Advertisement

Science China Information Sciences

, Volume 55, Issue 5, pp 983–993 | Cite as

L p shape deformation

  • Lin Gao
  • GuoXin Zhang
  • YuKun Lai
Research Paper Special Focus

Abstract

Shape deformation is a fundamental tool in geometric modeling. Existing methods consider preserving local details by minimizing some energy functional measuring local distortions in the L 2 norm. This strategy distributes distortions quite uniformly to all the vertices and penalizes outliers. However, there is no unique answer for a natural deformation as it depends on the nature of the objects. Inspired by recent sparse signal reconstruction work with non L 2 norm, we introduce general L p norms to shape deformation; the positive parameter p provides the user with a flexible control over the distribution of unavoidable distortions. Compared with the traditional L 2 norm, using smaller p, distortions tend to be distributed to a sparse set of vertices, typically in feature regions, thus making most areas less distorted and structures better preserved. On the other hand, using larger p tends to distribute distortions more evenly across the whole model. This flexibility is often desirable as it mimics objects made up with different materials. By specifying varying p over the shape, more flexible control can be achieved. We demonstrate the effectiveness of the proposed algorithm with various examples.

Keywords

shape deformation Lp norm geometric modeling 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Böttcher G, Allerkamp D, Wolter F E. Multi-rate coupling of physical simulations for haptic interaction with deformable objects. Vis Comput, 2010, 26: 903–914CrossRefGoogle Scholar
  2. 2.
    Chang J, Yang X S, Pan J J, et al. A fast hybrid computation model for rectum deformation. Vis Comput, 2011, 27: 97–107CrossRefGoogle Scholar
  3. 3.
    Candes E J, Romberg J, Tao T. Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information. IEEE Trans Inf Theory, 2006, 52: 489–509MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Donoho D L, Elad M, Temlyakov V N. Stable recovery of sparse overcomplete representations in the presence of noise. IEEE Trans Inf Theory, 2006, 52: 6–18MathSciNetCrossRefGoogle Scholar
  5. 5.
    Avron H, Sharf A, Greif C, et al. l 1-Sparse reconstruction of sharp point set surfaces. ACM Trans Graph, 2010, 29: 135CrossRefGoogle Scholar
  6. 6.
    Sorkine O, Alexa M. As-rigid-as-possible surface modeling. In: Proceedings of 5th Eurographics Symposium on Geometry Processing (SGP’ 07), Barcelona, 2007. 109–116Google Scholar
  7. 7.
    Boyd S, Vandenberghe L. Convex Optimization. Cambridge: Cambridge University Press, 2004zbMATHGoogle Scholar
  8. 8.
    Botsch M, Sorkine O. On linear variational surface deformation methods. IEEE Trans Vis Comput Graph, 2008, 14: 213–230CrossRefGoogle Scholar
  9. 9.
    Cohen-Or D. Space deformations, surface deformations and the opportunities in-between. J Comput Sci Tech, 2009, 24: 2–5CrossRefGoogle Scholar
  10. 10.
    Gain J, Bechmann D. A survey of spatial deformation from a user-centered perspective. ACM Trans Graph, 2008, 27: 107CrossRefGoogle Scholar
  11. 11.
    Terzopoulos D, Platt J, Barr A, et al. Elastically deformable models. In: Proceedings of the 14th Annual Conference on Computer Graphics and Interactive Techniques(SIGGRAPH’ 87), Anaheim, 1987. 205–214Google Scholar
  12. 12.
    James D L, Pai D K. Artdefo: accurate real time deformable objects. In: Proceedings of the 26th Annual Conference on Computer Graphics and Interactive Techniques(SIGGRAPH’ 99), Los Angeles, 1999. 65–72Google Scholar
  13. 13.
    Nealen A, Mueller M, Keiser R, et al. Physically based deformable models in computer graphics. Comput Graph Forum, 2006, 25: 809–836CrossRefGoogle Scholar
  14. 14.
    Barbic J, Zhao Y L. Real-time large-deformation substructuring. ACM Trans Graph, 2011, 30: 91CrossRefGoogle Scholar
  15. 15.
    Shen Y, Ma L Z, Liu H. An MLS-based cartoon deformation. Visual Comput, 2010, 26: 1229–1239CrossRefGoogle Scholar
  16. 16.
    Lewis J P, Cordner M, Fong N. Pose space deformation: A unified approach to shape interpolation and skeletondriven deformation. In: Proceedings of 27th Annual Conference on Computer Graphics and Interactive Techniques(SIGGRAPH’ 00), New Orleans, 2000. 165–172Google Scholar
  17. 17.
