Inverse problems (IPs) have been traditionally considered as mathematically challenging because they are intrinsically ill-posed. Imaging problems are a class of IPs with many practical applications in a variety of engineering disciplines, ranging from biomedical diagnostics to industrial non-destructive testing, up to geophysics and security screening, just to mention a few. Such IPs require suitable mathematical tools for their robust/stable solution in order to recover the well-posedness typical of forward/direct problems through suitable regularization and information-acquisition/exploitation techniques.

The course will review fundamentals and main issues of IPs, then focusing on classical/ state-of-the-art and recently introduced inverse solution procedures and algorithms, with main emphasis on the techniques for imaging and localization. Applicative examples including exercises will corroborate the theoretical concepts.

Course Topics

• Introduction and basics: motivations (methodological, applicative), imaging problems in engineering as IPs;
• Formulation of IPs and numerical techniques for dealing with their resolution;
• Non-linearity and ill-posedness: on the role of information in IPs;
• Non-linearity: physical meaning, degree of non linearity, the role of a-priori/available information;
• Ill-posedness and the need for regularization;
• Solution of IPs as minimization/maximization of a cost-function/functional;
• Multi-resolution and information-acquisition strategies as an effective recipe to counteract ill-posedness and non-linearity;
• Numerical techniques for imaging problem solving in biomedical and industrial contexts;
• Applicative examples including exercises regarding specific engineering applications.

Teaching Activities

• Theoretical Lessons
• e-Xam Self Assessment (on each teaching class or periodically)
• MATLAB Hands-On
• e-Xam Final Assessment