A triangle-based $C^1$ interpolation method

Renka, R.L.; Cline, Alan

doi:10.1216/rmj-1984-14-1-223

Cited by 167 publications

(86 citation statements)

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“…Typical methods involve doing local least squares with low degree polynomials or radial basis functions. For some specific methods and further references, see [4,14]. Remark 2.…”

Section: The Case Of Two-sided Constraintsmentioning

confidence: 99%

Nonnegativity preserving macro-element interpolation of scattered data

Schumaker

Speleers

2010

Computer Aided Geometric Design

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Abstract. Nonnegative bivariate interpolants to scattered data are constructed using some C 1 macro-element spline spaces. The methods are local, and rely on adjusting gradients at the data points to insure nonnegativity of the spline when the original data is nonnegative. More general range-restricted interpolation is also considered.Keywords: spline interpolation, shape preservation, nonnegative surfaces §1. IntroductionInterpolating scattered data is a common problem in a wide range of application areas. It is often required that the interpolant preserves known shape properties inherent to the data. Common shape properties are convexity, monotonicity and nonnegativity. In this paper we consider the nonnegativity-preserving interpolation problem.Problem 1.1. Given a set of scattered data points, find a function s defined on Ω such that s(x i , y i ) = f i for i = 1, . . . , n, and s(x, y) ≥ 0 for all (x, y) ∈ Ω.A standard approach for solving this problem is to work with either polynomial splines [2,5,6,9,16] or rational splines [1,8,11,12,15]. These splines are defined on a triangulation with its vertices at the data points. Problem 1.1 is often solved as an optimization problem in order to find a visually pleasing, nonnegative spline that interpolates the data. Such methods are mostly global in nature, i.e., the spline coefficients globally depend on all of the data. Our aim here is to focus on the construction of local and easy to use methods based on standard C 1 macro-element spline spaces.The paper is organized as follows. In Sects. 2 and 3 we describe nonnegativitypreserving interpolation methods using the C 1 Powell-Sabin and Clough-Tocher macro-elements, respectively. We discuss the approximation order of the methods in Sect. 4, and in the following section we explain how to treat certain rangerestricted interpolation problems. Sect. 6 is devoted to some numerical examples. We conclude with some remarks.

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“…Typical methods involve doing local least squares with low degree polynomials or radial basis functions. For some specific methods and further references, see [4,14]. Remark 2.…”

Section: The Case Of Two-sided Constraintsmentioning

confidence: 99%

Nonnegativity preserving macro-element interpolation of scattered data

Schumaker

Speleers

2010

Computer Aided Geometric Design

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“…Monthly values were interpolated at the centroid of each 0.5 ø grid cell using a triangle-based, piecewise linear interpolation designed for application on the surface of a sphere [Renka, 1982;Renka and Cline, 1984]. It is important for analyses of the impact of changing environmental conditions that the model be in equilibrium with climate conditions at the beginning of the simulation period.…”

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confidence: 99%

Historical variations in terrestrial biospheric carbon storage

Post¹,

King²,

Wullschleger³

1997

Global Biogeochemical Cycles

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“…The proof follows by a straightforward computation. generally used in the literature, (see, e.g., [24]). In Figure 6 …”

Section: Boolean Sum Operatorsmentioning

confidence: 99%

Extension of Some Positive and Linear Operators to Domains with Curved Sides

Cătinaş

2016

MATEC Web Conf.

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Abstract. There are constructed some Cheney-Sharma type operators defined on a square with one curved side. They are extensions of the Cheney-Sharma type operators of second kind, given by E.W. Cheney and A. Sharma in [14], to the case of a curved sided domain. There are constructed the univariate Cheney-Sharma type operators, their product and boolean sum operators and there are studied their properties, their orders of accuracy and the remainders of the corresponding approximation formulas. Finally, there are given some illustrative examples.

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A triangle-based $C^1$ interpolation method

Cited by 167 publications

References 4 publications

Nonnegativity preserving macro-element interpolation of scattered data

Nonnegativity preserving macro-element interpolation of scattered data

Historical variations in terrestrial biospheric carbon storage

Extension of Some Positive and Linear Operators to Domains with Curved Sides

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