This paper investigates a new stochastic algorithm to approximate semi-discrete optimal transport for large-scale problem, i.e., in high dimension and for a large number of points. The proposed technique relies on a hierarchical decomposition of the target discrete distribution and the transport map itself. A stochastic optimization algorithm is derived to estimate the parameters of the corresponding multi-layer weighted nearest neighbor model. This model allows for fast evaluation during synthesis and training, for which it exhibits faster empirical convergence. Several applications to patch-based image processing are investigated: texture synthesis, texture inpainting, and style transfer. The proposed models compare favorably to the state of the art, either in terms of image quality, computation time, or regarding the number of parameters. Additionally, they do not require any pixel-based optimization or training on a large dataset of natural images.

本文研究了一种新的随机算法，用于近似求解大型问题（即高维和大量点）的半离散最优输运。所提出的技术依赖于目标离散分布和传输图本身的分层分解。导出随机优化算法以估计相应的多层加权最近邻模型的参数。该模型允许在综合和训练期间进行快速评估，为此，它具有更快的经验收敛性。研究了基于补丁的图像处理的几种应用：纹理合成，纹理修复和样式转移。所提出的模型无论在图像质量，计算时间还是在参数数量方面都可以与现有技术相媲美。