Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1997. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 24)
HOFFMANN, M. and VÁRADY, L., Free-form curve design by neural networks
Free-form curve design by neural networks MIKLÓS HOFFMANN and LAJOS VÁRADY Abstract. This paper gives a new approach of the two dimensional scattered data manipulation. The standard approximation and interpolation methods which can only be used for non-scattered data will also be applicable for scattered input with the help of the neural network. The Kohonen network produces an ordering of the scattered input points and here the B-spline curve is used for the approximation and interpolation. Introduction The interpolation and the approximation of two dimensional scattered data are interesting problems of computer graphics. By scattered data we mean a set of points without any predefined order. Unfortunately all the standard interpolation and approximation methods —like Hermite interpolation, Bezier curves or B-spline curves —need a sequence of points, hence if we want to apply these methods we have to order the data. A good survey of the scattered data interpolation can be found in [1]. In this paper a completely new approach is given where the self-organizing ability of the neural networks will be used to order the points. The Kohonen network [2,3] can be trained by scattered data, that is the points will form the input of the network, while the weights of the network and their connections give us a polygon, the vertices, of which will be the input points. In this way the polygon we obtained can be used as the control polygon of a B-spline curve, so finally a standard approximation or interpolation method can be applied for the scattered data. We begin our discussion with the short definition of the B-spline curve and Kohonen's neural network. The B-spline curve The B-spline curve is the most common and widely used free-form representation method which can be used as an approximating and also as an interpolating curve [4]. If we have a'sequence of points P{ (i = 1,..., n ), then the curve approximating the plane polygon given by the points is defined as Research supported by the Hungarian National Research Science Foundation, Operating Grant Number OTKA F 019395.
