# Contribution to Image Restoration
and Sequence Analysis : Variational Approaches and
Viscosity Solutions

### Abstract

In this talk I will present the work I did during my phD at the computer
Vision lab Robotvis (INRIA) and at the Mathematical department of
Universite de Nice.
Our aim is to make a methodological contribution to the analysis of
three applications in Computer Vision. The methodologies being linked
with what we are looking for, our second aim will be to explicitely
write and justify the chosen algorithms and to show obtained results.

Applications are :

* Restore one image.

Among the great variety of nonlinear PDE based techniques, we begin
with a comparative study. We propose a common formalism for this
method and we proceed to a benchmark to test their capabilities. Then
we establish a convergent method for noise and blur removal. This
approach combines parabolic and hyperbolic operators. We will justify
this model using the theory of viscosity solutions and we will give a
suitable numerical scheme which take into account the specificity of
each term.

* Know the movement.

For a given sequence of images, our aim is to estimate the velocity
field, usually called the optical flow. We propose a variational
method based on the optical flow constraint. We study this problem on
the space of bounded variations. Two situations are considered,
depending on the regularity of the data :

- If the data is Lipschitz, we
prove that the problem on BV is well posed and we construct via
arguments from Gamma-convergence and duality, a convergent
algorithm.

- If the data is only BV, we establish the integral representation
of the relaxed problem for a suitable topology.

* Analyse sequence of images.

We propose an original variational approach to deal with sequences of
noisy images with static background. The specificity of this method is
to estimate simultaneously the background and the movement
detection. We justify theoretically this model and give a stable and
convergent algorithm. Numerous examples will illustrate the
capabilities of this original approach.

### On-line references

For more informations, you can have a look at my web page at INRIA :
http://www-sop.inria.fr/robotvis/personnel/pkornp/pkornp-eng.html
(Demonstrations and publications available)

Maintained by
Alexandre R.J. FRANÇOIS