# Welcome

I am a postdoctoral researcher at CRAN, Université de Lorraine in Nancy, France since October 2018. I work with David Brie and Sebastian Miron on source separation of polarized signals.

I defended my Ph.D. thesis on September, 27th 2018.

From October 2015 to September 2018, I was a Ph.D student in the beautiful city of Lille, France, under the supervision of Pierre Chainais and Nicolas Le Bihan. I was part of the SigMA group at CRIStAL.

My research interests are in signal processing theory and methods for physical applications.

# News

`Jun. 2019`

Received an *accessit* (runner up) for the PhD thesis prize in Signal, Image and Vision, from the Club EEA, GRETSI and GdR ISIS

`May 2019`

Two papers accepted at GRETSI, Lille, Aug. 2019.

# Publications

## Preprints

- J. Flamant, S. Miron, and D. Brie, “Quaternion Non-negative Matrix Factorization: definition, uniqueness and algorithm.” 2019.
This article introduces the notion of quaternion non-negative matrix factorization (Q-NMF), which extends the usual non-negative matrix factorization (NMF) to the case of bivariate or polarized signals. The Q-NMF relies on two key ingredients: (i) the algebraic representation of polarization information thanks to quaternions and (ii) the exploitation of physical polarization constraints that generalize non-negativity. Uniqueness conditions for the Q-NMF are presented. The relationship between Q-NMF and NMF highlights the key disambiguating role played by polarization information. A simple yet efficient algorithm called quaternion alternating least squares (Q-ALS) is introduced to solve the Q-NMF problem in practice. Numerical experiments on synthetic data demonstrate the relevance of the approach, which appears very promising, notably for blind source separation problems arising in spectro-polarimetric imaging.

@preprint{flamant19quaternion, author = {Flamant, Julien and Miron, Sebastian and Brie, David}, title = {Quaternion Non-negative Matrix Factorization: definition, uniqueness and algorithm}, year = {2019}, eprint = {arXiv:1903.10593}, arxiv = {https://arxiv.org/abs/1903.10593} }

## Journal Articles

- J. Flamant, P. Chainais, and N. Le Bihan, “A Complete Framework for Linear Filtering of Bivariate Signals,”
*IEEE Transactions on Signal Processing*, vol. 66, no. 17, pp. 4541–4552, Sep. 2018.A complete framework for the linear time-invariant (LTI) filtering theory of bivariate signals is proposed based on a tailored quaternion Fourier transform. This framework features a direct description of LTI filters in terms of their eigenproperties enabling compact calculus and physically interpretable filtering relations in the frequency domain. The design of filters exhibiting fondamental properties of polarization optics (birefringence, diattenuation) is straightforward. It yields an efficient spectral synthesis method and new insights on Wiener filtering for bivariate signals with prescribed frequency-dependent polarization properties. This generic framework facilitates original descriptions of bivariate signals in two components with specific geometric or statistical properties. Numerical experiments support our theoretical analysis and illustrate the relevance of the approach on synthetic data.

@article{flamant2018LTI, author = {Flamant, Julien and Chainais, Pierre and {Le Bihan}, Nicolas}, title = {A Complete Framework for Linear Filtering of Bivariate Signals}, year = {2018}, journal = {IEEE Transactions on Signal Processing}, volume = {66}, number = {17}, pages = {4541-4552}, eprint = {arXiv:1802.02469}, arxiv = {https://arxiv.org/abs/1802.02469}, doi = {http://dx.doi.org/10.1109/TSP.2018.2855659}, issn = {1053-587X}, month = sep, code = {https://github.com/jflamant/bispy} }

- R. Bardenet, J. Flamant, and P. Chainais, “On the zeros of the spectrogram of white noise,”
*manuscript accepted for publication in Applied and Computational Harmonic Analysis*, 2018.In a recent paper, Flandrin [2015] has proposed filtering based on the zeros of a spectrogram, using the short-time Fourier transform and a Gaussian window. His results are based on empirical observations on the distribution of the zeros of the spectrogram of white Gaussian noise. These zeros tend to be uniformly spread over the time-frequency plane, and not to clutter. Our contributions are threefold: we rigorously define the zeros of the spectrogram of continuous white Gaussian noise, we explicitly characterize their statistical distribution, and we investigate the computational and statistical underpinnings of the practical implementation of signal detection based on the statistics of spectrogram zeros. In particular, we stress that the zeros of spectrograms of white Gaussian noise correspond to zeros of Gaussian analytic functions, a topic of recent independent mathematical interest [Hough et al., 2009].

