A non-local gradient based approach of infinity Laplacian with $Γ$-convergence

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.