• 10h00 – 10h30 Accueil + café
• 10h30 – 11h15 Antoine Salmona
  Can push-forward models fit multimodal distributions ?
• 11h15 – 12h00 Anton François
  Topological Data Analysis for Train-Free Glioblastoma Segmentation in MRI: Advantages and Potential Impact
• 12h00 Pause déjeuner
• 14h00-14h45 Vadim Lebovici
  Hybrid transforms for topological data analysis.
• 14h45-15h30 Anna Song
  The geometry and topology of texture in shapes.
• 15h30-16h00 Pause café
• 16h00-16h45 Eddie Aamari
  Minimax Boundary Estimation and Estimation with Boundary.
• 16h45-19h00 Discussions libres
• 19h30 Diner au restaurant Chez Françoise (https://goo.gl/maps/MVAVqyUu7CQ1cVAa6)

• 9h30 – 10h15 Frédéric Richard
  Full inference for the anisotropic fractional Brownian field.
• 10h15 – 10h45 Pause café
• 10h45 – 11h30 Barbara Pascal
  Texture segmentation based on fractal attributes using convex functional minimization with generalized Stein formalism for automated regularization parameter selection.
• 11h30 – 12h15 Julien Rabin
  Semi-discrete Optimal Transport for Texture Generation and Image Stylization
• 12h15 Pause déjeuner
• 14h15-14h45 Jonathan Vacher
  Discussions : Images, Textures et Perception.
• 14h45-17h00 Discussions libres

Antoine Salmona 

Can push-forward models fit multimodal distributions? 

Les modèles génératifs sont aujourd’hui l’un des sujets de recherche les plus populaires en apprentissage automatique, notamment grâce à leur impressionnante capacité à générer des images synthétiques réalistes. Cependant, il reste souvent difficile de savoir si ces modèles s’approchent correctement de la distribution sous-jacente des données ou s’ils génèrent uniquement des échantillons qui semblent similaires aux données. Dans ce travail, nous nous concentrons sur la classe particulière des modèles génératifs push-forward, qui inclut les Variational Autoencoders, les Generative Adversarial Networks et les Normalizing Flows. Nous montrons que ces modèles doivent avoir de grandes constantes de Lipschitz afin de bien approcher les distributions multimodales. Ainsi, comme la majorité des méthodes pour stabiliser les modèles génératifs consistent à limiter, de façon plus ou moins directe,  la constante de Lipschitz des réseaux de neurones, il existe pour ces modèles un compromis entre la stabilité de leur apprentissage et leur expressivité.

Anton François

Topological Data Analysis for Train-Free Glioblastoma Segmentation in MRI: Advantages and Potential Impact

This presentation will describe a new method for glioblastoma segmentation in MRI scans using topological data analysis (TDA). Accurate segmentation of glioblastomas is critical for effective treatment planning and disease progression monitoring. Our TDA-based approach offers several advantages over traditional machine learning methods, such as the ability to perform segmentation without the need for large annotated datasets and the ability to adapt easily to different data sets and segmentation needs. TDA also provides a more interpretable and stable framework for segmentation by leveraging topological features. Overall, this TDA-based method has the potential to be a valuable tool for the medical imaging analysis.

Vadim Lebovici 

Hybrid transforms for topological data analysis

Euler calculus—integration of constructible functions with respect to the Euler characteristic—has led to important advances in topological data analysis.  

For instance, Schapira’s inversion theorem of the topological Radon transform is key to the following inverse problem: can one recover a shape embedded in the Euclidean space from the knowledge of the Euler characteristic of its intersections with all affine half-spaces? In this talk, I will introduce integral transforms combining Lebesgue integration and Euler calculus for constructible functions.

In this talk, I will present so-called « hybrid » transforms mixing Euler and Lebesgue calculus. These transforms provide more versatile shape descriptors than their topological analogues. I will show that these transforms output regular functions and are compatible with some topological operations on shapes. Finally, I will present an injectivity result for the hybrid Fourier transform.

