I have been working on time-frequency analysis and wavelet theory for a long time. In terms of signal processing applications,
my main interests were on uncertainty principles, the analysis and modeling of non-stationary signals, and time varying filtering.
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I'm currently interested in non-linear time-frequency and time-scale transformations, which adapt themselves
to transient or frequency dependent features in the analyzed signal. With P. Warion, we have designed prototypes
of such adaptive transforms and studied mathematical properties such as norm control. Current work focuses on non-linear inversion approaches.
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Earlier, my most recent works were dealing with the analysis and modeling of a class of non-stationary random signals,
that may be seen as stationary signals deformed by the action of a stationarity-breaking operator.
With H. Omer, and more recently A. Meynard, we have designed models and estimation algorithms to deal
with such signals. More precisely, we have studied algorithms that can estimate amplitude and frequency
modulations (AM-FM) and also amplitude modulation and time warping (AM-TW), in the case of narrow-band and wide-band signals.
We have also designed a Bayesian approach that estimates a random time-scale representation for such signals,
which allows signal synthesis.
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Still in the non-stationary world: with C. Chaux, V. Emiya and A. Krémé, we have introduced
a new class of time varying filters, defined through a variational formulation. Very much in the spirit of
the time invariant filter design problem, we have introduced the Time-Frequency Fading (TFF) problem, which
seeks an operator which optimally filters out a prescribed region in the time-frequency domain, while preserving
the complementary region. TFF naturally leads to Gabor multipliers, and is currently solved using a spectral approach,
combined with random projections to control the curse of dimensionality.
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In a previous project (UnLocX), B. Ricaud and I have studied uncertainty principles in general settings and obtained new
entropic and support uncertainty inequalities for representations by frame coefficients. These allow one to control the
ability of sparse representation algorithms to identify signals defined in unions of frames.
Analysis and processing of brain signals, and EEG/MEG inverse problems
I have been involved in several projects dealing with the analysis and processing of brain signals, such as EEG and MEG.
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Currently, I'm working on the neuro-electromagnetic inverse problem: reconstruct brain activity (described by dipole currents
on the cortical surface) from external measurements (electro-encephalography and electro-encephalography). Within the BMW's project
(funded by ANR, the french research agency), we focus on the difficult case of extended brain sources (as opposed to activities that
can be described by point sources), and develop approaches that combine wavelet systems defined on the cortical surface with Bayesian methods.
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With M.C. Roubaud and J.M. Lina, we had already addressed the MEG inverse problem using a spatio-temporal generalization of the maximum
entropy on the mean approach. The curse of dimensionality induced by space-time data is handled using various techniques, including
dimension reduction, time domain wavelet expansion and Kronecker matrix factorization for noise covariance matrices.
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In the framework of the analysis of brain signals, I have worked on various statistical models adapted to different contexts.
With Emilie Villaron, I have developed a time-frequency hidden Markov model that is able to characterize various brain states
from EEG signals, and detect jumps between states as a function of time. With B. Burle, M.C. Roubaud, and J. Spinnato, we have
developed a mixed model adapted to the decision problem in brain computer interfaces, and applied to P300 speller data. We have
also introduced matrix Gaussian models to handle dimensionality problems in this context.
Blind source separation, dictionary learning and application to NMR spectroscopy (and other domains)
I have been involved in several projects related to blind source separation, dictionary learning, and factorization techniques.
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The most recent project (BIFROST, funded by A*MIDEX) aimed at developing non-negative matrix factorization (NMF) techniques for unmixing
complex spectral NMR mixtures. We could study and evaluate the performances of several NMF algorithms in that particular situation
(joint work with S. Anthoine, C. Chaux, A. Cherni and colleagues from the ISM2 lab). Earlier, with S. Caldarelli and I. Toumi, we
introduced and studied NMF unmixing algorithms based upon beta divergences and sparsity constraints.
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In the past, we used independent component analysis for the analysis of gene expression data (RNA microarrays)