Transactions of the Korean Society of Mechanical Engineers A (대한기계학회논문집A)

The Korean Society of Mechanical Engineers (대한기계학회)
권호 표지
DOI QR코드

DOI QR Code

On B-spline Approximation for Representing Scattered Multivariate Data

비정렬 다변수 데이터의 B-스플라인 근사화 기법

  • Received : 2010.12.28
  • Accepted : 2011.05.23
  • Published : 2011.08.01
Download PDF
⟨ Previous Next ⟩

Abstract

This paper presents a data-fitting technique in which a B-spline hypervolume is used to approximate a given data set of scattered data samples. We describe the implementation of the data structure of a B-spline hypervolume, and we measure its memory size to show that the representation is compact. The proposed technique includes two algorithms. One is for the determination of the knot vectors of a B-spline hypervolume. The other is for the control points, which are determined by solving a linear least-squares minimization problem where the solution is independent of the data-set complexity. The proposed approach is demonstrated with various data-set configurations to reveal its performance in terms of approximation accuracy, memory use, and running time. In addition, we compare our approach with existing methods and present unconstrained optimization examples to show the potential for various applications.

본 연구는 B-스플라인 하이퍼볼륨을 사용하여 주어진 비정렬 데이터를 근사화하는 데이터 근사기법에 관한 것이다. 개발 구현을 위한 B-스플라인 하이퍼볼륨의 자료 구조가 기술되며 해당 메모리 크기의 측정을 통해 간결한 표현 모델임을 보인다. 제안하는 근사 기법은 두 가지 알고리즘으로 구성된다. 하나는 B-스플라인 하이퍼볼륨의 절점 벡터 결정에 관한 것이고, 다른 하나는 조정점 결정에 관한 것으로 최소자승 최소화 문제의 해를 구함으로써 얻게 된다. 여기서 구한 해는 데이터 복잡성에 의존하지 않는다. 본 연구 방식은 다양한 형태의 데이터 분포를 가지고 근사 정밀도, 메모리 사용량, 계산 시간 등의 근사화 성능(수준)을 평가한다. 더불어 기존 방법과의 비교를 통해 유용성을 보이며, 비구속 최적화 예제를 통하여 다양한 응용 분야로의 가능성을 보여준다.

Keywords

References

  1. Park, S., 2007, "Multiresidual Approximation of Scattered Volumetric Data With Volumetric Non-Uniform Rational B-Splines," Trans. of the Society of CAD/CAM Engineers, Vol. 12, No. 1, pp. 27-38.
  2. 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. https://doi.org/10.1109/VISUAL.2001.964530
  3. Kohavi, R., 1995, "A Study of Cross-Validation and Bootstrap for Accuracy Estimation and Model Selection," Proc. of International Joint Conference on Artificial Intelligence. Vol. 2, No. 12, pp. 1137-1143.
  4. Nielson, G. M., 1993, "Scattered Data Modeling," IEEE Computer Graphics and Applications, Vol. 13, No. 1, pp. 60-70. https://doi.org/10.1109/38.180119
  5. Shepard, D., 1968, "A Two Dimensional Interpolation Function for Irregularly Spaced Data," Proc. of ACM 23rd National Conference, pp. 517-524.
  6. 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.
  7. Franke, R. and Nielson, G. M., 1980, "Smooth Interpolation of Large Sets of Scattered Data," International Journal of Numerical Methods in Engineering, Vol. 15, pp. 1691-1704. https://doi.org/10.1002/nme.1620151110
  8. Wendland, H., 1995, "Piecewise Polynomial, Positive Definite and Compactly Supported Radial Functions of Minimal Degree," Advances in Computational Mathematics, Vol. 4, pp. 389-396. https://doi.org/10.1007/BF02123482
  9. Hardy, R., 1971, "Multiquadric Equations of Topography and Other Irregular Surfaces," J. Geophysical Research, Vol. 76, No. 8, pp. 1905-1915. https://doi.org/10.1029/JB076i008p01905
  10. Duchon, J., 1975, "Splines Minimizing Rotation-Invariant Semi-Norms in Sobolev Spaces," in Multivariate Approximation Theory, Basel, Switzerland: Birkhauser, pp. 85-100.
  11. Park, S., 2009, "A Rational B-spline Hypervolume for Multidimensional Multivariate Modeling," J. Mech. Sci. and Tech., Vol.23, pp. 1967-1981. https://doi.org/10.1007/s12206-009-0513-2
  12. Piegl, L. and Tiller, W., 1995, The NURBS Book, Springer-Verlag.
  13. De Boor, C., 1978, A Practical Guide to Splines, New York, Springer-Verlag.
  14. Farin, G., 1990, Curves and Surfaces for Computer Aided Geometric Design, Academic Press, San Diego.