Seamless surface mappings

Abstract

We introduce a method for computing seamless bijective mappings between two surface-meshes that interpolates a given set of correspondences.

A common approach for computing a map between surfaces is to cut the surfaces to disks, flatten them to the plane, and extract the mapping from the flattenings by composing one flattening with the inverse of the other. So far, a significant drawback in this class of techniques is that the choice of cuts introduces a bias in the computation of the map that often causes visible artifacts and wrong correspondences.

In this paper we develop a surface mapping technique that is indifferent to the particular cut choice. This is achieved by a novel type of surface flattenings that encodes this cut-invariance, and when optimized with a suitable energy functional results in a seamless surface-to-surface map.

We show the algorithm enables producing high-quality seamless bijective maps for pairs of surfaces with a wide range of shape variability and from a small number of prescribed correspondences. We also used this framework to produce three-way, consistent and seamless mappings for triplets of surfaces.

References

1. Aigerman, N., Poranne, R., and Lipman, Y. 2014. Lifted bijections for low distortion surface mappings. ACM Trans. Graph. 33, 4 (July), 69:1--69:12.
2. Andersen, E. D., and Andersen, K. D. 1999. The MOSEK interior point optimization for linear programming: an implementation of the homogeneous algorithm. Kluwer Academic Publishers, 197--232.
3. Bogo, F., Romero, J., Loper, M., and Black, M. J. 2014. FAUST: Dataset and evaluation for 3D mesh registration. In Proceedings IEEE Conf. on Computer Vision and Pattern Recognition (CVPR), IEEE, Piscataway, NJ, USA.
4. Bommes, D., Zimmer, H., and Kobbelt, L. 2009. Mixed-integer quadrangulation. ACM Trans. Graph. 28, 3 (July), 77:1--77:10.
5. Bommes, D., Lévy, B., Pietroni, N., Puppo, E., Silva, C., Tarini, M., and Zorin, D. 2013. Quad-Mesh Generation and Processing: A Survey. Computer Graphics Forum 32, 6, 51--76.
6. Bradley, D., Popa, T., Sheffer, A., Heidrich, W., and Boubekeur, T. 2008. Markerless garment capture. ACM Trans. Graph. 27, 3 (Aug.), 99:1--99:9.
7. Bronstein, A. M., Bronstein, M. M., and Kimmel, R. 2006. Generalized multidimensional scaling: a framework for isometry-invariant partial surface matching. Proc. National Academy of Sciences (PNAS).
8. Floater, M. S. 2003. Mean value coordinates. Computer Aided Geometric Design 20, 1, 19--27.
9. Giorgi, D., Biasotti, S., and Paraboschi, L. 2007. SHREC: SHape REtrieval Contest: Watertight models track. http://watertight.ge.imati.cnr.it/.
10. Hormann, K., Lévy, B., and Sheffer, A. 2007. Mesh parameterization: Theory and practice video files associated with this course are available from the citation page. In ACM SIGGRAPH 2007 Courses, ACM, New York, NY, USA, SIGGRAPH '07.
11. Huang, Q.-X., and Guibas, L. 2013. Consistent shape maps via semidefinite programming. In Proceedings of the Eleventh Eurographics/ACMSIGGRAPH Symposium on Geometry Processing, Eurographics Association, Aire-la-Ville, Switzerland, Switzerland, SGP '13, 177--186.
12. Jain, V., Zhang, H., and van Kaick, O. 2007. Non-rigid spectral correspondence of triangle meshes. International Journal on Shape Modeling 13, 1, 101--124.
13. Kälberer, F., Nieser, M., and Polthier, K. 2007. Quadcover - surface parameterization using branched coverings. 375--384.
14. Knöppel, F., Crane, K., Pinkall, U., and Schröder, P. 2013. Globally optimal direction fields. ACM Trans. Graph. 32, 4.
15. Kraevoy, V., and Sheffer, A. 2004. Cross-parameterization and compatible remeshing of 3d models. ACM Trans. Graph. 23, 3 (Aug.), 861--869.
16. Lee, A. W. F., Dobkin, D., Sweldens, W., and Schröder, P. 1999. Multiresolution mesh morphing. In Proceedings of the 26th Annual Conference on Computer Graphics and Interactive Techniques, ACM Press/Addison-Wesley Publishing Co., New York, NY, USA, SIGGRAPH '99, 343--350.
