Spherical Parameterization for Genus Zero Surfaces Using Laplace-Beltrami Eigenfunctions
In this work, we propose a fast and simple approach to obtain a spherical parameterization of a certain class of closed surfaces without holes. Our approach relies on empirical findings that can be mathematically investigated, to a certain extent, by using Laplace-Beltrami Operator and associated geometrical tools. The mapping proposed here is defined by considering only the three first non-trivial eigenfunctions of the Laplace-Beltrami Operator. Our approach requires a topological condition on those eigenfunctions, whose nodal domains must be 2. We show the efficiency of the approach through numerical experiments performed on cortical surface meshes.
KeywordsRiemannian manifold Laplace-Beltrami Operator Surface parameterization Nodal domains
We would like to thank the reviewers for their very constructive comments. In particular a careful observation by one the reviewer is at the origin of the Remark 4 and of important modifications in the structure of the manuscript.
- 9.Lefèvre, J., Intwali, V., Hertz-Pannier, L., Hüppi, P.S., Mangin, J.-F., Dubois, J., Germanaud, D.: Surface smoothing: a way back in early brain morphogenesis. In: Mori, K., Sakuma, I., Sato, Y., Barillot, C., Navab, N. (eds.) MICCAI 2013, Part I. LNCS, vol. 8149, pp. 590–597. Springer, Heidelberg (2013) CrossRefGoogle Scholar
- 11.Sheffer, A., Praun, E., Rose, K.: Mesh parameterization methods and their applications. Found. Trends\(\textregistered \) Comput. Graph. Vision 2(2), 105–171 (2006)Google Scholar
- 12.Zelditch, S.: Local and global analysis of eigenfunctions. arXiv preprint arXiv:0903.3420 (2009)