@article{bardenet2018zeros, author = {Bardenet, Rémi and Flamant, Julien and Chainais, Pierre}, title = {On the zeros of the spectrogram of white noise}, year = {2018}, journal = {manuscript accepted for publication in Applied and Computational Harmonic Analysis}, eprint = {arXiv:1708.00082}, arxiv = {https://arxiv.org/abs/1708.00082}, code = {https://github.com/jflamant/2018-zeros-spectrogram-white-noise} }

- J. Flamant, N. Le Bihan, and P. Chainais, “Time-frequency analysis of bivariate signals,”
*To appear in Applied and Computational Harmonic Analysis*, 2017.Many phenomena are described by bivariate signals or bidimensional vectors in applications ranging from radar to EEG, optics and oceanography. The time-frequency analysis of bivariate signals is usually carried out by analyzing two separate quantities, e.g. rotary components. We show that an adequate quaternion Fourier transform permits to build relevant time-frequency representations of bivariate signals that naturally identify geometrical or polarization properties. First, the quaternion embedding of bivariate signals is introduced, similar to the usual analytic signal of real signals. Then two fundamental theorems ensure that a quaternion short term Fourier transform and a quaternion continuous wavelet transform are well defined and obey desirable properties such as conservation laws and reconstruction formulas. The resulting spectrograms and scalograms provide meaningful representations of both the time-frequency and geometrical/polarization content of the signal. Moreover the numerical implementation remains simply based on the use of FFT. A toolbox is available for reproducibility. Synthetic and real-world examples illustrate the relevance and efficiency of the proposed approach.

@article{flamant2016time, title = {Time-frequency analysis of bivariate signals}, journal = {To appear in Applied and Computational Harmonic Analysis}, author = {Flamant, Julien and Le Bihan, Nicolas and Chainais, Pierre}, code = {https://github.com/jflamant/bispy}, arxiv = {http://arxiv.org/abs/1609.02463}, year = {2017}, doi = {http://doi.org/10.1016/j.acha.2017.05.007} }

- J. Flamant, N. Le Bihan, and P. Chainais, “Spectral analysis of stationary random bivariate signals,”
*IEEE Transactions on Signal Processing*, vol. 65, no. 23, pp. 6135–6145, 2017.IEEE A novel approach towards the spectral analysis of stationary random bivariate signals is proposed. Unlike existing approaches, the proposed framework exhibits a natural link between well-defined statistical objects and physical parameters for bivariate signals. Using the Quaternion Fourier Transform, we introduce a quaternion-valued spectral representation of random bivariate signals seen as complex-valued sequences. This makes possible the definition of a scalar quaternion-valued spectral density and the corresponding autocovariance for bivariate signals. This spectral density can be meaningfully interpreted in terms of frequency-dependent polarization attributes. A natural decomposition of the spectral density of any random bivariate signal in terms of unpolarized and polarized components is introduced. Nonparametric spectral density estimation is investigated, and we introduce the polarization periodogram of a random bivariate signal. Numerical experiments support our theoretical analysis, illustrating the relevance of the approach on synthetic data.

@article{flamant2017spectral, author = {Flamant, Julien and {Le Bihan}, Nicolas and Chainais, Pierre}, doi = {http://dx.doi.org/10.1109/TSP.2017.2736494}, issn = {1053587X}, journal = {IEEE Transactions on Signal Processing}, number = {23}, pages = {6135--6145}, code = {https://github.com/jflamant/bispy}, title = {{Spectral analysis of stationary random bivariate signals}}, volume = {65}, year = {2017}, arxiv = {https://arxiv.org/abs/1703.06417} }

- N. Le Bihan, J. Flamant, and J. H. Manton, “Density estimation on the rotation group using diffusive wavelets,”
*ISIF Journal of Advances in Information Fusion,*vol. 11, no. 2, pp. 173–185, Dec. 2016.This paper considers the problem of estimating probability density functions on the rotation group SO(3). Two distinct approaches are proposed, one based on characteristic functions and the other on wavelets using the heat kernel. Expressions are derived for their Mean Integrated Squared Errors. The performance of the estimators is studied numerically and compared with the performance of an existing technique using the De La Vallee Poussin kernel estimator. The heat-kernel wavelet approach appears to offer the best convergence, with faster convergence to the optimal bound and guaranteed positivity of the estimated probability density function.

