A result about scale transformation families in approximation: application to surface fitting from rapidly varying data

Abstract

Scale transformations are common in approximation. In surface approximation from rapidly varying data, one wants to suppress, or at least dampen the oscillations of the approximation near steep gradients implied by the data. In that case, scale transformations can be used to give some control over overshoot when the surface has large variations of its gradient. Conversely, in image analysis, scale transformations are used in preprocessing to enhance some features present on the image or to increase jumps of grey levels before segmentation of the image. In this paper, we establish the convergence of an approximation method which allows some control over the behavior of the approximation. More precisely, we study the convergence of an approximation from a data set $\{ x_i ,f(x_i )\}$of $\mathbb{R}^n \times \mathbb{R}$, while using scale transformations on the $f(x_i )$values before and after classical approximation. In addition, the construction of scale transformations is also given. The algorithm is presented with some numerical examples.