Sur la construction de surfaces de classe $C^k$ à partir d’un grand nombre de données de Lagrange

Apprato, D.; Arcangeli, Rémi; Manzanilla, Raúl

doi:10.1051/m2an/1987210405291

Cited by 17 publications

(4 citation statements)

References 3 publications

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“…[29]), finite elements [4] or iterative methods (see [9]) to process large data sets, 'discontinuity-preserving' smoothing by inf-convolution spline (see Laurent [23], Laurent and Utreras [24]), integral equations (eventually non-linear) (Wahba [35]), and SO on... It is well known that the use of only a few iterations of, for instance, the Richardson method to minimize (1.2), has also some smoothing properties (e.g.…”

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confidence: 99%

A fast ?Monte-Carlo cross-validation? procedure for large least squares problems with noisy data

1989

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Summary.We propose a fast Monte-Carlo algorithm for calculating reliable estimates of the trace of the influence matrix At involved in regularization of linear equations or data smoothing problems, where z is the regularization or smoothing parameter. This general algorithm is simply as follows: i) generate n pseudo-random values wl ..... Wn, from the standard normal distribution (where n is the number of data points) and let w=(wl, ..., w,) r, ii) compute the residual vector w-A~w, iii) take the 'normalized' inner-product (wT(w--A~w))/(wTw) as an approximation to (1/n)tr(l-A O. We show, both by theoretical bounds and by numerical simulations on some typical problems, that the expected relative precision of these estimates is very good when n is large enough, and that they can be used in practice for the minimization with respect to z of the well known Generalized Cross-Validation (GCV) function. This permits the use of the GCV method for choosing z in any particular large-scale application, with only a similar amount of work as the standard residual method. Numerical applications of this procedure to optimal spline smoothing in one or two dimensions show its efficiency.

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confidence: 99%

A fast ?Monte-Carlo cross-validation? procedure for large least squares problems with noisy data

1989

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“…In general, the type of representation considered is strongly related to the domain of application. For instance, triangulations, sudivision surfaces, discretized level sets, scalar spline functions, radial basis functions or point set surfaces have been investigated [5,6,11]. It is beyond the scope of this paper to review these approaches, but we refer the reader to the paper [27,35] for a survey of the topic of surface reconstruction.…”

Section: Related Workmentioning

confidence: 99%

Level set driven smooth curve approximation from unorganized or noisy point set

Claisse

Frey

2009

ESAIM: Proc.

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Abstract. In this paper, we propose a curve construction method for a non uniform point data set based on a minimal curve approximation model. Numerically, the level set method is used for curve reconstruction. We represent the shape of the curve through its distance function and formulate curve reconstruction as a constrained minimization problem. We solve the minimization problem on a highly anisotropic triangulation to improve the accuracy of the numerical scheme. This method can handle complex geometries and deal with arbitrary topologies as well as with noisy data sets. Several numerical examples are provided to show the efficiency of the proposed approach.Résumé. Dans ce papier, on propose un modèle de courbe d'approximation minimale pour construire une courbeà partir d'un nuage de points. Numériquement, la reconstruction de la courbe s'appuie sur une formulation de type ligne de niveau. On représente la forme de la courbe par sa fonction distance aux points et on exprime ce problème comme un problème de minimisation. Ce dernier est résolu sur une triangulation anisotrope qui permet d'améliorer la précision du schéma numérique. Cette méthode permet de traiter des géométries complexes et des topologies quelconques ainsi que des données bruitées. Des exemples de reconstruction sont proposés pour montrer l'éfficacité de cette approche.

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“…An explicit Lagrangian approach can be used to simulate (2). In this case, the topology and geometry of the model are explicitly represented during the simulation of the contour flow.…”

Section: Segmentation Processmentioning

confidence: 99%

Approximation of surfaces with fault(s) and/or rapidly varying data, using a segmentation process, D m -splines and the finite element method

et al. 2008

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In many problems of geophysical interest, one has to deal with data that exhibit complex fault structures. This occurs, for instance, when describing the topography of seafloor surfaces, mountain ranges, volcanoes, islands, or the shape of geological entities, as well as when dealing with reservoir characterization and modelling. In all these circumstances, due to the presence of large and rapid variations in the data, attempting a fitting using conventional approximation methods necessarily leads to instability 68 Numer Algor (2008) 48:67-92 phenomena or undesirable oscillations which can locally and even globally hinder the approximation. As will be shown in this paper, the right approach to get a good approximant consists, in effect, in applying first a segmentation process to precisely define the locations of large variations and faults, and exploiting then a discrete approximation technique. To perform the segmentation step, we propose a quasi-automatic algorithm that uses a level set method to obtain from the given (gridded or scattered) Lagrange data several patches delimited by large gradients (or faults). Then, with the knowledge of the location of the discontinuities of the surface, we generate a triangular mesh (which takes into account the identified set of discontinuities) on which a D m -spline approximant is constructed. To show the efficiency of this technique, we will present the results obtained by its application to synthetic datasets as well as real gridded datasets in Oceanography and Geosciences.

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Sur la construction de surfaces de classe $C^k$ à partir d’un grand nombre de données de Lagrange

Cited by 17 publications

References 3 publications

A fast ?Monte-Carlo cross-validation? procedure for large least squares problems with noisy data

A fast ?Monte-Carlo cross-validation? procedure for large least squares problems with noisy data

Level set driven smooth curve approximation from unorganized or noisy point set

Approximation of surfaces with fault(s) and/or rapidly varying data, using a segmentation process, D m -splines and the finite element method

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