[go: up one dir, main page]
More Web Proxy on the site http://driver.im/ Skip to main content
Springer Nature Link
Log in

Stochastic Geometric Iterative Method for Loop Subdivision Surface Fitting

  • Published:
Communications in Mathematics and Statistics Aims and scope Submit manuscript

Abstract

In this paper, we propose a stochastic geometric iterative method (S-GIM) to approximate the high-resolution 3D models by finite loop subdivision surfaces. Given an input mesh as the fitting target, the initial control mesh is generated using the mesh simplification algorithm. Then, our method adjusts the control mesh iteratively to make its finite loop subdivision surface approximate the input mesh. In each geometric iteration, we randomly select part of points on the subdivision surface to calculate the difference vectors and distribute the vectors to the control points. Finally, the control points are updated by adding the weighted average of these difference vectors. We prove the convergence of S-GIM and verify it by demonstrating error curves in the experiment. In addition, compared with existing geometric iterative methods, S-GIM has a shorter running time under the same number of iteration steps.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
£29.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (United Kingdom)

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  1. Cesa-Bianchi, N.: Analysis of two gradient-based algorithms for on-line regression. J. Comput. Syst. Sci. 59(3), 392–411 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  2. Cheng, F., Fan, F., Lai, S., Huang, C., Wang, J., Yong, J.H.: Loop subdivision surface based progressive interpolation. J. Comput. Sci. Tech-CH. 24(03), 39–46 (2009)

    Article  MathSciNet  Google Scholar 

  3. Deng, C., Lin, H.: Progressive and iterative approximation for least squares B-spline curve and surface fitting. Comput. Aid. Des. 47, 32–44 (2014)

    Article  MathSciNet  Google Scholar 

  4. Garland, M., Heckbert, P.: Surface simplification using quadric error metrics. Proceedings of the ACM SIGGRAPH Conference on computer graphics 1997(07) (1997)

  5. Halstead, M., Kass, M., Derose, T.: Efficient, fair interpolation using Catmull-Clark surfaces. Proceedings of the ACM SIGGRAPH Conference on computer graphics 27(12) (2000)

  6. Hoppe, H., Derose, T., Duchamp, T., Halstead, M., Jin, H., McDonald, J., Schweitzer, J., Stuetzle, W.: Piecewise smooth surface reconstruction. Proceedings of the 21st annual conference on Computer graphics and interactive techniques 295–302 (07 1994)

  7. Kivinen, J., Smola, A. J., Williamson, R. C.: Large margin classification for moving targets. In: algorithmic learning theory, pp. 113–127. Springer, Berlin, Heidelberg (2002)

    Book  MATH  Google Scholar 

  8. Lin, H., Bao, H., Wang, G.: Totally positive bases and progressive iteration approximation. Comput. Math. Appl. 50(3–4), 575–586 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  9. Lin, H., Chen, W., Wang, G.: Curve reconstruction based on an interval B-spline curve. Visual Comput. 21(07), 418–427 (2005)

    Article  Google Scholar 

  10. Lin, H., Jin, S., Liao, H., Jian, Q.: Quality guaranteed all-hex mesh generation by a constrained volume iterative fitting algorithm. Comput. Aid. Des. (2015)

  11. Lin, H., Maekawa, T., Deng, C.: Survey on geometric iterative methods and their applications. Comput. Aid. Des. 95, 40–51 (2018)

    Article  Google Scholar 

  12. Lin, H., Zhang, Z.: An efficient method for fitting large data sets using T-splines. SIAM J. Sci. Comput. (2013)

  13. Liu, H.T.D., Kim, V.G., Chaudhuri, S., Aigerman, N., Jacobson, A.: Neural subdivision. ACM Trans. Graph. 39(4), 124 (2020)

    Article  Google Scholar 

  14. Loop, C.: Smooth subdivision surfaces based on triangles. Master’s thesis ( 1987)

  15. Lu, L., Hu, Q., Wang, G.: An iterative algorithm for degree reduction of Bezier curves. J. Comput. Aid. Des. Comput. Graph. 21(12), 1689–1693 (2009)

    Google Scholar 

  16. Ma, W., Ma, X., Tso, S.K., Pan, Z.: A direct approach for subdivision surface fitting from a dense triangle mesh. Comput. Aid. Des. 36(6), 525–536 (2004)

    Article  Google Scholar 

  17. Maekawa, T., Matsumoto, Y., Namiki, K.: Interpolation by geometric algorithm. Comput. Aid. Des. 39(04), 313–323 (2007)

    Article  Google Scholar 

  18. Marinov, M., Kobbelt, L.: Optimization methods for scattered data approximation with subdivision surfaces. Graph. Models 67(09), 452–473 (2005)

    Article  Google Scholar 

  19. Marinov, M., Kobbelt, L.: Optimization methods for scattered data approximation with subdivision surfaces. Graph. Models 67(5), 452–473 (2005)

    Article  Google Scholar 

  20. Muja, M., Lowe, D.: Flann-fast library for approximate nearest neighbors user manual. University of British Columbia, Vancouver, BC, Canada, Computer Science Department (2009)

  21. Nishiyama, Y., Morioka, M., Maekawa, T.: Loop subdivision surface fitting by geometric algorithms. Poster proceedings of pacific graphics 67–74 (2008)

  22. Sasaki, Y., Takezawa, M., Kim, S., Kawaharada, H., Maekawa, T.: Adaptive direct slicing of volumetric attribute data represented by trivariate b-spline functions. Int. J. Adv. Manuf. Tech. (2017)

  23. Xu, C., Lin, H., Hu, H., He, Y.: Fast calculation of Laplace-Beltrami eigenproblems via subdivision linear subspace. Comput. Graph. 97, 236–247 (2021)

    Article  Google Scholar 

  24. Zhang, T.: Solving large scale linear prediction problems using stochastic gradient descent algorithms (01 2004)

Download references

Acknowledgements

This work is supported by the National Natural Science Foundation of China under Grant Nos. 61872316, 61932018, and the National Key R &D Plan of China under Grant No. 2020YFB1708900.

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Zhejiang University, Hangzhou, Zhejiang, People’s Republic of China

    Chenkai Xu, Hui Hu & Hongwei Lin

  2. State Key Laboratory of CAD &CG, Zhejiang University, Hangzhou, Zhejiang, People’s Republic of China

    Yaqi He & Hongwei Lin

Authors
  1. Chenkai Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yaqi He
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Hui Hu
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Hongwei Lin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hongwei Lin.

Rights and permissions

Springer Nature or its licensor 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, C., He, Y., Hu, H. et al. Stochastic Geometric Iterative Method for Loop Subdivision Surface Fitting. Commun. Math. Stat. (2023). https://doi.org/10.1007/s40304-022-00323-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40304-022-00323-5

Keywords

Mathematics Subject Classification

Search

Navigation