Journal of the Korea Computer Graphics Society (한국컴퓨터그래픽스학회논문지)

Korea Computer Graphics Society (한국컴퓨터그래픽스학회)
권호 표지

Multilevel Editing for Hierarchical B-spline Curves using Rotation Minimizing Frames

RMF을 이용한 계층적 B-spline 곡선의 다단계 편집기법

  • Zhang, Ci (Department of Multimedia Engineering, Dongguk University) ;
  • Yoon, Seung-Hyun (Department of Multimedia Engineering, Dongguk University) ;
  • Lee, Ji-Eun (School of Computer Engineering, Chosun University)
  • 장츠 (동국대학교 멀티미디어공학과) ;
  • 윤승현 (동국대학교 멀티미디어공학과) ;
  • 이지은 (조선대학교 컴퓨터공학부)
  • Received : 2010.04.28
  • Accepted : 2010.11.08
  • Published : 2010.12.01
Download PDF
⟨ Previous Next ⟩

Abstract

We present a new technique for multilevel editing of hierarchical B-spline curves. At each level, control points of a displacement function are expressed in the rotation minimizing frames (RMFs) [1] which are computed on nodal points of the curve at previous level. When the curve is edited at previous level, the corresponding RMFs are updated and the control points of the displacement function at current level are applied to the new RMFs, which maintains the relative details of the curve at current level to those of previous level. We demonstrate the effectiveness and robustness of the proposed technique using several experimental results.

본 논문에서는 계층적 B-spline곡선 (hierarchical B-spline curve)에 대한 새로운 다단계 편집 (multilevel editing)기법을 제안한다. 각 단계 변위함수 (displacement function)의 제어점 (control point)은 이전 단계 곡선위의 노드점 (nodal point)에서 계산되는 Rotation Minimizing Frame (RMF) [1]을 기준으로 표현된다. 이전 단계에서 곡선의 형상이 편집되면 해당노드 점에서 새로운 RMF가 계산되고, 현재 단계에서 변위함수의 제어점들은 새로운 RMF를 기준으로 적용되어, 현재 단계의 곡선은 이전 단계의 곡선에 대한 상대적인 세부 형상을 유지하게 된다. 본 논문에서는 다양한 형태의 곡선에 대한 다단계 편집실험을 통해 제안된 기법의 효율성과 안정성을 입증한다.

Keywords

References

  1. W. Wang, B. Juttler, D. Zheng, and Y. Liu, "Computation of rotation minimizing frames," ACM Transactions on Graphics, vol. 27, no. 1,2008.
  2. G. Farin, Curves and Surfaces for CAGD, 5th ed. Academic Press, 2002.
  3. L. Piegl and W. Tiller, The NURBS Book, 2nd ed. Springer, 1997.
  4. D. Salomon, Curves and Surfaces for Computer Graphics. Springer, 2006.
  5. K.-T. Miura, J. Sone, A. Yamashita, T. Kaneko, M. Ueda, and M. Osano, "Adaptive refinement for B-spline subdivision curves," Journal of Three Dimensional Images, vol. 16, no. 4, pp. 74-78, 2002.
  6. B. Joe, "Knot insertion for Beta-spline curves and surfaces," ACM Transactions on Graphics, vol. 9, no. 1, pp. 41-65, 1990. https://doi.org/10.1145/77635.77638
  7. Q.-X. Huang, S.-M. Hu, and R.-R. Martin, "Fast degree elevation and knot insertion for B-spline curves," Computer Aided Geometric Design, vol. 22, pp. 183-197,2004.
  8. S. Schaefer and R. Goldman, "Nonuniform subdivision for B-splines of arbitrary degree," Computer Aided Geometric Design, vol. 26, no. 1, pp. 75-81, 2009. https://doi.org/10.1016/j.cagd.2007.12.005
  9. G. Elber and C. Gotsman, "Multiresolution control for nonuniform B-spline curve. editing," in The 3rd Pacific Graphics Conference on Computer Graphics, 1995, pp. 267- 278.
  10. G. Elber, "Multiresolution curve editing with linear constraints," in ACM Symposium on Solid and Physical Modeling, 2001,pp.109-119.
  11. S. Hahmann, B. Sauvage, and G.-P. Bonneau, "Area preserving deformation of multiresolution curves," Computer Aided Geometric Design, vol. 22, no. 4, pp. 349-367, 2005. https://doi.org/10.1016/j.cagd.2005.01.006
  12. N.-A. Dodgson, M.-S. Floater, and M. Sabin, Advances in Multiresolution for Geometric Modeling. Springer, 2005.
  13. A. Finkelstein and D.-H. Salesin, "Multiresolution curves," in Computer Graphics, 1994.
  14. L. Olsen, F.-F. Samavati, and R.-H. Bartels, "Multiresolution B-spline based on wavelet constraints," in Eurographics Symposium on Geometric Processing, 2005, pp. 1-10.
  15. S. Hahmann and G. Elber, "Constrained multiresolution geometric modeling," in Advances in Multiresolution for Geometric Modeling, 2004, pp. 69-84.
  16. A. Dreger, M.-H. Gross, and J. Schlegel, "Multiresolution triangular B-spline surfaces," in Computer Graphics International, 1998, pp. 22-26.
  17. R Kazinnik and G. Elber, "Orthogonal decomposition of nonuniform B-spline spaces using wavelets," Computer Graphics Forum, vol. 16, no. 3, 1997.
  18. D.-R. Forsey and R-H. Bartels, "Hierarchical B-spline refinement," vol. 22, no. 4, pp. 205-212,1988.
  19. P. Prusinkiewicz, F. Samavati, F. Samavati, C. Smith, and R Karwowski, "L-system description of subdivision curves," International Journal of Shape Modeling, vol. 9, pp. 41-59, 2002.
  20. K. Poon, L. Bateman, R. Karwowski, P. Prusinkiewicz, and F. Samavati, "Abstract L-system implementation of multiresolution curves based on cubic B-spline subdivision," 2003.
  21. D. Li, K. Qin, and H. Sun, "Curve modeling with constrained B-spline wavelets," Computer Aided Geometric Design, vol. 22, no. 1, pp. 45-56, 2005. https://doi.org/10.1016/j.cagd.2004.08.004
  22. B. Sauvage, S. Hahmann, and G.-P. Bonneau, "Length preserving multiresolution editing of curves," Computing, vol. 72, no. 1-2,pp. 161-170, 2004. https://doi.org/10.1007/s00607-003-0054-y
  23. L.A.R. Escriba, P.C.P. Carvalho, and L. Velho, "Interactive manipulation of multiresolution curves." [Online]. Available: citeseer.ist.psu.edu/rivera99interactive.html
  24. R. Kazinnik and G. Elber, "Orthogonal decomposition of nonuniform B-spline spaces using wavelets," Computer Graphics Forum, vol. 16, no. 3,1997.
  25. W.-M. Hsu, J.-F. Hughes, and H. Kaufman, "Direct manipulation of freeform deformations," Computer Graphics, vol. 26, no. 2, 1992.