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Small Strictly Convex Quadrilateral Meshes of Point Sets
, 2004
"... In this paper we give upper and lower bounds on the number of Steiner points required to construct a strictly convex quadrilateral mesh for a planar point set. In particular, we show that 3⌊n/2 ⌋ internal Steiner points are always sufficient for a convex quadrilateral mesh of n points in the plane. ..."
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In this paper we give upper and lower bounds on the number of Steiner points required to construct a strictly convex quadrilateral mesh for a planar point set. In particular, we show that 3⌊n/2 ⌋ internal Steiner points are always sufficient for a convex quadrilateral mesh of n points in the plane. Furthermore, for any given n ≥ 4, there are point sets for which ⌈(n − 3)/2⌉−1 Steiner points are necessary for a convex quadrilateral mesh.
Optimal quasiinterpolation by quadratic C1 splines on fourdirectional meshes. Approximation Theory XI: Gatlinburg 2004
, 2005
"... Abstract. We describe a new scheme based on quadratic C1splines on type2 triangulations approximating gridded data. The quasiinterpolating splines are directly determined by setting the BernsteinBézier coefficients of the splines to appropriate combinations of the given data values. In this way ..."
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Abstract. We describe a new scheme based on quadratic C1splines on type2 triangulations approximating gridded data. The quasiinterpolating splines are directly determined by setting the BernsteinBézier coefficients of the splines to appropriate combinations of the given data values. In this way, each polynomial piece of the approximating spline is immediately available from local portions of the data, without using prescribed derivatives at any point of the domain. Since the BernsteinBézier coefficients of the splines are computed directly, an intermediate step making use of certain locally supported splines spanning the space is not needed. We prove that the splines yield optimal approximation order for smooth functions, where we provide explicit constants in the corresponding error bounds. The aim of this paper is to describe local methods which use quadratic C1splines on the type2 triangulation to approximate data on a
Energy minimization method for scattered data Hermite interpolation
 APPLIED NUMERICAL MATHEMATICS
, 2007
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Experimental Results on Quadrangulations of Sets of Fixed Points
"... Abstract We consider the problem of obtaining "nice " quadrangulations of planar sets of points. For many applications "nice " means that the quadrilaterals obtained are convex if possible and as"fat " or squarish as possible. For a given set of ..."
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Abstract We consider the problem of obtaining &quot;nice &quot; quadrangulations of planar sets of points. For many applications &quot;nice &quot; means that the quadrilaterals obtained are convex if possible and as&quot;fat &quot; or squarish as possible. For a given set of points a quadrangulation, if it exists, may not admit all its quadrilaterals to be convex. In such cases we desire that the quadrangulationshave as many convex quadrangles as possible. Solving this problem optimally is not practical. Therefore we propose and experimentally investigate a heuristic approach to solve this problem by converting &quot;nice &quot; triangulations to the desired quadrangulations with the aid of maximummatchings computed on the dual graph of the triangulations. We report experiments on several versions of this approach and provide theoretical justification for the good results obtained with one of these methods. The results of our experiments are particularly relevant for thoseapplications in scattered data interpolation which require quadrangulations that should stay faithful to the original data.
Experimental Results on Quadrangulations of Sets of Fixed Points
, 1997
"... We consider the problem of obtaining "nice" quadrangulations of planar sets of points. For many applications "nice" means that the quadrilaterals obtained are convex if possible and as "fat" or squarish as possible. For a given set of points a quadrangulation, if it e ..."
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We consider the problem of obtaining "nice" quadrangulations of planar sets of points. For many applications "nice" means that the quadrilaterals obtained are convex if possible and as "fat" or squarish as possible. For a given set of points a quadrangulation, if it exists, may not admit all its quadrilaterals to be convex. In such cases we desire that the quadrangulations have as many convex quadrangles as possible. Solving this problem optimally is not practical.
Scattered Data Interpolation by Bivariate Splines with Higher Approximation Order∗
, 2012
"... Given a set of scattered data, we usually use a minimal energy method to nd Lagrange interpolation based on bivariate spline spaces over a triangulation of the scattered data locations. It is known that the approximation order of the minimal energy spline interpolation is only 2 in terms of the size ..."
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Given a set of scattered data, we usually use a minimal energy method to nd Lagrange interpolation based on bivariate spline spaces over a triangulation of the scattered data locations. It is known that the approximation order of the minimal energy spline interpolation is only 2 in terms of the size of triangulation. To improve this order of approximation, we propose several new schemes in this paper. Mainly we follow the ideas of clamped cubic interpolatory splines and notaknot interpolatory splines in the univariate setting and extend them to the bivariate setting. In addition, instead of the energy functional of the second order, we propose to use higher order versions. We shall present some theoretical analysis as well as many numerical results to demonstrate that our bivariate spline interpolation schemes indeed have a higher order of approximation than the classic minimal energy interpolatory splines.