this involves solving a system of linear equations. Examples include several well-known methods for fitting scattered data, such as the minimal energy, least-squares, and penalized least-squares methods. Finite-element methods for solving boundary-value problems are also of this type. It is shown here that these types of globally-defined splines can be efficiently computed, provided we work with spline spaces with stable local minimal determining sets. §1. Introduction Bivariate splines defined over triangulations are important tools in several application areas including scattered data fitting and the numerical solution of boundaryvalue problems by the finite element method. Methods for computing spline approximations fall into two classes: 1) Local methods, where the coefficients of the spline are computed one at a time or in small groups, 2) Global methods, where all of the coefficients of the spline have to be computed simultaneously, usually as the solution of a single linear system of equations.In this paper we focus on global methods, and in particular those that arise from minimizing a quadratic form, possibly with some constraints. The purpose of this paper is to show how such minimization problems can be efficiently solved for spline spaces that possess stable local minimal determining sets (see Sect. 2). In particular, we show how our approach applies to three commmonly used scattered data fitting methods: the minimal energy method, the discrete least-squares method, and the penalized least-squares method. In addition, we discuss how it works for solving boundary-value problems involving partial differential equations. The standard approach to solving global variational problems involving piecewise polynomials on triangulations is to use Lagrange multipliers to enforce interpolation and smoothness conditions, see Remark 1. This results in a linear system

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“…The stencil for this basis is given by K m = fv 1 ; v 2 ; f 1 ; f 2 g (see Figure 5) and exploits the degrees of freedom implied to kill the functions x 2 , y 2 , and z 2 (and by implication the constant function [1]). Using the coordinates of the neighbors of the involved sites a small linear system results 0 B B @ Butterfly: This is the only basis which uses other than immediate neighbors (all the sites K m denoted in Figure 5).…”

Section: Quadraticmentioning

confidence: 99%

Spherical wavelets: Efficiently representing functions on a sphere

Schröder¹,

Sweldens²

Wavelets in the Geosciences

265

409

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Wavelets have proven to be powerful bases for use in numerical analysis and signal processing. Their power lies in the fact that they only require a small number of coefficients to represent general functions and large data sets accurately. This allows compression and efficient computations. Classical constructions have been limited to simple domains such as intervals and rectangles. In this paper we present a wavelet construction for scalar functions defined on the sphere. We show how biorthogonal wavelets with custom properties can be constructed with the lifting scheme. The bases are extremely easy to implement and allow fully adaptive subdivisions. We give examples of functions defined on the sphere, such as topographic data, bidirectional reflection distribution functions, and illumination, and show how they can be efficiently represented with spherical wavelets.

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Bernstein-Bézier polynomials on spheres and sphere-like surfaces

Cited by 105 publications

References 5 publications

Spherical Splines for Data Interpolation and Fitting

Spherical Splines for Data Interpolation and Fitting

Computing bivariate splines in scattered data fitting and the finite-element method

Spherical wavelets: Efficiently representing functions on a sphere

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