Skip to main content
SpringerLink
Log in

A variational approach for feature-aware B-spline curve design on surface meshes

  • Original article
  • Published:
The Visual Computer Aims and scope Submit manuscript

Abstract

Robust curve design on surface meshes with flexible controls is useful in a wide range of applications but remains challenging. Most existing methods fall into one of the two strategies: one is to discretize a curve into a polyline, which is then optimized, and the other is to directly design smooth splines on meshes. While the former approach usually needs a sufficiently dense sampling of curve points, which is computational costly, the latter approach relaxes the sampling requirement but suffers from the lack of user control. To tackle these problems, we proposed a variational method for designing feature-aware B-spline curves on surface meshes. Given the recent advances in shell space construction methods, we could relax the B-spline curve inside a simplified shell mesh and evaluate its distance to the surface using equipped bijective mapping. To effectively minimize the distance between the curve and the surface, with additional controls in the form of both internal and external constraints, we applied the interior point method and adaptively inserted knots of the spline to increase its freedom and adjust the weighting during the iterations. When the curve is close enough to the surface, it can be efficiently sampled at any resolution and robustly projected to the surface. Experiments show that our method is more robust, has higher flexibility, and generates smoother results than existing methods.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Algorithm 1
Fig. 1
Fig. 2
Fig. 3
Algorithm 2
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17

Similar content being viewed by others

References

  1. Cheng, L.T., Burchard, P., Merriman, B., Osher, S.: Motion of curves constrained on surfaces using a level-set approach. J. Comput. Phys. 175(2), 604–644 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  2. Dung, V.T., Tjahjowidodo, T.: A direct method to solve optimal knots of b-spline curves: An application for non-uniform b-spline curves fitting. PLoS ONE 12(3), 1–24 (2017)

    Article  Google Scholar 

  3. Estrada Sarlabous, J., Hernández Mederos, V., Martínez Morera, D., Velho, L., López Gil, N.: Conic-like subdivision curves on surfaces. Vis. Comput. 28, 971–982 (2012)

    Article  Google Scholar 

  4. Guennebaud, G., Jacob, B., et al.: Eigen. URl: http://eigen.tuxfamily.org3 (2010)

  5. Hofer, M.: Constrained optimization with energy-minimizing curves and curve networks: a survey. In: Proceedings of the 23rd Spring Conference on Computer Graphics, pp. 27–35 (2007)

  6. Hofer, M., Pottmann, H.: Energy-minimizing splines in manifolds. ACM Trans. Graph. (TOG) pp. 284–293 (2004)

  7. Ji, Z., Liu, L., Chen, Z., Wang, G.: Easy mesh cutting. Comput. Graph. Forum 25(3), 283–291 (2006)

    Article  Google Scholar 

  8. Jiang, Z., Schneider, T., Zorin, D., Panozzo, D.: Bijective projection in a shell. ACM Trans. Graph. 39(6), 1–18 (2020)

    Article  Google Scholar 

  9. Jin, Y., Song, D., Wang, T., Huang, J., Song, Y., He, L.: A shell space constrained approach for curve design on surface meshes. Comput. Aided Des. 113, 24–34 (2019)

    Article  MathSciNet  Google Scholar 

  10. Jung, M., Kim, H.: Snaking across 3d meshes. In: 12th Pacific Conference on Computer Graphics and Applications, 2004. PG 2004. Proceedings., pp. 87–93. IEEE (2004)

  11. Kang, H., Chen, F., Li, Y., Deng, J., Yang, Z.: Knot calculation for spline fitting via sparse optimization. Comput. Aided Des. 58, 179–188 (2015)

    Article  MathSciNet  Google Scholar 

  12. Lawonn, K., Gasteiger, R., Rössl, C., Preim, B.: Adaptive and robust curve smoothing on surface meshes. Comput. Graph. 40, 22–35 (2014)

    Article  Google Scholar 

  13. Lee, Y., Lee, S.: Geometric snakes for triangular meshes. Comput. Graph. Forum 21(3), 229–238 (2002)

    Article  Google Scholar 

  14. Lee, Y., Lee, S., Shamir, A., Cohen-Or, D., Seidel, H.P.: Mesh scissoring with minima rule and part salience. Comput. Aided Geom. Des. 22(5), 444–465 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  15. Liang, F., Zhao, J., Ji, S., Fan, C., Zhang, B.: A novel knot selection method for the error-bounded b-spline curve fitting of sampling points in the measuring process. Meas. Sci. Technol. 28(6), 065,015 (2017)

