Unifying Local and Nonlocal Processing with Partial Difference Operators on Weighted Graphs

In this paper, local and nonlocal image processing are unified, within the same framework, by defining discrete derivatives on weighted graphs. These discrete derivatives allow to transcribe continuous partial differential equations and energy functionals to partial difference equations and discrete functionals over weighted graphs. With this methodology, we consider two gradient-based problems: regularization and mathematical morphology. The gradient-based regularization framework allows to connect isotropic and anisotropic p-Laplacians diffusions, as well as neighborhood filtering. Within the same discrete framework, we present morphological operations that allow to recover and to extend well-known PDEs-based and algebraic operations to nonlocal configurations. Finally, experimental results show the ability and the flexibility of the proposed methodology in the context of image and unorganized data set processing.