Veuillez noter que le prochain séminaire aura lieu le 24 février à 10h30 en salle K071. Nous aurons le plaisir d’accueillir Diarra FALL, Maitresse de Conférences en Mathématiques Appliquées à l’Université d’Orélans dont les recherches se font dans le domaine de la Statistique Bayésienne appliquée à la santé.
Elle s’est également, depuis peu, intéressée à l’étude des réseaux de neurones.
Bayesian formulation of Regularization by denoising. Application to image restoration
Inverse problems are ubiquitous in signal and image processing. Canonical examples include
signal/image denoising (i.e, removing noise from a signal/image) and image reconstruction. As
inverse problems are known to be ill-posed or at least, ill-conditioned, they require regularization by
introducing additional constraints to mitigate the lack of information brought by the observations. A
common difficulty is to select an appropriate regularizer, which has a decisive influence on the
quality of the reconstruction. Another challenge is the confidence we may have in the reconstructed
signal/image. To put it another way, it is desirable for a method to be able to quantify the
uncertainty associated with the reconstructed image in order to encourage more principled
decision-making. These two tasks (regularization and uncertainty quantification) can be overcome
at the same time by addressing the problem within the Bayesian statistical framework. It allows to
include additional information by specifying a marginal distribution for the signal/image, known as
prior distribution. The traditional approach consists in defining the prior analytically, as a hand-
crafted explicit function chosen to encourage specific desired properties of the recovered
signal/image. Following the up-to-date surge of deep learning, data-driven regularization using
priors specified by neural networks has become ubiquitous in signal and image inverse problems.
Popular approaches within this methodology are Plug & Play (PnP) and regularization by
denoising (RED) methods.
We have proposed a probabilistic approach to the RED method, defining a new probability
distribution based on a RED potential, which can be chosen as the prior distribution in a Bayesian
inversion task. We have also proposed a dedicated Markov chain Monte Carlo sampling algorithm
that is particularly well suited to high-dimensional sampling of the resulting posterior distribution. In
addition, we provide a theoretical analysis that guarantees convergence to the target distribution,
and quantify the speed of convergence. The power of the proposed approach is illustrated in
various restoration tasks such as image deblurring, inpainting, and super-resolution.
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