Nonlocal Multiscale Hierarchical Decomposition on Graphs

The decomposition of images into their meaningful components is one of the major tasks in computer vision. Tadmor, Nezzar and Vese [1] have proposed a general approach for multiscale hierarchical decomposition of images. On the basis of this work, we propose a multiscale hierarchical decomposition of functions on graphs. The decomposition is based on a discrete variational framework that makes it possible to process arbitrary discrete data sets with the natural introduction of nonlocal interactions. This leads to an approach that can be used for the decomposition of images, meshes, or arbitrary data sets by taking advantage of the graph structure. To have a fully automatic decomposition, the issue of parameter selection is fully addressed. We illustrate our approach with numerous decomposition results on images, meshes, and point clouds and show the benefits.