最近, X. Zhao等人[33]基于Monge-Brenier理论提出了一个用最优传输设计的保面积映射算法. 该算法理论上严谨、可靠, 计算过程中并行计算效率高, 应用广泛. 本文致力于这一美丽工作的梳理与解释, 期望能够更好地理解当今医学、信息学等领域中图像处理的主流方法. 与传统的Monge-Kantorovich方法相比, 该计算方法有以下改进部分: 首先, 将变量的数量从O(n^2)到O(n), 增加了计算的可行性; 第二, 将最优传输问题转化为凸优化问题, 使其可以通过牛顿迭代法有效实现; 第三,参考顾险峰的想法引入功率图以及面积权重策略法, 使得在算法中,可以定量而精确地控制和调整各个胞腔的面积大小, 显着降低了问题的复杂性, 并提高了胞腔分解结果可视化的效率、灵活性以及可扩展性. 最后提供了几种不同类型曲面模型的实验结果, 用于检验算法的高效性、稳健性和实用性.

Recently, based on Monge-Brenier theory, X. Zhao et al. presented a novel area-preservation mapping using the optimal mass transport technique in [33]. This　optimal transport map approach was rigorous and solid in theory, efficient and parallel in computation, and general for various applications. Our aim was to comb and explain this beautiful work so that we could have a better understanding for current mainstream methods of image processing in medicine and informatics. Compared with traditional Monge-Kantorovich methods, the improvements of this algorithm were as follows: Firstly, the number of variables were decreased from O(n^2) to O(n), so that the calculation was more feasible. Secondly, the OMT problem was transformed into a convex function optimization problem, which could be effectively solved with Newton’s iteration method. Thirdly, with reference to Gu Xianfeng's idea, the power diagram theory was introduced into the algorithm framework, and each size of the cells could be accurately controlled and adjusted by users. Finally, several experimental results for different types of surface models were provided to test the efficiency and robustness of the algorithm.