Graph Wedgelets: Adaptive Data Compression on Graphs Based on Binary Wedge Partitioning Trees and Geometric Wavelets

We introduce graph wedgelets - a tool for data compression on graphs based on the representation of signals by piecewise constant functions on adaptively generated binary graph partitionings. The adaptivity of the partitionings, a key ingredient to obtain sparse representations of a graph signal, is realized in terms of recursive wedge splits adapted to the signal. For this, we transfer adaptive partitioning and compression techniques known for 2D images to general graph structures and develop discrete variants of continuous wedgelets and binary space partitionings. We prove that continuous results on best m -term approximation with geometric wavelets can be transferred to the discrete graph setting and show that our wedgelet representation of graph signals can be encoded and implemented in a simple way. Finally, we illustrate that this graph-based method can be applied for the compression of images as well.

Introduction

In Line with the extraordinarily fast growth of stored and transmitted digital information, and the increase in complexity and interdependency of this Big Data, there is a strong need of novel compression techniques that are able to efficiently compress large data sets on unstructured or semi-structured domains. In many cases, these data sets and their interrelations can be organized in terms of networks or graphs as underlying domains. Adaptive algorithms able to compress data based on its intrinsic content as well as on the topological structure of the surrounding graph environment are therefore of main importance.