一类具有3组参数的分形插值迭代函数系及其性质
|
摘要
对于给定的插值点集,构造了一类具有3组参数的迭代函数系。与经典的迭代函数系的构造不同,文章采用一般形式的抽象函数来构造迭代函数系,无需考虑函数的具体结构,并将迭代函数系的插值条件归结为选取合适的参数。所构造的迭代函数系涵盖了一些已有的迭代函数系作为特殊情形,研究了迭代函数系中自由参数和数据集中纵坐标发生扰动时,相应的分形插值函数的变化情况,给出了扰动误差的估计式。
With a data set selected before, a class of iterated function system with three systems of parameters is constructed on. It is different from classic constructions of IFS that in this paper we construct IFS by generally abstract functions and neglect the specific structure of functions in IFS. Hence the interpolation conditions of FIF are attributed to the selections of parameters. IFS constructed in previous studies can be regarded as special situations of IFS constructed in this paper. The changes of the corresponding FIFs are studied when the free parameter in IFS and the ordinate of data set has a perturbation, and the formulas of error estimating are obtained.
With a data set selected before, a class of iterated function system with three systems of parameters is constructed on. It is different from classic constructions of IFS that in this paper we construct IFS by generally abstract functions and neglect the specific structure of functions in IFS. Hence the interpolation conditions of FIF are attributed to the selections of parameters. IFS constructed in previous studies can be regarded as special situations of IFS constructed in this paper. The changes of the corresponding FIFs are studied when the free parameter in IFS and the ordinate of data set has a perturbation, and the formulas of error estimating are obtained.
引文
[1]BARNSLEY M F.Fractal function and interpolation[J]Construc tive Approximation,1986(2):303-329.
[2]BIRDI K S.Fractals in chemistry,geochemistry,and biophysics[M].New York:Plenum Press,1995.
[3]P IE TRONERO L.Fractals in physics:applications and theoreticaldevelopments[J].PhysicaA,1992,191(1):85-94.
[4]ALI M,CLAKSON T G.Using linear fractal interpolation function to compress video images[J].Fractals,1994,2(3):417-421.
[5]VEHEL J L,DAUODI K,LUTTON E.Fractal modelling of speech signals[J].Fractals,1994,2(3):379-382.
[6]MASSOPUST P R.Fractal functions,fractal surfaces and Wavelets[M].Manhattan:Academic Press,1994.
[7]SHEPARD M K,CAMBELL B A.Shadows on a planetary surface and impications for photo metric roughness[J].I-CARUS,1998,134(2):279-291.
[8]XIE H P,SUN H Q,Zu Y,et al.Study on generation of rock fracture surfaces by using fractal interpolation[J].International Journal of Solids Structures,2001,38(3):5765-5787.
[9]YOKOYA N,YAMAMOTO K,FUNAKUBO N.Fractal-based analysis and interpolation of 3d natural surface shapes and their application to terrain modeling[J].Computer Vision Graphics Image Process,1989,45(3):284-302.
[10]BARNSLEY M F,ELTON J H&HARDIN D P.Recurrent iterated function sysytems[J].Construc tive Approximation,1989,5:3-31.
[11]BARNSLEY M F,ELTON J H,Hardin D P,et al.Hidden variable fractal interpolation functions[J].SIAM Journal on Mathematical Analysis,1989,20(5):1218-1242.
[12]DALLA L,DRAKOPOULOS V,PRODROMOU M.On the box dimension for a class of non-affine fractal interpolation functions[J].Analysis in Theory and Applications,2003,19(3):220-233.
[13]王宏勇.一类具有双参数的迭代函数系及其吸引子[J].厦门大学学报,2007,46(2):157-160.
[14]WANG H Y,YU J S.Fractal interpolation functions with variable parameters and their analytical properties[J].Journal of Approximation Theory,2013(175):1-18.
[15]章立亮.一种带自动参数的迭代函数系统[J].工程图学学报,2006(2):171-175.
[16]樊昭磊,王宏勇.基于参数扰动的二元分形插值函数误差分析[J].安徽大学学报(自然科学版),2010,34(2):48-53.
[17]WANG H Y.Error estimates of fitting for bivariate fractal interpolation[J].Journal of Mathematical Research and Exposition,2009,29(3):551-557.
[18]沙震,阮火军.分形与拟合[M].杭州:浙江大学出版社,2005.
[19]孙洪泉.矩形域上分形插值研究[J].数学物理学报,2009,29(3):773-783.
[2]BIRDI K S.Fractals in chemistry,geochemistry,and biophysics[M].New York:Plenum Press,1995.
[3]P IE TRONERO L.Fractals in physics:applications and theoreticaldevelopments[J].PhysicaA,1992,191(1):85-94.
[4]ALI M,CLAKSON T G.Using linear fractal interpolation function to compress video images[J].Fractals,1994,2(3):417-421.
[5]VEHEL J L,DAUODI K,LUTTON E.Fractal modelling of speech signals[J].Fractals,1994,2(3):379-382.
[6]MASSOPUST P R.Fractal functions,fractal surfaces and Wavelets[M].Manhattan:Academic Press,1994.
[7]SHEPARD M K,CAMBELL B A.Shadows on a planetary surface and impications for photo metric roughness[J].I-CARUS,1998,134(2):279-291.
[8]XIE H P,SUN H Q,Zu Y,et al.Study on generation of rock fracture surfaces by using fractal interpolation[J].International Journal of Solids Structures,2001,38(3):5765-5787.
[9]YOKOYA N,YAMAMOTO K,FUNAKUBO N.Fractal-based analysis and interpolation of 3d natural surface shapes and their application to terrain modeling[J].Computer Vision Graphics Image Process,1989,45(3):284-302.
[10]BARNSLEY M F,ELTON J H&HARDIN D P.Recurrent iterated function sysytems[J].Construc tive Approximation,1989,5:3-31.
[11]BARNSLEY M F,ELTON J H,Hardin D P,et al.Hidden variable fractal interpolation functions[J].SIAM Journal on Mathematical Analysis,1989,20(5):1218-1242.
[12]DALLA L,DRAKOPOULOS V,PRODROMOU M.On the box dimension for a class of non-affine fractal interpolation functions[J].Analysis in Theory and Applications,2003,19(3):220-233.
[13]王宏勇.一类具有双参数的迭代函数系及其吸引子[J].厦门大学学报,2007,46(2):157-160.
[14]WANG H Y,YU J S.Fractal interpolation functions with variable parameters and their analytical properties[J].Journal of Approximation Theory,2013(175):1-18.
[15]章立亮.一种带自动参数的迭代函数系统[J].工程图学学报,2006(2):171-175.
[16]樊昭磊,王宏勇.基于参数扰动的二元分形插值函数误差分析[J].安徽大学学报(自然科学版),2010,34(2):48-53.
[17]WANG H Y.Error estimates of fitting for bivariate fractal interpolation[J].Journal of Mathematical Research and Exposition,2009,29(3):551-557.
[18]沙震,阮火军.分形与拟合[M].杭州:浙江大学出版社,2005.
[19]孙洪泉.矩形域上分形插值研究[J].数学物理学报,2009,29(3):773-783.