A volume-preserving parameterization is a bijective mapping that maps a 3-manifold onto a specified canonical domain that preserves the local volume. This paper formulates the computation of ball-shaped volume-preserving parameterizations as an isovolumetric energy minimization (IEM) problem with the boundary points constrained on a unit sphere. In addition, we develop a new preconditioned nonlinear conjugate gradient algorithm for solving the IEM problem with guaranteed theoretical convergence and significantly improved accuracy and computational efficiency compared to other state-of-the-art algorithms. Applications to solid shape registration and deformation are presented to highlight the usefulness of the proposed algorithm.

翻译：保体积参数化是一种将三维流形双射映射到特定规范域并保持局部体积的映射。本文将球面保体积参数化的计算表述为一个边界点约束在单位球面上的等容能量最小化问题。此外，我们提出了一种新的预处理非线性共轭梯度算法来求解该等容能量最小化问题，该算法在理论上保证收敛，并且在精度和计算效率上相比其他最先进算法均有显著提升。本文通过实体形状配准与变形等应用实例，展示了所提算法的实用价值。