“Controlled-distortion constrained global parametrization” by Myles and Zorin

  • ©Ashish Myles and Denis Zorin




    Controlled-distortion constrained global parametrization

Session/Category Title: Deformation & Distortion




    The quality of a global parametrization is determined by a number of factors, including amount of distortion, number of singularities (cones), and alignment with features and boundaries. Placement of cones plays a decisive role in determining the overall distortion of the parametrization; at the same time, feature and boundary alignment also affect the cone placement. A number of methods were proposed for automatic choice of cone positions, either based on singularities of cross-fields and emphasizing alignment, or based on distortion optimization.In this paper we describe a method for placing cones for seamless global parametrizations with alignment constraints. We use a close relation between variation-minimizing cross-fields and related 1-forms and conformal maps, and demonstrate how it leads to a constrained optimization problem formulation. We show for boundary-aligned parametrizations metric distortion may be reduced by cone chains, sometimes to an arbitrarily small value, and the trade-off between the distortion and the number of cones can be controlled by a regularization term. Constrained parametrizations computed using our method have significantly lower distortion compared to the state-of-the art field-based method, yet maintain feature and boundary alignment. In the most extreme cases, parametrization collapse due to alignment constraints is eliminated.


    1. Ben-Chen, M., Gotsman, C., and Bunin, G. 2008. Conformal flattening by curvature prescription and metric scaling. Computer Graphics Forum 27, 2, 449–458.Google ScholarCross Ref
    2. Bommes, D., Zimmer, H., and Kobbelt, L. 2009. Mixed-integer quadrangulation. ACM Trans. Graph. 28, 3, 77. Google ScholarDigital Library
    3. Bommes, D., Lvy, B., Pietroni, N., Puppo, E., Silva, C., Tarini, M., and Zorin, D. 2012. Quad Meshing. Eurographics Association, Cagliari, Sardinia, Italy, M.-P. Cani and F. Ganovelli, Eds., 159–182.Google Scholar
    4. Bunin, G. 2008. Towards unstructured mesh generation using the inverse poisson problem. arXiv preprint arXiv:0802.2399.Google Scholar
    5. Bunin, G. 2008. A continuum theory for unstructured mesh generation in two dimensions. CAGD 25, 14–40. Google ScholarDigital Library
    6. Cappell, S., DeTurck, D., Gluck, H., and Miller, E. 2006. Cohomology of harmonic forms on riemannian manifolds with boundary. In Forum Mathematicum, vol. 18, 923–932.Google ScholarCross Ref
    7. Carr, N., Hoberock, J., Crane, K., and Hart, J. 2006. Rectangular multi-chart geometry images. In Symposium on Geometry Processing, Eurographics Association, 190. Google ScholarDigital Library
    8. Crane, K., Desbrun, M., and Schröder, P. 2010. Trivial connections on discrete surfaces. Computer Graphics Forum 29, 5 (July), 1525–1533.Google ScholarCross Ref
    9. Daniels, J., Silva, C. T., and Cohen, E. 2009. Localized quadrilateral coarsening. Computer Graphics Forum 28, 5, 1437–1444. Google ScholarDigital Library
    10. Daniels II, J., Silva, C. T., and Cohen, E. 2009. Semiregular quadrilateralonly remeshing from simplified base domains. Computer Graphics Forum 28, 5 (July), 1427–1435. Google ScholarDigital Library
    11. de Goes, F., and Crane, K., 2010. Trivial connections on discrete surfaces revisited: A simplied algorithm for simply-connected surfaces.Google Scholar
    12. Dindoš, M. 2008. Hardy Spaces and Potential Theory on C1 in Riemannian Manifolds. American Mathematical Soc.Google Scholar
    13. Dong, S., Bremer, P., Garland, M., Pascucci, V., and Hart, J. 2006. Spectral surface quadrangulation. ACM Trans. Graph. 25, 3, 1057–1066. Google ScholarDigital Library
    14. Eck, M., DeRose, T., Duchamp, T., Hoppe, H., Lounsbery, M., and Stuetzle, W. 1995. Multiresolution analysis of arbitrary meshes. SIGGRAPH 1995, 173–182. Google ScholarDigital Library
