Infinity Laplacian on graphs with gradient terms for image and data clustering

In this paper, we introduce a new family of graph-based operators for semi-supervised and unsupervised classification. These operators interpolate between two morphological gradient operators introduced on graphs, and are linked with the discrete infinity Laplacian. Then, we consider semi-supervised classification as the Dirichlet problem associated with this new family of operators. We show the proof of existence and uniqueness of the solution of this problem and propose an implementation. Similarly, we consider unsupervised classification as a diffusion problem associated with this new family of operators to handle it. We finally illustrate these two approaches on image segmentation and data clustering.