scattered data approximation refers to the computation of a multi-dimensional function from measurements obtained from scattered spatial locations. For this problem, the class of methods that adopt a roughness minimiz...
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scattered data approximation refers to the computation of a multi-dimensional function from measurements obtained from scattered spatial locations. For this problem, the class of methods that adopt a roughness minimization are the best performing ones. These methods are called variational methods and they are capable of handling contrasting levels of sample density. These methods express the required solution as a continuous model containing a weighted sum of thin-plate spline or radial basis functions with centres aligned to the measurement locations, and the weights are specified by a linear system of equations. The main hurdle in this type of method is that the linear system is ill-conditioned. Further, getting the weights that are parameters of the continuous model representing the solution is only a part of the effort. Getting a regular grid image requires re-sampling of the continuous model, which is typically expensive. We develop a computationally efficient and numerically stable method based on roughness minimization. The method leads to an algorithm that uses standard regular grid array operations only, which makes it attractive for parallelization. We demonstrate experimentally that we get these computational advantages only with a little compromise in performance when compared with thin-plate spline methods.
One common problem encountered in many fields is the generation of surfaces based on values at irregularly distributed nodes. To tackle such problems, we present a modified, robust moving least squares (MLS) method fo...
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One common problem encountered in many fields is the generation of surfaces based on values at irregularly distributed nodes. To tackle such problems, we present a modified, robust moving least squares (MLS) method for scattereddata smoothing and approximation. The error functional used in the derivation of the classical MLS approximation is augmented with additional terms based on the coefficients of the polynomial base functions. This allows quadratic base functions to be used with the same size of the support domain as linear base functions, resulting in better approximation capability. The increased robustness of the modified MLS method to irregular nodal distributions makes it suitable for use across many fields. The analysis is supported by several univariate and bivariate examples. Crown Copyright (C) 2015 Published by Elsevier Inc. All rights reserved.
A new multilevel approximation scheme for scattereddata is proposed. The scheme relies on an adaptive domain decomposition strategy using quadtree techniques (and their higher-dimensional generalizations). It is show...
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A new multilevel approximation scheme for scattereddata is proposed. The scheme relies on an adaptive domain decomposition strategy using quadtree techniques (and their higher-dimensional generalizations). It is shown in the numerical examples that the new method achieves an improvement on the approximation quality of previous well-established multilevel interpolation schemes.
A Detection Algorithm for the localisation of unknown fault lines of a surface from scattereddata is given. The method is based on a local approximation scheme using thin plate splines, and we show that this yields a...
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A Detection Algorithm for the localisation of unknown fault lines of a surface from scattereddata is given. The method is based on a local approximation scheme using thin plate splines, and we show that this yields approximation of second order accuracy instead of first order as in the global case. Furthermore, the Detection Algorithm works with triangulation methods, and we show their utility for the approximation df the fault lines. The output of our method provides polygonal curves which can be used for the purpose of constrained surface approximation.
We construct generalized inverses to solve least squares problems with partially prescribed kernel and image spaces. To this end we parameterize a special subset of all (1, 3)-generalized inverses, and analyze their p...
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We construct generalized inverses to solve least squares problems with partially prescribed kernel and image spaces. To this end we parameterize a special subset of all (1, 3)-generalized inverses, and analyze their properties. Furthermore, we discuss an application to scattered data approximation where certain (1, 3)-generalized inverses are more adequate than the Moore-Penrose inverse. (C) 2012 Elsevier Inc. All rights reserved.
For scattered data approximation with multilevel B-spline(MBS) method, accuracy could be enhanced by densifying control lattice. Nevertheless, when control lattice density reaches to some extent, approximation accurac...
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ISBN:
(纸本)9783037851937
For scattered data approximation with multilevel B-spline(MBS) method, accuracy could be enhanced by densifying control lattice. Nevertheless, when control lattice density reaches to some extent, approximation accuracy could not be enhanced further. A strategy based on integration of moving least squares(MLS) and multilevel B-spline(MBS) is presented. Experimental results demonstrate that the presented strategy has higher approximation accuracy.
Choosing models from a hypothesis space is a frequent task in approximation theory and inverse problems. Cross-validation is a classical tool in the learner's repertoire to compare the goodness of fit for differen...
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Choosing models from a hypothesis space is a frequent task in approximation theory and inverse problems. Cross-validation is a classical tool in the learner's repertoire to compare the goodness of fit for different reconstruction models. Much work has been dedicated to computing this quantity in a fast manner but tackling its theoretical properties occurs to be difficult. So far, most optimality results are stated in an asymptotic fashion. In this paper we propose a concentration inequality on the difference of cross-validation score and the risk functional with respect to the squared error. This gives a preasymptotic bound which holds with high probability. For the assumptions we rely on bounds on the uniform error of the model which allow for a broadly applicable *** support our claims by applying this machinery to Shepard's model, where we are able to determine precise constants of the concentration inequality. Numerical experiments in combination with fast algorithms indicate the applicability of our results.(c) 2022 Elsevier Inc. All rights reserved.
We consider scattered data approximation problems on SO(3). To this end, we construct a new operator for polynomial approximation on the rotation group. This operator reproduces Wigner-D functions up to a given degree...
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We consider scattered data approximation problems on SO(3). To this end, we construct a new operator for polynomial approximation on the rotation group. This operator reproduces Wigner-D functions up to a given degree and has uniformly bounded L-p-operator norm for all 1 <= p <= infinity. The operator provides a polynomial approximation with the same approximation degree of the best polynomial approximation. Moreover, the operator together with a Markov type inequality for Wigner-D functions enables us to derive scattereddata L-p-Marcinkiewicz-Zygmund inequalities for these functions for all 1 <= p <= infinity. As a major application of such inequalities, we consider the stability of the weighted least squares approximation problem on SO(3).
We present a three-stage scheme for constructing smooth grid functions approximating data defined over scattered point sets in R(S). The scheme is useful for approximating large scattereddata sets and is particularly...
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We present a three-stage scheme for constructing smooth grid functions approximating data defined over scattered point sets in R(S). The scheme is useful for approximating large scattereddata sets and is particularly successful when the data points are unevenly distributed. The paper includes several examples of grid surface construction over the plane.
A new multivariate approximation scheme to scattereddata on arbitrary bounded domains in R(d) is developed. The approximant is selected from a space spanned (essentially) by corresponding translates of the 'shift...
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A new multivariate approximation scheme to scattereddata on arbitrary bounded domains in R(d) is developed. The approximant is selected from a space spanned (essentially) by corresponding translates of the 'shifted' thin-plate spline ('essentially,' since the space is augmented by certain functions in order to eliminate boundary effects). This scheme applies to noisy data as well as to noiseless data, but its main advantage seems to be in the former case. We suggest an algorithm for the new approximation scheme with a detailed description (in a MATLAB-like program). Some numerical examples are presented along with comparisons with thin-plate spline interpolation and Wahba's thin-plate smoothing spline approximation.
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