97269 - Applied Inverse Problems in Imaging

Academic Year 2022/2023

  • Teaching Mode: Traditional lectures
  • 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/or 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: Unconstrained and constrained, multivariate (large/huge size), convex, differentiable and 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 exam.

Given the type of activity and teaching methods adopted, the attendance of this course requires the prior participation of all students in the training modules 1 and 2 on safety in the study places ( https://elearning-sicurezza.unibo. it/ ) in e-learning mode.

Assessment methods

Attendance to the lessons is not mandatory nor will it be considered for the final evaluation, but it is highly recommended to maximize the teaching impact and to easy the students' learning.

The final exam consists in the oral discussion of both the theoretical topics presented during the course and the results obtained by the student in carrying out a project assigned at the end of the course and involving the implementation of the solution in Matlab.

The exam 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 teacher, and other material (Matlab source codes, etc.). The teaching material will be available on the University of Bologna e-learning platform (https://virtuale.unibo.it).

Office hours

See the website of Alessandro Lanza

SDGs

Quality education

This teaching activity contributes to the achievement of the Sustainable Development Goals of the UN 2030 Agenda.