Accès direct au contenu


Version anglaise


Accueil > Formations > Master MVA > Présentation des cours

The partial differential equations of image processing (and their surprising applications)

Lecturer : Jean-Michel MOREL, CMLA, ENS de Cachan

Objective of the course :

As hinted by the prominent part that evolution has given to visual perception in our brain, most human activities rely on visual perception.
In human information media images and video dominate. It is even more so for scientific and technical activities, which rely on the creation or acquisition of images and on their analysis. They are for example indispensable in medicine, astronomy, material science and biology.
 In this booming context, imaging has developed as an autonomous science in the past 40 years.  Its goal is to define the structure of digital images, to acquire them, to improve their quality, to compare them, and to extract information from them.
Unsurprisingly, each image being a continuous medium, calculus plays a prominent role in these operations. The most basic operations on images rely on variants of the simplest partial differential equations that also appear in continuum mechanics.                                                                                     While the equations are simple, their use is both subtle and funny in the realm of image processing. In this course, I'll try to develop some of these uses. I'll give for each usage the theory, but also the complete algorithms, and demonstrate their use on real images.  Matlab sessions may be organized as a supplement to the course.   The tentative program follows.

Topics :

Fourier analysis and the fundamentals of its application to digital images: sampling theory. Poisson editing: how to perform seamless copy-paste on images and other applications. Retinex theory: from color perception theory to the restoration of photographs with backlight. Heat equation, sampling and scale invariance: simulating image zooms. The SIFT method comparing any two images and deciding if they see the same objects. Affine invariant image recognition: the ASIFT method

Prerequisites :

Although the course will be self-contained, students will find it easier if they have some notions of Fourier Analysis, partial differential equations (Laplace, Poisson, heat equations). But the viewpoint being different, all notions will be introduced from scratch anyway.

Organization of courses :

  • Each session will imply a combination of course, experiments, exercises and computer practice.

Validation :

One written examination, and a paper report supplemented with numerical  trials. IPOL project: for a selected and motivated subgroup of students, validation of a second semester module will be offered, preparing an IPOL publication on  one of the ground-breaking papers on the subject.

Lectures Notes

My lecture notes can be downloaded here

    References :