A Novel Bijective Angle and Volume-preservation Balanced Parameterization for $n$-dimensional Manifolds

Abstract

We propose a unified framework for bijective parameterizations of $n$-dimensional manifolds that jointly control angular and volumetric distortion through a weighted combination of conformal and volume-preserving energies. A logarithmic barrier based on signed simplex Jacobians prevents element inversion and degeneration during energy minimization. On oriented simplicial manifolds, the gradients of all three discrete energies admit a unified cotangent Laplacian-type representation that preserves the sparsity of the mesh connectivity. Based on these formulations, we develop a three-stage algorithm that constructs an initial parameterization, repairs foldings to restore strict feasibility, and subsequently minimizes the weighted distortion energy while maintaining local injectivity. The framework is formulated in a general setting, with particular attention to parameterizations onto the unit sphere ${\mathbb{S}}^n$ and the unit ball ${\mathbb{B}}^n$. We derive sufficient conditions for discrete spherical maps to have and retain degree one, and apply classical topological criteria to establish global bijectivity. Numerical experiments demonstrate that the proposed framework achieves simultaneous control of angular and volumetric distortion, while maintaining bijectivity and eliminating surface and volumetric folds. In a brain-tumor label-transfer study, the method improves tumor-label reconstruction compared with a non-bijective ablation.

30 pages