“Simple Formulas For Quasiconformal Plane Deformations” by Lipman, Kim and Funkhouser

We introduce a simple formula for 4-point planar warping that produces provably good 2D deformations. In contrast to previous work, the new deformations minimize the maximum conformal distortion and spread the distortion equally across the domain. We derive closed-form formulas for computing the 4-point interpolant and analyze its properties. We further explore applications to 2D shape deformations by building local deformation operators that use thin-plate splines to further deform the 4-point interpolant to satisfy certain boundary conditions. Although this modification no longer has any theoretical guarantees, we demonstrate that, practically, these local operators can be used to create compound deformations with fewer control points and smaller worst-case distortions in comparisons to the state-of-the-art.

Ahlfors, L. 1966. Complex Analysis.Google Scholar
Ahlfors, L. V. 2006. Lectures on Quasiconformal Mappings. University Lecture Series, vol. 38.Google Scholar
Bookstein, F. L. 1989. Principal warps: Thin-Plate splines and the decomposition of deformations. IEEE Trans. Pattern Anal. Mach. Intell. 11, 567–585. Google ScholarDigital Library
Desbrun, M., Meyer, M., and Alliez, P. 2002. Intrinsic parameterizations of surface meshes. Comput. Graph. Forum 21.Google Scholar
Fletcher, A. and Marković, V. 2007. Quasiconformal Maps and Teichmüller Theory. Oxford Graduate Texts in Mathematics, Oxford University Press.Google Scholar
Floater, M. S. 2003. Mean value coordinates. Comput. Aid. Geom. Des. 20, 19–27. Google ScholarDigital Library
Igarashi, T., Moscovich, T., and Hughes, F. J. 2005. As-Rigid-as-Possible shape manipulation. ACM Trans. Graph 24, 1134–1141. Google ScholarDigital Library
Ju, T., Schaefer, S., and Warren, J. 2005. Mean value coordinates for closed triangular meshes. ACM Trans. Graph. 24, 561–566. Google ScholarDigital Library
Lévy, B., Petitjean, S., Ray, N., and Maillo t, J. 2002. Least squares conformal maps for automatic texture atlas generation. In Proceedings of ACM SIGGRAPH Conference, ACM, New York. Google ScholarDigital Library
Lipman, Y., Levin, D., and Cohen-Or, D. 2008. Green coordinates. ACM Trans. Graph. 27, 3. Google ScholarDigital Library
Schaefer, S., McPhail, T., and Warren, J. 2006. Image deformation using moving least squares. ACM Trans. Graph. 25, 533–540. Google ScholarDigital Library
Sederberg, T. and Parry, S. 1986. Free-Form deformation of solid geometric models. SIGGRAPH Comput. Graph. 20, 151–160. Google ScholarDigital Library
Weber, O., G. C. 2010. Controllable conformal maps for shape deformation and interpolation. ACM Trans. Graph. 29, 78:1–78:11. Google ScholarDigital Library
Weber, O., Ben-Chen, M., and Gotsman, C. 2009. Complex barycentric coordinates with applications to planar shape deformation. Comput. Graph. Forum 28.Google Scholar
Wendland, H. 2005. Scattered Data Approximation. Cambridge Monographs on Applied and Computational Mathematics, No. 17.Google Scholar
Zeng, W. and Gu, X. D. 2011. Registration for 3d surfaces with large deformations using quasi-conformal curvature flow. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR). 2457–2464. Google ScholarDigital Library
Zeng, W., Luo, F., Yau, S. T., and Gu, X. D. 2009. Surface quasi-conformal mapping by solving beltrami equations. In Proceedings of the 13th IMA International Conference on Mathematics of Surfaces XIII. Springer, Berlin. 391–408. Google ScholarDigital Library
Zeng, W., Marino, J., Chaitanya Gurijala, K., Gu, X., and Kaufman, A. 2010. Supine and prone colon registration using quasi-conformal mapping. IEEE Trans. Vis. Comput. Graph. 16, 6, 1348–1357. Google ScholarDigital Library