@article{lebihan2016diffusive, author = {Le Bihan, Nicolas and Flamant, Julien and Manton, Jonathan H.}, journal = {ISIF Journal of Advances in Information Fusion,}, keywords = {Probability density estimation, Rotation group SO(3), Diffusive wavelets, Characteristic function, Kernel estimators, Mean Integrated Square Error (MISE), Mixture of densities}, title = {Density estimation on the rotation group using diffusive wavelets}, volume = {11}, issue = {2}, pages = {173-185}, year = {2016}, month = dec, arxiv = {http://arxiv.org/abs/1512.06023}, doi = {http://isif.org/publications/jaif-articles-appear-future-issue} }

- J. Flamant, N. Le Bihan, A. V. Martin, and J. H. Manton, “Expansion-maximization-compression algorithm with spherical harmonics for single particle imaging with x-ray lasers,”
*Phys. Rev. E*, vol. 93, no. 5, p. 053302, May 2016.In 3D single particle imaging with X-ray free-electron lasers, particle orientation is not recorded during measurement but is instead recovered as a necessary step in the reconstruction of a 3D image from the diffraction data. Here we use harmonic analysis on the sphere to cleanly separate the angular and radial degrees of freedom of this problem, providing new opportunities to efficiently use data and computational resources. We develop the Expansion-Maximization-Compression algorithm into a shell-by-shell approach and implement an angular bandwidth limit that can be gradually raised during the reconstruction. We study the minimum number of patterns and minimum rotation sampling required for a desired angular and radial resolution. These extensions provide new av- enues to improve computational efficiency and speed of convergence, which are critically important considering the very large datasets expected from experiment.

@article{flamant2016expansion, title = {Expansion-maximization-compression algorithm with spherical harmonics for single particle imaging with x-ray lasers}, author = {Flamant, Julien and Le Bihan, Nicolas and Martin, Andrew V. and Manton, Jonathan H.}, journal = {Phys. Rev. E}, volume = {93}, issue = {5}, pages = {053302}, numpages = {14}, year = {2016}, month = may, publisher = {American Physical Society}, doi = {http://dx.doi.org/10.1103/PhysRevE.93.053302}, code = {http://github.com/jflamant/sphericalEMC}, arxiv = {http://arxiv.org/abs/1602.01301} }

## Conference Articles

- J. Flamant, P. Chainais, and N. Le Bihan, “Linear Filtering of Bivariate Signals Using Quaternions,” in
*2018 IEEE Statistical Signal Processing Workshop (SSP)*, 2018, pp. 154–158.A new approach towards linear time-invariant (LTI) filtering of bivariate signals is proposed using a tailored quaternion Fourier transform. In the proposed framework LTI filters are naturally described by their eigenproperties providing economical, physically interpretable and straightforward filtering definitions in the frequency domain. It enables an easy design of LTI filters and a simple method for spectral synthesis of bivariate signals with prescribed frequency polarization properties. It also yields various natural decompositions of bivariate signals. Numerical experiments illustrate the approach.

@inproceedings{flamant18ssp, author = {Flamant, Julien and Chainais, Pierre and {Le Bihan}, Nicolas}, booktitle = {2018 IEEE Statistical Signal Processing Workshop (SSP)}, title = {Linear Filtering of Bivariate Signals Using Quaternions}, year = {2018}, volume = {}, number = {}, pages = {154-158}, doi = {http://dx.doi.org/10.1109/SSP.2018.8450687}, issn = {}, month = jun }

- J. Flamant, P. Chainais, E. Chassande-Mottin, F. Feng, and N. Le Bihan, “Non-parametric characterization of
gravitational-wave polarizations,” in
*26th European Signal Processing Conference (EUSIPCO)*, 2018.@inproceedings{flamant18eusipco, author = {Flamant, Julien and Chainais, Pierre and Chassande-Mottin, Eric and Feng, Fangcheng and {Le Bihan}, Nicolas}, title = {Non-parametric characterization of gravitational-wave polarizations}, booktitle = {26th European Signal Processing Conference (EUSIPCO)}, year = {2018} }

