Inverse problems

We study inverse problems, emphasizing the concept of a "well-posed problem."

First, inverse problems are presented in finite dimensions (concepts of conditioning, singular values, etc.), then in infinite dimensions (stability of the problem, regularization, pseudo-inverse, etc.).

Next, the properties of the Fourier transform and the convolution operation are reviewed in Lp spaces and their usual subspaces. We examine how these two operations lead to well-posed or ill-posed problems. For convolution in particular, the concepts of "filter" and "deconvolution" are discussed in the context of signal and image analysis.

Finally, these concepts are applied to the study and inversion of Radon transforms, or related transforms, derived from medical imaging techniques.

• Learn about linear inverse problems.
• Master Fourier transforms and convolution in common functional spaces, and understand their practical applications.
• Study some applications in signal and image analysis and reconstruction.

Differential and integral calculus at L3 level, functional analysis at M1 level.

Recommended prerequisites: This course builds on the concepts of functional analysis developed in the first year of the program.

Continuous assessment throughout the course.

An indicative program for the course is as follows:

1) Introduction to linear inverse problems

* Concept of a well-posed linear inverse problem

* Linear inverse problem in finite dimensions

* Inverse linear problem in infinite dimensions

* Pseudo-inverse and regularization

2) Fourier transform and convolution: two tools and two inverse problems

* The Fourier transform of signals in 1-dimensional space

* Convolution and its link to the Fourier transform. The concept of filters.

* Fourier transform and convolution of images in n-dimensional space

3) Radon transform and image reconstruction

* Radon transform and 2D image reconstruction

* Converted into X-rays and 3D image reconstruction

* Some insights into medical imaging techniques

Hourly volumes:

            CM: 21

            TD: 0

            TP: 0

            Land: 0