- Scientific Context :
Multicomponent nonstationary signals (MCSs) are ubiquitous: for instance, audio (music, speech), medical (electrocardiogram, electroencephalogram), astronomical (gravitational waves) or echolocation (bats, marine mammals) signals can be modeled as the superposition of amplitude/frequency modulated (AM/FM) modes. To identify and separate these constituent modes is a challenging task due to the variety of MCSs encountered and time-frequency (TF) analysis is often used for that purpose .
There exist many different methods to represent MCSs in the TF plane among which wavelet and short-time Fourier transforms are the most popular. However, these techniques smear the information in the TF plane, and reassignment techniques, such as the synchrosqueezing transform (SST) , were introduced to sharpen such TF representations while enabling the retrieval of the modes. These approaches, though interesting, are somewhat limited in that the theoretical setting in which they are proved to behave well is quite restricted . One of the goal of this PhD thesis will be to broaden the type of signals these transforms can be applied to. Furthermore, as the definition of the modes of an MCS varies from one application to another, their extraction should involve techniques as adaptive as possible. From this perspective, non parametric techniques offer a greater flexibility than parametric ones, at least when very little is known about the modes to extract.
In many situations however, like in speech or biological signals like ECG or EEG, mode definition is very tricky in that each event can be associated with many different harmonics, and a single event can therefore be viewed as an MCS. To study these types of signals, it is therefore of interest to take into account the structure of the studied signal so as to get a sparser representation, and the separation of the signal into different modes may not be relevant. Indeed, the studied events are associated with TF patterns one expects the analysis to uncover, and to identify the latter is one of the goal of source separation for which Non-Negative Matrix Factorization (NMF) is commonly used on the spectrogram. To our knowledge, such a technique has not yet been combined with SST, although a preliminary recent study  has shown that the former could benefit from the latter. Indeed, we believe that SST, since it is more "compact" than standard TF representations, should be more easily processed by NMF. This also raises the question of what is the optimal representation for source separation, and how one can then reconstruct the sources. In line with this, a recent approach showed that it is profitable to mix a TF representation based on so-called contours and NMF for phonocardiogram denoising . With this in mind, one of the goal of this thesis will also be to investigate the behavior of NMF on reassigned transforms like SST in the context of applications to biomedical signals.
Indeed, several biomedical applications can be considered based on this methodological question. For instance, phonocardiogram (PCG) denoising may benefit of such propositions . Heart rate estimations from ECG or PCG signals are also of interest. In that context, specific approaches to the use of NMF must be developed to take into account the harmonic structure of such signals and then allow for the estimation of time events along with their variability. In this regard, real database of physiological signals are available at TIMC-IMAG laboratory.
In spite of the intrinsic limitations of SST mentioned above, it has recently been extended to the multivariate case, where one has to deal with signals corresponding to the same phenomenon recorded with different sensors, a typical example being ECG or EEG recordings. The notion of mode in this multivariate setting is significantly different since one has to take into account dependencies between channels in the transform domain . As it is, we emphasize that SST in the multivariate setting still needs further clarifications from a mathematical point of view and potentially very interesting practical problems can be tackled with this new method , as illustrated a bit later. To study the multivariate SST and to broaden its application to a wider class of signals is also part of the present project.
An interesting biomedical application of such technique will focus on the recovery of fetal ECG waveforms to improve the diagnosis of fetal cardiac pathologies. The proposed technique should help doctors to analyze some typical ECG segments at early stages of pregnancy. Real data from ECG electrodes networks will be available in 2019 thanks to a clinical protocol with the fetal cardiology service of the University Grenoble Alpes Hospital, in the context of SurFAO ANR project (‘Surveillance Foetale Assistée par Ordinateur’).
These are three-fold :
To extend SST to a wider class of signals than AM/FM signals.
To investigate sourc separation using NMF applied to the most up-to-date reassigned transform like SST and variants
To extend this analysis to the multivariate setting, focusing on some specific applications like EEG and ECG recordings.
Scientific Program :
Among non parametric methods based on TF representations used to deal with MCSs, reassignment techniques such as synchrosqueezing (SST) are very popular . Initially designed to handle modes with small frequency modulations, variants of SST were recently developed to circumvent these limitations [6,7,8,9,10], mainly based on the design of new reassignment operators. In spite of undeniable achievements, these new developments have also raised several questions, in particular regarding the sensitivity to noise of these improved versions of SST. Understanding this aspect and then adapting the transform to properly deal with impulses or colored noises is an open question we aim to tackle in the present project. Furthermore, two other inherent limitations of SST are that the sought modes cannot vanish and have to be separated in the TF plane. In particular, SST cannot deal with two modes colliding in the TF plane. To adapt SST to such cases, we propose to investigate how to better estimate the instantaneous frequency (IF) of these modes at crossing locations by better analyzing the very nature of interference in the TF plane, and then develop new robust IF estimators. To this end, we propose to investigate several signal models, especially Dirac impulses, hyperbolic chirps and also stochastic models that can handle a large variety of realistic configurations.
In another direction, and looking at the identification of the modes of an MCS from the perspective of source separation, Non-Negative Matrix Factorization (NMF) will be combined with SST and variants. Indeed, we expect that, since these representations are sparser than the spectrogram, better results should be obtained as suggested by a recent preliminary study . Not limiting ourselves to investigate the behavoir of NMF on SST, we will try to answer the question of what are the features TF representations should share to fit well into the framework of source separation.