- J. Flamant, N. Le Bihan, and P. Chainais, “Spectrogramme de polarisation pour l’analyse des signaux bivariés,” in
*GRETSI*, Juan-les-Pins, France, 2017.@inproceedings{flamantGretsi2, title = {{Spectrogramme de polarisation pour l'analyse des signaux bivari{\'e}s}}, author = {Flamant, Julien and Le Bihan, Nicolas and Chainais, Pierre}, booktitle = {{GRETSI}}, address = {Juan-les-Pins, France}, year = {2017}, month = sep, hal_id = {hal-01691274}, hal_version = {v1}, doi = {https://hal.archives-ouvertes.fr/hal-01691276} }

- J. Flamant, N. Le Bihan, and P. Chainais, “Polarization spectrogram of bivariate signals,” in
*IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP), 2017, New Orleans, USA*, 2017.Bivariate signals are commonly processed with the usual Fourier transform, using methods such as the rotary spectrum analysis. We show that bivariate signals can be efficiently processed using the Quaternion Fourier transform. A bivari- ate counterpart of the analytic signal is introduced, the quater- nion embedding of a complex signal. It leads to identify natu- ral parameters describing polarization properties, amplitude and phase of the signal. The properties of the quaternion short-term Fourier transform are studied and the polarization spectrogram is introduced. A synthetic example illustrates the relevance of the proposed approach.

@inproceedings{flamant2017polarization, title = {Polarization spectrogram of bivariate signals}, author = {Flamant, Julien and Le Bihan, Nicolas and Chainais, Pierre}, booktitle = {IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP), 2017, New Orleans, USA}, year = {2017}, doi = {https://doi.org/10.1109/ICASSP.2017.7952905} }

- J. Flamant, N. Le Bihan, and P. Chainais, “Analyse spectrale des signaux aléatoires bivariés,” in
*GRETSI*, Juan-les-Pins, France, 2017.@inproceedings{flamant:hal-01691274, title = {{Analyse spectrale des signaux al{\'e}atoires bivari{\'e}s}}, author = {Flamant, Julien and Le Bihan, Nicolas and Chainais, Pierre}, doi = {https://hal.archives-ouvertes.fr/hal-01691274}, booktitle = {{GRETSI}}, address = {Juan-les-Pins, France}, year = {2017}, month = sep, hal_id = {hal-01691274}, hal_version = {v1} }

- J. Flamant, N. Le Bihan, A. V. Martin, and J. H. Manton, “Low-Resolution reconstruction of intensity functions on the sphere for single-particle diffraction imaging,” in
*IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP), 2016, Shanghai, China*, 2016.Single-particle imaging experiments using X-ray Free-Electron Lasers (XFEL) belong to a new generation of X-ray imaging techniques potentially allowing high resolution images of non- crystallizable molecules to be obtained. One of the challenges of single-particle imaging is the reconstruction of the 3D intensity function from only a few samples collected on a planar detector after the interaction of a free falling molecule and the X-ray beam. In this paper, we take advantage of the symmetries of the intensity function to propose an original low-resolution reconstruction algorithm based on an Expansion Maximization Compression (EMC) approach. We study the problem of adequate sampling of the rotation group via simulation to illustrate the potential of the approach.

@inproceedings{flamant2016lowres, title = {Low-Resolution reconstruction of intensity functions on the sphere for single-particle diffraction imaging}, author = {Flamant, Julien and Le Bihan, Nicolas and Martin, Andrew V and Manton, Jonathan H}, booktitle = {IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP), 2016, Shanghai, China}, year = {2016}, doi = {http://doi.org/10.1109/ICASSP.2016.7471758} }

