  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.
