• (Members involved: S. Anthoine, C. Chaux, P. Escande, C. Mélot, F. Richard, K. Schneider)

  The signal/image team has been working closely with the LIS Qarma learning team for several years. Collaborations initially centered on parsimonious optimization problems common to both fields of research (e.g. the study of gluttonous algorithms) have been enriched around themes such as the completion of missing data in audio or the learning of efficient representations.

  Team members are also working on operator family learning problems (e.g. in microscopy), as well as the integration of deep neurons to estimate texture models (c.f. the “Statistical analysis of Brownian textures” section).

  The team is also committed to developing efficient numerical methods for large-scale problems. This requires complete mastery of the method design chain. That is, from its modeling to its implementation on highly parallel architectures, via its theoretical study using tools from different horizons such as variety learning, concentration inequalities, non-convex optimization methods…

  Analysis, modeling and numerical simulation of fluid and plasma turbulence using multiscale techniques, with applications in plasma physics, fluid-structure interaction and bio-fluids, are another of the team’s themes. Extreme-resolution scientific computing on supercomputers and GPUs plays an important role. Annual consumption is on the order of 25,000,000 CPU hours on IDRIS and the CEA’s TGC. We also use the Marseille mesocenter for code development and data storage.

  For the analysis of charged particle turbulence, calculated by high-resolution direct numerical simulation, we use machine learning techniques, such as DBscan, to identify particle clusters. Methods for synthesizing artificial turbulent flow fields and inertial particle distributions based on the maximum entropy principle are developed. Gradient descent algorithms are then used to find a field that corresponds to a set of predefined moments (i.e. higher-order statistics). Scattering transforms are essential building blocks. The results are compared with data generated by generative antagonistic networks (GANs).

• (Members involved: S. Anthoine, C. Chaux, P. Escande, K. Schneider)

  A classic approach to inverse problem solving is to define an estimate of the target object as the minimizer of a composite criterion, the sum of a data attachment term linked to the observation model and regularization or constraint terms reinforcing certain properties of the target object. Once the criterion has been formulated, iterative algorithms (e.g. proximal algorithms, Majorization-Minimization (MM) techniques, etc.) are implemented. Applications include biomedical image reconstruction and restoration, restoration of missing audio data, spectral unmixing, reconstruction of fast camera images in tokamaks (in the context of magnetic confinement fusion) by tomography (with wavelet-wavelet regularization)…

(Members involved: P. Escande, F. Richard, K. Schneider, B. Torrésani)

• Statistical analysis of Brownian textures

We are working on the statistical analysis of the properties of irregular image micro-textures that can be described using extensions of the fractional Brownian field model. The properties we are interested in are irregularity, anisotropy and heterogeneity. We use mono- or multi-fractional anisotropic Brownian field models. We build estimators to infer or test the properties of these fields from one of their realizations. These estimators are mainly based on field increments and their quadratic variations. We study their asymptotic theoretical and numerical properties. Statistical methodology is applied to all kinds of fields (bio-medical imaging, photographic conservation, defect detection, etc.). This work is currently being pursued with a Bayesian approach that incorporates the definition of deep neural networks to estimate models. Learning these networks requires the use of the mesocenter’s computing resources.

• Statistical analysis of random surfaces represented by currents

We are working on the statistical analysis of surfaces represented by current models. Inspired by the theory of generalized processes, we have proposed a stochastic modeling of surfaces by immersing them in a space of random linear forms defined on a space of vector fields. In this context, we have proposed methods for estimating average representatives of surface populations and individual variability. For the discretization of estimators, we relied mainly on Hilbertian bases (Fourier, wavelets, …) of the vector field space on which random linear forms are defined. We have studied the asymptotic properties of these estimators. These methods are mainly applied to surfaces extracted from brain images. This work continues with the implementation of strategies for adapting the representation to the data.

• Analysis and synthesis of locally stationary processes, separation of non-stationary sources

We are developing algorithms for the analysis of non-stationary processes obtained by local deformation of stationary processes, the deformation being approximated by a translation in a well-chosen representation domain (wavelet, time-frequency, etc.). The most recent results include a generalization to the spectral analysis of multivariate non-stationary signal classes (seen from the angle of source separation), and a synthesis-type approach that we consider particularly promising. The main field of application is the analysis of non-stationary sounds, described below.

• Transformation analysis tools

Transformations are objects that appear, often implicitly, in various fields such as image processing, shape analysis, statistical learning, … These transformations are expensive to store, apply and interpret geometrically. We are developing a range of tools to study them. It is based on a multi-scale decomposition of transformations on dictionaries of elementary transformations.

• Concentration of optimization problem solutions

Some methods of statistical analysis can be formalized as trying to solve an optimization problem on a variety from a finite set of data. These data result from a random sampling process of an underlying probability measure. We study the behavior of solutions to these optimization problems.

