B3093 - MATHEMATICAL AND MACHINE LEARNING METHODS IN IMAGING

Learning outcomes

At the end of the course, students have numerical and computational knowledge on variational regularization and machine learning methods for imaging problems, both from a theoretical, and a computational point of view. In particular, students are able to implement algorithmic strategies for challenging imaging applications, by combining optimization and machine learning tools, and to critically evaluate the numerical results.

Course contents

The course offers a computational view of Imaging problems through both Variational Methods and Machine Learning techniques. The image processing applications that will be covered range from image deblurring and denoising, to inpainting and super-resolution, to segmentation and classification for the extraction of meaningful information/structures, to reconstruction from different imaging modalities (e.g., Computed Tomography, CT) in biomedicine and geophysics.

The course covers the entire chain of resolution of Imaging problems, that is problem identification → modeling and discretization → analysis → numerical optimization / Machine Learning approach.By the end of the course, students will be able to tackle mathematical and Machine Learning 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:

1. Problem identification: identification of Imaging problems as inverse operator problems (for example, integral equations);
2. Modeling and discretization: formulation of problems occurring in imaging applications through linear and nonlinear variational methods; use of Bayesian modeling to account for data and model uncertainty; continuous versus discrete modeling;
3. Analysis: understanding of the main concepts of linear and nonlinear regularization theory and how it affects the existence and uniqueness of solutions;
4. Numerical optimization: numerical solution of large-scale, convex and non-convex, differentiable and non-differentiable, constrained and unconstrained optimization problems;
5. Machine Learning approach: basic introduction to Machine Learning and some of its underlying mathematics; Machine Learning in the context of inverse Imaging problems; convolutional neural networks (CNN), learning and post-processing priors, learned iterative optimization schemes.

Readings/Bibliography

• P. C. Hansen, Discrete Inverse Problems: Insight and Algorithms, SIAM, 2010.

• Nocedal, Jorge, Wright, S., Numerical Optimization, Springer, 2006.
• http://www.deeplearningbook.org/ - Ian Goodfellow and Yoshua Bengio and Aaron Courville, "Deep Learning", 2016.

Teaching methods

Theoretical lectures 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 (possibly) discussed by the students during the final 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

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

The final exam aims to verify the achievement of the following learning objectives:- student's ability to solve application problems requiring knowledge of advanced numerical tools - both theoretical and computational - such as, in particular, ill-posed inverse problems for signal/image processing, numerical optimization problems;- student's ability to critically analyze the results obtained from the implemented programs, in the light of the theoretical knowledge acquired during the course.

Teaching tools

Slides and notes from the teachers, 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 Serena Morigi

See the website of Alessandro Lanza