Nondegeneracy of Harmonic Volumetric Parameterization on Star-shaped Domains

sectionHarmonic Volumetric Parameterization using MFS After the decomposition of a given object M , we get a set of star shapes {Mi}, each region being guarded by a point gi. Then we can parameterize each subregion onto a solid sphere. A key property that we will show shortly is that such a harmonic map is guaranteed to be bijective. The harmonic map can be computed using the method of fundamental solutions (MFS) [4], by setting the boundary condition of the volumetric map to be the spherical parameterization of ∂Mi. We use the harmonic spherical parameterization [2] to get the harmonic surface map f ′ : ∂Mi → S . [2] takes the normal map as the initial mapping and conduct the optimization on local tangential plane before projecting the adjusted position back to the sphere. If the initial map (like the normal map) has large flip-over regions, the optimization will be slow and could trap locally. Since each Mi now is a star shape, the following approach efficiently gets a bijective initial spherical mapping. Figure 1 illustrates our idea. In the left picture, full visibility of the local region guarantees a bijective projection from the boundary points onto the sphere. When the boundary mapping is decided, we only need to compute the third