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A Paper by Basile de Loynes and Fabien Navarro Published in Journal of Computational and Applied Mathematics

“Data-driven thresholding in denoising with Spectral Graph Wavelet Transform”, a paper by Basile de Loynes, Professor of Statistics and researcher at ENSAI, Fabien Navarro, Professor of Statistics at ENSAI and researcher at CREST, and Baptiste Olivier, research scientist at Orange Labs, was published in the Journal of Computational and Applied Mathematics.

This article is the fulfillment of collaborative work by the three researchers, as part of the partnership between ENSAI and Orange.

Since June 2018, the Data Science school and the telecom giant maintain a privileged relationship in terms of research projects, recruitment for interns and recent graduates, of educational implication in the form of classes, as well as professional seminars or student projects.


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 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.


Spectral graph theory / Denoising / Stein unbiased risk estimation / Spectral Graph Wavelet Transform / Tight frame / Variance estimation

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Find out more about Basile de Loynes, Fabien Navarro and research at ENSAI