Inverse problems

Inverse problems are studied, emphasizing the notion of a "well-posed problem".

A presentation is first made of inverse problems in finite dimension (notion of conditioning, singular values, ...), then in infinite dimension (stability of the problem, regularization, pseudo-inverse...).

In a second step, the properties of the Fourier transform and the convolution operation are recalled in Lp spaces and their usual subspaces. We examine how these two operations lead to well or badly posed problems. For convolution, in particular, the notions of "filter" and "deconvolution" are discussed in the context of signal and image analysis.

Finally, we apply these notions to the study and inversion of the Radon transform, or related transforms, from medical imaging techniques.

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

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

Recommended prerequisites: This course is based on the concepts of Functional Analysis developed in the first year of the course.

Evaluation by continuous assessment.

An indicative program of the course is as follows:

1) Introduction to linear inverse problems

* Notion of a well-posed linear inverse problem

* Linear inverse problem in finite dimension

* Linear inverse problem in infinite dimension

* Pseudo-inverse and regularization

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

* The Fourier transform of signals in dimension 1 of space

* The convolution and its link with the Fourier transform. Notion of filter.

* The Fourier transform and convolution of n-dimensional images

3) Radon transform and image reconstruction

* Radon transform and 2D image reconstruction

* X-ray transform and 3D image reconstruction

* Some overviews of medical imaging techniques

Hourly volumes:

            CM :21

            TD :0

            TP :0

            Land : 0