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

TL;DR

This paper introduces graph wedgelets, a novel adaptive data compression method for signals on graphs using binary partitioning trees and geometric wavelets, extending techniques from image processing to graph structures.

Contribution

It develops a new graph-based wedgelet representation for signal compression, adapting continuous geometric wavelet approximation results to discrete graphs.

Findings

Effective graph signal compression demonstrated

Method applicable to image data compression

Theoretical guarantees for approximation quality

Abstract

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.