[论文解读] An efficient numerical algorithm for the L2 optimal transport problem with applications to image processing

Louis-Philippe Saumier Demers|arXiv (Cornell University)|Sep 30, 2010

Geometric Analysis and Curvature Flows参考文献 13被引用 9

一句话总结

本文提出了一种高效的数值算法，用于求解具有周期性、非均匀概率密度的L2最优传输问题，通过将阻尼牛顿法扩展至处理Monge-Ampère方程。该方法结合基于FFT的线性求解器与四阶有限差分法，实现了高精度与计算效率，在包括多发性硬化症检测在内的图像处理任务中表现出稳健性能。

ABSTRACT

We present a numerical method to solve the optimal transport problem with a quadratic cost when the source and target measures are periodic probability densities. This method is based on a numerical resolution of the corresponding Monge-Amp\`ere equation. We extend the damped Newton algorithm of Loeper and Rapetti \cite{LR} to the more general case of a non uniform density which is relevant to the optimal transport problem, and we show that our algorithm converges for sufficiently large damping coefficients. The main idea consists of designing an iterative scheme where the fully nonlinear equation is approximated by a non-constant coefficient linear elliptic PDE that we solve numerically. We introduce several improvements and some new techniques for the numerical resolution of the corresponding linear system. Namely, we use a Fast Fourier Transform (FFT) method by Strain \cite{St}, which allows to increase the efficiency of our algorithm against the standard finite difference method. Moreover, we use a fourth order finite difference scheme to approximate the partial derivatives involved in the nonlinear terms of the Newton algorithm, which are evaluated once at each iteration; this leads to a significant improvement of the accuracy of the method, but does not sacrifice its efficiency. Finally, we present some numerical experiments which demonstrate the robustness and efficiency of our method on several examples of image processing, including an application to multiple sclerosis disease detection.

研究动机与目标

• 开发一种数值上高效且准确的方法，用于求解具有周期性、非均匀概率测度的L2最优传输问题。
• 将Loeper和Rapetti提出的阻尼牛顿算法扩展至处理与最优传输相关的非均匀密度。
• 通过先进的数值技术，提高求解Monge-Ampère方程的计算效率与精度。
• 在实际图像处理场景（包括医学影像）中，展示该方法的稳健性与适用性。

提出的方法

• 该方法通过迭代阻尼牛顿格式求解Monge-Ampère方程，在每一步对完全非线性PDE进行线性化。
• 在每次牛顿迭代中，将非线性方程近似为具有变系数的线性椭圆PDE。
• 采用四阶有限差分格式对非线性项中的空间导数进行离散化，提升精度而不损失速度。
• 在每次迭代中生成的线性系统，通过基于Strain方法的快速傅里叶变换（FFT）方法高效求解。
• 算法采用阻尼系数策略，确保在足够大的值下实现收敛。
• 该方法被调整以处理周期性边界条件以及非均匀的源密度与目标密度。

实验结果

研究问题

• RQ1 能否有效将阻尼牛顿法扩展至L2最优传输问题中的非均匀密度？
• RQ2 如何在保持高精度的同时降低求解Monge-Ampère方程的计算成本？
• RQ3 将FFT求解器与高阶有限差分法结合，能在多大程度上提升最优传输算法的性能？
• RQ4 所提出的方法能否在现实世界图像处理应用中实现稳健且高效的求解？

主要发现

• 当阻尼系数足够大时，算法表现出收敛性，确保了数值稳定性。
• 基于FFT的线性求解器相比标准有限差分方法显著提升了计算效率。
• 四阶有限差分格式在不增加计算开销的前提下提高了求解精度。
• 在涉及图像处理任务的数值实验中，该方法实现了高精度与强鲁棒性。
• 该方法成功应用于医学图像分析，具体用于基于最优传输的图像配准技术检测多发性硬化症。