Mathematical Morphology and Eikonal Equations on Graphs for Nonlocal Image and Data Processing

Mathematical morphology (MM) offers a wide range of operators to address various image processing problems. These operators can be defined in terms of algebraic (discrete) sets or as partial differential equations (PDEs). In this paper, we introduce a novel formulation of MM formalized as a framework of partial difference equations (PdEs) over weighted graphs of arbitrary topology. Then, we present and analyze a new family of morphological operators defined on weighted graphs. The proposed framework recovers local algebraic and PDEs-based formulations of MM. It also introduces nonlocal configurations for morphological image processing and extends PDEs-based methods to process any discrete data that can be described by a graph such as high dimensional data defined on irregular domains. Moreover, based on same ideas, we propose an adaptation of the eikonal equation on weighted graphs. Our formulation of the eikonal leads to novel applications of this equation such as nonlocal image segmentation and weighted distances or data clustering. Index Terms Mathematical morphology, partial difference equations (PdEs), weighted graphs, arbitrary discrete data sets, eikonal equation, nonlocal image segmentation, data clustering.