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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 ..."
Abstract

Cited by 108 (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.
Data Mining With Sparse Grids
 Computing
, 2001
"... We present a new approach to the classification problem arising in data mining. It is based on the regularization network approach but, in contrast to the other methods which employ ansatz functions associated to data points, we use a grid in the usually highdimensional feature space for the min ..."
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Cited by 18 (9 self)
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We present a new approach to the classification problem arising in data mining. It is based on the regularization network approach but, in contrast to the other methods which employ ansatz functions associated to data points, we use a grid in the usually highdimensional feature space for the minimization process. To cope with the curse of dimensionality, we employ sparse grids. Thus, only O(h \Gamma1 n n d\Gamma1 ) instead of O(h \Gammad n ) grid points and unknowns are involved. Here d denotes the dimension of the feature space and h n = 2 \Gamman gives the mesh size. To be precise, we suggest to use the sparse grid combination technique where the classification problem is discretized and solved on a certain sequence of conventional grids with uniform mesh sizes in each coordinate direction. The sparse grid solution is then obtained from the solutions on these different grids by linear combination. In contrast to other sparse grid techniques, the combination metho...
Smooth approximation and rendering of large scattered data sets
 In Proceedings of the conference on Visualization ’01
, 2001
"... We present an efficient method to automatically compute a smooth approximation of large functional scattered data sets given over arbitrarily shaped planar domains. Our approach is based on the construction of a ¢¤ £continuous bivariate cubic spline and our method offers optimal approximation order ..."
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Cited by 16 (4 self)
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We present an efficient method to automatically compute a smooth approximation of large functional scattered data sets given over arbitrarily shaped planar domains. Our approach is based on the construction of a ¢¤ £continuous bivariate cubic spline and our method offers optimal approximation order. Both local variation and nonuniform distribution of the data are taken into account by using local polynomial least squares approximations of varying degree. Since we only need to solve small linear systems and no triangulation of the scattered data points is required, the overall complexity of the algorithm is linear in the total number of points. Numerical examples dealing with several real world scattered data sets with up to millions of points demonstrate the efficiency of our method. The resulting spline surface is of high visual quality and can be efficiently evaluated for rendering and modeling. In our implementation we achieve realtime frame rates for typical flythrough sequences and interactive frame rates for recomputing and rendering a locally modified spline surface. CR Categories: G.1.2 [Numerical Analysis]: Approximation— approximation of surfaces, least squares approximation, spline and piecewise polynomial approximation; I.3.3 [Computer Graphics]: Picture/Image Generation—display algorithms, viewing algorithms; I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling—surface representation, splines; E.4 [Coding
Adaptive smooth scattereddata approximation for largescale terrain visualization
 IN PROCEEDINGS OF THE SYMPOSIUM ON DATA VISUALISATION 2003
, 2003
"... We present a fast method that adaptively approximates largescale functional scattered data sets with hierarchical Bsplines. The scheme is memory efficient, easy to implement and produces smooth surfaces. It combines adaptive clustering based on quadtrees with piecewise polynomial least squares app ..."
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Cited by 11 (5 self)
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We present a fast method that adaptively approximates largescale functional scattered data sets with hierarchical Bsplines. The scheme is memory efficient, easy to implement and produces smooth surfaces. It combines adaptive clustering based on quadtrees with piecewise polynomial least squares approximations. The resulting surface components are locally approximated by a smooth Bspline surface obtained by knot removal. Residuals are computed with respect to this surface approximation, determining the clusters that need to be recursively refined, in order to satisfy a prescribed error bound. We provide numerical results for two terrain data sets, demonstrating that our algorithm works efficiently and accurate for large data sets with highly nonuniform sampling densities.
Classification With Sparse Grids Using Simplicial Basis Functions
, 2002
"... Recently we presented a new approach [20] to the classification problem arising in data mining. It is based on the regularization network approach but in contrast to other methods, which employ ansatz functions associated to data points, we use a grid in the usually highdimensional feature space fo ..."
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Cited by 7 (7 self)
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Recently we presented a new approach [20] to the classification problem arising in data mining. It is based on the regularization network approach but in contrast to other methods, which employ ansatz functions associated to data points, we use a grid in the usually highdimensional feature space for the minimization process. To cope with the curse of dimensionality, we employ sparse grids [52]. Thus, only O(h 1 n n d 1 ) instead of O(h d n ) grid points and unknowns are involved. Here d denotes the dimension of the feature space and hn = 2 n gives the mesh size. We use the sparse grid combination technique [30] where the classification problem is discretized and solved on a sequence of conventional grids with uniform mesh sizes in each dimension. The sparse grid solution is then obtained by linear combination. The method computes a nonlinear classifier but scales only linearly with the number of data points and is well suited for data mining applications where the amount of data is very large, but where the dimension of the feature space is moderately high. In contrast to our former work, where dlinear functions were used, we now apply linear basis functions based on a simplicial discretization. This allows to handle more dimensions and the algorithm needs less operations per data point. We further extend the method to socalled anisotropic sparse grids, where now different apriori chosen mesh sizes can be used for the discretization of each attribute. This can improve the run time of the method and the approximation results in the case of data sets with different importance of the attributes. We describe the sparse grid combination technique for the classification problem, give implementational details and discuss the complexity of the algorithm. It turns out that...
Data mining with sparse grids using simplicial basis functions
 In Proceedings of the Seventh ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, 2001. also as SFB 256 Preprint 713, Universitat
, 2001
"... basis functions ..."
Introductory Overview of Surface Reconstruction Methods,” Lulea
 University of Technology, Dept
, 1999
"... We consider the issue of constructing a digital model of a physical surface. An overview of fundamental methods is given. The methods use measurements of points on the surface in order to compute a mathematical surface representation that approximates the physical surface. Different surface represen ..."
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Cited by 3 (0 self)
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We consider the issue of constructing a digital model of a physical surface. An overview of fundamental methods is given. The methods use measurements of points on the surface in order to compute a mathematical surface representation that approximates the physical surface. Different surface representations are presented and algorithms that fit such representations to the measurements are outlined. We identify some of the difficulties and pointers to the literature in the field are
Scattered Data Interpolation Using Data Dependant Optimization Techniques
, 1995
"... Interpolation of scattered data has many applications in different areas. Recently, this problem has gained a lot of interest for CAD applications, in combination with the process of "reverse engineering", i.e. the construction of CAD models for existing objects. Up until now, no method for Scattere ..."
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Cited by 2 (1 self)
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Interpolation of scattered data has many applications in different areas. Recently, this problem has gained a lot of interest for CAD applications, in combination with the process of "reverse engineering", i.e. the construction of CAD models for existing objects. Up until now, no method for Scattered Data Interpolation with a bivariate function produces surface formats that can be directly integrated into a CAD system. Additionally many of the existing interpolation schemes exhibit undesirable curvature distribution of the reconstructed surface. In this paper we present a method for Scattered Data Interpolation producing tensorproduct Bsplines with high quality curvature distribution. This method first determines the knot vectors in a way that guarantees the existence of an interpolating Bspline. In a second step the degrees of freedom not specified by the interpolation constraints are automatically set using a data dependent optimization technique. Examples