Thomas Oberlin

Time-Frequency Reassignment and Synchrosqueezing: An Overview

This article provides a general overview of time-frequency (T-F) reassignment and synchrosqueezing techniques applied to multicomponent signals, covering the theoretical background and applications. We explain how synchrosqueezing can be viewed as a special case of reassignment enabling mode reconstruction and place emphasis on the interest of using such T-F distributions throughout with illustrative examples.

A New Algorithm for Multicomponent Signals Analysis Based on SynchroSqueezing: With an Application to Signal Sampling and Denoising

In this paper, we address the problem of the retrieval of the components from a multicomponent signal using ideas from the synchrosqueezing framework. The emphasis is on the wavelet choice and we propose a novel algorithm based first on the detection of components followed by their reconstruction. Simulations illustrate how the proposed procedure compares with the empirical mode decomposition and other related methods in terms of mode-mixing. We conclude the paper by studying the sensitivity of the proposed technique to sampling and an application to signal denoising.

Second-Order Synchrosqueezing Transform or Invertible Reassignment? Towards Ideal Time-Frequency Representations

This paper considers the analysis of multicomponent signals, defined as superpositions of real or complex modulated waves. It introduces two new post-transformations for the short-time Fourier transform that achieve a compact time-frequency representation while allowing for the separation and the reconstruction of the modes. These two new transformations thus benefit from both the synchrosqueezing transform (which allows for reconstruction) and the reassignment method (which achieves a compact time-frequency representation). Numerical experiments on real and synthetic signals demonstrate the efficiency of these new transformations, and illustrate their differences.

An Alternative Formulation for the Empirical Mode Decomposition

The Empirical Mode Decomposition (EMD) is a relatively new adaptive method for multicomponent signal representation which allows for analyzing nonlinear and nonstationary signals. In spite of its lack of mathematical foundations, very few papers are dedicated to defining new decompositions that would preserve the interesting properties of the EMD while improving the mathematical setting. The new decomposition based on direct constrained optimization we introduce in this article is an attempt in that direction.

The fourier-based synchrosqueezing transform

The short-time Fourier transform (STFT) and the continuous wavelet transform (CWT) are extensively used to analyze and process multicomponent signals, i.e. superpositions of modulated waves. The synchrosqueezing is a post-processing method which circumvents the uncertainty relation inherent to these linear transforms, by reassigning the coefficients in scale or frequency. Originally introduced in the setting of the CWT, it provides a sharp, concentrated representation, while remaining invertible. This technique received a renewed interest with the recent publication of an approximation result related to the application of the synchrosqueezing to multi-component signals. In the current paper, we adapt the formulation of the synchrosqueezing to the STFT and state a similar theoretical result to that obtained in the CWT framework. The emphasis is put on the differences with the CWT-based synchrosqueezing with numerical experiments illustrating our statements.

Adaptive multimode signal reconstruction from time-frequency representations.

This paper discusses methods for the adaptive reconstruction of the modes of multicomponent AM–FM signals by their time–frequency (TF) representation derived from their short-time Fourier transform (STFT). The STFT of an AM–FM component or mode spreads the information relative to that mode in the TF plane around curves commonly called ridges. An alternative view is to consider a mode as a particular TF domain termed a basin of attraction. Here we discuss two new approaches to mode reconstruction. The first determines the ridge associated with a mode by considering the location where the direction of the reassignment vector sharply changes, the technique used to determine the basin of attraction being directly derived from that used for ridge extraction. A second uses the fact that the STFT of a signal is fully characterized by its zeros (and then the particular distribution of these zeros for Gaussian noise) to deduce an algorithm to compute the mode domains. For both techniques, mode reconstruction is then carried out by simply integrating the information inside these basins of attraction or domains.

Bayesian EEG source localization using a structured sparsity prior

ABSTRACT This paper deals with EEG source localization. The aim is to perform spatially coherent focal localization and recover temporal EEG waveforms, which can be useful in certain clinical applications. A new hierarchical Bayesian model is proposed with a multivariate Bernoulli Laplacian structured sparsity prior for brain activity. This distribution approximates a mixed ∂20 pseudo norm regularization in a Bayesian framework. A partially collapsed Gibbs sampler is proposed to draw samples asymptotically distributed according to the posterior of the proposed Bayesian model. The generated samples are used to estimate the brain activity and the model hyperparameters jointly in an unsupervised framework. Two different kinds of Metropolis–Hastings moves are introduced to accelerate the convergence of the Gibbs sampler. The first move is based on multiple dipole shifts within each MCMC chain, whereas the second exploits proposals associated with different MCMC chains. Experiments with focal synthetic data shows that the proposed algorithm is more robust and has a higher recovery rate than the weighted ∂21 mixed norm regularization. Using real data, the proposed algorithm finds sources that are spatially coherent with state of the art methods, namely a multiple sparse prior approach and the Champagne algorithm. In addition, the method estimates waveforms showing peaks at meaningful timestamps. This information can be valuable for activity spread characterization. HIGHLIGHTSA Bayesian model for solving the EEG source localization problem is proposed.The model promotes sparsity using a multivariate Bernoulli Laplacian prior.A Gibbs sampler is used to generate samples according to the posterior distribution.Metropolis–Hastings moves are used to improve the convergence rate of the sampler.The method is compared with synthetic and real data to state of the art algorithms.

Multicomponent signal denoising with synchrosqueezing

In this paper, we develop a new technique based on the synchrosqueezing method to denoise multicomponent signals. The approach proposed is based on a two step strategy: a mode detection step followed by a reconstruction one. The emphasis is put on the robustness of the detection step in a noisy context, a key issue in the implementation of the method. Numerical applications show the improvement in terms of multicomponent signal denoising brought about by the proposed method over the translation-invariant wavelet thresholding.

Analysis of strongly modulated multicomponent signals with the short-time Fourier transform

This paper addresses the issue of the retrieval of the components of a multicomponent signal from its short-time Fourier transform. It recalls two popular reconstruction methods, and extends each of them for the case of strong frequency modulation, by taking into account the second derivative of the phase. Numerical experiments illustrate the improvement and compare the methods.

The ASTRES toolbox for mode extraction of non-stationary multicomponent signals

In this paper, we introduce the ASTRES∗ toolbox which offers a set of Matlab functions for non-stationary multi-component signal processing. The main purposes of this proposal is to offer efficient tools for analysis, synthesis and transformation of any signal made of physically meaningful components (e.g. sinusoid, trend or noise). The proposed techniques contain some recent and new contributions, which are now unified and theoretically strengthened. They can provide efficient time-frequency or time-scale representations and they allow elementary components extraction. Usage and description of each method are then detailed and numerically illustrated.