NIELSON, G. M. Coordinate-free scattered data interpolation. In Topics in Multivariate Approximation, L. Schumaker, C. Chui, and F. Utreras, Eds. Academic Press, Inc., New York, New York, 1987, pp. 175--184.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....data sites .x i ; y i ; z i might be situated on the boundary of a three dimensional object. This is how the surface reconstruction problem occurs in pattern recognition,computer vision and computer graphics. The task is to construct a mathematical model of the three dimensional object [CvD94a, CvD94b, Far83, Nie79, Nie87a, Nie87b, Pet88a, Pet88b, Pet89a, Pet89b, Pet90a, Pet90b, Sch90]. A slightly different problem comes from microscopy [Cah67, DR68, DeH81, DeH82, DAC71, DW90, Eli67, Und70] where the three dimensional structure has to be reconstructed from a sequence of two dimensional images. The direct functional approach does not work in above applications, one of the ....

G. M. Nielson. Coordinate free scattered data interpolation. In C.K. Chui et. al., editor, Topics in Multivariate Approximation, pages 175--184. Academic Press, 1987.

....Distance Metrics As most scattered data interpolation methods define the interpolation function depending on the distance to the data points, particular applications often call for the definition of a suitable metric. For example, an affine invariant metric has been developed by Nielson [8, 9]. This metric is particularly useful when the components of the data sites are not comparable, e.g. when a bivariate function depends on a length and a temperature. Similarly, for the application described here, it appears useful to define a distance metric describing the distance of a point from ....

Gregory M. Nielson. Coordinate free scattered data interpolation. In L.L. Schumaker, C.C. Chui, and F.Utreras et al., editors, Topics in Multivariate Approximation, pages 175--184. Academic Press, New York, 1987.

....of the original one, and thus the shape of the resulting interpolant is different. Unfortunately, this means that the parameterization (and thus the resulting interpolant) is not invariant under affine transformations. To overcome this problem, an affine invariant metric has been proposed in [11] [18] and [19] The metric can be used to measure the distance of the data points in a way that is invariant under affine transformations, and especially under non uniform scalings. 4.6. An Affine Invariant Metric 33 If we are given a set of data points as V = 2 6 6 6 6 6 6 4 x 0 y 0 z 0 x 1 y 1 ....

....can not be found. This means that, for a 3 dimensional interpolation problem, not all the data points must be in one plane, and for a 2 dimensional one not all must be on one line. If such a case happens the parameterization problem has to be solved in a lower dimension. We refer the reader to [18] and [19] for a discussion of why DV (c; d) is a norm, and why it leads to an affine invariant metric. When we replace the Euclidean distance in the chord length parameterization and in the centripetal parameterization by this affine invariant metric, it is immediately clear that the resulting ....

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Gregory M. Nielson. Coordinate free scattered data interpolation. In Larry. L. Schumaker, C. Chui, and F. Utreras, editors, Topics in Multivariate Approximation, pages 175--184. Academic Press, 1987.

....Geometric Meaning of Nielson s Affine Invariant Norm Wendelin L. F. Degen and Volker Milbrandt Mathematisches Institut B, Universitat Stuttgart, Pfaffenwaldring 57, D 70550 Stuttgart, Germany This paper presents a geometric interpretation of the affine invariant metric introduced in (Nielson, 1987) and (Nielson and Foley, 1989) The norm allows the modification of several methods of scattered data interpolation to achieve affine invariance of the interpolating surfaces. Key words: Affine invariant norm, Methods of scattered data interpolation 1 Introduction In his paper, Nielson, 1987) ....

....in (Nielson, 1987) and (Nielson and Foley, 1989) The norm allows the modification of several methods of scattered data interpolation to achieve affine invariance of the interpolating surfaces. Key words: Affine invariant norm, Methods of scattered data interpolation 1 Introduction In his paper, (Nielson, 1987) introduced an affine invariant metric in the plane to achieve affine invariance (invariance with respect to translation, rotation, scalar and scale multiplication) of some widely used methods of scattered data interpolation. Most methods of scattered data interpolation are a priori not affine ....

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Nielson, G. M. (1987), Coordinate Free Scattered Data Interpolation, in: Schumaker, L. L., Chui, C. and Utreras, F., eds., Topics in Multivariate Approximation.

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NIELSON, G. M. Coordinate-free scattered data interpolation. In Topics in Multivariate Approximation, L. Schumaker, C. Chui, and F. Utreras, Eds. Academic Press, Inc., New York, New York, 1987, pp. 175--184.

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Nielson, G. 1987. Coordinate Free Scattered Data Interpolation. Topics in Multivariate Approximation.

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