Korean Journal of Computational Design and Engineering (한국CDE학회논문집)

Society for Computational Design and Engineering (한국CDE학회)
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An Error-Bounded B-spline Fitting Technique to Approximate Unorganized Data

무작위 데이터 근사화를 위한 유계오차 B-스플라인 근사법

  • Park, Sang-Kun (Department of Mechanical Engineering, Korea National University of Transportation)
  • 박상근 (한국교통대학교 기계공학과)
  • Received : 2011.11.22
  • Accepted : 2012.07.03
  • Published : 2012.08.01
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Abstract

This paper presents an error-bounded B-spline fitting technique to approximate unorganized data within a prescribed error tolerance. The proposed approach includes two main steps: leastsquares minimization and error-bounded approximation. A B-spline hypervolume is first described as a data representation model, which includes its mathematical definition and the data structure for implementation. Then we present the least-squares minimization technique for the generation of an approximate B-spline model from the given data set, which provides a unique solution to the problem: overdetermined, underdetermined, or ill-conditioned problem. We also explain an algorithm for the error-bounded approximation which recursively refines the initial base model obtained from the least-squares minimization until the Euclidean distance between the model and the given data is within the given error tolerance. The proposed approach is demonstrated with some examples to show its usefulness and a good possibility for various applications.

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References

  1. Haber, J., Zeilfelder, F., Davydov, O. and Seidel, H.-P., 2001, Smooth Approximation and Rendering of Large Scattered Data Sets, 12th IEEE Visualization 2001, pp. 341-571.
  2. Nielson, G.M., Scattered Data Modeling, 1993, IEEE Computer Graphics and Applications, 13(1), pp. 60-70. https://doi.org/10.1109/38.180119
  3. Shepard, D., 1968, A Two Dimensional Interpolation Function for Irregularly Spaced Data, Proceedings of ACM 23rd National Conference, pp. 517-524.
  4. Schaback, R., 1995, Multivariate Interpolation and Approximation by Translates of a Basis Function in Approximation Theory VIII, Vol. 1: Approximation and Interpolation, World Scientific Publishing, Singapore, pp. 491-514.
  5. Franke, R. and Nielson, G.M., 1980, Smooth Interpolation of Large Sets of Scattered Data, International Journal of Numerical Methods in Engineering, 15, pp. 1,691-1,704. https://doi.org/10.1002/nme.1620151110
  6. Wendland, H., 1995, Piecewise Polynomial, Positive Definite and Compactly Supported Radial Functions of Minimal Degree, Advances in Computational Mathematics, 4, pp. 389-396. https://doi.org/10.1007/BF02123482
  7. Hardy, R., 1971, Multiquadric Equations of Topography and Other Irregular Surfaces, J. Geophysical Research, 76(8), pp. 1,905-1,915. https://doi.org/10.1029/JB076i008p01905
  8. Duchon, J., 1975, Splines Minimizing Rotation-Invariant Semi-Norms in Sobolev Spaces in Multivariate Approximation Theory, Basel, Switzerland: Birkhauser, pp. 85-100.
  9. Park, S., 2009, A Rational B-spline Hypervolume for Multidimensional Multivariate Modeling, Journal of Mechanical Science and Technology, 23, pp. 1967-1981. https://doi.org/10.1007/s12206-009-0513-2
  10. Park, S., 2011, On B-spline Approximation for Representing Scattered Multivariate Data, Transactions of the KSME A, 35(8), pp. 921-931.
  11. Piegl, L. and Tiller, W., 1995, The NURBS Book, Springer-Verlag.
  12. De Boor, C., 1978, A Practical Guide to Splines, New York, Springer-Verlag.
  13. Farin, G., 1990, Curves and Surfaces for Computer Aided Geometric Design, Academic Press, San Diego.
  14. Sethian, J.A., 1999, Level Set Methods and Fast Marching Methods, Cambridge University Press, Cambridge, UK.
  15. Lorensen, W.E. and Cline, H.E., 1987, Marching Cubes: A High Resolution 3D Surface Construction Algorithm, Computer Graphics (Proceedings of SIGGRAPH 87), 21(4), pp. 163-169. https://doi.org/10.1145/37402.37422