LIP: Learning for Inverse Problems

 Description

Inverse problems require to determine the cause from a set of observations. Such problems appear in many real-world situations, but the most known applications are biomedical imaging, geoscience, astronomy, and computer vision to mention but a few. Therefore their deepened mathematical study represents breakthroughs in the applications.

Inverse problems are often ill-posed (no-uniqueness and/or very weak stability estimates hold), hence regularization techniques must be used to solve such problems. To name but a few, traditional methods are: Variational, Iterative, and Bayesian Methods.  These approaches are known as knowledge-driven modeling because the physics of the data acquisitation, mathematically represented by the so-called "forward operator",  is used in the procedure.

Incorporating additional information and a-priori assumptions on the parameters/objects to be determined and interpolating known available data are some of the strategies used in the community to get good reconstructions, bringing to a reduction of the artifacts in the reconstruction process. However, the success of these algorithms strongly depends on how severe the ill-posedness of the problem is.

In the last few years, methods based on data-driven modeling (learning approaches) and that partially discard forward modeling have emerged. Data-driven methods, such as deep learning theory, are increasingly attracting the attention of researchers in the field of inverse problems, thanks to their recognized ability at least to match and very often exceed state-of-the-art results.

Despite the wide and successful range of application of data-driven methods, numerous aspects call for a deeper investigation. For example, the performance of neural networks, one of most common tools in machine learning, which are used to approximate nonlinear relations between inputs and outputs, significantly depend on the choice of the network architecture. Especially in applications, the choice of such architectures is often arbitrary, or defined by means of trial-and-error procedures. Moving from black-box models to networks that are (fully or partially) theoretically justified, guaranteeing comparable numerical results, is a field of growing interest. Further aspects of theoretical investigation are the dependence of the learned method from the data set used to generate it, and its stability with respect to perturbations of the input data.

It is now a fairly accepted idea that a combination of knowledge/physics-driven and data-driven approaches allows to overcome the drawbacks of the two different methods, so it represents the right path of research to improve the reconstruction schemes for inverse problems. This very challenging task is nowadays a very attractive field of research, and is one of the main focuses of several international research groups. In fact, in the last years, numerical reconstruction procedures that combine the two approaches are appearing and provide results exceeding the state-of-the-art. At the same time, it must be noted that a solid theoretical framework is still almost missing, except for rare exceptions, hence a lot of work should be done for theoretical justifications. 

The primary goal of the workshop is to bring together theoreticians and practitioners in order to illustrate recent advances and to discuss new directions in data+physics-based approaches for solving inverse problems in imaging and related fields of application.