    15. Eppstein, D. 2003. Dynamic generators of topologically embedded graphs. In Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms, Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, SODA ’03, 599–608. Google ScholarDigital Library
    16. Gu, X., and Yau, S.-T. 2003. Global conformal surface parameterization. In Proc. 2003 Eurographics/ACM SIGGRAPH Symposium on Geometry Processing, SGP ’03, 127–137. Google ScholarDigital Library
    17. Jin, M., Wang, Y., Yau, S., and Gu, X. 2004. Optimal global conformal surface parameterization. In Proc. IEEE Visualiza-tion’04, 267–274. Google ScholarDigital Library
    18. Jin, M., Kim, J., Luo, F., and Gu, X. 2008. Discrete surface ricci flow. IEEE Trans. Visualization and Computer Graphics 14, 1030–1043. Google ScholarDigital Library
    19. Kälberer, F., Nieser, M., and Polthier, K. 2007. Quad-Cover: Surface Parameterization using Branched Coverings. Computer Graphics Forum 26, 3, 375–384.Google ScholarCross Ref
    20. Khodakovsky, A., Litke, N., and Schröder, P. 2003. Globally smooth parameterizations with low distortion. ACM Trans. Graph. 22, 3, 350–357. Google ScholarDigital Library
    21. Lai, Y., Jin, M., Xie, X., He, Y., Palacios, J., Zhang, E., Hu, S., and Gu, X. 2009. Metric-driven rosy field design and remeshing. IEEE Trans. Visualization and Computer Graphics, 95–108. Google ScholarDigital Library
    22. Lee, A., Sweldens, W., Schröder, P., Cowsar, L., and Dobkin, D. 1998. MAPS: multiresolution adaptive parameterization of surfaces. In SIGGRAPH 1998, 95–104. Google ScholarDigital Library
    23. Liu, L., Zhang, L., Xu, Y., Gotsman, C., and Gortler, S. J. 2008. A Local/Global approach to mesh parameterization. Computer Graphics Forum 27, 5 (July), 1495–1504. Google ScholarDigital Library
    24. Marinov, M., and Kobbelt, L. 2005. Automatic generation of structure preserving multiresolution models. Computer Graphics Forum 24, 3 (Sept.), 479–486.Google ScholarCross Ref
    25. Myles, A., and Zorin, D. 2012. Global parametrization by incremental flattening. ACM Transactions on Graphics (TOG) 31, 4, 109. Google ScholarDigital Library
    26. Myles, A., Pietroni, N., Kovacs, D., and Zorin, D. 2010. Feature-aligned T-meshes. ACM Trans. Graph. 29, 4, 1–11. Google ScholarDigital Library
    27. O’Neill, B. 2006. Elementary Differential Geometry, Revised 2nd Edition. Elementary Differential Geometry Series. Elsevier Science.Google Scholar
    28. Palacios, J., and Zhang, E. 2007. Rotational symmetry field design on surfaces. ACM Trans. Graph. 26, 3 (July). Google ScholarDigital Library
    29. Pietroni, N., Tarini, M., and Cignoni, P. 2009. Almost isometric mesh parameterization through abstract domains. IEEE Trans. Visualization and Computer Graphics 99, RapidPosts. Google ScholarDigital Library
    30. Pinkall, U., and Polthier, K. 1993. Computing discrete minimal surfaces and their conjugates. Experimental mathematics 2, 1, 15–36.Google Scholar
    31. Polthier, K. 2000. Conjugate harmonic maps and minimal surfaces. Preprint No. 446, TU-Berlin, SFB 288, 2000.Google Scholar
    32. Ray, N., Li, W., Lévy, B., Sheffer, A., and Alliez, P. 2006. Periodic global parameterization. ACM Trans. Graph. 25, 4, 1460–1485. Google ScholarDigital Library
    33. Ray, N., Vallet, B., Li, W., and Lévy, B. 2008. N-Symmetry direction field design. ACM Trans. Graph. 27, 2. Google ScholarDigital Library
    34. Ray, N., Vallet, B., Alonso, L., and Levy, B. 2009. Geometry-aware direction field processing. ACM Trans. Graph. 29, 1, 1–11. Google ScholarDigital Library
    35. Springborn, B., Schröder, P., and Pinkall, U. 2008. Conformal equivalence of triangle meshes. ACM Trans. Graph. 27 (August), 77:1–77:11. Google ScholarDigital Library
    36. Tarini, M., Pietroni, N., Cignoni, P., Panozzo, D., and Puppo, E. 2010. Practical quad mesh simplification. Computer Graphics Forum 29, 2.Google ScholarCross Ref
    37. Tong, Y., Alliez, P., Cohen-Steiner, D., and Desbrun, M. 2006. Designing quadrangulations with discrete harmonic forms. Symposium on Geometry Processing, 201–210. Google ScholarDigital Library

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