Variational reconstruction using subdivision surfaces with continuous sharpness control

Abstract

We present a variational method for subdivision surface reconstruction from a noisy dense mesh. A new set of subdivision rules with continuous sharpness control is introduced into Loop subdivision for better modeling subdivision surface features such as semi-sharp creases, creases, and corners. The key idea is to assign a sharpness value to each edge of the control mesh to continuously control the surface features. Based on the new subdivision rules, a variational model with L1 norm is formulated to find the control mesh and the corresponding sharpness values of the subdivision surface that best fits the input mesh. An iterative solver based on the augmented Lagrangian method and particle swarm optimization is used to solve the resulting non-linear, non-differentiable optimization problem. Our experimental results show that our method can handle meshes well with sharp/semi-sharp features and noise.

References

  1. [1]

    MPEG-4. ISO-IEC 14496-16. Coding of audio-visual objects: Animation framework extension (AFX). Available at http://mpeg.chiariglione.org/standards/ mpeg-4/animation-framework-extension-afx.

  2. [2]

    Hoppe, H.; DeRose, T.; Duchamp, T.; Halstead, M.; Jin, H.; McDonald, J.; Schweitzer, J.; Stuetzle, W. Piecewise smooth surface reconstruction. In: Proceedings of the 21st Annual Conference on Computer Graphics and Interactive Techniques, 295–302, 1994.

    Google Scholar 

  3. [3]

    DeRose, T.; Kass, M.; Truon, T. Subdivision surfaces in character animation. In: Proceedings of the 25th Annual Conference on Computer Graphics and Interactive Techniques, 85–94, 1998.

    Google Scholar 

  4. [4]

    Stam, J. On subdivision schemes generalizing uniform B-spline surfaces of arbitrary degree. Computer Aided Geometric Design Vol. 18, No. 5, 383–396, 2001.

    MathSciNet  Article  MATH  Google Scholar 

  5. [5]

    Kosinka, J.; Sabin, M. A.; Dodgson, N. A. Semi-sharp creases on subdivision curves and surfaces. Computer Graphics Forum Vol. 33, No. 5, 217–226, 2014.

    Article  Google Scholar 

  6. [6]

    Zhang, L.; Liu, L.; Gotsman, C.; Huang, H. Mesh reconstruction by meshless denoising and parameterization. Computer & Graphics Vol. 34, No. 3, 198–208, 2010.

    Article  Google Scholar 

  7. [7]

    Catmull, E.; Clark, J. Recursively generated B-spline surfaces on arbitrary topological meshes. Computer-Aided Design Vol. 10, No. 6, 350–355, 1978.

    Article  Google Scholar 

  8. [8]

    Doo, D.; Sabin, M. Behaviour of recursive division surfaces near extraordinary points. Computer-Aided Design Vol. 10, No. 6, 356–360, 1978.

    Article  Google Scholar 

  9. [9]

    Loop, C. T. Smooth subdivision surfaces based on triangles. Master Thesis. University of Utah, 1987.

    Google Scholar 

  10. [10]

    Stam, J.; Loop, C. Quad/triangle subdivision. Computer Graphics Forum Vol. 22, No. 1, 79–85, 2003.

    Article  Google Scholar 

  11. [11]

    Habib, A.; Warren, J. Edge and vertex insertion for a class of C1 subdivision surfaces. Computer Aided Geometric Design Vol. 16, No. 4, 223–247, 1999.

    MathSciNet  Article  MATH  Google Scholar 

  12. [12]

    Peters, J.; Reif, U. The simplest subdivision scheme for smoothing polyhedra. ACM Transactions on Graphics Vol. 16, No. 4, 420–431, 1997.

    Article  Google Scholar 

  13. [13]

    Kobbelt, L. \(\sqrt 3 \) -subdivisionn. In: Proceedings of the 27th Annual Conference on Computer Graphics and Interactive Techniques, 103–112, 2000.

    Google Scholar 

  14. [14]

    Stam, J. Exact evaluation of Catmull–Clark subdivision surfaces at arbitrary parameter values. In: Proceedings of the 25th Annual Conference on Computer Graphics and Interactive Techniques, 395–404, 1998.

    Google Scholar 

  15. [15]

    Sederberg, T. W.; Zheng, J.; Sewell, D.; Sabin, M. Non-uniform recursive subdivision surfaces. In: Proceedings of the 25th Annual Conference on Computer Graphics and Interactive Techniques, 387–394, 1998.

    Google Scholar 

  16. [16]

    Yang, X. Matrix weighted rational curves and surfaces. Computer Aided Geometric Design Vol. 42, 40–53, 2016.

  17. [17]

    Pan, Q.; Xu, G.; Zhang, Y. A unified method for hybrid subdivision surface design using geometric partial differential equations. Computer-Aided Design Vol. 46, 110–119, 2014.

    Article  Google Scholar 

  18. [18]

    Kosinka, J.; Sabin, M. A.; Dodgson, N. A. Creases and boundary conditions for subdivision curves. Graphical Models Vol. 76, No. 5, 240–251, 2014.

    Article  Google Scholar 

  19. [19]

    Lee, A.; Moreton, H.; Hoppe, H. Displaced subdivision surfaces. In: Proceedings of the 27th Annual Conference on Computer Graphics and Interactive Techniques, 85–94, 2000.

