New interpolation method with fractal curves

• Authors: Andrzej Cader;Marcin Krupski
• Venue: ICAISC'06 Proceedings of the 8th international conference on Artificial Intelligence and Soft Computing
• Year: 2006

Classic interpolation methods using polynominals or other functions for reconstruction complex dependences or contours perform their role well in the case of smooth and relatively regular curves. However, many shapes found in nature or dynamic relations corresponding to real process are of very irregular character and the appropriate characteristics are rough and demonstrate a complex structure at difference scales. This type of curves are numbered among fractals or stochastic fractals – multifractals. In practice it is impossible to approximate them with the help of classic methods. It is necessary to use fractal methods for the interpolation. At present the only group of this type of methods are the ones based on fractal interpolation functions (FIFs) suggested by Barnsley [1]. However, these methods are burdened with numerous inadequacies making it difficult to use them in practice. The study presents another alternative method of using fractal curves for complex curves approximation. This method is more adequate than FIF for multifractal structures interpolation. It generalizes classic notion of an interpolation knot and introduces non-local values for its description, as for instance fractal dimension. It also suggests continuous, as regards fractal dimension, family of fractal curves as a set of base elements of approximation – an equivalent of base splines. In this aspect the method is similar to the classic B-splines method and does not use Iterated Function Systems (IFS), as Barnsey's method does. It may be determined as a hard interpolation method aiming at working out an algorithm providing its effective application in practice, whereas to a lesser degree attention is paid to mathematical elegance.