Le calcul d’Euler, c’est-à-dire l’intégration des fonctions constructibles par rapport à la caractéristique d’Euler, a permis d’importantes avancées dans le domaine de l’analyse topologique de données. Par exemple, le théorème d’inversion de la transformée de Radon topologique dû à Schapira est la clé du problème inverse suivant : une forme de l’espace euclidien est-elle déterminée par la caractéristique d’Euler de ses intersections avec tous les demi-espaces affines ? 

Dans cet exposé, je présenterai des transformées intégrales dites « hybrides » mêlant le calcul d’Euler et l’intégrale de Lebesgue. Ces transformées fournissent des descripteurs de formes plus versatiles que leurs analogues purement topologiques. Je montrerai que ces transformées produisent des fonctions régulières et conservent une compatibilité à certaines opérations topologiques sur les formes. Je terminerai la présentation par un résultat d’injectivité pour la transformée de Fourier hybride.

Anna Song

The geometry and topology of texture in shapes

In biology and materials science, porous structures display a wide morphological variety. Their 3D shape texture constitutes an important reservoir of information, and is for instance a marker of disease progression in a vascular network. The disparity of these structures calls for a unified mathematical modelling, which poses real challenges.

First, I will present curvatubes, a generative geometric model for 3D porous shapes. A surface adopts an optimal shape with respect to a curvature energy which makes it fold into tubular or membranous shapes. This model is framed as a novel phase-field formulation compatible with the GPU and satisfying a Gamma-limsup property. This results in geometrically random but statistically controlled textures which has many applications.

Second, I will present a topological method that quantifies a shape by its components, loops and cavities, at several spatial scales. We will give an interpretation of the resulting persistence diagram in terms of thicknesses and gaps, and especially of shape texture. Inhomogeneous mixtures of textures are thus detectable.

Finally, I will show how to combine both approaches to study the remodelling of bone marrow blood vessels by leukaemia.

Eddie Aamarie 

Minimax Boundary Estimation and Estimation with Boundary

In this talk, we study the non-asymptotic minimax rates for the Hausdorff estimation of 𝑑-dimensional manifolds 𝑀 with (possibly) non-empty boundary 𝜕𝑀. The class of target sets that we consider reunites and extends the most prevalent C²-type models: manifolds without boundary, and full-dimensional domains. We will consider both the estimation of the manifold 𝑀 itself and that of its boundary 𝜕𝑀 if non-empty. In the process, we will present a Voronoi-based procedure that allows to identify enough points close to 𝜕𝑀 for reconstructing it. Explicit constant derivations are given, showing that these rates do not depend on the ambient dimension. If time permits, we will talk about possible extensions of the estimation procedure to smoother manifolds with corners.

Pascal Barbara

Texture segmentation based on fractal attributes using convex functional minimization with generalized Stein formalism for automated regularization parameter selection.

Texture segmentation still constitutes an ongoing challenge, especially when processing large-size real world images. The aim of this work is twofold.

First, we provide a variational model for simultaneously extracting and regularizing local texture fractal features, namely the local regularity and the local variance. For this purpose, a scale-free model, based on wavelet leaders, penalized by a Total Variation regularizer, is embedded into a convex optimisation framework. The resulting functional is shown to be strongly-convex, leading to a fast minimization scheme.

Second, we investigate Stein-like strategies for the selection of regularization parameters. A generalized Stein estimator of the quadratic risk is built, taking into account the covariance structure of leader coefficients. Then it is minimized via a quasi-Newton algorithm relying on a proposed generalized estimator of the gradient of the risk with respect to hyperparameters, leading to an automated and data-driven tuning of regularization parameters.

The overall procedure is illustrated on multiphasic flow images, analyzed as part of a long-term collaboration with physicists from the Laboratoire de Physique of ENS Lyon. 

Julien Rabin

Semi-discrete Optimal Transport for Texture Generation and Image Stylization

In this presentation we focus on the problem of image generation in the specific setting of textures.

Various techniques have been proposed in the last decades to synthesize realistic images from a single example, from patch-based copy approaches to neural network training with perceptual features.

After a short overview, we introduce a new model which aims at taking advantages of these techniques by optimizing the optimal transport cost between the distributions of synthetic and exemplar features.

We will show that such model is capable of interpolating between characteristics from several textures.