Convergence Analysis of Volumetric Stretch Energy Minimization and Its Associated Optimal Mass Transport

Volumetric stretch energy has been widely applied to the computation of volume-/mass-preserving parameterizations of simply connected tetrahedral mesh models M. However, this approach still lacks theoretical support. In this paper, we provide a theoretical foundation for volumetric stretch energy minimization (VSEM) to show that a map is a precise volume-/mass-preserving parameterization from M to a region of a specified shape if and only if its volumetric stretch energy reaches 3/2 μ(M), where μ(M) is the total mass of M. We use VSEM to compute an ε-volume-/mass-preserving map f* from M to a unit ball, where ε is the gap between the energy of f* and 3/2 μ(M). In addition, we prove the efficiency of the VSEM algorithm with guaranteed asymptotic R-linear convergence. Furthermore, based on the VSEM algorithm, we propose a projected gradient method for the computation of the ε-volume-/mass-preserving optimal mass transport map with a guaranteed convergence rate of O(1/m), and combined with Nesterov-based acceleration, the guaranteed convergence rate becomes O(1/m²). Numerical experiments are presented to justify the theoretical convergence behavior for various examples drawn from known benchmark models. Moreover, these numerical experiments show the effectiveness of the proposed algorithm, particularly in the processing of 3D medical MRI brain images.