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Scattered Data Interpolation with Multilevel Splines
 IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS
, 1997
"... This paper describes a fast algorithm for scattered data interpolation and approximation. Multilevel Bsplines are introduced to compute a C²continuous surface through a set of irregularly spaced points. The algorithm makes use of a coarsetofine hierarchy of control lattices to generate a sequen ..."
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Cited by 109 (9 self)
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This paper describes a fast algorithm for scattered data interpolation and approximation. Multilevel Bsplines are introduced to compute a C²continuous surface through a set of irregularly spaced points. The algorithm makes use of a coarsetofine hierarchy of control lattices to generate a sequence of bicubic Bspline functions whose sum approaches the desired interpolation function. Large performance gains are realized by using Bspline refinement to reduce the sum of these functions into one equivalent Bspline function. Experimental results demonstrate that highfidelity reconstruction is possible from a selected set of sparse and irregular samples.
Multistep scattered data interpolation using compactly supported radial basis functions
 J. Comp. Appl. Math
, 1996
"... Abstract. A hierarchical scheme is presented for smoothly interpolating scattered data with radial basis functions of compact support. A nested sequence of subsets of the data is computed efficiently using successive Delaunay triangulations. The scale of the basis function at each level is determine ..."
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Cited by 65 (12 self)
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Abstract. A hierarchical scheme is presented for smoothly interpolating scattered data with radial basis functions of compact support. A nested sequence of subsets of the data is computed efficiently using successive Delaunay triangulations. The scale of the basis function at each level is determined from the current density of the points using information from the triangulation. The method is rotationally invariant and has good reproduction properties. Moreover the solution can be calculated and evaluated in acceptable computing time. During the last two decades radial basis functions have become a well established tool for multivariate interpolation of both scattered and gridded data; see [2,7,8,22,25] for some surveys. The major part
Image Warping by Radial Basis Functions: Application to Facial Expressions
 CVGIP: Graphical Models and Image Processing
, 1994
"... The human face is an elastic object. A natural paradigm for representing facial expressions is to form a complete 3D model of facial muscles and tissues. However, determining the actual parameter values for synthesizing and animating facial expressions is tedious; evaluating these parameters for fac ..."
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Cited by 64 (3 self)
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The human face is an elastic object. A natural paradigm for representing facial expressions is to form a complete 3D model of facial muscles and tissues. However, determining the actual parameter values for synthesizing and animating facial expressions is tedious; evaluating these parameters for facial expression analysis out of greylevel images is ahead of the state of the art in computer vision. Using only 2D face images and a small number of anchor points, we show that the method of radial basis functions provides a powerful mechanism for processing facial expressions. Although constructed specifically for facial expressions, our method is applicable to other elastic objects as well.
Notes on ScatteredData RadialFunction Interpolation
"... We will be considering two types of interpolation problems. Given a continuous function h: Rn → R, a set of vectors X = {xj} N j=1 in Rn and scalars {yj} N j=1, one version of the scattered data interpolation problem consists in finding a function f such that the system of equations f(xj) = yj, j = ..."
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Cited by 1 (0 self)
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We will be considering two types of interpolation problems. Given a continuous function h: Rn → R, a set of vectors X = {xj} N j=1 in Rn and scalars {yj} N j=1, one version of the scattered data interpolation problem consists in finding a function f such that the system of equations f(xj) = yj, j = 1,...,N has a solution of the form (1.1) f(x) = N∑ cjh(x − xj). j=1 Equivalently, one wishes to know when the N ×N matrix A with entries Aj,k = h(xj −xk) is invertible. In the second version of the scattered data interpolation problem, we require polynomial reproduction. Let πm−1(Rn) be the set of polynomials in n variables having degree m − 1 or less. In multiindex notation, p ∈ πm−1(Rn) has the form p(x) = pαx α,
Negative Observations Concerning Approximations From Spaces Generated By Scattered Shifts of Functions Vanishing At Infinity
"... : Approximation by scattered shifts fOE(\Delta \Gamma ff)g ff2A of a basis function OE are considered, and different methods for localizing these translates are compared. It is argued in the note that the superior localization processes are those that employ the original translates only. AMS (MOS) ..."
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Cited by 1 (0 self)
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: Approximation by scattered shifts fOE(\Delta \Gamma ff)g ff2A of a basis function OE are considered, and different methods for localizing these translates are compared. It is argued in the note that the superior localization processes are those that employ the original translates only. AMS (MOS) Subject Classifications: 41A15, 41A25, 41A63 Key Words and phrases: approximation order, StrangFix conditions, scattered shifts, radial basis functions, localization of splines. Supported in part by the U.S. Army (Contract DAAL03G900090), by the National Science Foundation (grant DMS9102857), and by the IsraelU.S. Binational Science Foundation (grant 9000220) Negative observations concerning approximations from spaces generated by scattered shifts of functions vanishing at 1 Amos Ron 1. Introduction In recent years, approximation from spaces spanned by integer translates of one or several functions became a major theme in various areas of approximation theory (and in other field...
and Theoretical Physics
, 1990
"... Approximation properties of the dilations of the integer translates of a smooth function, with some derivatives vanishing at infinity, are studied. The results apply to fundamental solutions of homogeneous elliptic operators and to “shifted ” fundamental solutions of the iterated Laplacian. Followin ..."
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Approximation properties of the dilations of the integer translates of a smooth function, with some derivatives vanishing at infinity, are studied. The results apply to fundamental solutions of homogeneous elliptic operators and to “shifted ” fundamental solutions of the iterated Laplacian. Following the approach from spline theory, the question of polynomial reproduction by quasiinterpolation is addressed first. The analysis makes an essential use of the structure of the generalized Fourier transform of the basis function. In contrast with spline theory, polynomial reproduction is not sufficient for the derivation of exact order of convergence by dilated quasiinterpolants. These convergence orders are established by a careful and quite involved examination of the decay rates of the basis function. Furthermore, it is shown that the same approximation orders are obtained with quasiinterpolants defined on a bounded domain.
spaces generated by scattered shifts of functions vanishing at ∞
"... observations concerning approximations from spaces generated by scattered shifts of functions vanishing at ∞ ..."
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observations concerning approximations from spaces generated by scattered shifts of functions vanishing at ∞