The "classical" approach to imaging problems relies on tools and notions in the field of regularisation theory for ill-posed inverse problems. This approach allows for a formulation of these problems in suitable function spaces where the use of appropriate energy functionals typically enforces the well-posedness of the new problems.

In this context, the definition and the theoretical study of the analytical properties of the desired solution (for instance, its sparsity with respect to some basis or dictionary) and of the underlying forward degradation operator are crucial and often require the use of advanced tools in the field of calculus of variations and functional/harmonic analysis. In the case of Bayesian-type approaches, where the solution of the inverse problem is defined in terms of a probability density, the choice of a-priori information assumed on the solution are on the other hand imposed in terms of statistical properties and (hyper-)parameters to estimate in a often non-trivial way.

On the other hand, the study of optimisation algorithms with theoretical convergence and stability guarantees is fundamental for the practical implementation and the numerical solution of these models. This is often a challenging task due to the non-smoothness and non-convexity of the models considered, which make necessary the use of generalisation of standard iterative gradient-type algorithms. For a solution of these problems compatible with the limited computational resources available it is furthermore necessary to exploit the structure of the operators describing the discretised problem whose inversion, although regularised, is needed. By means of suitable pre-conditioning techniques and advanced linear algebra approaches concerned with the spectral properties of the operators, it is indeed possible to limit considerably the computational complexity.

The main research axes of this theme are:

• Variational models, PDEs, optimal transport

• Regularisation theory and spectral analysis of operators

• Sparsity

• Modelling, assimilation and data analysis in high dimensions

• Optimisation

• Machine learning and deep learning for imaging

Several tools of harmonic and geometric analysis, variational calculus and differential equations (ODEs and PDEs) are often useful to study the underlying mechanisms of human vision. Recent theories also show characteristics very similar to those encountered in mathematical physics.

The physiological phenomena of neuronal activation localised over the different layers of the visual cortex can be indeed described by means of standard mathematical operations, such as convolutions and diffusion/transport operators. In its variational form, the problem here is to minimise the redundancy of perceived information, in agreement to well-known phenomena which regulate the visual process (neural coding). In order to integrate within these models the different features of the perceived image, it is thus necessary to introduce bio-inspired functional spaces (for instance, color spaces), non-Euclidean geometries (sub-Riemannian and hyperbolic) and the use of suitable filters (Gabor, wavelets) describing the neural response to the observed input. The theoretical advances in the field of the mathematical modelling for human vision is also crucial for the study of neuronal interactions which cannot be explained by means of 'classical' tools, as it happens in the case of modern approaches based on the use of neural networks solving several imaging problems.

The research axes of this theme are the follwoing:

• Bio-inspired models for vision

• Psycho-physics and neurogeometry of vision

Several 2D and 3D biomedical imaging problems (tomography, MRI, ultrasounds, microscopy) require the use of advanced image reconstruction techniques for the retrieve of information often non-visible in the given data and/or for the removal of several acquisition artefacts (blur, noise) in view of a precise and reliable quantitative analysis. In addition to these problems, other challenges aim to overcome the `physical' limits of the problem considered (such as the diffraction barrier for super-resolution problems) and/or to distinguish and track objects in an image in terms of their distinctive features. It is thus necessary to precisely model the properties that need to be favoured in the reconstruction process (typically, some type of sparsity) along with the physical/statistical properties of the problem (noise type, degradation process..) so as to model the acquisition process in a consistent way.

Analogous problems are often encountered also in astronomical imaging applications, where the reconstruction of the desired images is performed from given satellite measurements acquired in different modalities (such as high-energy X rays, for instance). In this case, the study is fundamental to identify astronomical phenomena of interest happening in the extra-atmosphere space.

Another area of application is the use of advanced large-scale techniques for the digital restoration of cultural heritage artworks which can be damaged from several external (previous restoration, censorship) factors and internal (wear and tear) factors. The physical restoration of this type of data is often impossible due to the fragility of the materials employed, hence, a digital processing is often the only possibility available.

The main applicative axes are:

• Biomedical imaging

• Astronomical imaging

• Cultural heritage imaging