The extension of SST to the multivariate case will also be investigated, when considering signals corresponding to the same phenomenon recorded with different sensors, a typical example being ECG or EEG recordings. A particular example is of interest to us and involves the study of emotional states from multivariate SST : It has actually been recently shown that the TF patterns associated with multivariate SST could be informative on the emotional state of a patient. There is still a lot to do in that direction and in particular, it would be of interest to develop multivariate SST based on higher order approximation of the phase as done in the one-dimensional case to see whether better results could be achieved. In this regard, a clear mathematical analysis of the multivariate synchrosqueezing transform is still missing and should be carried out.
Another interesting aspect of recordings such as ECG signals is that, measured on different patients, they exhibit similarities in their TF representations, in particular their so-called TF signatures are often very much alike. It would therefore be of interest to investigate in what way these signatures could be used as features to help detect abnormal behaviors. This aspect could of course be generalized to many different types of TF representations used to analyze biomedical signals.
Expected results :
The expected results should be of two different kind, both theoretical regarding the use of SST on complex signals, and applicative, in relation with the subject of ECG, more particularly on multi sensors approaches and PCG denoising.
Apart from the PhD student, the project will involve 3 permanent researchers, from three different laboratories in Grenoble who have already collaborated on the subject of enhancing phonocardiogram by using synchronous ECG, a project called Optisens (Signal processing for enhanced recording), and which was founded by the exploratory project LabEx Persyval. This project has enabled us to write 4 different contributions in the past year (ICASSP 2018 ,EUSIPCO 2018 , IEEE signal processing letters 2018, SAM 2018).
Hereafter follows a brief description of the scientific environments of the three partners involved in the project:
1) LJK is a major laboratory in applied mathematics. CVGI team develops theoretical methods on multi-resolution (empirical mode decomposition, synchrosqueezing) and calculus of variations for partial differential equations. Sylvain Meignen (Ass. Prof, Grenoble INP) will bring his expertise for modeling signals.
2) GIPSA-lab is a major laboratory in signal processing. ViBS team develops theoretical methods (blind source separation, deconvolution and non-linear filtering) applied to many problems. Bertrand Rivet (Ass. Prof., Grenoble INP) will bring skills and expertise in statistical SP and multimodal data processing .
3) TIMC-IMAG is a major laboratory in biomedical engineering. PRETA covers fundamental and applied domains of physiology. Julie Fontecave (Ass. Prof., University Grenoble Alpes, UGA) will bring her expertise in physiology, human non-invasive recordings, mathematical modeling of physiological systems, and SP, mainly based on time-scale approaches.
- Bibliography :
 P. Flandrin, Explorations in Time-Frequency Analysis. Cambridge University Press, 2018.
 I. Daubechies, J. Lu, and H.-T. Wu, Synchrosqueezed wavelet transforms: An empirical mode decomposition-like tool," Applied and computational harmonic analysis, vol. 30, no. 2, pp. 243-261, 2011.
 D. Fourer and G. Peeters, Fast and adaptive blind audio source separation using recursive levenberg-marquardt synchrosqueezing," in Proc. IEEE ICASSP, Calgary, Canada, Apr. 2018.
 D.-H. Pham, S. Meignen, N. Dia, J. Fontecave-Jallon, and B. Rivet, Phonocardiogram signal denoising based on nonnegative matrix factorization and adaptive contour representation computation," IEEE Signal Process. Lett., vol. 25, no. 10, pp. 1475-1479, 2018.
 L. Stankovic, D. Mandic, M. Dakovic, and M. Brajovic, Time-frequency decomposition of multivariate multicomponent signals," IEEE Signal Process. Lett., vol. 142, pp. 468-479, 2018.
 R. Behera, S. Meignen, and T. Oberlin, Theoretical analysis of the second-order synchrosqueezing transform, Applied and Computational Harmonic Analysis, vol. 45, no. 2, pp. 379- 404, Sep. 2018.
 D.-H. Pham and S. Meignen, High-order synchrosqueezing transform for multicomponent signals analysis – with an application to gravitational-wave signal, IEEE Trans. Signal Process., vol. 65, no. 12, pp. 3168-3178, 2017.
 T. Oberlin, S. Meignen, and V. Perrier, Second-order synchrosqueezing transform or invertible reassignment? Towards ideal time-frequency representations, IEEE Trans. Signal Processing, vol. 63, no. 5, pp. 1335-1344, 2015.
 D. Fourer, F. Auger, K. Czarnecki, S. Meignen, and P. Flandrin, Chirp rate and instantaneous frequency estimation: Application to recursive vertical synchrosqueezing, IEEE Signal Process. Lett., vol. 24, no. Issue 11, pp. 1724-1728, Nov. 2017.
 D. Fourer, F. Auger, and P. Flandrin, Recursive versions of the Levenberg-Marquardt reassigned spectrogram and of the synchrosqueezed STFT, in Proc. IEEE ICASSP, Shanghai, China, Mar. 2016, pp. 4880-4884.
 D-H. Pham and S. Meignen, A novel thresholding technique for the denoising of multicomponent signals, ICASSP 2018.
 N Dia, J Fontecave-Jallon, P-Y Guméry, B Rivet. Denoising Phonocardiogram signals with Non-negative Matrix Factorization informed by synchronous Electrocardiogram. EUSIPCO 2018.
 N Dia, J Fontecave-Jallon, P-Y Guméry, B Rivet. Quasi-Periodic Non-negative Matrix Factorization for Phonocardiographic signals denoising. 2018 IEEE Sensor Array and Multichannel Signal Processing Workshop.
(c) GdR 720 ISIS - CNRS - 2011-2019.