• Multiscale statistical analysis of turbulent flows

We are developing statistical tools for the analysis of turbulent flows that quantify scaling, directional and geometric information. Applications include turbulent flow data, either Eulerian or Lagrangian, to quantify, for example, intermittency, scale-dependent helicity and scale-dependent angles of curvature or torsion.

(Members involved: P. Escande, C. Mélot, K. Schneider, B. Torrésani)

• Multiresolution analysis of Markov chain graphs

Data on graphs do not allow the use of certain classic signal processing tools, in particular multiscale wavelet analysis. As dilation and translation operations are poorly defined on a graph, classical multi-resolution decomposition algorithms are difficult to apply. However, since a weighted graph is naturally associated with a Markov chain, we have been working to implement a multiscale decomposition algorithm based on a filtering-subsampling decomposition scheme, using probabilistic tools, for signal processing of graph data.

• Multiresolution analysis and wavelet bases on surfaces

Motivated by applications in electromagnetic neuroimaging (EEG, MEG, SEEG), we are studying new constructions of biorthogonal wavelet bases on surfaces (target application: data processing on cortical surfaces). We have developed an initial construction based on the lifting scheme, which we are currently studying and improving. Inverse problem applications are described below.

• Multifractal analysis, point regularity

Multifractal analysis concerns the study of signals whose point regularity varies from point to point. We are working on different point regularity criteria (point regularity for non-locally bounded signals, anisotropic regularity), and their characterization using coefficients on wavelet-type systems, as well as on models in which it is possible to clearly understand the links between different point regularity exponents and the global quantities that we can seek to calculate to characterize the signal.

The study of the decay of wavelet coefficients makes it possible to analyze the local regularity of turbulent flow fields, and can be used to detect possible singularities in flows, such as the dissipation rate or the reconnection of current sheets in MHD, calculated at high resolution.

• Time-frequency analysis, uncertainty principles

We are developing techniques for representing and approximating signals in the joint time-frequency domain. The main results concern adaptive window and sampling network optimization techniques in the Gabor transform, layered signal decomposition exploiting concepts of parsimony and structured parsimony (see also Variational methods and optimization, inverse problems above) and the use of empirical mode decomposition for audio coding. We also study theoretical bounds given by uncertainty inequalities of the Heisenberg, Hirschman-Babenko-Beckner, Elad-Bruckstein or Dembo-Maassen-Uffink type, refinements of which we have obtained.

• Representation of operators in time-frequency frames

We are developing new methods for approximating operators in frames, in particular time-frequency and wavelets. In particular, we have obtained explicit expressions for the optimal approximation of certain classes of operators by time-frequency multipliers, exploiting techniques related to Plancherel theory. We also study operator estimation and identification problems, in the context of audio signal analysis and transformation.

• Fast operator decomposition

Multi-resolution methods for approximating operators have led to significant acceleration of algorithms, particularly in inverse problems. However, these methods rely on a change in operator representation. The complexity of this step is quadratic or even cubic in relation to the discretization of the problem. This makes them unusable even for medium-sized problems. We have developed fast decomposition methods in quasi-linear time. For example, when restoring images degraded by variable blur, an operator working on 16 million pixel images takes 4 years on a small computing server. Our methods make it possible to achieve this decomposition in just 4 hours.

• Adaptive numerical methods for PDEs

Motivated by turbulent fluid and plasma flows, we propose a semi-Lagrangian method for solving incompressible Euler equations on a time and space adaptive grid. This new approach advances the application of time flow with a “gradient-augmented level set” method. Multi-resolution analysis makes it possible to introduce an adaptive grid that reduces memory and computing time, while controlling the accuracy of the pattern. Complex geometries are handled by a penalty method. We run very high-resolution parallel numerical simulations to study the multiscale structure of turbulence.

A second approach is the development of an adaptive wavelet-based solver for incompressible Navier-Stokes equations using block-structured grids. Adaptability is introduced using interpolatory wavelets, and the lifting scheme is applied to achieve better scale separation. The code developed for WABBIT (Wavelet Adaptive Block-Based solver for Interactions with Turbulence) is freely available and implemented on massively parallel supercomputers. Simulations of beating insect flows have demonstrated its applicability to complex, bio-inspired problems.

The team works with different types of data:

(Members involved: S. Anthoine, C. Chaux, P. Escande, F. Richard)

• Breast imaging

We are working on texture analysis of breast images, with two main objectives: to automatically detect lesions present in the images, and to assess the risk of cancer development before it occurs. The analysis methods we use are based on modeling images using extended fractional Brownian fields.

