Inverse Problems and Imaging

ABSTRACT

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 CONTENT

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

 

FURTHER READINGS

  1. M. Bertero and P. Boccacci, “Introduction to Inverse Problems in Imaging”. IoP Press, 1998.
  2. D. Colton and R. Kress, “Inverse Acoustic and Electromagnetic Scattering Theory”, Springer-Verlag, 1998.
  3. F. D. Moura Neto, A. J. da Silva Neto, “An Introduction to Inverse Problems with Applications”. Springer, 2013.
  4. M. Pastorino and A. Randazzo, “Microwave Imaging – Methods and Applications”. Artech House, 2018.
  5. G. Franceschetti, “Electromagnetics Theory, Techniques, and Engineering Paradigms”, Kluwer Academic/Plenum Publishers, 1997.
  6. W. C. Chew, “Waves and Fields in Inhomogeneous Media”, Oxford University Press, 1996.