Hostname: page-component-7b7567c8c-jswmv Total loading time: 0 Render date: 2026-11-15T20:59:52.525Z Has data issue: false hasContentIssue false
  • English
  • Français

FRACTAL INTERPOLATION SURFACES ON RECTANGULAR GRIDS

Part of:Classical measure theoryApproximations and expansions

Published online by Cambridge University Press:  26 February 2015

HUO-JUN RUAN and
QIANG XU
HUO-JUN RUAN*
Affiliation:
Department of Mathematics, Zhejiang University, Hangzhou, 310027, PR China email ruanhj@zju.edu.cn
QIANG XU
Affiliation:
School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, 221116, PR China email xuqiangwang@jsnu.edu.cn
*
✉ruanhj@zju.edu.cn
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

In this paper, we present a general framework to construct fractal interpolation surfaces (FISs) on rectangular grids. Then we introduce bilinear FISs, which can be defined without any restriction on interpolation points and vertical scaling factors.

Keywords

fractal interpolation surfacesiterated function systemsvertical scaling factorsrectangular grids

MSC classification

Primary:28A80: Fractals
Secondary:41A30: Approximation by other special function classes

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 91 , Issue 3 , June 2015, pp. 435 - 446
Copyright
© 2015 Australian Mathematical Publishing Association Inc. 

References

Barnsley, M. F., ‘Fractal functions and interpolation’, Constr. Approx. 2 (1986), 303–329.Google Scholar
Barnsley, M. F., Fractals Everywhere, 2nd edn (Academic Press Professional, Boston, 1993), 27–41.Google Scholar
Barnsley, M. F., Elton, J., Hardin, D. and Massopust, P., ‘Hidden variable fractal interpolation functions’, SIAM J. Math. Anal. 20 (1989), 1218–1242.CrossRefGoogle Scholar
Barnsley, M. F. and Harrington, A. N., ‘The calculus of fractal interpolation functions’, J. Approx. Theory 57 (1989), 14–34.Google Scholar
Barnsley, M. F. and Massopust, P. R., Bilinear fractal interpolation and box dimension, Preprint, arXiv:1209.3139v1.Google Scholar
Bouboulis, P. and Dalla, L., ‘A general construction of fractal interpolation functions on grids of ℝn’, Eur. J. Appl. Math. 18 (2007), 449–476.CrossRefGoogle Scholar
Bouboulis, P., Dalla, L. and Drakopoulos, V., ‘Construction of recurrent bivariate fractal interpolation surfaces and computation of their box-counting dimension’, J. Approx. Theory 141 (2006), 99–117.Google Scholar
Chand, A. K. B. and Kapoor, G. P., ‘Hidden variable bivariate fractal interpolation surfaces’, Fractals 11 (2003), 277–288.CrossRefGoogle Scholar
Dalla, L., ‘Bivariate fractal interpolation functions on grids’, Fractals 10 (2002), 53–58.CrossRefGoogle Scholar
Falconer, K. J., The Geometry of Fractal Sets (Cambridge University Press, Cambridge, 1985), 36–37.Google Scholar
Feng, Z., ‘Variation and Minkowski dimension of fractal interpolation surfaces’, J. Math. Anal. Appl. 345 (2008), 322–334.Google Scholar
Geronimo, J. S. and Hardin, D., ‘Fractal interpolation surfaces and a related 2D multiresolution analysis’, J. Math. Anal. Appl. 176 (1993), 561–586.Google Scholar
Haupt, R. L. and Haupt, S. E., Practical Genetic Algorithms, 2nd edn (John Wiley & Sons, Hoboken, NJ, 2004), 27–50.Google Scholar
Małysz, R., ‘The Minkowski dimension of the bivariate fractal interpolation surfaces’, Chaos Solitons Fractals 27 (2006), 1147–1156.CrossRefGoogle Scholar
Massopust, P. R., ‘Fractal surfaces’, J. Math. Anal. Appl. 151 (1990), 275–290.Google Scholar
Mazel, D. S. and Hayes, M. H., ‘Using iterated function systems to model discrete sequences’, IEEE Trans. Signal Process. 40 (1992), 1724–1734.Google Scholar
Metzler, W. and Yun, C. H., ‘Constuction of fractal interpolation surfaces on rectangular grids’, Internat. J. Bifur. Chaos 20 (2010), 4079–4086.CrossRefGoogle Scholar
Ruan, H.-J., Sha, Z. and Su, W.-Y., ‘Counterexamples in parameter identification problem of the fractal interpolation functions’, J. Approx. Theory 122 (2003), 121–128.CrossRefGoogle Scholar
Ruan, H.-J., Su, W.-Y. and Yao, K., ‘Box dimension and fractional integral of linear fractal interpolation functions’, J. Approx. Theory 161 (2009), 187–197.CrossRefGoogle Scholar
Véhel, J. L., Daoudi, K. and Lutton, E., ‘Fractal modeling of speech signals’, Fractals 2 (1994), 379–382.Google Scholar
Wang, H.-Y. and Yu, J.-S., ‘Fractal interpolation functions with variable parameters and their analytical properties’, J. Approx. Theory 175 (2013), 1–18.CrossRefGoogle Scholar
Xie, H. and Sun, H., ‘The study on bivariate fractal interpolation functions and creation of fractal interpolated surfaces’, Fractals 5 (1997), 625–634.CrossRefGoogle Scholar
Zhao, N. , ‘Construction and application of fractal interpolation surfaces’, Visual Computer 12 (1996), 132–146.CrossRefGoogle Scholar