Abstract
Parameterizations have a wide range of applications in computer graphics, geometric design and many other fields of science and engineering. Although surface parameterizations have been widely studied and are well developed, little research exists on the volumetric data due to the intrinsic difficulties in extending surface parameterization algorithms to volumetric domain. In this paper, we present a technique for parameterizing star-shaped volumes using the Green’s functions. We first show that the Green’s function on the star shape has a unique critical point. Then we prove that the Green’s functions can induce a diffeomorphism between two star-shaped volumes. We develop algorithms to parameterize star shapes to simple domains such as balls and star-shaped polycubes, and also demonstrate the volume parameterization applications: volumetric morphing, anisotropic solid texture transfer and GPU-based volumetric computation.
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References
Floater, M.S., Hormann, K.: Surface parameterization: a tutorial and survey. In: Advances in Multiresolution for Geometric Modelling, pp. 157–186. Springer, Heidelberg (2005)
Sheffer, A., Praun, E., Rose, K.: Mesh parameterization methods and their applications. Foundations and Trends® in Computer Graphics and Vision 2(2) (2006)
Labelle, F., Shewchuk, J.R.: Isosurface stuffing: fast tetrahedral meshes with good dihedral angles. ACM Trans. Graph. 26(3), 57 (2007)
Alliez, P., Cohen-Steiner, D., Yvinec, M., Desbrun, M.: Variational tetrahedral meshing. ACM Trans. Graph. 24(3), 617–625 (2005)
Attali, D., Cohen-Steiner, D., Edelsbrunner, H.: Extraction and simplification of iso-surfaces in tandem. In: Symposium on Geometry Processing, pp. 139–148 (2005)
Zhou, K., Huang, J., Snyder, J., Liu, X., Bao, H., Guo, B., Shum, H.Y.: Large mesh deformation using the volumetric graph laplacian. ACM Trans. Graph. 24(3), 496–503 (2005)
Wang, Y., Gu, X., Thompson, P.M., Yau, S.T.: 3d harmonic mapping and tetrahedral meshing of brain imaging data. In: MICCAI (2004)
Li, X., Guo, X., Wang, H., He, Y., Gu, X., Qin, H.: Harmonic volumetric mapping for solid modeling applications. In: SPM, pp. 109–120 (2007)
Martin, T., Cohen, E., Kirby, M.: Volumetric parameterization and trivariate b-spline fitting using harmonic functions. In: Proceeding of Symposium on Solid and Physical Modeling (2008)
Ju, T., Schaefer, S., Warren, J.D.: Mean value coordinates for closed triangular meshes. ACM Trans. Graph. 24(3), 561–566 (2005)
Floater, M.S., Kós, G., Reimers, M.: Mean value coordinates in 3d. Computer Aided Geometric Design 22(7), 623–631 (2005)
Fairweather, G., Karageorghis, A.: The method of fundamental solution for elliptic boundary value problems. Advances in Computational Mathematics 9(1-2), 69–95 (1998)
Li, X., Guo, X., Wang, H., He, Y., Gu, X., Qin, H.: Meshless harmonic volumetric mapping using fundamental solution methods. IEEE Transactions on Automation Science and Engineering 6(3), 409–422 (2009)
Tarini, M., Hormann, K., Cignoni, P., Montani, C.: Polycube-maps. ACM Trans. Graph. 23(3), 853–860 (2004)
Wang, H., He, Y., Li, X., Gu, X., Qin, H.: Polycube splines. Computer-Aided Design 40(6), 721–733 (2008)
Wang, H., Jin, M., He, Y., Gu, X., Qin, H.: User-controllable polycube map for manifold spline construction. In: ACM Symposium on Solid and Physical Modeling (SPM 2008), pp. 397–404 (2008)
Lin, J., Jin, X., Fan, Z., Wang, C.C.L.: Automatic polycube-maps. In: Chen, F., Jüttler, B. (eds.) GMP 2008. LNCS, vol. 4975, pp. 3–16. Springer, Heidelberg (2008)
He, Y., Wang, H., Fu, C.W., Qin, H.: A divide-and-conquer approach for automatic polycube map construction. Comput. Graph. 33(3), 369–380 (2009)
Weber, H.J., Arfken, G.B.: Mathematical Methods For Physicists. Academic Press, London (2005)
Gergen, J.J.: Note on the green function of a star-shaped three dimensional region. American Journal of Mathematics 53(4), 746–752 (1931)
MOSEK, http://www.mosek.com/
Gu, X., Wang, Y., Chan, T.F., Thompson, P.M., Yau, S.T.: Genus zero surface conformal mapping and its application to brain surface mapping. TMI 23(8), 949–958 (2004)
Si, H.: Tetgen: A quality tetrahedral mesh generator and three-dimensional delaunay triangulator, http://tetgen.berlios.de/
Kopf, J., Fu, C.W., Cohen-Or, D., Deussen, O., Lischinski, D., Wong, T.T.: Solid texture synthesis from 2d exemplars. ACM Transactions on Graphics (Proceedings of SIGGRAPH 2007) 26(3), 2:1–2:9 (2007)
Takayama, K., Okabe, M., Ijiri, T., Igarashi, T.: Lapped solid textures: filling a model with anisotropic textures. ACM Trans. Graph. 27(3), 1–9 (2008)
Praun, E., Finkelstein, A., Hoppe, H.: Lapped textures. In: SIGGRAPH, pp. 465–470 (2000)
Turing, A.: The chemical basis of morphogenesis. Royal Society of London Philosophical Transactions Series B 237, 37–72 (1952)
Turk, G.: Generating textures on arbitrary surfaces using reaction-diffusion. In: SIGGRAPH, pp. 289–298 (1991)
Witkin, A.P., Kass, M.: Reaction-diffusion textures. In: SIGGRAPH, pp. 299–308 (1991)
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Xia, J., He, Y., Han, S., Fu, CW., Luo, F., Gu, X. (2010). Parameterization of Star-Shaped Volumes Using Green’s Functions. In: Mourrain, B., Schaefer, S., Xu, G. (eds) Advances in Geometric Modeling and Processing. GMP 2010. Lecture Notes in Computer Science, vol 6130. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-13411-1_15
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DOI: https://doi.org/10.1007/978-3-642-13411-1_15
Publisher Name: Springer, Berlin, Heidelberg
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