Variational methods are widely used in image processing. They allow us to propose models taking into account specificities of the tackled problems. They also enable the study of the properties of solutions. The proposed functional in image processing are not smooth (to account for the presence of interfaces) and not necessarily convex. This naturally raises questions of existence of solutions, uniqueness, and (fast) computation. The choice of the regularization is often based on an assumption of sparsity (eg, of the gradient, or in a transformed domain). Non-local interactions and the concept of patch are typically involved in the design of functional. Finally, all these approaches are based in principle on the weighting of the data fidelity term and the regularization term, which leads to the question of the reliable estimation of this parameter.

# Axis 2: Optimal transport for image and signal processing

This axis focuses on the development of new methodologies for large data analysis such as histograms, images, or point clouds, based on concepts from the optimal transport theory. This methodology results in the use of non-Euclidean metric (such as Wasserstein distances) to extract the geometrical information in the presence of non-linear sources of variability in the data. In this context, a new method of Principal Component Analysis based on the Wasserstein distance has recently been proposed with applications to statistical analysis of histograms.

The use of optimal transport has also been proposed for various image processing problems. By generalizing transport distances by regularizing the associated transport plans, new image interpolation methods were developed for applications in oceanography. The Wasserstein distance was also considered to more traditional problems such as image segmentation or color transfer.

# Axis 3: Guiding therapy imaging

These activities fall within the scope of interventional radiology, a booming discipline, for which France ranks in the forefront in the world. The possibility of depositing an energy locally and non-invasively opens new perspectives for more reliable therapeutic strategies of malignant tumors, less aggressive for the patients, which will allow a reduction of time and costs of hospitalization. In this context, an activity of the IOP team is oriented around image processing methods in real time, allowing the guiding imaging (ultrasound or MRI) of a mini or non-invasive treatment (thermal ablation or radiotherapy), also allowing local deposition of medications.

# Axis 4: Sparse, non-local, patch-based approaches and harmonic analysis

This research axis is motivated by the problems of data analysis in high dimension (signals and images) organized in the form of vectors or matrices of large size.

The first part of this axis focuses on sparse-based approaches for the analysis of high-dimensional data when they can be well approximated in a space of low dimension. We are interested in the development of variational models which naturally promote this sparsity to a preselected representation (wavelets, gradient fields, etc.) or learned on an external dataset. This representation (the space of small size) can also be selected by the model itself, it is called dictionary learning which is usually expressed as a matrix factorization problem. For examples, we focus on approximation methods based on thresholding wavelets in image processing, as well as variable selection problems with the Lasso (Least Absolute Shrinkage and Selection Operator). Beyond the development of such models, we are seeking to understand their behavior. In particular, in the first example, we are interested in conditions ensuring that a signal can be recovered, or that its reconstruction error can be bounded (robustness). In the second example, we focus on the conditions ensuring the identification of good variables.

A second part concerns the analysis of repeating patterns in the case of images or multidimensional signals. These repetitions indeed provide key information for the processing or interpretation of the signal. In particular, we are interested in non-local approaches used in image processing that rely on the similarity of patterns using *patches* (typically small 8 by 8 rectangular windows). The term *non-local* means here that only the content of the patches is relevant, regardless of the spatial location. Part of our work involves the design and choice of metrics ensuring a robust matching of patterns as well as the implementation of efficient search algorithms. We are also interested in the analysis of these repetitions, typically on the connectivity graph. Finally, we aim at developing and studying new variational models taking into account these non-local information.

# Axis 5: Stochastic calculus, probabilities and statistics on manifolds

This axis relates to the use of all methods of stochastic calculus, particularly the detailed analysis of process trajectories, their probabilities, their variation, their couplings, in order to:

- analyse the diffusion semigroups and evolution equations in manifolds (heat equation, mean curvature equation, Ricci flow), and their use in signal and image processing,
- get functional inequalities,
- study the Poisson boundaries,
- computing price sensitivity in financial models,
- get transport inequality,
- design search and optimization algorithms in manifolds for images and signals.

Existence and uniqueness problems are also studied for martingales with given terminal value in manifolds. Several contributions also cover the notion of Fréchet mean which is an extension of the usual Euclidean barycenter to spaces equiped with non-Euclidean distances. In this context, many statistical properties of the Fréchet mean were established for deformable models of signals.

# Axis 6: Optimization algorithms: deterministic and stochastic methods

In information geometry, statistical data take values in sets equipped with a Riemannian structure, with finite or infinite dimension. The estimators of quantities relating to these data are averages, medians, or more generally p-averages of these data. Stochastic algorithms to find these p-averages are very useful for all of the practical applications that have been developed in the publications listed below. It may be mentioned in particular applications in processing stationary radar signals. A very important issue is to consider non-stationary signals. To this end, we need to work on paths spaces in Riemannian manifolds, and develop a good notion of metric, distance, and average for these paths.

New stochastic algorithms of Robbins-Monro type have also been proposed to efficiently estimate the unknown parameters of deformation models. These estimation procedures are implemented on real ECG data to detect cardiac arrhythmia problems.

Proximal methods have been extremely successful in image processing to provide efficient algorithms to compute the solutions of the considered problems. A major theme of the team is the study of the convergence of such algorithms, their speed, and robustness to errors.

# Axis 7: PDE: stochastic approches

This research axis consists in studying the properties of some classes of PDEs (existence, uniqueness, long time behavior, regularity …) using stochastic processes. The study of systems of progressive retrograde stochastic differential equations (SDE-RSDEs) allows for example to obtain a probabilistic representation for such PDEs, representation commonly called Feynman-Kac formula. This representation also helps to build and to study the convergence of probabilistic algorithms to numerically solve these PDEs.

The SDE-RSDEs also allow us to model the equations of hydrodynamics and their approximate solution give new simulation methods. Their combination with variational methods to answer questions of existence of generalized flux with initial and final conditions. RSDEs are also a promising tool for smoothing and denoising signals via the design of martingales with given terminal value.

# Axis 8: Large deviations and concentration inequalities

This research axis consists of two parts. The first part deals with large deviations properties of quadratic forms of Gaussian processes and Brownian diffusion. One can also cite recent work on large deviations of least squares estimators of the unknown parameters of Ornstein-Uhlenbeck process with shift. The second part is dedicated to concentration inequalities for sums of independent random variables and martingales. A book is forthcoming, with some applications of concentration inequalities in probabilities and statistics, particularly on the autoregressive process, random permutations and random matrix spectrum.

# Axis 9: Nonparametric estimation and statistical process

In this axis, one of the research direction is focused on nonparametric and semi-parametric statistics for the design of optimal estimators (in the minimax sense, or from Oracle inequalities) for statistical inference problems in large dimension (deformable models in signal processing, covariance matrix estimation, inverse problems).

In this context, the first part focuses on the minimization of the Stein unbiased risk estimator (SURE) for variational models. A first theoretical difficulty is to design such estimators when the targeted functionals are non-smooth, non-convex or discontinuous. A second difficulty relates to the development of efficient algorithms for the calculation and minimization of the SURE when solutions of these models are themselves derived from an optimization algorithm. Finally, a last difficulty concerns the extension of the SURE to complex inference problems (ill-posed problems, non white Gaussian noise, etc.).

Another part of this axis concerns semi-parametric regression models when the regression function is estimated by a recursive Nadaraya-Watson type estimator. In this context, a “région Aquitaine” contract was obtained in 2014 for 3 years. It covers the development of new non-parametric estimation methods with applications in valvometry and environmental sciences.