    Google Scholar 

  20. [20]

    Panozzo, D.; Puppo, E.; Tarini, M.; Pietroni, N.; Cignoni, P. Automatic construction of quad-based subdivision surfaces using fitmaps. IEEE Transactions on Visualization and Computer Graphics Vol. 17, No. 10, 1510–1520, 2011.

    Article  Google Scholar 

  21. [21]

    Suzuki, H.; Takeuchi, S.; Kimura, F.; Kanai, T. Subdivision surface fitting to a range of points. In: Proceedings of the 7th Pacific Conference on Computer Graphics and Applications, 158–167, 1999.

    Google Scholar 

  22. [22]

    Kanai, T. MeshToSS: Converting subdivision surfaces from dense meshes. In: Proceedings of the Vision Modeling andVisulization Conference, 325–332, 2001.

    Google Scholar 

  23. [23]

    Litke, N.; Levin, A.; Schröder, P. Fitting subdivision surfaces. In: Proceedings of the Conference on Visualization, 319–324, 2001.

    Google Scholar 

  24. [24]

    Ma, X.; Keates, S.; Jiang, Y.; Kosinka, J. Subdivision surface fitting to a dense mesh using ridges and umbilics. Computer Aided Geometric Design Vol. 32, 5–21, 2015.

    MathSciNet  Article  Google Scholar 

  25. [25]

    Marinov, M.; Kobbelt, L. Optimization methods for scattered data approximation with subdivision surfaces. Graphical Models Vol. 67, No. 5, 452–473, 2005.

    Article  Google Scholar 

  26. [26]

    Cheng, K. S.; Wang, W.; Qin, H.; Wong, K. Y.; Yang, H.; Liu, Y. Design and analysis of optimization methods for subdivision surface fitting. IEEE Transactions on Visualization and Computer Graphics Vol. 13, No. 5, 878–890, 2007.

    Article  Google Scholar 

  27. [27]

    Ma, W.; Ma, X.; Tso, S.-K.; Pan, Z. A direct approach for subdivision surface fitting from a dense triangle mesh. Computer-Aided Design Vol. 36, No. 6, 525–536, 2004.

    Article  Google Scholar 

  28. [28]

    Ling, R.; Wang, W.; Yan, D. Fitting sharp features with Loop subdivision surfaces. Computer Graphics Forum Vol. 27, No. 5, 1383–1391, 2008.

    Article  Google Scholar 

  29. [29]

    Lavoué, G.; Dupont, F. Semi-sharp subdivision surface fitting based on feature lines approximation. Computers & Graphics Vol. 33, No. 2, 151–161, 2009.

    Article  Google Scholar 

  30. [30]

    Garland, M.; Heckbert, P. S. Surface simplification using quadric error metrics. In: Proceedings of the 24th Annual Conference on Computer Graphics and Interactive Techniques, 209–216, 1997.

    Google Scholar 

  31. [31]

    Kennedy, J.; Eberhart, R. Particle swarm optimization. In: Proceedings of the IEEE International Conference on Neural Networks, Vol. 4, 1942–1948, 1995.

    Article  Google Scholar 

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Acknowledgements

The main idea of this paper was presented in the Computational Visual Media Conference 2017. This research was partially supported by the National Natural Science Foundation of China (No. 61602015), an MOE AcRF Tier 1 Grant of Singapore (RG26/15), Beijing Natural Science Foundation (No. 4162019), open funding project of State Key Lab of Virtual Reality Technology and Systems at Beihang University (No. BUAAVR-16KF-06), and the Research Foundation for Young Scholars of Beijing Technology and Business University.

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Correspondence to Xiaoqun Wu.

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This article is published with open access at Springerlink.com

Xiaoqun Wu is now an assistant professor in the School of Computer and Information Engineering, Beijing Technology and Business University, China. She received her B.S. and M.S. degrees from Zhejiang University, China, in 2007 and 2009, respectively, and her Ph.D. degree from Nanyang Technological University, Singapore, in 2014. Her research focuses on digital geometry processing and applications.

Jianmin Zheng is an associate professor in the School of Computer Science and Engineering, Nanyang Technological University, Singapore. He received his B.S. and Ph.D. degrees from Zhejiang University, China. His recent research focuses on T-spline technologies, digital geometry processing, virtual reality, visualization, interactive digital media, and applications. He has published more than 150 technical papers in international conferences and journals. He was the conference co-chair of Geometric Modeling and Processing 2014 and has served on the program committees of many international conferences.

Yiyu Cai is an associate professor in the School of Mechanical and Aerospace Engineering, Nanyang Technological University, Singapore. He received his B.S. and M.S. degrees from Zhejiang University, China, and his Ph.D. degree from the National University of Singapore. His research interests include 3D-based design, simulation, serious games, virtual reality,etc. He is also active in IDM application research for engineering, bio- and medical sciences, education, arts, etc.

Haisheng Li is a professor in the School of Computer and Information Engineering, Beijing Technology and Business University, China. He received his Ph.D. degree from Beihang University, China, in 2002. He is the director of the Network Center and the discipline leader in Computer Science and Technology in Beijing Technology and Business University, China. His current research interests include computer graphics, scientific visualization, 3D model retrieval, etc.

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Wu, X., Zheng, J., Cai, Y. et al. Variational reconstruction using subdivision surfaces with continuous sharpness control. Comp. Visual Media 3, 217–228 (2017). https://doi.org/10.1007/s41095-017-0088-2

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Keywords

  • variational model
  • subdivision surface
  • sharpness
  • surface reconstruction
  • L1norm