Volume-Preserving Parameterizations via Preconditioned Nonlinear Conjugate Gradient Method

A volume-preserving parameterization is a bijective mapping that maps a 3-manifold onto a canonical domain while preserving local volume. We formulate this problem as an unconstrained nonlinear optimization problem by introducing an isovolumetric energy that quantifies volumetric distortion. This energy is minimized using a preconditioned nonlinear conjugate gradient method, which is globally convergent when the line search satisfies the strong Wolfe conditions. Numerical experiments demonstrate that the proposed method achieves improved accuracy and efficiency, and it supports applications in shape registration and volumetric deformation.