Weighted averages on surfaces

Abstract

We consider the problem of generalizing affine combinations in Euclidean spaces to triangle meshes: computing weighted averages of points on surfaces. We address both the forward problem, namely computing an average of given anchor points on the mesh with given weights, and the inverse problem, which is computing the weights given anchor points and a target point. Solving the forward problem on a mesh enables applications such as splines on surfaces, Laplacian smoothing and remeshing. Combining the forward and inverse problems allows us to define a correspondence mapping between two different meshes based on provided corresponding point pairs, enabling texture transfer, compatible remeshing, morphing and more. Our algorithm solves a single instance of a forward or an inverse problem in a few microseconds. We demonstrate that anchor points in the above applications can be added/removed and moved around on the meshes at interactive framerates, giving the user an immediate result as feedback.

References

1. Alexa, M. 2002. Linear combination of transformations. ACM Trans. Graph. 21, 3, 380--387.
2. Baran, I., Vlasic, D., Grinspun, E., and Popović, J. 2009. Semantic deformation transfer. ACM Trans. Graph. 28, 3.
3. Bonneel, N., van de Panne, M., Paris, S., and Heidrich, W. 2011. Displacement interpolation using lagrangian mass transport. ACM Trans. Graph. 30, 6.
4. Botsch, M., and Sorkine, O. 2008. On linear variational surface deformation methods. IEEE Trans. Visualization and Computer Graphics 14, 1, 213--230.
5. Botsch, M., Steinberg, S., Bischoff, S., and Kobbelt, L. 2002. OpenMesh - a generic and efficient polygon mesh data structure. In Proc. OpenSG Symposium.
6. Boubekeur, T., and Alexa, M. 2008. Phong tessellation. ACM Trans. Graph. 27, 5, 141:1--141:5.
7. Buss, S. R., and Fillmore, J. P. 2001. Spherical averages and applications to spherical splines and interpolation. ACM Trans. Graph. 20, 2, 95--126.
8. Cartan, É. 1929. Groupes simples clos et ouverts et géométrie riemannienne. J. Math. Pures Appl. 8, 1--33.
9. Chen, Y., and Medioni, G. 1991. Object modeling by registration of multiple range images. In Proc. IEEE International Conference on Robotics and Automation, 2724--2729.
10. Cox, T. F., and Cox, M. A. A. 2000. Multidimensional Scaling, Second Edition. Chapman & Hall/CRC, Sept.
11. Crane, K., Weischedel, C., and Wardetzky, M. 2013. Geodesics in heat. ACM Trans. Graph.. to appear.
12. de Silva, V., and Tenenbaum, J. B. 2002. Global versus local methods in nonlinear dimensionality reduction. In Proc. NIPS, 705--712.
13. Eckstein, I., Surazhsky, V., and Gotsman, C. 2001. Texture mapping with hard constraints. Comput. Graph. Forum 20, 3, 95--104.
14. Floater, M. S. 2003. Mean value coordinates. Computer Aided Geometric Design 20, 1, 19--27.
15. Fréchet, M. 1948. Les éléments alétoires de nature quelconque dans un espace distancié. Ann. Inst. H. Poincaré 10, 4, 215--310.
16. Hofer, M., and Pottmann, H. 2004. Energy-minimizing splines in manifolds. ACM Trans. Graph. 23, 3, 284--293.
17. Hormann, K., and Sukumar, N. 2008. Maximum entropy coordinates for arbitrary polytopes. In Proc. SGP, 1513--1520.
18. Hormann, K., Polthier, K., and Sheffer, A. 2008. Mesh parameterization: Theory and practice. In SIGGRAPH ASIA 2008 Course Notes.
19. Jin, J., Garland, M., and Ramos, E. A. 2009. MLS-based scalar fields over triangle meshes and their application in mesh processing. In Proc. ACM I3D, 145--153.
20. Joshi, P., Meyer, M., DeRose, T., Green, B., and Sanocki, T. 2007. Harmonic coordinates for character articulation. ACM Trans. Graph. 26, 3, 71:1--71:9.
21. Ju, T., Schaefer, S., and Warren, J. 2005. Mean value coordinates for closed triangular meshes. ACM Trans. Graph. 24, 3, 561--566.
22. Karcher, H. 1977. Riemannian center of mass and mollifier smoothing. Communications on pure and applied mathematics 30, 5, 509--541.
23. Kendall, W. 1990. Probability, convexity, and harmonic maps with small image I: uniqueness and fine existence. Proceedings of the London Mathematical Society 3, 2, 371.
