Parisian Seminar on the Mathematics of Imaging

Welcome on the website of the Paris Seminar on the Mathematics of Imaging.

The goal of this seminar is to cover the fields of the mathematics of imaging in a very wide sense (including for instance signal processing, image processing, computer graphics, computer vision, various applications and connexion with statistics and machine learning). It is open to everyone. It takes place the first Thursday of each month at IHP, from 14:00 to 16:00. Each seminar is composed of two presentations.

You can access the list of past seminars. The list of seminars prior to sept. 2016 is available on the website SMATI.

You can also subscribe to the mailing list of the seminar.

8 november 2018, 14h-15h, room 235A, 29 rue de l'Ulm.

Rémi Gribonval (INRIA, Panama project-team)

**Title:** *Approximation with sparsely connected deep networks*

**Abstract:** Many of the data analysis and processing pipelines that have been carefully engineered by generations of mathematicians and practitioners can in fact be implemented as deep networks. Allowing the parameters of these networks to be automatically trained (or even randomized) allows to revisit certain classical constructions.
The talk first describes an empirical approach to approximate a given matrix by a fast linear transform through numerical optimization. The main idea is to write fast linear transforms as products of few sparse factors, and to iteratively optimize over the factors. This corresponds to training a sparsely connected, linear, deep neural network. Learning algorithms exploiting iterative hard-thresholding have been shown to perform well in practice, a striking example being their ability to somehow “reverse engineer” the fast Hadamard transform. Yet, developing a solid understanding of their conditions of success remains an open challenge.
In a second part, we study the expressivity of sparsely connected deep networks. Measuring a network's complexity by its number of connections, we consider the class of functions which error of best approximation with networks of a given complexity decays at a certain rate. Using classical approximation theory, we show that this class can be endowed with a norm that makes it a nice function space, called approximation space. We establish that the presence of certain “skip connections” has no impact of the approximation space, and discuss the role of the network's nonlinearity (also known as activation function) on the resulting spaces, as well as the benefits of depth. For the popular ReLU nonlinearity (as well as its powers), we relate the newly identified spaces to classical Besov spaces, which have a long history as image models associated to sparse wavelet decompositions. The sharp embeddings that we establish highlight how depth enables sparsely connected networks to approximate functions of increased “roughness” (decreased Besov smoothness) compared to shallow networks and wavelets.
Joint work with Luc Le Magoarou (Inria), Gitta Kutyniok (TU Berlin), Morten Nielsen (Aalborg University) and Felix Voigtlaender (KU Eichstätt).

8 november 2018, 15h-16h, room 235A, 29 rue de l'Ulm.

Antoine Houdard (Telecom ParisTech & Universite Paris Descartes)

**Title:** *Some advances in patch-based image denoising*

**Abstract:** In this talk I will present my PhD thesis work on non-local methods for image processing, and their application to various tasks such as denoising. Natural images contain redundant structures, and this property can be used for restoration purposes. A common way to consider this self-similarity is to separate the image into patches. These patches can then be grouped, compared and filtered together. The main part of this talk will be dedicated to the study of Gaussian priors for patch-based image denoising. Such priors are widely used for image restoration. We propose some ideas to answer the following questions: Why are Gaussian priors so widely used? What information do they encode about the image? Next I shall propose a probabilistic high-dimensional mixture model on the noisy patches. This model adopts a sparse modeling which assumes that the data lie on group-specific subspaces of low dimensionalities. This yields a denoising algorithm that demonstrates state-of-the-art performance. The last part of the talk explores different ways of aggregating the patches together. A framework that expresses the patch aggregation in the form of a least squares problem is proposed.

6 december 2018, 14h-15h, room 314.

Hughes Talbot (CentraleSupelec)

**Title:**

**Abstract:**

6 december 2018, 15h-16h, room 314.

Denis Fortun (iCUBE, CNRS, Université de Strasbourg)

**Title:**

**Abstract:**

- thursday oct 4th: Salle 314
- thursday nov 8th: TBA
- thursday dec 6th: Salle 314

The seminar will stop in its usual form during the IHP trimester “The Mathematics of Imaging”.

After that the seminar will start again in october 2019.

- Andrés Almansa (CNRS and Paris 5)
- Julie Delon (Paris 5)
- Agnès Desolneux (CNRS and ENS Cachan)
- Jalal Fadili (ENSICAEN)
- Bruno Galerne (Paris 5)
- Yann Gousseau (Telecom ParisTech)
- Gabriel Peyré (CNRS and ENS)

The seminar is hosted by IHP, and is labelled by the SIGMA group of the SMAI and the GdR MIA.

Bienvenu sur le site du Séminaire Parisien des Mathématiques Appliquées à l’Imagerie.

Le but de ce séminaire est de couvrir le domaine des mathématiques de l’imagerie. Il est ouvert à tous. Le séminaire a lieu le premier jeudi de chaque mois à l’IHP, de 14h à 16h. Chaque séance est composée de deux exposés.

Vous pouvez consulter la liste des séminaires à venir ainsi que celle des séminaire passés. La liste de séminaires antérieurs à septembre 2016 sont disponibles dans le site SMATI.

Vous pouvez également vous abonner ou désabonner à la liste de diffusion du séminaire.