• Original Articles •

### FITTING C1 SURFACES TO SCATTERED DATA WITH S21 (Δm,n(2))

Kai Qu1,2, Renhong Wang3

1. 1. School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China;
2. Department of Mathematics, Dalian Maritime University, Dalian 116026, China;
3. School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China
• Received:2009-09-07 Revised:2010-12-22 Online:2011-07-15 Published:2011-07-15
• Supported by:

This work was supported by the National Natural Science Foundation of China (Nos. U0935004,11071031,11071037,10801024), and the Fundamental Funds for the Central Universities. should be changed to Acknowledgments. This work is partly supported by the National Natural Science Foundation of China (Nos. U0935004,11071031,10801024), the Fundamental Funds for the Central Universities (DUT10ZD112, DUT11LK34), and National Engineering Research Center of Digital Life, Guangzhou 510006, China.

Kai Qu, Renhong Wang, Chungang Zhu. FITTING C1 SURFACES TO SCATTERED DATA WITH S21m,n(2))[J]. Journal of Computational Mathematics, 2011, 29(4): 396-414.

This paper presents a fast algorithm (BS2 Algorithm) for fitting C1 surfaces to scattered data points. By using energy minimization, the bivariate spline space S21m,n(2)) is introduced to construct a C1-continuous piecewise quadratic surface through a set of irregularly 3D points. Moreover, a multilevel method is also presented. Some experimental results show that the accuracy is satisfactory. Furthermore, the BS2 Algorithm is more suitable for fitting surfaces if the given data points have some measurement errors.

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 [1] R.E. Barnhill, Representation and Approximation of Surfaces, in: Mathematical Software III(J. R. Rice, Ed.), Academic Press, Newyork, 1977, 69-120.[2] C. de Boor and J.R. Rice, Least Squares Cubic Splines Approximation. I: Fixed Knots, CSD TR20, Purdue Univ., Lafayette, 1968.[3] C. de Boor and G.J. Fix, Spline approximation by quasi-interpolants, J. Approx. Theory, 8 (1973),19-45.[4] M.D. Buhmann, Radial functions on compact support, Proc. Edinb. Math. Soc., 41 (1998), 33-46.[5] M.D. Buhmann, A new class of radial basis functions with compact support, Math. Comput., 70(2001), 307-318.[6] F. Cheng and B.A. Barsky, Interproximation: interpolation and approximation using cubic splinecurves, Comput. Aided Design, 10 (1991), 700-706.[7] R. Franke and G.M. Nielson, Smooth interpolation of large sets of scattered data, Int. J. NumericalMethods in Eng., 15 (1980), 1691-1704.[8] R. Franke and G.M. Nielson, Scattered Data Interpolation and Applications: A Tutorial andSurvey, in: H. Hagen and D. Roller (Eds.), Geometric Modelling: Methods and Their Application,Springer-Verlag, Berlin, 1991, 131-160.[9] H. Hagen and G. Schulze, Automatic smoothing with geometric surface patches, Computer AidedGeometric Design, 4 (1987), 231-236.[10] R. Hardy, Multiquadratic equations of topography and other irregular surfaces, J. Geophys. Res.,76:8 (1971), 1905-1915.[11] J. Hoschek and D. Lasser, Computer Aided Geometric Design, A. K. Peters, 1993.[12] M. Kallay, A method to approximate the space curve of minimal energy and prescribed length,Comput. Aided Design, 19:2 (1987), 73-76.[13] M. Kallay and B. Rarani, Optimal twist vectors as a tool for interpolating a network of curveswith a minimum energy surface, Comput. Aided Geom. D., 7 (1990), 465-473.[14] C. Lawson, Software for C1 surface interpolation. in: Mathematical Software III (J. R. Rice, Ed.),Academic Press, Newyork, 1977, 161-194.[15] S. Lee, G. Wolberg, and S.Y. Shin, Scattered data interpolation with multilevel B-splines, IEEET. Vis. Comput. Gr., 3:3 (1997), 228-244.[16] T. Lyche and L. Schumaker, Local spline approximation methods, J. Approx. Theory, 15 (1975),294-325.[17] E. Quak and L.L. Schumaker, Calculation of the energy of a piecewise polynomial surface, in:M. G. Cox and J. C. Mason (Eds.), Algorithms for Approximation II, Clarendon Press, Oxford,1989, 134-143.[18] E. Quak and L.L. Schumaker, Cubic spline fitting using data dependent triangulations, ComputAided Geom. D., 7 (1990), 293-301.[19] D. Shepard, A two dimentional interpolation function for irregularly spaced data. in: Proc. ACM23rd Nat'l Conf., 1968, 517-524.[20] R. Szeliski, Fast surface interpolation using hierarchial basis functions, IEEE Trans. Pattern Anal.Machine Intell., 12 (1990), 513-528.[21] S. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells, Mcgraw-Hill InternationalBook Company, New York, 1959.[22] R.H. Wang, The structural characterization and interpolation for multivariate splines, Acta MathSin., 18:2 (1975), 91-106. (English transl., ibid. 18 (1975), 10-39.[23] R.H. Wang, The dimension and basis of spaces of multivariate splines, J. Comput. Appl. Math.,12-13 (1985), 163-177.[24] R.H. Wang, Multivariate Spline Function and Their Applications, Science Press/Kluwer Acad.Pub., Beijing/New York, 2001.[25] R.H. Wang, C.J. Li and C.G. Zhu, A Course for Computational Geometry, Science Press, Beijing,2008.[26] X.F. Wang, F. Cheng and B.A. Barsky, Energy and B-spline interproximation, Computer-AidedDesign, 29 (1997), 485-496.[27] H. Wendland, Piecewise polynomial, positive definite and compactly supported radial functionsof minimal degree. Adv. Comput. Math., 4 (1995), 389-396.[28] H. Wendland, Scattered Data Approximation, Cambridge Monographs on Applied and ComputationalMathematics, Cambridge University Press, Cambridge, 2005.[29] Z.M. Wu, Compactly supported positive definite radial functions, Adv. Comput. Math., 4 (1995),283-292.
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