As-rigid-as-possible surface modeling

ABSTRACT

Modeling tasks, such as surface deformation and editing, can be analyzed by observing the local behavior of the surface. We argue that defining a modeling operation by asking for rigidity of the local transformations is useful in various settings. Such formulation leads to a non-linear, yet conceptually simple energy formulation, which is to be minimized by the deformed surface under particular modeling constraints. We devise a simple iterative mesh editing scheme based on this principle, that leads to detail-preserving and intuitive deformations. Our algorithm is effective and notably easy to implement, making it attractive for practical modeling applications.

References

1. {ACOL00} Alexa M., Cohen-Or D., Levin D.: As-rigid-aspossible shape interpolation. In Proceedings of ACM SIGGRAPH (2000), pp. 157--164.
2. {Ale01} Alexa M.: Local control for mesh morphing. In Proceedings of SMI (2001), pp. 209--215.
3. {ATLF06} Au O. K.-C., Tai C.-L., Liu L., Fu H.: Dual Laplacian editing for meshes. IEEE TVCG 12, 3 (2006), 386--395.
4. {BK04} Botsch M., Kobbelt L.: An intuitive framework for real-time freeform modeling. ACM TOG 23, 3 (2004), 630--634.
5. {BPGK06} Botsch M., Pauly M., Gross M., Kobbelt L.: PriMo: Coupled prisms for intuitive surface modeling. In Proceedings of SGP (2006), pp. 11--20.
6. {BS07} Botsch M., Sorkine O.: On linear variational surface deformation methods. IEEE TVCG (2007). To appear.
7. {BSPG06} Botsch M., Sumner R., Pauly M., Gross M.: Deformation transfer for detail-preserving surface editing. In Proceedings of VMV (2006), pp. 357--364.
8. {CCOST05} Chen D., Cohen-Or D., Sorkine O., Toledo S.: Algebraic analysis of high-pass quantization. ACM TOG 24, 4 (2005), 1259--1282.
9. {GSS99} Guskov I., Sweldens W., Schröder P.: Multiresolution signal processing for meshes. In Proceedings of ACM SIGGRAPH (1999), pp. 325--334.
10. {Hor87} Horn B. K. P.: Closed-form solution of absolute orientation using unit quaternions. Journal of the Optical Society of America 4, 4 (1987).
11. {HSL*06} Huang J., Shi X., Liu X., Zhou K., Wei L.-Y., Teng S., Bao H., Guo B., Shum H.-Y.: Subspace gradient domain mesh deformation. ACM TOG 25, 3 (2006), 1126--1134.
12. {IMH05} Igarashi T., Moscovich T., Hughes J. F.: Asrigid-as-possible shape manipulation. ACM TOG 24, 3 (2005), 1134--1141.
13. {JSW05} Ju T., Schaefer S., Warren J.: Mean value coordinates for closed triangular meshes. ACM TOG 24, 3 (2005), 561--566.
14. {KCVS98} Kobbelt L., Campagna S., Vorsatz J., Seidel H.-P.: Interactive multi-resolution modeling on arbitrary meshes. In Proceedings of ACM SIGGRAPH (1998), ACM Press, pp. 105--114.
15. {KS06} Kraevoy V., Sheffer A.: Mean-value geometry encoding. IJSM 12, 1 (2006), 29--46.
16. {LCOGL07} Lipman Y., Cohen-Or D., Gal R., Levin D.: Volume and shape preservation via moving frame manipulation. ACM TOG 26, 1 (2007).
17. {LSCO*04} Lipman Y., Sorkine O., Cohen-Or D., Levin D., Rössl C., Seidel H.-P.: Differential coordinates for interactive mesh editing. In Proceedings of SMI (2004), pp. 181--190.
18. {LSLCO05} Lipman Y., Sorkine O., Levin D., Cohen-Or D.: Linear rotation-invariant coordinates for meshes. ACM TOG 24, 3 (2005), 479--487.
19. {MDSB03} Meyer M., Desbrun M., Schröder P., Barr A. H.: Discrete differential-geometry operators for triangulated 2-manifolds. In Visualization and Mathematics III, Hege H.-C., Polthier K., (Eds.). Springer-Verlag, Heidelberg, 2003, pp. 35--57.
20. {PP93} Pinkall U., Polthier K.: Computing discrete minimal surfaces and their conjugates. Experiment. Math. 2, 1 (1993), 15--36.
21. {PTVF92} Press W. H., Teukolsky S. A., Vetterling W. T., Flannery B. P.: Numerical Recipes in C: The Art of Scientific Computing. Cambridge University Press, 1992.
22. {SLCO*04} Sorkine O., Lipman Y., Cohen-Or D., Alexa M., Rössl C., Seidel H.-P.: Laplacian surface editing. In Proceedings of SGP (2004), pp. 179--188.
23. {SMW06} Schaefer S., McPhail T., Warren J.: Image deformation using moving least squares. ACM TOG 25, 3 (2006), 533--540.
24. {Sor06} Sorkine O.: Differential representations for mesh processing. Computer Graphics Forum 25, 4 (2006), 789--807.
25. {Tol03} Toledo S.: Taucs: A Library of Sparse Linear Solvers, version 2.2. Tel-Aviv University, Available online at http://www.tau.ac.il/~stoledo/taucs/, Sept. 2003.
26. {TPBF87} Terzopoulos D., Platt J., Barr A., Fleischer K.: Elastically deformable models. In Proceedings of ACM SIGGRAPH (1987), pp. 205--214.
27. {WBH*07} Wardetzky M., Bergou M., Harmon D., Zorin D., Grinspun E.: Discrete quadratic curvature energies. CAGD (2007). To appear.
28. {XZWB05} Xu D., Zhang H., Wang Q., Bao H.: Poisson shape interpolation. In Proceedings of SPM (2005), pp. 267--274.
29. {YZX*04} Yu Y., Zhou K., Xu D., Shi X., Bao H., Guo B., Shum H.-Y.: Mesh editing with Poisson-based gradient field manipulation. ACM TOG 23, 3 (2004), 644--651.
30. {ZRKS05} Zayer R., Rössl C., Karni Z., Seidel H.-P.: Harmonic guidance for surface deformation. In Computer Graphics Forum (Proceedings of Eurographics) (2005), pp. 601--609.
31. {ZSS97} Zorin D., Schröder P., Sweldens W.: Interactive multiresolution mesh editing. In Proceedings of ACM SIGGRAPH (1997), pp. 259--268.

Index Terms

• Computing methodologies

  • Computer graphics

    • Shape modeling

• Theory of computation

  • Randomness, geometry and discrete structures