Center-weighted median graph filters

Linear graph filters are a class of graph-signal operators that combine mathematical tractability with practical relevance. However, a number of meaningful problems cannot be satisfactory addressed within the linear domain. Motivated by this, in this paper we introduce the concept of nonlinear weighted median graph filters (WMGF). The goal is to generalize the definition of classical weighted median filters to operate over graph signals not defined in regular domains such as time or a rectangular lattice. Under the proposed definition the output of a WMGF can be viewed as a nonlinear (median) combination of shifted versions of the input. Additionally, the value of each of these shifted versions at a particular node is found iteratively by computing the weighted median of the values of the previous shifted input within the one-hop neighborhood of the node. Within the class of WMGFs we pay special attention to center-WMGF - where each node can select its own weight but must weight all its neighbors evenly -, and analyze some of their properties, including the characterization of their roots. Simulations illustrating how our definitions can be used to model and analyze nonlinear network diffusion processes close the paper.