17. Lévy, B., Petitjean, S., Ray, N., and Maillot, J. 2002. Least squares conformal maps for automatic texture atlas generation. ACM Trans. Graph. 21, 3 (July), 362--371.
18. Lin, J. L., Chuang, J. H., Lin, C. C., and Chen, C. C. 2003. Consistent parametrization by quinary subdivision for remeshing and mesh metamorphosis. In Proceedings of the 1st International Conference on Computer Graphics and Interactive Techniques in Australasia and South East Asia, ACM, New York, NY, USA, GRAPHITE '03, 151--158.
19. Lipman, Y. 2012. Bounded distortion mapping spaces for triangular meshes. ACM Trans. Graph. 31, 4 (July), 108:1--108:13.
20. Löfberg, J. 2004. Yalmip : A toolbox for modeling and optimization in MATLAB. In Proceedings of the CACSD Conference.
21. Mémoli, F., and Sapiro, G. 2005. A theoretical and computational framework for isometry invariant recognition of point cloud data. Foundations of Computational Mathematics 5, 3, 313--347.
22. Michikawa, T., Kanai, T., Fujita, M., and Chiyokura, H. 2001. Multiresolution interpolation meshes. In Computer Graphics and Applications, 2001. Proceedings. Ninth Pacific Conference on, 60--69.
23. Myles, A., and Zorin, D. 2012. Global parametrization by incremental flattening. ACM Trans. Graph. 31, 4 (July), 109:1--109:11.
24. Myles, A., and Zorin, D. 2013. Controlled-distortion constrained global parametrization. ACM Trans. Graph. 32, 4 (July), 105:1--105:14.
25. Nguyen, A., Ben-Chen, M., Welnicka, K., Ye, Y., and Guibas, L. 2011. An optimization approach to improving collections of shape maps. Computer Graphics Forum 30, 5, 1481--1491.
26. Ovsjanikov, M., Mérigot, Q., Mémoli, F., and Guibas, L. 2010. One point isometric matching with the heat kernel. In Computer Graphics Forum (Proc. of SGP).
27. Ovsjanikov, M., Ben-Chen, M., Solomon, J., Butscher, A., and Guibas, L. 2012. Functional maps: A flexible representation of maps between shapes. ACM Trans. Graph. 31, 4 (July), 30:1--30:11.
28. Panozzo, D., Baran, I., Diamanti, O., and Sorkine-Hornung, O. 2013. Weighted averages on surfaces. ACM Trans. Graph. 32, 4 (July), 60:1--60:12.
29. Praun, E., Sweldens, W., and Schröder, P. 2001. Consistent mesh parameterizations. In Proceedings of the 28th annual conference on Computer graphics and interactive techniques, ACM, 179--184.
30. Ray, N., Li, W. C., Lévy, B., Sheffer, A., and Alliez, P. 2006. Periodic global parameterization. ACM Trans. Graph. 25, 4 (Oct.), 1460--1485.
31. Ray, N., Vallet, B., Li, W. C., and Lévy, B. 2008. N-symmetry direction field design. ACM Trans. Graph. 27, 2 (May), 10:1--10:13.
32. Schreiner, J., Asirvatham, A., Praun, E., and Hoppe, H. 2004. Inter-surface mapping. ACM Trans. Graph. 23, 3 (Aug.), 870--877.
33. Sheffer, A., Praun, E., and Rose, K. 2006. Mesh parameterization methods and their applications. Found. Trends. Comput. Graph. Vis. 2, 2 (Jan.), 105--171.
34. Steiner, D., and Fischer, A. 2005. Planar parameterization for closed manifold genus-g meshes using any type of positive weights. Journal of Computing and Information Science in Engineering 5, 2, 118--125.
35. Tong, Y., Alliez, P., Cohen-Steiner, D., and Desbrun, M. 2006. Designing quadrangulations with discrete harmonic forms. In Proceedings of the Fourth Eurographics Symposium on Geometry Processing, Eurographics Association, Aire-la-Ville, Switzerland, Switzerland, SGP '06, 201--210.
36. Tsui, A., Fenton, D., Vuong, P., Hass, J., Koehl, P., Amenta, N., Coeurjolly, D., DeCarli, C., and Carmichael, O. 2013. Globally optimal cortical surface matching with exact landmark correspondence. In Proceedings of the 23rd International Conference on Information Processing in Medical Imaging, Springer-Verlag, Berlin, Heidelberg, IPMI'13, 487--498.
37. van Kaick, O., Zhang, H., Hamarneh, G., and Cohen-Or, D. 2011. A survey on shape correspondence. Computer Graphics Forum 30, 6, 1681--1707.

Index Terms

• Computing methodologies

  • Computer graphics

    • Shape modeling

• Theory of computation

  • Randomness, geometry and discrete structures

    • Computational geometry