  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.