  16. Liu, Z., Zhang, H., Wu, C.: On geodesic curvature flow with level set formulation over triangulated surfaces. J. Sci. Comput. 70(2), 631–661 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  17. Livesu, M.: A heat flow based relaxation scheme for n dimensional discrete hyper surfaces. Comput. Graph. 71, 124–131 (2018)

    Article  Google Scholar 

  18. Mancinelli, C., Nazzaro, G., Pellacini, F., Puppo, E.: b/surf: Interactive b\(\backslash \)’ezier splines on surface meshes. IEEE Trans. Comput. Graph. Vis. (to be appear) (2023)

  19. Mohanty, S.D., Fahnestock, E.: Adaptive spline fitting with particle swarm optimization. Comput. Stat. 36(1), 155–191 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  20. Morera, D.M., Carvalho, P.C., Velho, L.: Modeling on triangulations with geodesic curves. Vis. Comput. 24(12), 1025–1037 (2008)

    Article  Google Scholar 

  21. Panozzo, D., Baran, I., Diamanti, O., Sorkine-Hornung, O.: Weighted averages on surfaces. ACM Trans. Graph. 32(4), 1–12 (2013)

    MATH  Google Scholar 

  22. Park, F.C., Ravani, B.: Bézier curves on Riemannian manifolds and lie groups with kinematics applications. In: International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, vol. 12846, pp. 15–21. American Society of Mechanical Engineers (1994)

  23. Pottmann, H., Hofer, M.: A variational approach to spline curves on surfaces. Comput. Aided Geom. Design 22(7), 693–709 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  24. Sharp, N., Crane, K.: You can find geodesic paths in triangle meshes by just flipping edges. ACM Trans. Graph. 39(6), 1–15 (2020)

    Article  Google Scholar 

  25. Wallner, J., Pottmann, H.: Intrinsic subdivision with smooth limits for graphics and animation. ACM Trans. Graph. 25(2), 356–74 (2006)

    Article  Google Scholar 

  26. Yeh, R., Nashed, Y.S., Peterka, T., Tricoche, X.: Fast automatic knot placement method for accurate b-spline curve fitting. Comput.-Aided Design 128, 102,905 (2020)

  27. Zheng, W., Bo, P., Liu, Y., Wang, W.: Fast B-spline curve fitting by L-BFGS. Comput. Aided Geom. Design 29(7), 448–462 (2012)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We would like to thank all the anonymous reviewers for their valuable comments. We would also like to thank Wanqiang Shen from Jiannan University for her helpful discussions. This work was supported by Key R &D Programs of Zhejiang Province (No. 2022C01220,2023C01224) and National Natural Science Foundation of China (No. 61702458) and the Fundamental Research Funds of Zhejiang Sci-Tech University (No. 23232106-Y). Yun Zhang was partially supported by the Zhejiang Province Public Welfare Technology Application Research (No. LGG22F020009), Key Lab of Film and TV Media Technology of Zhejiang Province (No. 2020E10015), and Teaching Reform Project of Communication University of Zhejiang (No. jgxm202131).

Author information

Authors and Affiliations

  1. School of Computer Science and Technology, Zhejiang Sci-Tech University, Hangzhou, 310018, China

    Rongyan Xu, Yao Jin & Huaxiong Zhang

  2. Zhejiang Provincial Innovation Center of Advanced Textile Technology, Shaoxing, 312000, China

    Yao Jin & Huaxiong Zhang

  3. College of Media Engineering, Communication University of Zhejiang, Hangzhou, 310018, China

    Yun Zhang

  4. School of Computer Science and Informatics, Cardiff University, Cardiff, Wales, CF24 4AG, UK

    Yu-kun Lai

  5. Mathworks, Natick, MA, 01760, USA

    Zhe Zhu

  6. School of Engineering and Computer Science, Victoria University of Wellington, Wellington, 6012, New Zealand

    Fang-Lue Zhang

Authors
  1. Rongyan Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yao Jin
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Huaxiong Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Yun Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Yu-kun Lai
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Zhe Zhu
    View author publications

    You can also search for this author in PubMed Google Scholar

  7. Fang-Lue Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Huaxiong Zhang.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Xu, R., Jin, Y., Zhang, H. et al. A variational approach for feature-aware B-spline curve design on surface meshes. Vis Comput 39, 3767–3781 (2023). https://doi.org/10.1007/s00371-023-03001-x

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00371-023-03001-x

Keywords

Search

Navigation