非静态4点二重混合细分法
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摘要
为了得到插值与逼近相统一的非静态细分法,根据非静态插值4点细分法和三次指数B-样条细分法之间的联系,构造了3类非静态4点二重混合细分法:基于非静态插值细分的非静态逼近细分法,基于非静态逼近细分的非静态插值细分法,非静态插值与逼近混合细分法.诸多已有的插值细分法和逼近细分法都是所提混合细分法的特例.最后给出了这3类混合细分法的几何解释,分析了其Ck连续性、指数多项式生成性和再生性.数值实例表明,利用文中的混合细分法,通过适当选取参数可以实现对极限曲线的形状控制.
In order to obtain the non-stationary subdivision scheme unifying interpolation and approximation,three different non-stationary four-point binary blending subdivision schemes are constructed according to the relationship between the non-stationary interpolating four-point subdivision scheme and cubic exponential B-spline subdivision scheme. Among them are a non-stationary approximating subdivision scheme based on non-stationary interpolating subdivision, a non-stationary interpolating subdivision scheme based on non-stationary approximating subdivision, and a non-stationary blending subdivision scheme that integrates interpolating and approximating. Many existing interpolating subdivision schemes and approximating subdivision schemes are special cases of the proposed blending subdivision schemes. The schemes are explained geometrically, and some properties of the schemes are analyzed such as Ck continuity, the exponential polynomial generation and reproduction. Numerical examples show that the proposed blending subdivision scheme can be used to control the shape of limit curves by selecting appropriate parameters.
In order to obtain the non-stationary subdivision scheme unifying interpolation and approximation,three different non-stationary four-point binary blending subdivision schemes are constructed according to the relationship between the non-stationary interpolating four-point subdivision scheme and cubic exponential B-spline subdivision scheme. Among them are a non-stationary approximating subdivision scheme based on non-stationary interpolating subdivision, a non-stationary interpolating subdivision scheme based on non-stationary approximating subdivision, and a non-stationary blending subdivision scheme that integrates interpolating and approximating. Many existing interpolating subdivision schemes and approximating subdivision schemes are special cases of the proposed blending subdivision schemes. The schemes are explained geometrically, and some properties of the schemes are analyzed such as Ck continuity, the exponential polynomial generation and reproduction. Numerical examples show that the proposed blending subdivision scheme can be used to control the shape of limit curves by selecting appropriate parameters.
引文
[1]Dyn N,Levin D,Gregory J A.A 4-point interpolatory subdivision scheme for curve design[J].Computer Aided Geometric Design,1987,4(4):257-268
[2]Deslauriers G,Dubuc S.Symmetric iterative interpolation processes[J].Constructive Approximation,1989,5(1):49-68
[3]Hassan M F,Ivrissimitzis I P,Dodgson N A,et al.An interpolating 4-point C2 ternary stationary subdivision scheme[J].Computer Aided Geometric Design,2002,19(1):1-18
[4]Hassan M F,Dodgson N A.Ternary and three-point univariate subdivision schemes[C]//Proceedings of Curve and Surface Fitting.Brentwood:Nashboro Press,2003:199-208
[5]Siddiqi S S,Ahmad N.A C6 approximating subdivision scheme[J].Applied Mathematics Letters,2008,21(7):722-728
[6]Beccari C,Casciola G,Romani L.A non-stationary uniform tension controlled interpolating 4-point scheme reproducing conics[J].Computer Aided Geometric Design,2007,24(1):1-9
[7]Beccari C,Casciola G,Romani L.An interpolating 4-point2C ternary non-stationary subdivision scheme with tension control[J].Computer Aided Geometric Design,2007,24(4):210-219
[8]Maillot J,Stam J.A unified subdivision scheme for polygonal modeling[J].Computer Graphics Forum,2001,20(3):471-479
[9]Li G,Ma W.A method for constructing interpolatory subdivision schemes and blending subdivisions[J].Computer Graphics Forum,2007,26(2):185-201
[10]Lin S J,You F,Luo X N,et al.Deducing interpolating subdivision schemes from approximating subdivision schemes[J].ACM Transactions on Graphics,2008,27(5):Article No.146
[11]Luo Z X,Qi W F.On interpolatory subdivision from approximating subdivision scheme[J].Applied Mathematics and Computation,2013,220:339-349
