Course: 2021/2022

Inverse and Image Reconstruction

(12479)

Requirements (Subjects that are assumed to be known)

Partial Differential Equations.
Numerical Analysis.

COMPETENCES and SKILLS
- Acquiring theoretical knowledge that allows the student to develop original ideas, in a research context, knowing how to translate industrial needs into I+D+i projects in the Industrial Mathematics field.
- Being able to explain the results, along with the acquired knowledge, to experts and non-experts.
- Being able to get deeper into a subject in an autonomous way, which will be very useful to obtain a Ph.D.
- Being able to get quantitative and qualitative information from experimental data using numerical techniques.
- Knowing how to select the appropriate techniques to solve a specific problem.

Description of contents: programme

Introduction and Basic Notions
- Direct and inverse problems
- Well and ill-posed problems
- Existence and uniqueness of the solution
- Stability
Least Squares
- Motivation and general idea
- Applications
Regularization
- Motivation and general idea
- Tikhonov, Lardy and Landweber algorithms
- Morozov's discrepancy principle
Singular Value Decomposition
- Theoretical background, meaning and properties
- Noise filtering and data reconstruction
- Linear systems and regularization
- Extensions
Computed Axial Tomography
- Radon transform and sinogram
- Methods: back-projection and algebraic reconstruction
Topological Derivative
- Theoretical background
- Defects detection
- Methods: multifrequency and iterative
- Applications

Learning activities and methodology

Methodology:
- In-person classes.
- Homeworks and presentations.
Tutorials:
The students can ask questions via e-mail or during the classes.

Assessment System

- % end-of-term-examination 0
- % of continuous assessment (assigments, laboratory, practicals...) 100

Basic Bibliography

- A. Kirsch. An Introduction to the Mathematical Theory of Inverse Problems. Springer-Verlag New York. 2011
- Frank Natterer, Frank Wübbeling. Mathematical Methods in Image Reconstruction. SIAM. 2001
- J. Mueller, S. Siltanen. Linear and Nonlinear Inverse Problems with Practical Applications. SIAM Computational Science and Engineering. 2012
- M. Bertero, P. Boccacci. Introduction to Inverse Problems in Imaging. CRC Press. 1998