    Yan H B, Hu S M, Martin R R, et al. Shape deformation using a skeleton to drive simplex transformations. IEEE Trans Vis Comput Graph, 2008, 14: 693–706CrossRefGoogle Scholar
  18. 18.
    Jacobson A, Sorkine O. Stretchable and Twistable bones for skeletal shape deformation. ACM Trans Graph, 2011, 30: 165Google Scholar
  19. 19.
    Kim B U, Feng W W, Yu Y Z. Real-time data driven deformation with affine bones. Vis Comput, 2010, 26: 487–495CrossRefGoogle Scholar
  20. 20.
    Ju T, Schaefer S, Warren J. Mean value coordinates for closed triangular meshes. ACM Trans Graph, 2005, 24: 561–566CrossRefGoogle Scholar
  21. 21.
    Joshi P, Meyer M, DeRose T, et al. Harmonic coordinates for character articulation. ACM Trans Graph, 2007, 26: 71CrossRefGoogle Scholar
  22. 22.
    Lipman Y, Kopf J, Cohen-Or D, et al. GPU-assisted positive mean value coordinates for mesh deformations. In: Proceedings of Eurographics Symposium on Geometry Processing (SGP’ 07), Barcelona, 2007. 117–123Google Scholar
  23. 23.
    Ju T, Zhou Q Y, Panne M V D, et al. Reusable skinning templates using cage-based deformations. ACM Trans Graph, 2008, 27: 122CrossRefGoogle Scholar
  24. 24.
    Lipman Y, Levin D, Cohen-Or D. Green coordinates. ACM Trans Graph, 2008, 27: 78CrossRefGoogle Scholar
  25. 25.
    Li Z, Levin D, Deng Z J. Cage-free local deformations using green coordinates. Vis Comput, 2010, 26: 1027–1036CrossRefGoogle Scholar
  26. 26.
    Jacobson A, Baran I, Popovis J, et al. Bounded biharmonic weights for real-time deformation. ACM Trans Graph, 2011, 30: 78Google Scholar
  27. 27.
    Lipman Y, Sorkine O, Levin D, et al. Linear rotation-invariant coordinates for meshes. ACM Trans Graph, 2005, 24: 479–487CrossRefGoogle Scholar
  28. 28.
    Sorkine O, Cohen-Or D, Lipman Y, et al. Laplacian surface editing. In: Proceedings of Eurographics Symposium on Geometry Processing (SGP’ 04), Nice, 2004. 175–184Google Scholar
  29. 29.
    Yu Y Z, Zhou K, Xu D, et al. Mesh editing with poisson-based gradient field manipulation. ACM Trans Graph, 2004, 23: 644–651CrossRefGoogle Scholar
  30. 30.
    Au O K C, Tai C L, Liu L, et al. Dual laplacian editing for meshes. IEEE Trans Vis Comput Graph, 2006, 12: 386–395CrossRefGoogle Scholar
  31. 31.
    Liao S H, Tong R F, Dong J X, et al. Gradient field based inhomogeneous volumetric mesh deformation for maxillofacial surgery simulation. Comput Graph, 2009, 33: 424–432CrossRefGoogle Scholar
  32. 32.
    Liao S H, Tong R F, Geng J P, et al. Inhomogeneous volumetric laplacian deformation for rhinoplasty planning and simulation system. Comput Anim Virt Worlds, 2010, 21: 331–341Google Scholar
  33. 33.
    Alexa M, Cohen-Or D, Levin D. As-rigid-as-possible shape interpolation. In: Proceedings of 27th Annual Conference on Computer Graphics and Interactive Techniques(SIGGRAPH’ 00), New Orleans, 2000. 157–164Google Scholar
  34. 34.
    Igarashi T, Moscovich T, Hughes J F. As-rigid-as-possible shape manipulation. ACM Trans Graph, 2005, 24: 1134–1141CrossRefGoogle Scholar
  35. 35.
    Bougleux S. Local and nonlocal discrete regularization on weighted graphs for image and mesh processing. Int J Comput Vis, 2009, 84: 220–236CrossRefGoogle Scholar
  36. 36.
    Popa T, Julius D, Sheffer A. Material-aware mesh deformations. In: Proceedings of the IEEE International Conference on Shape Modeling and Applications 2006(SMI’ 06), Matsushima, 2006. 22–30Google Scholar
  37. 37.
    Gal R, Sorkine O, Mitra N J, et al. iWIRES: An analyze-and-edit approach to shape manipulation. ACM Trans Graph, 2009, 28: 33CrossRefGoogle Scholar
  38. 38.
    Meyer M, Desbrun M, Schröder P, et al. Discrete differential-geometry operators for triangulated 2-manifolds. Vis Math, 2002, 3: 34–57Google Scholar
  39. 39.
    Toh K C, Todd M J, Tutuncu R H. SDPT3 Version 4.0 — A MATLAB Software for Semidefinite-Quadratic-Linear Programming, 2009Google Scholar
  40. 40.
    Chen L, Meng X X. Anisotropic resizing and deformation preserving geometric textures. Sci China Inf Sci, 2010, 53: 2441–2451MathSciNetCrossRefGoogle Scholar

Copyright information

© Science China Press and Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Department of Computer Science and TechnologyTsinghua UniversityBeijingChina
  2. 2.School of Computer Science and InformaticsCardiff UniversityWalesUK

Personalised recommendations