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.
翻译:体积伸展能量被广泛应用于计算简单的连接四面网格模型的体积/质量保存参数的计算中,但这种方法仍然缺乏理论支持。在本文件中,我们为体积伸展能量最小化(VSEM)提供了理论基础,以计算体积/质量保护参数。此外,我们开发了一种相关的高效VSEM算法,保证无症状R线趋同。此外,根据VSEM算法,我们提出了一种预测梯度方法,用于计算体积/质量保存最佳大众运输图,保证汇合率为$\mathcal{O}(1/m)$,并与Nesterov的加速结合,保证汇合率变为$\mathcal{O}(1/m ⁇ 2)$。数字实验为从已知基准模型中提取的各种例子的理论趋同行为提供了依据。此外,这些数字实验显示了拟议的计算法的有效性和准确性,特别是在处理3D医疗MRI脑图时。