Journal of KIISE:Computer Systems and Theory (한국정보과학회논문지:시스템및이론)

Korean Institute of Information Scientists and Engineers (한국정보과학회)
권호 표지

Geodesics-based Shape-preserving Mesh Parameterization

직선형 측지선에 기초한 원형보전형 메쉬 파라미터화

  • 이혜영 (홍익대학교 컴퓨터공학과)
  • Published : 2004.08.01
Download PDF
⟨ Previous Next ⟩

Abstract

Among the desirable properties of a piecewise linear parameterization, guaranteeing a one-to-one mapping (i.e., no triangle flips in the parameter plane) is often sought. A one-to-one mapping is accomplished by non-negative coefficients in the affine transformation. In the Floater's method, the coefficients were computed after the 3D mesh was flattened by geodesic polar-mapping. But using this geodesic polar map introduces unnecessary local distortion. In this paper, a simple variant of the original shape-preserving mapping technique by Floater is introduced. A new simple method for calculating barycentric coordinates by using straightest geodesics is proposed. With this method, the non-negative coefficients are computed directly on the mesh, reducing the shape distortion introduced by the previously-used polar mapping. The parameterization is then found by solving a sparse linear system, and it provides a simple and visually-smooth piecewise linear mapping, without foldovers.

구분선형 파라미터화의 특성 중 파라미터 평면상에서 중복되는 삼각형이 발생하지 않도록 하는 일대일 맵핑이 특히 강조된다. 일대일 맵핑은 아핀변환식의 비음수 계수 값으로 보장된다. Floater는 3차원 메쉬를 geodesic polar-mapping으로 평면화한 후 무게중심 좌표를 이용, 비음수 계수 값을 산출하였다. 그러나 평면화 된 삼각형은 이미 3차원상의 원형이 왜곡된 상태로 이 계수를 사용한 파라미터화는 원형왜곡을 심화시킨다. 본 논문에서는 기존의 Floater 방법을 개선한, 새로운 구분 선형 파라미터화 방법을 제안하고자 한다. 메쉬상의 직선형 측지선 길이를 이용하여 무게중심 좌표를 간단히 산출할 수 있는 새로운 방법으로 계산의 과부하 없이 비음수 계수 값을 3차원 메쉬상에서 직접 계산한다. 위의 비음수 계수로 구성된 선형시스템을 사용하여 삼각형의 중복이 없이 일대일 맵핑이 보장되는 구분선형 파라미터화를 제공한다. 본 방법은 기존 Floater방법의 평면화 단계를 제거함으로써, 이로 인한 원형왜곡을 감소시키고 파라미터화 전체 과정도 단순화하였다.

Keywords

References

  1. M. Floater and C. Gotsman, 'How to Morph Tilings Injectively,' J Compo Appl. Math., pp. 117-129 1999 https://doi.org/10.1016/S0377-0427(98)00202-7
  2. V. Surazhsky and C. Gotsman, 'High quality compatible triangulations,' Proceedings of the International Meshing Roundtable(Ithaca, NJ), 2002
  3. M. Eck, T. DeRose, T. Duchamp, H. Hoppe, M. Lounsbery, and W. Stuetzle, 'Multiresolution Analysis of Arbitrary Meshes,' Proceedigns of SIGGRAPH 95, pp. 173-182, 1995 https://doi.org/10.1145/218380.218440
  4. P. Alliez, M. Meyer, and M. Desbrun, 'Interactive geometry rerneshing,' Proceedings of SIGGRAPH 2002, ACM Transactions on Graphics, Vol. 21, 3, pp. 347-354, 2002 https://doi.org/10.1145/566570.566588
  5. X. Gu, S. J. Gortler, and H. Hoppe, 'Geometry images,' Proceedings of SIGGRAPH 02, ACM Transactions on Graphics, pp. 355-361, 2002 https://doi.org/10.1145/566570.566589
  6. U. Pinkall and K. Polthier, 'Computing discrete minimal surfaces,' Experimental Mathematics, 2(1), pp. 15-36, 1993 https://doi.org/10.1080/10586458.1993.10504266
  7. S. Haker, S. Angenent, A. Tannenbaum, R. Kikinis, G. Sapiro, and M. Halle, 'Conformal Surface Parameterization for Texture Mapping,' IEEE Transactions on Visualization and Computer Graphics, 6(2), pp. 181-189, 2000 https://doi.org/10.1109/2945.856998
  8. M. Desbrun, M. Meyer, and P. Alliez, 'Intrinsic parameterizations of surface meshes,' Eurographics 2002 Conference Proceedings, 2002
  9. B. Levy, S. Petitjean, and J Maillot, 'Least Squares Conformal Maps for Automatic Texture Atlas Generation,' ACM SIGGRAPH Proceedings, 2002
  10. A. Sheffer and E. de Struler, 'Surface Parameterization For Meshing by Triangulation Flattening,' Proceedings of the 9th International Meshing Roundtable, Sandia National Labs, pp. 161-172, 2000
  11. P. V. Sanders, J. Snyder, S. J. Gortler, and H. Hoppe, 'Texture mapping progressive meshes,' Proceedings of SIGGRAPH 2001, pp. 409-416, 2001 https://doi.org/10.1145/383259.383307
  12. W. T. Tutte, 'How to Draw A Graph,' Proc. London Math. Soc., pp. 743-768, 1963 https://doi.org/10.1112/plms/s3-13.1.743
  13. M.S. Floater, 'Parameterization and smooth approximation of surface triangulartions,' Computer Aided Geometric Design, 14(3), pp. 231-250, 1997 https://doi.org/10.1016/S0167-8396(96)00031-3
  14. M. Floater, 'Mean Value Coordinates,' CAGD (20), pp.19-27, 2003 https://doi.org/10.1016/S0167-8396(03)00002-5
  15. W. Welch and A. Witkin, 'Free-form shape design using triangulated surfaces,' Proceedings of SIGGRAPH; 94, pp. 247-256, 1994 https://doi.org/10.1145/192161.192216
  16. M. Meyer, H. Lee A. Barr, and M. Desbrun, 'Generalizaed barycentric coordinates to irregular n-gons,' Journal of Graphics Tools, 7(1), pp, 13-22, 2002 https://doi.org/10.1080/10867651.2002.10487551
  17. K. Polthier and M. Schmies, 'Straightest geodesics on polyhedral surfaces,' Mathematical Visualization, pp. 135-150, 1998 https://doi.org/10.1007/978-3-662-03567-2_11
  18. K. Polthier and M. Schmies, 'Geodesic flow on polyhedral surfaces,' Proceedings of Eurographics-IEEE Symposium on Scientific Visualization '99, 1999
  19. H. Lee, L. Kim, M. Meyer, and M. Desbrun, 'Meshes on Fire,' Computer Animation and Simulation 2001 (Eurographics Workshop), pp. 75-84, 2001