[12]Tan Jieqing,Tong Guangyue,Zhang Li.An approximating subdivision based on interpolating subdivision scheme[J].Journal of Computer-Aided Design&Computer Graphics,2015,27(7):1162-1166(in Chinese)(檀结庆,童广悦,张莉.基于插值细分的逼近细分法[J].计算机辅助设计与图形学学报,2015,27(7):1162-1166)
[13]Pan J,Lin S J,Luo X N.A combined approximating and interpolating subdivision scheme with C2 continuity[J].Applied Mathematics Letters,2012,25(12):2140-2146
[14]Romani L.From approximating subdivision schemes for exponential splines to high-performance interpolating algorithms[J].Journal of Computational and Applied Mathematics,2009,224(1):383-396
[15]Yan Feiyi,Zheng Hongchan.Non-stationary blending subdivision scheme[J].Journal of Graphics,2015,36(2):178-185(in Chinese)(闫飞一,郑红婵.非静态混合细分法[J].图学学报,2015,36(2):178-185)
[16]Zheng H C,Zhang B X.A non-stationary combined subdivision scheme generating exponential polynomials[J].Applied Mathematics and Computation,2017,313:209-221
[17]Dyn N,Levin D.Analysis of asymptotically equivalent binary subdivision schemes[J].Journal of Mathematical Analysis and Application,1995,193(2):594-621
[18]Charina M,Contin C,Romani L.Reproduction of exponential polynomials by multivariate non-stationary subdivision schemes with a general dilation matrix[J].Numerical Mathematics,2014,127(2):223-254
[2]Deslauriers G,Dubuc S.Symmetric iterative interpolation processes[J].Constructive Approximation,1989,5(1):49-68
[3]Hassan M F,Ivrissimitzis I P,Dodgson N A,et al.An interpolating 4-point C2 ternary stationary subdivision scheme[J].Computer Aided Geometric Design,2002,19(1):1-18
[4]Hassan M F,Dodgson N A.Ternary and three-point univariate subdivision schemes[C]//Proceedings of Curve and Surface Fitting.Brentwood:Nashboro Press,2003:199-208
[5]Siddiqi S S,Ahmad N.A C6 approximating subdivision scheme[J].Applied Mathematics Letters,2008,21(7):722-728
[6]Beccari C,Casciola G,Romani L.A non-stationary uniform tension controlled interpolating 4-point scheme reproducing conics[J].Computer Aided Geometric Design,2007,24(1):1-9
[7]Beccari C,Casciola G,Romani L.An interpolating 4-point2C ternary non-stationary subdivision scheme with tension control[J].Computer Aided Geometric Design,2007,24(4):210-219
[8]Maillot J,Stam J.A unified subdivision scheme for polygonal modeling[J].Computer Graphics Forum,2001,20(3):471-479
[9]Li G,Ma W.A method for constructing interpolatory subdivision schemes and blending subdivisions[J].Computer Graphics Forum,2007,26(2):185-201
[10]Lin S J,You F,Luo X N,et al.Deducing interpolating subdivision schemes from approximating subdivision schemes[J].ACM Transactions on Graphics,2008,27(5):Article No.146
[11]Luo Z X,Qi W F.On interpolatory subdivision from approximating subdivision scheme[J].Applied Mathematics and Computation,2013,220:339-349
[12]Tan Jieqing,Tong Guangyue,Zhang Li.An approximating subdivision based on interpolating subdivision scheme[J].Journal of Computer-Aided Design&Computer Graphics,2015,27(7):1162-1166(in Chinese)(檀结庆,童广悦,张莉.基于插值细分的逼近细分法[J].计算机辅助设计与图形学学报,2015,27(7):1162-1166)
[13]Pan J,Lin S J,Luo X N.A combined approximating and interpolating subdivision scheme with C2 continuity[J].Applied Mathematics Letters,2012,25(12):2140-2146
[14]Romani L.From approximating subdivision schemes for exponential splines to high-performance interpolating algorithms[J].Journal of Computational and Applied Mathematics,2009,224(1):383-396
[15]Yan Feiyi,Zheng Hongchan.Non-stationary blending subdivision scheme[J].Journal of Graphics,2015,36(2):178-185(in Chinese)(闫飞一,郑红婵.非静态混合细分法[J].图学学报,2015,36(2):178-185)
[16]Zheng H C,Zhang B X.A non-stationary combined subdivision scheme generating exponential polynomials[J].Applied Mathematics and Computation,2017,313:209-221
[17]Dyn N,Levin D.Analysis of asymptotically equivalent binary subdivision schemes[J].Journal of Mathematical Analysis and Application,1995,193(2):594-621
[18]Charina M,Contin C,Romani L.Reproduction of exponential polynomials by multivariate non-stationary subdivision schemes with a general dilation matrix[J].Numerical Mathematics,2014,127(2):223-254