Inverse problems

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

Inverse problems are first presented in finite dimension (notion of conditioning, singular values, etc.), then in infinite dimension (problem stability, regularization, pseudo-inverse, etc.).

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

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

• Learn about linear inverse problems.
• Master the Fourier transform and convolution in common 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 builds on the Functional Analysis concepts developed in the first year of the course.

Assessment by continuous assessment.

An indicative course schedule 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

* Fourier transform of signals in dimension 1 space

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

* 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

* Overview of medical imaging techniques

Hourly volumes :

            CM:21

            TD:0

            TP:0

            Land: 0