Positivity-Preserving Scattered Data Interpolation

Piah, Abd Rahni Mt; Goodman, T. N. T.; Unsworth, K.

doi:10.1007/11537908_20

Cited by 27 publications

(35 citation statements)

References 6 publications

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“…A sufficient criterion for nonnegativity is to require that all Bcoefficients of the polynomial are nonnegative. Weaker sufficient conditions were given in [1,11]. For some related optimization strategies, see [9].…”

Section: Remarkmentioning

confidence: 99%

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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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Section: Remarkmentioning

confidence: 99%

“…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.…”

Section: §1 Introductionmentioning

confidence: 99%

Nonnegativity preserving macro-element interpolation of scattered data

Schumaker

Speleers

2010

Computer Aided Geometric Design

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“…In [3] and [5], a lower bound on all Bézier ordinates except at the vertices is derived to preserve the non-negativity for the patch. Such lower bound is motivated by the one given in [4] for cubic Bézier patch. It depends on the values at the triangle vertices.…”

Section: Sufficient Range Restriction Conditionmentioning

confidence: 99%

“…Besides, the data that lie on one side of a constraint surface need to have an interpolant on the same side of the constraint. In recent years, a decent number of preservation methods have been published, such as [1], [2], [3], [4] and [5].…”

Section: Introductionmentioning

confidence: 99%

Constrained C1 scattered data interpolation using rational blend

Chua¹,

Pang²

2014

AIP Conference Proceedings

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“…Particularly, when data from some scientific observation is studied, a user may be interested to visualize it graphically. The properties that are most often used to quantify "shape" are convexity, monotonicity (for non-parametric data) and positivity [2]. The problem of positivity preserving interpolation, i.e.…”

Section: Introductionmentioning

confidence: 99%