## Theses

- J. Flamant, “A general approach for the analysis and filtering of bivariate signals,” 2018.
Bivariate signals appear in a broad range of applications where the joint analysis of two real-valued signals is required: polarized waveforms in seismology and optics, eastward and northward current velocities in oceanography, pairs of electrode recordings in EEG or MEG or even gravitational waves emitted by coalescing compact binaries. Simple bivariate signals take the form of an ellipse, whose properties (size, shape, orientation) may evolve with time. This geometric feature of bivariate signals has a natural physical interpretation called polarization. This notion is fundamental to the analysis and understanding of bivariate signals. However, existing approaches do not provide straightforward descriptions of bivariate signals or filtering operations in terms of polarization or ellipse properties. To this purpose, this thesis introduces a new and generic approach for the analysis and filtering of bivariate signals. It essentially relies on two key ingredients: (i) the natural embedding of bivariate signals – viewed as complex-valued signals – into the set of quaternions H and (ii) the definition of a dedicated quaternion Fourier transform to enable a meaningful spectral representation of bivariate signals. The proposed approach features the definition of standard signal processing quantities such as spectral densities, linear time-invariant filters or spectrograms that are directly interpretable in terms of polarization attributes. These geometric and physical interpretations are made possible by the use of quaternion algebra. More importantly, the framework does not sacrifice any mathematical guarantee and the newly introduced tools admit computationally fast implementations. By revealing the specificity of bivariate signals, the proposed framework greatly simplifies the design of analysis and filtering operations. Numerical experiment support throughout our theoretical developments. We demonstrate the potential of the approach for the characterization of (polarized) gravitational waves emitted by compact coalescing binaries. A companion Python package called BiSPy implements our findings for the sake of reproducibility.

@thesis{flamant2018phdthesis, title = {A general approach for the analysis and filtering of bivariate signals}, author = {Flamant, Julien}, year = {2018}, code = {https://github.com/jflamant/bispy} }

- J. Flamant, “Three-dimensional intensity reconstruction in single-particle experiments: a spherical symmetry approach,” 2015.
The ability to decipher the three-dimensional structures of biomolecules at high resolution will greatly improve our understanding of the biological machinery. To this aim, X-ray crystallography has been used by scientists for several decades with tremendous results. This imaging method however requires a crystal to be grown, and for most interesting biomolecules (proteins, viruses) this may not be possible. The single-particle experiment was proposed to address these limitations, and the recent advent of ultra-bright X-ray Free Electron Lasers (XFELs) opens a new set of opportunities in biomolecular imaging. In the single-particle experiment, thousands of diffraction patterns are recorded, where each image corresponds to an unknown, random orientation of individual copies of the biomolecule. These noisy, unoriented two-dimensional diffraction patterns need to be then assembled in three-dimensional space to form the three-dimensional intensity function, which characterizes completely the three-dimensional structure of the biomolecule. This work focuses on geometrical variations of an existing algorithm, the Expansion-Maximization-Compression (EMC) algorithm introduced by Loh and Elser. The algorithm relies upon an expec-tation-maximization method, by maximizing the likelihood of an intensity model with respect to the diffraction patterns. The contributions of this work are (i) the redefinition of the EMC algorithm in a spherical design, motivated by the intrinsic properties of the intensity function, (ii) the utilisation of an orthonormal harmonic basis on the three-dimensional ball which allows a sparse representation of the intensity function, (iii) the scaling of the EMC parameters with the desired resolution, increasing computational speed and (iv) the intensity error is analysed with respect to the EMC parameters.

@thesis{flamant2015masterthesis, title = {Three-dimensional intensity reconstruction in single-particle experiments: a spherical symmetry approach}, author = {Flamant, Julien}, year = {2015}, doi = {http://hdl.handle.net/11343/58637} }

# Vita

`Oct. 2018 - Currently`

**Postdoctoral Researcher**- CRAN, Université de Lorraine
*Blind source separation of polarized signals* `Oct. 2015 - Sept. 2018`

**PhD, Signal Processing**- Ecole Centrale de Lille
*A general approach for the analysis and filtering of bivariate signals* `2014 - 2015`

**MPhil, Applied Physics**- The University of Melbourne, Australia
*Three-dimensional intensity reconstruction in single-particle experiments: a spherical symmetry approach* `2011 - 2015`

**MSc, Applied Physics and Electrical Engineering**- Ecole Normale Supérieure de Cachan (Paris-Saclay)
*Including one-year of intensive training for teaching in higher education*

Detailed CV and track record available here .

# BiSPy

An open-source python package for signal processing of bivariate signals, under ongoing development. Available on GitHub with documentation.

*Time-frequency Stokes parameters from polarization spectrogram*