Surface parameterization is fundamental to computational geometry and plays an important role in various applications ranging from texturing in computer graphics to the analysis and comparison of anatomical shapes in evolutionary biology. In many cases, it is desired, or even essential, that such a parameterization is invertible. The focus of this talk is invertibility, which while desirable, often presents a challenge in the computation of parameterizations.
Remarkably, in the special case of harmonic parameterization onto a convex subset, invertibility is guaranteed by Tutte’s graph embedding and its continuous analog, the Rado-Kneser-Choquet Theorem. I will discuss our work on generalizing these results to the non-convex case and present simple geometric conditions for invertibility.
Joint work with Noam Aigerman, Misha Kazhdan, Ingrid Daubechies, Jianfeng Lu and Stefan Steinerberger