Smooth interpolation of large sets of scattered data

Abstract

Methods for solving the following data‐fitting problems are discussed: given the data (xi, yi, fi),i = 1,…,N construct a smooth bivariate function S with the property that S(xi, yi) = fii = 1,…,N. Because the desire to fit this type of data is encountered frequently in many areas of scientific applications, an investigation of the available methods for solving this problem was undertaken. Several aspects, such as computational efficiency, fitting characteristics and ease of implementation, were analysed and compared. Within the context of a general‐purpose method for large sets of data, two of these methods emerged as being generally superior to the others. It is the purpose of this paper to describe these two methods and present examples illustrating their use and application.

Primerjava metod prostorske interpolacije in večslojnih nevronskih mrež za različne geometrijske razporeditve točk na digitalnem modelu višin

“…The Modified Shepard's Method can be either an exact or a smoothing interpolator. Franke and Nielson (1980) developed a modification that eliminates the insufficiency of the Shepard's method, and does not tend to generate "bull's eye" patterns.…”

Regularized implicit surface reconstruction from points and normals

“…A partition of unity (PU) [1], [4] is a mathematical tool very useful to combine local approximation in order to define a global one. Important properties such as the global maximal error and the convergence order could be inherited from the local approximation.…”

Surfaces Generated by Moving Least Squares Methods

“…Particular versions of the latter process have previously been applied in such programs as the SURFACE II package of the Kansas Geological Survey. After the preparation of this paper, the recent work of Franke and Nielson [3] came to our attention. In [3] too, the projector P is derived as an alternative to G, and some practical guidance to actual computation is given.…”