Inverse Problems

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

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

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

• An Introduction to Inverse Linear Problems.
• Master the Fourier transform and convolution in common functional spaces, and understand their practical applications.
• Examine some applications in signal and image analysis and reconstruction.

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

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

Assessment based entirely on continuous evaluation.

A sample course schedule is as follows:

1) Introduction to Linear Inverse Problems

* The concept of a well-posed linear inverse problem

* Finite-dimensional linear inverse problem

* Linear inverse problem in infinite dimensions

* Pseudo-inverse and regularization

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

* The Fourier transform of one-dimensional signals

* Convolution and its connection to the Fourier transform. The concept of a filter.

* The Fourier transform and convolution of n-dimensional images

3) Radon transform and image reconstruction

* Radon transform and 2D image reconstruction

* Converted to X-rays and 3D image reconstruction

* A Brief Overview of Medical Imaging Techniques

Hourly volumes:

            CM: 21

            TD: 0

            TP: 0

            Land: 0