Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Barnsley, M.F.: Fractal Functions and Interpolation. Constructive Approximation 2, 303–332 (1986)
Mandelbrot, B.: The Fractal Geometry of Nature. Freeman, San Francisco (1982)
Barnsley, M.F.: Fractals Everywhere, 2nd edn. Academic Press, Inc., Boston (1993)
Hearn, D., Baker, M.P.: Computer Graphics. Prentice-Hall, Englewood Cliffs (1997)
Stanley, H.E., Ostrosky, N.: On Growth and Form: Fractal and Non-Fractal Patterns in Physics. Nijhoff, Boston (1986)
Nittman, J., Daccord, G., Stanley, H.E.: Fractal growth of viscous fingers: quantative characterization of a fluid instability phenomenon. Nature, London 314, 141 (1985)
Lovejoy, S.: Area-perimeter relation for rain and cloud areas. Science 216, 185–187 (1982)
Russ, J.C.: Fractal Surfaces. Plenum Press, New York (1984)
Peitgen, H., Saupe, D.: The science of f ractal images. Springer, New York (1988)
Frisch, U., Parisi, G.: Fully developed turbulence and intermittency. In: Ghil, M., et al. (eds.) Turbulence and Predictability in Geophysical Fluid Dynamics and Climate Dynamics, North-Holland, Amsterdam (1985)
Kawaguchi, Y.: A morphological study of the form of nature. Comput. Graph. 16, 3 (1982)
Matsushita, M.: Experimental Observation of Aggregations. In: Avnir, D. (ed.) The Fractal Approach to Heterogeneous Chemistry: Surfaces, Colloids, Polymers, J. Wiley and Sons, Chichester (1989)
West, B.: Fractal Physiology and Chaos in Medicine. World Scientific Publ. Co, Singapore (1990)
Feder, J.: Fractals. Plenum Press, New York (1988)
Voss, R.: Random fractals: characterisation and measurement. In: Pynn, R., Skjeltorp, A. (eds.) Scaling Phenomena in Disordered Systems, Plenum Press, New York (1986)
Calvet, L., Fisher, A.: Multifractality in Asset Returns: Theory and Evidence. The Review of Economics and Statistics 84(3), 381–406 (2002)
Hutchinson, J.E.: fractals and Self Similarity. Ind. Univ. J. Math. 30, 713–747 (1981)
Barnsley, M.F., Harrington, A.N.: The calculus of fractal interpolation functions. Journal of Appoximation Theory 57, 14–34 (1989)
Navascues, M.A., Sebastian, M.V.: Some Results of Convergence of Cubic Spline Fractal Interpolation Functions. Fractals 11(1), 1–7 (2003)
Barnsley, M.F., Elton, J., Hardin, D., Massopust, P.: Hidden Variable Fractal Interpolation Functions. SIAM J. Math. Anal. 20, 1218–1242 (1989)
Massopust, P.R.: Fractal Functions, Fractal Surfaces and Wavelets. Academic Press, San Diego (1994)
Navascues, M.A.: Fractal Trigonometric Interpolation. Electronic Transactions on Numerical Analysis 20, 64–74 (2005)
Wittenbrink, C.M.: IFS fractal interpolation for 2D and 3D visualization. In: Proc. IEEE Visualation 1994, pp. 77–83 (1995)
Zhao, N.: Construction and application of fractal interpolation surfaces. The Visual Computer 12, 132–146 (1996)
Billingsley, P.: Ergodic Theory and Information. John Wiley, Chichester (1965)
Mandelbrot, B.: Les Objects Fractals: Forme, Hasard et Dimension, Flammarion, Paris (1975)
Mandelbrot, B.B., Frame, M.: Fractals. In: Encyclopedia of Physical Science and Technology, June 28, 2001, vol. 6. Academic Press, Yale University (2002)
Theiler, J.: Estimating fractal dimension. J. Opt. Soc. Am. A 7, 6 (1990)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Cader, A., Krupski, M. (2006). New Interpolation Method with Fractal Curves. In: Rutkowski, L., Tadeusiewicz, R., Zadeh, L.A., Żurada, J.M. (eds) Artificial Intelligence and Soft Computing – ICAISC 2006. ICAISC 2006. Lecture Notes in Computer Science(), vol 4029. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11785231_112
Download citation
DOI: https://doi.org/10.1007/11785231_112
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-35748-3
Online ISBN: 978-3-540-35750-6
eBook Packages: Computer ScienceComputer Science (R0)