Abstract
We present a framework for the computation of harmonic and conformal mappings in the plane with mathematical guarantees that the computed mappings are C∞, locally injective and satisfy strict bounds on the conformal and isometric distortion. Such mappings are very desirable in many computer graphics and geometry processing applications.
We establish the sufficient and necessary conditions for a harmonic planar mapping to have bounded distortion. Our key observation is that these conditions relate solely to the boundary behavior of the mapping. This leads to an efficient and accurate algorithm that supports handle-based interactive shape-and-image deformation and is demonstrated to outperform other state-of-the-art methods.
Supplemental Material
Available for Download
Supplemental files
- Ahlfors, L. 1966. Lectures on quasiconformal mappings, vol. 38. Amer. Mathematical Society.Google Scholar
- Aigerman, N., Poranne, R., and Lipman, Y. 2014. Lifted bijections for low distortion surface mappings. ACM Transactions on Graphics (TOG) 33, 4, 69. Google ScholarDigital Library
- Bell, S. R. 1992. The Cauchy transform, potential theory and conformal mapping, vol. 7. CRC press.Google Scholar
- Ben-Chen, M., Weber, O., and Gotsman, C. 2009. Variational harmonic maps for space deformation. ACM Transactions on Graphics (TOG) 28, 3, 34. Google ScholarDigital Library
- Duren, P. 2004. Harmonic mappings in the plane. Cambridge University Press.Google Scholar
- Floater, M. S., and Kosinka, J. 2010. On the injectivity of wachspress and mean value mappings between convex polygons. Advances in Computational Mathematics 32, 2, 163--174.Google ScholarCross Ref
- Floater, M. S., Kós, G., and Reimers, M. 2005. Mean value coordinates in 3d. Computer Aided Geometric Design 22, 7, 623--631. Google ScholarDigital Library
- Hormann, K., and Floater, M. S. 2006. Mean value coordinates for arbitrary planar polygons. ACM Transactions on Graphics (TOG) 25, 4, 1424--1441. Google ScholarDigital Library
- Igarashi, T., Moscovich, T., and Hughes, J. F. 2005. As-rigid-as-possible shape manipulation. ACM transactions on Graphics (TOG) 24, 3, 1134--1141. Google ScholarDigital Library
- Jacobson, A. 2013. Bijective mappings with generalized barycentric coordinates: a counterexample. Journal of Graphics Tools 17, 1-2, 1--4.Google ScholarCross Ref
- Joshi, P., Meyer, M., DeRose, T., and Green, B. 2007. Harmonic coordinates for character articulation. In ACM Transactions on Graphics (TOG), vol. 26, 71. Google ScholarDigital Library
- Ju, T., Schaefer, S., and Warren, J. 2005. Mean value coordinates for closed triangular meshes. In ACM Transactions on Graphics (TOG), vol. 24, 561--566. Google ScholarDigital Library
- Kharevych, L., Springborn, B., and Schröder, P. 2006. Discrete conformal mappings via circle patterns. ACM Trans. Graph. 25, 2, 412--438. Google ScholarDigital Library
- Kovalsky, S. Z., Aigerman, N., Basri, R., and Lipman, Y. 2014. Controlling singular values with semidefinite programming. ACM Transactions on Graphics (TOG), 4. Google ScholarDigital Library
- Levi, Z., and Zorin, D. 2014. Strict minimizers for geometric optimization. ACM Transactions on Graphics (TOG) 33, 6, 185. Google ScholarDigital Library
- Lévy, B., Petitjean, S., Ray, N., and Maillot, J. 2002. Least squares conformal maps for automatic texture atlas generation. ACM Transactions on Graphics (TOG) 21, 3, 362--371. Google ScholarDigital Library
- Lipman, Y., Levin, D., and Cohen-Or, D. 2008. Green coordinates. In ACM Transactions on Graphics (TOG), vol. 27, 78. Google ScholarDigital Library
- Lipman, Y. 2012. Bounded distortion mapping spaces for triangular meshes. ACM Transactions on Graphics (TOG) 31, 4, 108. Google ScholarDigital Library