24. Kim, V. G., Lipman, Y., and Funkhouser, T. 2011. Blended intrinsic maps. ACM Trans. Graph. 30, 4.
25. Kobbelt, L., Vorsatz, J., and Seidel, H.-P. 1999. Multiresolution hierarchies on unstructured triangle meshes. Comput. Geom. Theory Appl. 14, 1--3, 5--24.
26. Kraevoy, V., and Sheffer, A. 2004. Cross-parameterization and compatible remeshing of 3D models. ACM Trans. Graph. 23, 3, 861--869.
27. Langer, T., Belyaev, A., and Seidel, H.-P. 2006. Spherical barycentric coordinates. In Proc. SGP, 81--88.
28. Lipman, Y., Kopf, J., Cohen-Or, D., and Levin, D. 2007. GPU-assisted positive mean value coordinates for mesh deformations. In Proc. SGP, 117--124.
29. Lipman, Y., Rustamov, R. M., and Funkhouser, T. A. 2010. Biharmonic distance. ACM Trans. Graph. 29, 3.
30. Loop, C. 1987. Smooth subdivision surfaces based on triangles. Master's thesis, Department of Mathematics, University of Utah.
31. Ovsjanikov, M., Mérigot, Q., Mémoli, F., and Guibas, L. J. 2010. One point isometric matching with the heat kernel. Comput. Graph. Forum 29, 5, 1555--1564.
32. Pálfia, M. 2009. The Riemann barycenter computation and means of several matrices. Int. J. Comput. Math. Sci. 3, 3, 128--133.
33. Pennec, X. 1998. Computing the mean of geometric features: Application to the mean rotation. Rapport de Recherche RR--3371, INRIA - Epidaure project, Sophia Antipolis, France, March.
34. Phong, B. 1975. Illumination for computer generated pictures. Communications of the ACM 18, 6, 311--317.
35. Ritschel, T., Thormählen, T., Dachsbacher, C., Kautz, J., and Seidel, H.-P. 2010. Interactive on-surface signal deformation. ACM Trans. Graph. 29, 4.
36. Rustamov, R., Lipman, Y., and Funkhouser, T. 2009. Interior distance using barycentric coordinates. Comput. Graph. Forum 28, 5.
37. Rustamov, R. 2010. Barycentric coordinates on surfaces. Comput. Graph. Forum 29, 5, 1507--1516.
38. Sander, P. V., Gu, X., Gortler, S. J., Hoppe, H., and Snyder, J. 2000. Silhouette clipping. In Proc. ACM SIGGRAPH, 327--334.
39. Schmidt, R., Grimm, C., and Wyvill, B. 2006. Interactive decal compositing with discrete exponential maps. ACM Trans. Graph. 25, 3, 605--613.
40. Schreiner, J., Asirvatham, A., Praun, E., and Hoppe, H. 2004. Inter-surface mapping. ACM Trans. Graph. 23, 3.
41. Sethian, J. A. 1996. A fast marching level set method for monotonically advancing fronts. In Proc. Nat. Acad. Sci, 1591--1595.
42. Sorkine, O., and Cohen-Or, D. 2004. Least-squares meshes. In Proc. Shape Modeling International, 191--199.
43. Sorkine, O., Cohen-Or, D., Goldenthal, R., and Lischinski, D. 2002. Bounded-distortion piecewise mesh parameterization. In Proc. IEEE Visualization, 355--362.
44. Sumner, R. W., and Popović, J. 2004. Deformation transfer for triangle meshes. ACM Trans. Graph. 23, 3, 399--405.
45. Surazhsky, V., Surazhsky, T., Kirsanov, D., Gortler, S. J., and Hoppe, H. 2005. Fast exact and approximate geodesics on meshes. ACM Trans. Graph. 24, 3, 553--560.
46. Tzur, Y., and Tal, A. 2009. FlexiStickers: Photogrammetric texture mapping using casual images. ACM Trans. Graph. 28, 3.
47. Waldron, S. 2011. Affine generalised barycentric coordinates. Jaen Journal on Approximation 3, 2.
48. Wallner, J., and Pottmann, H. 2006. Intrinsic subdivision with smooth limits for graphics and animation. ACM Trans. Graph. 25, 2, 356--374.
49. Xin, S.-Q., Ying, X., and He, Y. 2012. Constant-time all-pairs geodesic distance query on triangle meshes. In Proc. ACM I3D.
50. Yeh, I.-C., Lin, C.-H., Sorkine, O., and Lee, T.-Y. 2011. Template-based 3D model fitting using dual-domain relaxation. IEEE Trans. Vis. Comput. Graph. 17, 8, 1178--1190.
51. Zhou, K., Synder, J., Guo, B., and Shum, H.-Y. 2004. Isocharts: stretch-driven mesh parameterization using spectral analysis. In Proc. SGP, ACM, New York, NY, USA, 45--54.

Index Terms

• Computing methodologies

  • Computer graphics

    • Animation

    • Image manipulation

    • Shape modeling

      • Parametric curve and surface models

• Mathematics of computing

  • Mathematical analysis

    • Numerical analysis

      • Interpolation

• Theory of computation

  • Randomness, geometry and discrete structures

    • Computational geometry