n-Dimensional Volumetric Stretch Energy Minimization for Volume-/Mass-Preserving Parameterizations

In this paper, we develop an n dimensional volumetric stretch energy (n-VSE) functional for the volume-/mass-preserving parameterization of the n-manifolds topologically equivalent to n-ball. The n-VSE has a lower bound and equal to it if and only if the map is volume-/mass-preserving. This motivates us to minimize the n-VSE to achieve the ideal volume-/mass-preserving parameterization. In the discrete case, we also guarantee the relation between the lower bound and the volume-/mass-preservation, and propose the spherical and ball volume-/mass-preserving parameterization algorithms. The numerical experiments indicate the accuracy and robustness of the proposed algorithms. The modified algorithms are applied to the manifold registration and deformation, showing the versatility of n-VSE.