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

TL;DR

This paper establishes a theoretical foundation for volumetric stretch energy minimization (VSEM), introduces an efficient algorithm with guaranteed convergence, and applies it to compute volume-preserving maps, including in medical imaging.

Contribution

It provides the first theoretical support for VSEM, develops a convergent algorithm, and extends it to optimal mass transport with accelerated convergence rates.

Findings

The VSEM algorithm has guaranteed asymptotic R-linear convergence.

The proposed method achieves a convergence rate of O(1/m) and O(1/m^2) with acceleration.

Numerical experiments confirm the effectiveness and accuracy in medical MRI brain image processing.

Abstract

The volumetric stretch energy has been widely applied to the computation of volume-/mass-preserving parameterizations of simply connected tetrahedral mesh models. However, this approach still lacks theoretical support. In this paper, we provide the theoretical foundation for volumetric stretch energy minimization (VSEM) to compute volume-/mass-preserving parameterizations. In addition, we develop an associated efficient 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 volume/mass-preserving optimal mass transport map with a guaranteed convergence rate of $\mathcal{O}(1/m)$, and combined with Nesterov-based acceleration, the guaranteed convergence rate becomes $\mathcal{O}(1/m^2)$. 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 and accuracy of the proposed algorithm, particularly in the processing of 3D medical MRI brain images.