Weak∗^* dentability index of spaces C([0,α])C([0,\alpha])

We compute the weak$^*$-dentability index of the spaces $C(K)$ where $K$ is a countable compact space. Namely ${Dz}(C([0,\omega^{\omega^\alpha}])) = \omega^{1+\alpha+1}$, whenever $0\le\alpha<\omega_1$. More generally, ${Dz}(C(K))=\omega^{1+\alpha+1}$ if $K$ is a scattered compact whose height $\eta(K)$ satisfies $\omega^\alpha<\eta(K)\leq \omega^{\alpha+1}$ with an $\alpha$ countable

Szlenk indices of convex hulls

We study the general measures of non-compactness defined on subsets of a dual Banach space, their associated derivations and their $\omega$-iterates. We introduce the notions of convexifiable and sublinear measure of non-compactness and investigate the properties of its associated fragment and slice derivations. We apply our results to the Kuratowski measure of non-compactness and to the study of the Szlenk index of a Banach space. As a consequence, we obtain that the Szlenk index and the convex Szlenk index of a separable Banach space are always equal. We also give, for any countable ordinal $\alpha$, a characterization of the Banach spaces with Szlenk index bounded by $\omega^{\alpha+1}$ in terms of the existence of an equivalent renorming. This extends a result by Knaust, Odell and Schlumprecht on Banach spaces with Szlenk index equal to $\omega$.Comment: This is the final revised version of this pape

Génération Numérique d'Hologrammes : État de l'Art

National audienceThis paper reviews Computer Generated Holography techniques applied to 3D video. Using these methods, it is possible to acquire holograms of synthetic or existing scenes without physical interference between light waves. Most limitations characterizing conventional holography, namely the need for a powerful, highly coherent laser and extreme stability of the optical system are thus avoided.Cet article présente un état de l'art des méthodes de génération numérique d'hologrammes appliquées à la vidéo 3D. Les méthodes présentées permettent de générer l'hologramme d'une scène synthétique ou réelle sans passer par le processus physique réel d'interférence entre deux ondes lumineuses. Les principales limitations liées à l'holographie conventionnelle, qui sont (1) la nécessité d'utiliser une source laser cohérente et (2) l'obligation d'avoir un système optique extrêmement stable, peuvent ainsi être évitées

Revisiting energy release rates in brittle fracture

International audienceWe revisit in a 2d setting the notion of energy release rate, which plays a pivotal role in brittle fracture. Through a blow-up method, we extend that notion to crack patterns which are merely closed sets connected to the crack tip. As an application, we demonstrate that, modulo a simple meta-stability principle, a moving crack cannot generically kink while growing continuously in time. This last result potentially renders obsolete in our opinion a longstanding debate in fracture mechanics on the correct criterion for kinking

Image decomposition: application to textured images and SAR images

In this report, we present a new algorithm to split an image f into a component u belonging to BV and a component v made of textures and noise of the initial image. We introduce a functional adapted to this problem. The minimum of this functional corresponds to the image decomposition we want to get. We compute this minimum by minimizing successively our functional with respect to u and v. We carry out the mathematical study of our algorithm. We present some numerical results. On the one hand, we show how the v component can be used to classify textured images, and on the other hand, we show how the u component can be used in SAR image restoration

Unsupervised Object Localization: Observing the Background to Discover Objects

Recent advances in self-supervised visual representation learning have paved the way for unsupervised methods tackling tasks such as object discovery and instance segmentation. However, discovering objects in an image with no supervision is a very hard task; what are the desired objects, when to separate them into parts, how many are there, and of what classes? The answers to these questions depend on the tasks and datasets of evaluation. In this work, we take a different approach and propose to look for the background instead. This way, the salient objects emerge as a by-product without any strong assumption on what an object should be. We propose FOUND, a simple model made of a single $conv1\times1$ initialized with coarse background masks extracted from self-supervised patch-based representations. After fast training and refining these seed masks, the model reaches state-of-the-art results on unsupervised saliency detection and object discovery benchmarks. Moreover, we show that our approach yields good results in the unsupervised semantic segmentation retrieval task. The code to reproduce our results is available at https://github.com/valeoai/FOUND.Comment: CVPR 202

Génération Numérique d'Hologrammes : État de l'Art

National audienceThis paper reviews Computer Generated Holography techniques applied to 3D video. Using these methods, it is possible to acquire holograms of synthetic or existing scenes without physical interference between light waves. Most limitations characterizing conventional holography, namely the need for a powerful, highly coherent laser and extreme stability of the optical system are thus avoided.Cet article présente un état de l'art des méthodes de génération numérique d'hologrammes appliquées à la vidéo 3D. Les méthodes présentées permettent de générer l'hologramme d'une scène synthétique ou réelle sans passer par le processus physique réel d'interférence entre deux ondes lumineuses. Les principales limitations liées à l'holographie conventionnelle, qui sont (1) la nécessité d'utiliser une source laser cohérente et (2) l'obligation d'avoir un système optique extrêmement stable, peuvent ainsi être évitées

Proprioceptive and tactile processing in individuals with Friedreich ataxia: an fMRI study

ObjectiveFriedreich ataxia (FA) neuropathology affects dorsal root ganglia, posterior columns in the spinal cord, the spinocerebellar tracts, and cerebellar dentate nuclei. The impact of the somatosensory system on ataxic symptoms remains debated. This study aims to better evaluate the contribution of somatosensory processing to ataxia clinical severity by simultaneously investigating passive movement and tactile pneumatic stimulation in individuals with FA.MethodsTwenty patients with FA and 20 healthy participants were included. All subjects underwent two 6 min block-design functional magnetic resonance imaging (fMRI) paradigms consisting of twelve 30 s alternating blocks (10 brain volumes per block, 120 brain volumes per paradigm) of a tactile oddball paradigm and a passive movement paradigm. Spearman rank correlation tests were used for correlations between BOLD levels and ataxia severity.ResultsThe passive movement paradigm led to the lower activation of primary (cSI) and secondary somatosensory cortices (cSII) in FA compared with healthy subjects (respectively 1.1 ± 0.78 vs. 0.61 ± 1.02, p = 0.04, and 0.69 ± 0.5 vs. 0.3 ± 0.41, p = 0.005). In the tactile paradigm, there was no significant difference between cSI and cSII activation levels in healthy controls and FA (respectively 0.88 ± 0.73 vs. 1.14 ± 0.99, p = 0.33, and 0.54 ± 0.37 vs. 0.55 ± 0.54, p = 0.93). Correlation analysis showed a significant correlation between cSI activation levels in the tactile paradigm and the clinical severity (R = 0.481, p = 0.032).InterpretationOur study captured the difference between tactile and proprioceptive impairments in FA using somatosensory fMRI paradigms. The lack of correlation between the proprioceptive paradigm and ataxia clinical parameters supports a low contribution of afferent ataxia to FA clinical severity

Décomposition d'images : Application aux images RSO

Nous construisons un algorithme pour séparer une image f en une somme u + v d'une composante à variation bornée et d'une composante contenant les textures et le bruit. Ce type de décomposition a été récemment proposée par Y. Meyer [1]. On obtient cette décomposition en minimisant une fonctionnelle convexe qui dépend de deux variables u et v, alternativement dans chaque direction. Chaque minimisation est basée sur un algorithme de projection pour minimiser la variation totale. Nous effectuons l'étude mathématique de notre méthode, et nous présentons des résultats numériques. En particulier, nous montrons comment la composante u peut être utilisée en restauration d'image RSO