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L p shape deformation

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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.

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References

  1. Böttcher G, Allerkamp D, Wolter F E. Multi-rate coupling of physical simulations for haptic interaction with deformable objects. Vis Comput, 2010, 26: 903–914

    Article  Google Scholar 

  2. Chang J, Yang X S, Pan J J, et al. A fast hybrid computation model for rectum deformation. Vis Comput, 2011, 27: 97–107

    Article  Google Scholar 

  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: 489–509

    Article  MathSciNet  MATH  Google Scholar 

  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: 6–18

    Article  MathSciNet  Google Scholar 

  5. Avron H, Sharf A, Greif C, et al. l 1-Sparse reconstruction of sharp point set surfaces. ACM Trans Graph, 2010, 29: 135

    Article  Google Scholar 

  6. Sorkine O, Alexa M. As-rigid-as-possible surface modeling. In: Proceedings of 5th Eurographics Symposium on Geometry Processing (SGP’ 07), Barcelona, 2007. 109–116

  7. Boyd S, Vandenberghe L. Convex Optimization. Cambridge: Cambridge University Press, 2004

    MATH  Google Scholar 

  8. Botsch M, Sorkine O. On linear variational surface deformation methods. IEEE Trans Vis Comput Graph, 2008, 14: 213–230

    Article  Google Scholar 

  9. Cohen-Or D. Space deformations, surface deformations and the opportunities in-between. J Comput Sci Tech, 2009, 24: 2–5

    Article  Google Scholar 

  10. Gain J, Bechmann D. A survey of spatial deformation from a user-centered perspective. ACM Trans Graph, 2008, 27: 107

    Article  Google Scholar 

  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. 205–214

  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. 65–72

  13. Nealen A, Mueller M, Keiser R, et al. Physically based deformable models in computer graphics. Comput Graph Forum, 2006, 25: 809–836

    Article  Google Scholar 

  14. Barbic J, Zhao Y L. Real-time large-deformation substructuring. ACM Trans Graph, 2011, 30: 91

    Article  Google Scholar 

  15. Shen Y, Ma L Z, Liu H. An MLS-based cartoon deformation. Visual Comput, 2010, 26: 1229–1239

    Article  Google Scholar 

  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. 165–172

  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: 693–706

    Article  Google Scholar 

  18. Jacobson A, Sorkine O. Stretchable and Twistable bones for skeletal shape deformation. ACM Trans Graph, 2011, 30: 165

    Google Scholar 

  19. Kim B U, Feng W W, Yu Y Z. Real-time data driven deformation with affine bones. Vis Comput, 2010, 26: 487–495

    Article  Google Scholar 

  20. Ju T, Schaefer S, Warren J. Mean value coordinates for closed triangular meshes. ACM Trans Graph, 2005, 24: 561–566

    Article  Google Scholar 

  21. Joshi P, Meyer M, DeRose T, et al. Harmonic coordinates for character articulation. ACM Trans Graph, 2007, 26: 71

    Article  Google Scholar 

  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. 117–123

  23. Ju T, Zhou Q Y, Panne M V D, et al. Reusable skinning templates using cage-based deformations. ACM Trans Graph, 2008, 27: 122

    Article  Google Scholar 

  24. Lipman Y, Levin D, Cohen-Or D. Green coordinates. ACM Trans Graph, 2008, 27: 78

    Article  Google Scholar 

  25. Li Z, Levin D, Deng Z J. Cage-free local deformations using green coordinates. Vis Comput, 2010, 26: 1027–1036

    Article  Google Scholar 

  26. Jacobson A, Baran I, Popovis J, et al. Bounded biharmonic weights for real-time deformation. ACM Trans Graph, 2011, 30: 78

    Google Scholar 

  27. Lipman Y, Sorkine O, Levin D, et al. Linear rotation-invariant coordinates for meshes. ACM Trans Graph, 2005, 24: 479–487

    Article  Google Scholar 

  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. 175–184

  29. Yu Y Z, Zhou K, Xu D, et al. Mesh editing with poisson-based gradient field manipulation. ACM Trans Graph, 2004, 23: 644–651

    Article  Google Scholar 

  30. Au O K C, Tai C L, Liu L, et al. Dual laplacian editing for meshes. IEEE Trans Vis Comput Graph, 2006, 12: 386–395

    Article  Google Scholar 

  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: 424–432

    Article  Google Scholar 

  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: 331–341

    Google Scholar 

  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. 157–164

  34. Igarashi T, Moscovich T, Hughes J F. As-rigid-as-possible shape manipulation. ACM Trans Graph, 2005, 24: 1134–1141

    Article  Google Scholar 

  35. Bougleux S. Local and nonlocal discrete regularization on weighted graphs for image and mesh processing. Int J Comput Vis, 2009, 84: 220–236

    Article  Google Scholar 

  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. 22–30

  37. Gal R, Sorkine O, Mitra N J, et al. iWIRES: An analyze-and-edit approach to shape manipulation. ACM Trans Graph, 2009, 28: 33

    Article  Google Scholar 

  38. Meyer M, Desbrun M, Schröder P, et al. Discrete differential-geometry operators for triangulated 2-manifolds. Vis Math, 2002, 3: 34–57

    Google Scholar 

  39. Toh K C, Todd M J, Tutuncu R H. SDPT3 Version 4.0 — A MATLAB Software for Semidefinite-Quadratic-Linear Programming, 2009

  40. Chen L, Meng X X. Anisotropic resizing and deformation preserving geometric textures. Sci China Inf Sci, 2010, 53: 2441–2451

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Computer Science and Technology, Tsinghua University, Beijing, 100084, China

    Lin Gao & GuoXin Zhang

  2. School of Computer Science and Informatics, Cardiff University, Wales, CF24 3AA, UK

    YuKun Lai

Authors
  1. Lin Gao
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  2. GuoXin Zhang
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  3. YuKun Lai
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Correspondence to Lin Gao.

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Gao, L., Zhang, G. & Lai, Y. L p shape deformation. Sci. China Inf. Sci. 55, 983–993 (2012). https://doi.org/10.1007/s11432-012-4574-y

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  • DOI: https://doi.org/10.1007/s11432-012-4574-y

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