# Welcome

I am a PhD student since October, 2015 in the beautiful city of Lille, France. I work under the supervision of Pierre Chainais and Nicolas Le Bihan, and I am part of the SigMA group at CRIStAL.

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

# Publications

## Preprints

- J. Flamant, P. Chainais, and N. L. Bihan, “A complete framework for linear filtering of bivariate signals,”
*submitted*. 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.

@preprint{flamant2018LTI, author = {Flamant, Julien and Chainais, Pierre and Bihan, Nicolas Le}, title = {A complete framework for linear filtering of bivariate signals}, year = {2018}, journal = {submitted}, eprint = {arXiv:1802.02469}, arxiv = {https://arxiv.org/abs/1802.02469} }

- R. Bardenet, J. Flamant, and P. Chainais, “On the zeros of the spectrogram of white noise,”
*submitted*. 2017.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].

@preprint{bardenet2017zeros, author = {Bardenet, Rémi and Flamant, Julien and Chainais, Pierre}, title = {On the zeros of the spectrogram of white noise}, year = {2017}, journal = {submitted}, eprint = {arXiv:1708.00082}, arxiv = {https://arxiv.org/abs/1708.00082} }

## Journal Articles

- 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} }

- 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, 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} }

- 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} }

## Conference Articles

- 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, 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, “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

`2015 - currently`

**PhD, Signal Processing**- Ecole Centrale de Lille
*Theory and methods for bivariate signal processing* `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*

# 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*