- Liu, L., Zhang, L., Xu, Y., Gotsman, C., and Gortler, S. 2008. A local/global approach to mesh parameterization. In Computer Graphics Forum, vol. 27, Wiley Online Library, 1495--1504. Google ScholarDigital Library
- Poranne, R., and Lipman, Y. 2014. Provably good planar mappings. ACM Transactions on Graphics (TOG) 33, 4, 76. Google ScholarDigital Library
- Schneider, T., Hormann, K., and Floater, M. S. 2013. Bijective composite mean value mappings. In Computer Graphics Forum, vol. 32, Wiley Online Library, 137--146. Google ScholarDigital Library
- Schüller, C., Kavan, L., Panozzo, D., and Sorkine-Hornung, O. 2013. Locally injective mappings. In Computer Graphics Forum, vol. 32, Wiley Online Library, 125--135. Google ScholarDigital Library
- Sheffer, A., and de Sturler, E. 2001. Parameterization of Faceted Surfaces for Meshing using Angle-Based Flattening. Engineering with Computers 17, 3, 326--337.Google ScholarCross Ref
- Sorkine, O., and Alexa, M. 2007. As-rigid-as-possible surface modeling. In Proceedings of EUROGRAPHICS/ACM SIGGRAPH Symposium on Geometry Processing, 109--116. Google ScholarDigital Library
- Springborn, B., Schröder, P., and Pinkall, U. 2008. Conformal equivalence of triangle meshes. ACM Transactions on Graphics (TOG) 27, 3, 77. Google ScholarDigital Library
- Weber, O., and Gotsman, C. 2010. Controllable conformal maps for shape deformation and interpolation. ACM Transactions on Graphics (TOG) 29, 4, 78. Google ScholarDigital Library
- Weber, O., and Zorin, D. 2014. Locally injective parametrization with arbitrary fixed boundaries. ACM Transactions on Graphics (TOG) 33, 4, 75. Google ScholarDigital Library
- Weber, O., Sorkine, O., Lipman, Y., and Gotsman, C. 2007. Context-aware skeletal shape deformation. In Computer Graphics Forum, vol. 26, Wiley Online Library, 265--274.Google Scholar
- Weber, O., Ben-Chen, M., and Gotsman, C. 2009. Complex barycentric coordinates with applications to planar shape deformation. In Computer Graphics Forum, vol. 28, Wiley Online Library, 587--597.Google Scholar
- Weber, O., Ben-Chen, M., Gotsman, C., and Hormann, K. 2011. A complex view of barycentric mappings. In Computer Graphics Forum, vol. 30, Wiley Online Library, 1533--1542.Google Scholar
- Weber, O., Myles, A., and Zorin, D. 2012. Computing extremal quasiconformal maps. In Computer Graphics Forum, vol. 31, Wiley Online Library, 1679--1689. Google ScholarDigital Library
- Weber, O., Poranne, R., and Gotsman, C. 2012. Biharmonic coordinates. In Computer Graphics Forum, vol. 31,Wiley Online Library, 2409--2422. Google ScholarDigital Library
- Weber, O. 2010. Hybrid Methods for Interactive Shape Manipulation. PhD thesis, Technion - Israel Institute of Technology.Google Scholar
- Zayer, R., Rössl, C., Karni, Z., and Seidel, H.-P. 2005. Harmonic guidance for surface deformation. In Computer Graphics Forum, vol. 24, Wiley Online Library, 601--609.Google Scholar
Index Terms
- Bounded distortion harmonic mappings in the plane
Recommendations
Bounded distortion mapping spaces for triangular meshes
The problem of mapping triangular meshes into the plane is fundamental in geometric modeling, where planar deformations and surface parameterizations are two prominent examples. Current methods for triangular mesh mappings cannot, in general, control ...
On the convexity and feasibility of the bounded distortion harmonic mapping problem
Computation of mappings is a central building block in many geometry processing and graphics applications. The pursuit to compute mappings that are injective and have a controllable amount of conformal and isometric distortion is a long endeavor which ...
Bounded distortion harmonic shape interpolation
Planar shape interpolation is a classic problem in computer graphics. We present a novel shape interpolation method that blends C∞ planar harmonic mappings represented in closed-form. The intermediate mappings in the blending are guaranteed to be ...
Comments