具有变量自由参数的分形插值曲面的构造与性质

王宏勇, 杨守志

数学学报 ›› 2014, Vol. 57 ›› Issue (2) : 223-234.

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数学学报 ›› 2014, Vol. 57 ›› Issue (2) : 223-234. DOI: 10.12386/A2014sxxb0023
论文

具有变量自由参数的分形插值曲面的构造与性质

    王宏勇1, 杨守志2
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Construction and Properties of Fractal Interpolation Surfaces with Variable Free Parameters

    Hong Yong WANG1, Shou Zhi YANG2
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摘要

在矩形区域上使用具有变量自由参数的迭代函数系，构造了一类新的分形插值曲面，并研究了这类曲面的若干性质.证明了这类变参数的迭代函数系能生成连续的二元分形插值曲面，指出该类曲面可用作任意数据点上的自仿射和非自仿射的分形插值模型.在两种度量意义下，导出了能刻画这类分形曲面敏感性的一些不等式.给出了相应的分形插值函数与数据生成函数之间的误差估计. 最后，在一定条件下，证明了这类分形插值函数序列一致收敛于数据生成函数.

Abstract

A new construction of fractal interpolation surfaces (FISs) on rectangular grids using iterated function systems (IFSs) is proposed and some of their properties are studied in this paper. The IFSs employed are endowed with variable free parameters and are proved that they can generate continuous bivariate FISs, which can be used as self-affine and non-self-affine fractal interpolation models on arbitrary data points. Some inequalities concerning the sensitivity of the resulting bivariate fractal interpolation functions (FIFs) are deduced in two metric senses. The estimation of the error between the bivariate FIF and a bivariate data generating function is given. Finally, under certain conditions, the uniform convergence of the bivariate FIFs to the data generating function is proved.

关键词

分形插值曲面 / 迭代函数系 / 变量自由参数 / 敏感性 / 收敛性

Key words

fractal interpolation surfaces / iterated function systems / variable free parameters / sensitivity / convergence

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王宏勇, 杨守志. 具有变量自由参数的分形插值曲面的构造与性质. 数学学报, 2014, 57(2): 223-234 https://doi.org/10.12386/A2014sxxb0023
Hong Yong WANG, Shou Zhi YANG. Construction and Properties of Fractal Interpolation Surfaces with Variable Free Parameters. Acta Mathematica Sinica, Chinese Series, 2014, 57(2): 223-234 https://doi.org/10.12386/A2014sxxb0023

参考文献

[1] Barnsley M. F., Fractal functions and interpolation, Constr. Approx., 1986, 2: 303-329.
[2] Bouboulis P., Dalla L., A general construction of fractal interpolation functions on grids of Rn, Euro. J. Appl. Math., 2007, 18: 449-476.
[3] Bouboulis P., Dalla L., Fractal interpolation surfaces derived from fractal interpolation functions, J. Math. Anal. Appl., 2007, 336: 919-936.
[4] Bouboulis P., Dalla L., Drakopoulos V., Construction of recurrent bivariate fractal interpolation surfaces and computation of their box-counting dimension, J. Approx. Theory, 2006, 141(2): 99-117.
[5] Chand A. K. B., Natural cubic spline coalescence hidden variable fractal interpolation surfaces, Fractals, 2012, 20(2): 117-131.
[6] Chand A. K. B., Kapoor G. P., Hidden variable bivariate fractal interpolation surfaces, Fractals, 2003, 11(3): 277-288.
[7] Dalla L., Bivariate fractal interpolation functions on grids, Fractals, 2002, 10(1): 53-58.
[8] Falconer K., Fractal Geometry: Mathematical Foundations and Applications, Wiley, New York, 1990.
[9] Feng Z., Variation and Minkowski dimension of fractal interpolation surfaces, J. Math. Anal. Appl., 2008, 345: 322-334.
[10] Feng Z., Feng Y., Yuan Z., Fractal interpolation surfaces with function vertical scaling factors, Appl. Math. Lett., 2012, 25: 1896-1900.
[11] Geronimo J. S., Hardin D. P., Fractal interpolation surfaces and a related 2-D multiresolution analysis, J. Math. Anal. Appl., 1993, 176: 561-586.
[12] Kapoor G. P., Prasad S. A., Smoothness of coalescence hidden-variable fractal interpolation surfaces, Int. J. Bifurcat. Chaos, 2009, 19(7): 2321-2333.
[13] Malysz R., The Minkowski dimension of the bivariate fractal interpolation surfaces, Chaos Solitons Fractals, 2006, 27: 1147-1156.
[14] Mandelbrot B. B., Passoja D. E., Paullay A. J., Fractal character of fracture surfaces of metals, Nature, 1984, 308: 721-722.
[15] Massopust P. R., Fractal surfaces, J. Math. Anal. Appl., 1990, 151: 275-290.
[16] Metzler W., Yun C. H., Construction of fractal interpolation surfaces on rectangular grids, Int. J. Bifurcat. Chaos, 2010, 20(12): 4079-4086.
[17] Vehél J. L., Lutton E., Tricot C., Fractals in Engineering: From Theory to Industrial Applications, Springer-Verlag, New York, 1997.
[18] Voss R. F., Random Fractal Forgeries in Fundamental Algorithms in Computer Graphics, Springer-Verlag, Berlin, 1985.
[19] Wang H. Y., On smoothness for a class of fractal interpolation surfaces, Fractals, 2006, 14(3): 223-230.
[20] Wang H. Y., Fan Z. L., Analytical characteristics of fractal interpolation functions with function vertical scaling factors, Acta Math. Sinica, Chinese Series, 2011, 54(1): 147-158 (in Chinese).
[21] Wang H. Y., Xu Z. B., A class of rough surfaces and their fractal dimensions, J. Math. Anal. Appl., 2001, 259: 537-553.
[22] Wong P., Howard J., Li J., Surfaces roughening and the fractal nature of rocks, Phys. Rev. Lett., 1986, 57: 637-640.
[23] Xie H., Sun H., The study on bivariate fractal interpolation functions and creation of fractal interpolation surfaces, Fractals, 1997, 5(4): 625-634.
[24] Xie H., Sun H., Ju Y., Feng Z., Study on generation of rock fracture surfaces by using fractal interpolation, Int. J. Solids Struct., 2001, 38: 5765-5787.
[25] Zhao N., Construction and application of fractal interpolation surfaces, Vis. Comput., 1996, 12: 132-146.

基金

国家自然科学基金资助项目（11071152）；南京财经大学预研究基金（A2011019）
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