# Data-driven Thresholding in Denoising with Spectral Graph Wavelet Transform

## TL;DR

This paper introduces a data-driven thresholding method for denoising signals in Graph Signal Processing using Spectral Graph Wavelet Transform, optimizing parameters via SURE to improve performance over existing methods.

## Contribution

It proposes a novel adaptive thresholding approach in SGWT-based denoising, accounting for correlated noise and providing a multi-scale analysis with improved results.

## Key findings

- Better denoising performance than recent methods
- Effective thresholding strategies for over-complete SGWT
- Numerical validation on various graph signals

## Abstract

This paper is devoted to adaptive signal denoising in the context of Graph Signal Processing (GSP) using Spectral Graph Wavelet Transform (SGWT). This issue is addressed \emph{via} a data-driven thresholding process in the transformed domain by optimizing the parameters in the sense of the Mean Square Error (MSE) using the Stein's Unbiased Risk Estimator (SURE). The SGWT considered is built upon a partition of unity making the transform semi-orthogonal so that the optimization can be performed in the transformed domain. However, since the SGWT is over-complete, the divergence term in the SURE needs to be computed in the context of correlated noise. Two thresholding strategies called coordinatewise and block thresholding process are investigated. For each of them, the SURE is derived for a whole family of elementary thresholding functions among which the soft threshold and the James-Stein threshold. This multi-scales analysis shows better performance than the most recent methods from the literature. That is illustrated numerically for a series of signals on different graphs.