Cubic mean value coordinates

Abstract

We present a new method for interpolating both boundary values and gradients over a 2D polygonal domain. Despite various previous efforts, it remains challenging to define a closed-form interpolant that produces natural-looking functions while allowing flexible control of boundary constraints. Our method builds on an existing transfinite interpolant over a continuous domain, which in turn extends the classical mean value interpolant. We re-derive the interpolant from the mean value property of biharmonic functions, and prove that the interpolant indeed matches the gradient constraints when the boundary is piece-wise linear. We then give closed-form formula (as generalized barycentric coordinates) for boundary constraints represented as polynomials up to degree 3 (for values) and 1 (for normal derivatives) over each polygon edge. We demonstrate the flexibility and efficiency of our coordinates in two novel applications, smooth image deformation using curved cage networks and adaptive simplification of gradient meshes.

References

1. Baran, I., and Popović, J. 2007. Automatic rigging and animation of 3d characters. ACM Transactions on Graphics 26, 3, 72:1--8.
2. Botsch, M., and Kobbelt, L. 2004. An intuitive framework for real-time freeform modeling. ACM Transactions on Graphics 23, 3, 630--634.
3. Botsch, M., Kobbelt, L., Pauly, M., Alliez, P., and Lévy, B. 2010. Polygon Mesh Processing. A. K. Peters, Natick.
4. Dyken, C., and Floater, M. S. 2009. Transfinite mean value interpolation. Computer Aided Geometric Design (CAGD) 26, 1, 117--134.
5. Farbman, Z., Hoffer, G., Lipman, Y., Cohen-Or, D., and Lischinski, D. 2009. Coordinates for instant image cloning. ACM Transactions on Graphics 28, 3, 67:1--9.
6. Ferguson, J. 1964. Multivariable curve interpolation. Journal of the ACM 11, 2, 221--228.
7. Floater, M. S., and Schulz, C. 2008. Pointwise radial minimization: Hermite interpolation on arbitrary domains. Computer Graphics Forum 27, 5, 1505--1512.
8. Floater, M. S., Hormann, K., and Kós, G. 2006. A general construction of barycentric coordinates over convex polygons. Advances in Computational Mathematics 24, 3, 311--331.
9. Floater, M. S. 2003. Mean value coordinates. Computer Aided Geometric Design (CAGD) 20, 1, 19--27.
10. Goyal, S., and Goyal, V. B. 2012. Mean value results for second and higher order partial differential equations. Applied Mathematical Sciences 6, 77-80, 3941--3957.
11. Hormann, K., and Floater, M. S. 2006. Mean value coordinates for arbitrary planar polygons. ACM Transactions on Graphics 25, 4, 1424--1441.
12. Hormann, K., and Sukumar, N. 2008. Maximum entropy coordinates for arbitrary polytopes. Computer Graphics Forum 27, 5, 1513--1520.
13. Jacobson, A., Baran, I., Popović, J., and Sorkine, O. 2011. Bounded biharmonic weights for real-time deformation. ACM Transactions on Graphics 30, 4, 78:1--8.
14. Joshi, P., Meyer, M., DeRose, T., Green, B., and Sanocki, T. 2007. Harmonic coordinates for character articulation. ACM Transactions on Graphics 26, 3, 71:1--9.
15. Ju, T., Schaefer, S., and Warren, J. 2005. Mean value coordinates for closed triangular meshes. ACM Transactions on Graphics 24, 3, 561--566.
16. Lai, Y.-K., Hu, S.-M., and Martin, R. R. 2009. Automatic and topology-preserving gradient mesh generation for image vectorization. ACM Transactions on Graphics 28, 3, 85:1--8.
17. Langer, T., and Seidel, H.-P. 2008. Higher order barycentric coordinates. Computer Graphics Forum 27, 2, 459--466.
18. Lipman, Y., Levin, D., and Cohen-Or, D. 2008. Green coordinates. ACM Transactions on Graphics 27, 3, 78:1--10.
19. Manson, J., and Schaefer, S. 2010. Moving least squares coordinates. Computer Graphics Forum 29, 5, 1517--1524.
20. Manson, J., Li, K., and Schaefer, S. 2011. Positive Gordon-Wixom coordinates. Computer Aided Design 43, 11, 1422--1426.
21. Meyer, M., Lee, H., Barr, A., and Desbrun, M. 2002. Generalized barycentric coordinates on irregular polygons. Journal of Graphics Tools 7, 1, 13--22.
22. Orzan, A., Bousseau, A., Winnemöller, H., Barla, P., Thollot, J., and Salesin, D. 2008. Diffusion curves: a vector representation for smooth-shaded images. ACM Transactions on Graphics 27, 3, 92:1--8.
23. Pinkall, U., and Polthier, K. 1993. Computing discrete minimal surfaces and their conjugates. Experimental Mathematics 2, 1, 15--36.
24. Polyanin, A. D. 2002. Handbook of linear partial differential equations for engineers and scientists. Chapman & Hall/CRC, London.
25. Sun, J., Liang, L., Wen, F., and Shum, H.-Y. 2007. Image vectorization using optimized gradient meshes. ACM Transactions on Graphics 26, 3, 11:1--7.
26. Wachspress, E. 1975. A Rational Finite Element Basis. London: Academic Press.
27. Weber, O., and Gotsman, C. 2010. Controllable conformal maps for shape deformation and interpolation. ACM Transactions on Graphics 29, 4, 78:1--11.
28. Weber, O., Ben-Chen, M., and Gotsman, C. 2009. Complex barycentric coordinates with applications to planar shape deformation. Computer Graphics Forum 28, 2, 587--597.
29. Weber, O., Ben-Chen, M., Gotsman, C., and Hormann, K. 2011. A complex view of barycentric mappings. Computer Graphics Forum 30, 5, 1533--1542.
30. Weber, O., Poranne, R., and Gotsman, C. 2012. Biharmonic coordinates. Computer Graphics Forum 31, 8, 2409--2422.
31. Xia, T., Liao, B., and Yu, Y. 2009. Patch-based image vectorization with automatic curvilinear feature alignment. ACM Transactions on Graphics 28, 5, 115:1--10.

Index Terms

• Computing methodologies

  • Computer graphics

    • Shape modeling

• Mathematics of computing

  • Mathematical analysis

    • Numerical analysis

      • Interpolation

• Theory of computation

  • Randomness, geometry and discrete structures