97269 - Applied Inverse Problems in Imaging

Academic Year 2021/2022

  • Docente: Serena Morigi
  • Credits: 6
  • SSD: MAT/08
  • Language: English
  • Moduli: Serena Morigi (Modulo 1) Alessandro Lanza (Modulo 2)
  • Teaching Mode: Traditional lectures (Modulo 1) Traditional lectures (Modulo 2)
  • Campus: Bologna
  • Corso: Second cycle degree programme (LM) in Mathematics (cod. 5827)

Learning outcomes

At the end of the course, students know linear and nonlinear variational regularization methods for inverse problem imaging, their theoretical properties and the implementation aspects. In particular, students are able to implement numerical methods for optimization problems arising in imaging applications and to critically evaluate the numerical results.

Course contents

This course covers inverse imaging problems. The reconstruction and mathematical processing of images is of fundamental importance in medical, industrial and geophysical applications. In many cases, the underlying inverse problems can be formulated and solved using variational methods and partial derivative equations. This course offers a computational view of inverse problems and variational models for mathematical imaging. It deals with problems of image deblurring and denoising, reconstruction from different imaging modalities (for example CT) in biomedicine and geophysics, segmentation for the extraction of significant structures. The course covers the entire chain of resolution of inverse imaging problems, i.e. Problem identification → Modeling and discretization → Analysis → Numerical optimization.


At the end of the course the students will be able to tackle inverse problems for imaging with a new repertoire of numerical cutting edge tools.
In particular, at the end of the course, students will have achieved the following learning objectives:
Identification of the problem: identification of imaging problems as mathematical inverse operator problems (e.g. integral equations);
Modeling and discretization: formulation of problems that occur in applications that use nonlinear variational methods and partial derivative equations; use Bayesian modeling to take data into account and model uncertainty; continuous modeling against discrete;
Analysis: Understanding of the main concepts of linear and nonlinear regularization theory and how it affects existence and uniqueness of the results;
Numerical optimization: - Constrained, unconstrained, multivariate, convex, non-differentiable numerical optimization methods.

Readings/Bibliography

  • P. C. Hansen, J. G. Nagy, D. P. O'Leary, Deblurring Images: Matrices, Spectra, and Filtering, SIAM, 2006.
  • P. C. Hansen, Discrete Inverse Problems: Insight and Algorithms, SIAM, 2010.
  • Nocedal, Jorge, Wright, S., Numerical Optimization, Springer , 2006

Teaching methods

Theoretical lessons and exercises in the computer laboratory using the Matlab software. The laboratory exercises will be partly guided by the teacher, partly solved by the students (either individually or in groups). The results of the exercises will be analyzed during the lessons and discussed by the students during the oral examination.

Assessment methods

The examination consists of the oral discussion on the theoretical part and some project assignments carried out during the labs.

The oral examination is aimed at evaluating the ability of the student to solve numerical problems in imaging and to critically analyze the obtained results.

Teaching tools

Slides and notes from the teachers, and other material (Matlab source codes, etc.)

Office hours

See the website of Serena Morigi

See the website of Alessandro Lanza