Imagerie et problèmes inverses

1) Direct and inverse problems in imaging (a direct problem in X-ray imaging and the structure of DNA, the phase problem as an example of an inverse problem, the alternating projections algorithm, ...).

2) Time and band-limited functions (Paley-Wiener theorem, the problem of time-frequency concentration, properties of Slepian functions, out of band extrapolation, ...)

3) Regularization of inverse problems (pseudo-inverse, regularization methods, deconvolution, ...)

4) Medical imaging and the Radon transform (the Radon transform, properties, the inverse Radon transform, ...)

5) Discrete wavelets (the cosine transform, Haar wavelets, orthogonal wavelets, filter banks, biorthogonal wavelets, applications, ...).

6) Wavelets in L2(R) (multiresolution analysis, wavelet decomposition, construction of a multiresolution analysis, properties and numerical computation of wavelet functions, biorthogoanl wavelets, ...).

This course offers an introduction to data inversion methods and their application to imaging and other problems.

At the end of this course a student will be capable of
1) giving some examples of ill-posed inverse problems
2) regularising ill-posed inverse problems
3) describing several ways of inverting the Radon transform
4) identifying the properties of discrete and continuous wavelet transforms

In the context of the course subjects, a student will be able to interpret and produce mathematical content (such as texts, diagrams, and formulas) using a variety of notations (lowercase and uppercase letters, italics, boldface, Latin and Greek letters, numbers, symbols, etc.), presented in common fonts and sizes.

Required and Corequired knowledge and skills

Some functional and harmonic analysis and some linear algebra

Oral lectures and/or personal work

References, bibliography, and recommended reading

Charles L. Epstein. Introduction to the mathematics of medical imaging. SIAM, 2 edition, 2007.
G. Strang and T. Nguyen. Wavelets and filter banks. Cambridge, 1996.
S. Mallat. A Wavelet Tour of Signal Processing : The Sparse Way. Academic Press, third edition edition, 2009.
K. Bredies and D. Lorenz. Mathematical image processing, Springer, 2018.

Course notes

• Syllabus
• Université virtuelle

Additional information

Lecture notes are available in pdf form (French only)

Contacts

mail (Ignace.Loris@ulb.be), Teams or in person in my office (campus Plaine, building NO, office 2.O7.107)

Campus

Plaine

Method(s) of evaluation

• Oral examination

Oral examination

The course is evaluated with an oral exam in session . Typically ~30 minutes per student.

Mark calculation method (including weighting of intermediary marks)

Typically two main questions (two chapters). 10 points out of 20 per question. A global mark (out of 20) will be given for the whole course. There are no partial marks.

The purpose of any assessment is to verify the student’s personal mastery of the learning outcomes. It is therefore the student’s responsibility to demonstrate such mastery. This principle applies in all circumstances and at every stage of the assessment process, both during and after the examination, including in the event of a challenge to the grade by the student, an irregularity, suspected academic misconduct, etc. The grade exclusively reflects the personal mastery that the course instructor can reasonably consider to have been established in light of all the available information.

Language(s) of evaluation

• french
• (if applicable english )