• Standard and spectral X-ray tomography

In recent years, X-ray tomography, or CT scanning, has benefited from a technological breakthrough thanks to the latest generation of hybrid pixel X-ray detectors developed at CPPM. They enable us to work at very low flux and capture energy information on the imaged object. The first part of this study concerns tomographic reconstruction methods incorporating the new detection physics and enabling low-flux reconstruction. The relative inverse problem models Poisson noise, the only noise inherent in this type of measurement, as well as parsimonious a priori assumptions about the object of study. The second part of this study concerns the development of a methodology for reconstructing the object of study and mapping its main constituents of interest using the energy (i.e. spectral) information acquired by hybrid pixel detectors. This problem combines tomographic reconstruction with source separation in a non-linear context.

• Biphoton imaging

We are working on the restoration of biphotonic images, a fluorescence microscopy modality that has a number of advantages, including its ability to image at greater depths, limit phototoxicity (enabling in vivo experiments), and reduce acquisition noise.

Various acquisition experiments show that the PSF (instrument impulse response) is non-stationary along the Z axis (depth).

The problem is formulated in the form of a criterion to be minimized (taking into account the direct model and a priori information) and optimization algorithms (proximal approaches, Majoration-Minimization (MM) strategy) are implemented to solve the optimization problem.

• Light sheet imaging

Like biphoton imaging, light sheet imaging modalities are fluorescence microscopy. In contrast, the PSF here varies in all three volume directions. This specificity gives rise to additional difficulties in i) image restoration. Faced with the explosion in calculation times, we need to develop new methods to make them useful for biologists, ii) identification of the PSF in the entire field of view. In practice, image degradations are induced by the sample being imaged and must therefore be estimated. This is an important problem that requires the development of efficient methods for estimating large datasets.

(Members involved: C. Mélot, F. Richard, B. Torrésani)

• Brain imaging (neuroscience)

We are working on the characterization of cortical folds based on surfaces extracted from brain imaging. We use spectral graph representation tools (Fourier Transform with or without window, wavelets, etc.). Based on these representations, we propose surface analysis tools to characterize or compare populations.

Another area of research is the application of new multi-resolution analysis methods based on Markov chain data modeling. We plan to apply them to problems involving the classification, segmentation and regularity analysis of brain data.

• Neuroelectromagnetic inverse problem (EEG/MEG/SEEG) and applications in cognitive neuroscience

We are developing new approaches to the inverse EEG/MEG problem. The two main features of the approaches we are developing are the precise consideration of the time dependence of brain signals, and the aim of obtaining truly quantitative source reconstructions, unlike the statistical maps commonly used. The main ingredients in our approach are the use of wavelets (in time, and more original in space, i.e. on the cortical surface), the use of maximum entropy methods combined with Gaussian mixture models, and factorization techniques to control the dimensional explosion resulting from spatiotemporal modeling.

• Statistical modeling of electrophysiological signals

We are developing models for brain signals and corresponding estimation and classification algorithms, notably in the context of brain-machine interfaces, in collaboration with the Laboratoire de Neurosciences Cognitives(LNC), the DYCOG team at the Centre de Recherche en Neurosciences de Lyon (CRNL) and the INRIA Athena project.

We are interested in the implementation of models integrating various sources of variability and their validation on real data. The methods we are developing combine decomposition on Hilbertian bases (wavelets, time-frequency, etc.) and statistical modeling in the coefficient domain (mixed models, hidden Markov models). This is in the context of the characterization and discrimination of different types of brain waves.

(Members involved: S. Anthoine, C. Chaux, F. Richard, B. Torrésani)

In the context of NMR spectroscopy, we are interested in non-negative matrix decomposition techniques for unmixing spectra. In the case of fluorescence spectroscopy, we’re interested in non-negative tensor decompositions for image unmixing.

(Members involved: C. Chaux, B. Torrésani)

• Time-frequency analysis and synthesis of non-stationary sounds

In the context of audio applications, we are developing an original approach to the analysis and synthesis of classes of non-stationary random processes, modeled as deformations of stationary processes, as described above. In terms of audio applications, the models we are developing fall within the “action-object” paradigm developed by theUMR PRISM, in which a non-stationary action is performed on an object whose linear response is stationary. Emerging applications include the synthesis of sound textures, or environmental noise.

• Time-frequency filtering (in collaboration with LIS)

We are developing an original approach to time-frequency signal filtering, in other words the modulation or suppression of non-stationary components in signals whose temporal and frequency locations are known. The approach is based on a simple variational formulation, whose explicit solution involves time-frequency multipliers. We tackle the problems associated with the scourge of dimension using random projection techniques.

• Time-frequency inpainting

A joint project with LIS is to restore missing audio data. This “inpainting” theme has been studied more extensively in images. The aim here is to work within the general framework of audio inpainting for tasks involving missing audio data (declipping, packet loss …).