We propose an infinity Laplacian method to address the problem of interpolation on an unstructured point cloud. In doing so, we find the labeling function with the smallest infinity norm of its gradient. By introducing the non-local gradient, the continuous functional is approximated with a discrete form. The discrete problem is convex and can be solved efficiently with the split Bregman method. Experimental results indicate that our approach provides consistent interpolations and the labeling functions obtained are globally smooth, even in the case of extreme low sampling rate. More importantly, convergence of the discrete minimizer to the optimal continuous labeling function is proved using $\Gamma$-convergence and compactness, which guarantees the reliability of the infinity Laplacian method in various potential applications.

翻译：我们提出一个无限拉普拉西亚方法来解决非结构化点云层的内插问题。 在这样做的时候,我们发现其梯度的最小无限规范的标签功能。 通过引入非本地梯度,连续功能与离散的形态相近。 离散问题是一个共和问题,可以通过分裂的布雷格曼方法有效解决。 实验结果表明,我们的方法提供了一致的内插,所获得的标签功能是全球均匀的,即使在取样率极低的情况下也是如此。 更重要的是,离散最小化器与最佳连续标签功能的汇合使用$\Gamma$-converggggence和紧凑性来证明,这就保证了各种潜在应用中无限拉普利西亚方法的可靠性。