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The approximation power of moving leastsquares
 Math. Comp
, 1998
"... Abstract. A general method for nearbest approximations to functionals on Rd, using scattereddata information is discussed. The method is actually the moving leastsquares method, presented by the BackusGilbert approach. It is shown that the method works very well for interpolation, smoothing and ..."
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Cited by 108 (6 self)
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Abstract. A general method for nearbest approximations to functionals on Rd, using scattereddata information is discussed. The method is actually the moving leastsquares method, presented by the BackusGilbert approach. It is shown that the method works very well for interpolation, smoothing and derivatives ’ approximations. For the interpolation problem this approach gives Mclain’s method. The method is nearbest in the sense that the local error is bounded in terms of the error of a local best polynomial approximation. The interpolation approximation in Rd is shown to be a C ∞ function, and an approximation order result is proven for quasiuniform sets of data points. 1.
Image Warping with Scattered Data Interpolation Methods
, 1992
"... Image warping has many applications in art as well as in image processing. Usually, displacements are computed with mathematical functions or by transformations of a triangulation of control points. Here, different approaches based on scattered data interpolation methods are presented. These methods ..."
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Cited by 75 (3 self)
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Image warping has many applications in art as well as in image processing. Usually, displacements are computed with mathematical functions or by transformations of a triangulation of control points. Here, different approaches based on scattered data interpolation methods are presented. These methods provide smooth deformations with easily controllable behavior. The usefulness and performance of some selected classes of scattered data interpolation methods in this context is analyzed.
Global Optimization with NonAnalytical Constraints
"... This paper presents an approach for the global optimization of constrained nonlinear programming problems in which some of the constraints are nonanalytical (nonfactorable), defined by a computational model for which no explicit analytical representation is available. ..."
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This paper presents an approach for the global optimization of constrained nonlinear programming problems in which some of the constraints are nonanalytical (nonfactorable), defined by a computational model for which no explicit analytical representation is available.
Applications Of Spatial Databases And Structures To The Study Of Miocene Deposits Of Borod Basin
, 2003
"... In this paper we apply spatial database and structure to render the paleorelief of the Borod Basin. The boreholes data are stored in a spatial database. For the effective surface rendering we use localShepard interpolation with variable radius, based on a spatial grid and Delaunay triangulation. T ..."
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In this paper we apply spatial database and structure to render the paleorelief of the Borod Basin. The boreholes data are stored in a spatial database. For the effective surface rendering we use localShepard interpolation with variable radius, based on a spatial grid and Delaunay triangulation. The generated pictures are more realistic compared to pictures generated by means of other methods.
Article electronically published on August 8, 2006 SHEPARD–BERNOULLI OPERATORS
"... Abstract. We introduce the Shepard–Bernoulli operator as a combination of the Shepard operator with a new univariate interpolation operator: the generalized Taylor polynomial. Some properties and the rate of convergence of the new combined operator are studied and compared with those given for class ..."
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Abstract. We introduce the Shepard–Bernoulli operator as a combination of the Shepard operator with a new univariate interpolation operator: the generalized Taylor polynomial. Some properties and the rate of convergence of the new combined operator are studied and compared with those given for classical combined Shepard operators. An application to the interpolation of discrete solutions of initial value problems is given. 1. The problem Let X = {x1,...,xN} be a set of N distinct points of Rs,s ∈ N, andletfbe a function defined on a domain D containing X. The classical Shepard operators (first introduced in [24] in the particular case s = 2) are defined by N∑ (1.1) SN,µ [f](x) = Aµ,i (x) f (xi) , µ> 0, i=1 where the weight functions Aµ,i (x) in barycentric form are (1.2) Aµ,i (x) = −µ x − xi N∑ x − xk  −µ k=1 and ·  denotes the Euclidean norm in R s. The interpolation operator SN,µ [f] is stable, in the sense that min f (xi) ≤ SN,µ [f](x) ≤ max i i f (xi), but for µ>1 the interpolating function SN,µ [f](x) hasflatspotsintheneighborhood of all data points. Also, the degree of exactness of the operator SN,µ [·] is0,in the sense that if it is restricted to the polynomial space P m: = {p:deg(p) ≤ m}, then SN,µ [·] P m = IdP m (the identity function on P m)onlyform=0. These drawbacks can be avoided by replacing each value f (xi) in (1.1) with an interpolation operator in xi, applied to f, having a certain degree of exactness m>0. More precisely, if for each i =1,...,N P [·,xi] denotes such an interpolation operator in xi, then the related combined Shepard operator is N∑ (1.3) SN,µP [f](x) = Aµ,i (x) P [f,xi](x).