Fonction à variation bornée

Résumé

En analyse, une fonction est dite à variation bornée quand elle vérifie une certaine condition de régularité. Cette condition a été introduite en 1881 par le mathématicien Camille Jordan pour étendre le théorème de Dirichlet sur la convergence des séries de Fourier. Définition Soit f une fonction définie sur un ensemble totalement ordonné T et à valeurs dans un espace métrique (E, d). Pour toute subdivision σ = (x, x, … , x) d'un intervalle quelconque de T, on définit V(f, σ) par : V(f,\sigma)=\sum_{i=1}^n d(f(x_{i-1}),f(x_i)). On appelle variation totale de f sur T la valeur V(f) ∈ définie par : V_T(f)=\sup_{\sigma}V(f,\sigma). On dit que f est à variation bornée si cette borne supérieure V(f) est finie, autrement dit si l'« arc » (non nécessairement continu) défini par f est rectifiable au sens de Jordan. Intérêt de la notion Les fonctions monotones forment une classe importante de fonctions en analyse. Cepe

Source officielle

https://fr.wikipedia.org/wiki/Fonction_%C3%A0_variation_born%C3%A9e

À propos de ce résultat

Cette page est générée automatiquement et peut contenir des informations qui ne sont pas correctes, complètes, à jour ou pertinentes par rapport à votre recherche. Il en va de même pour toutes les autres pages de ce site. Veillez à vérifier les informations auprès des sources officielles de l'EPFL.

Publications associées (11)

Chargement

Afficher plus

Publications associées

Chargement

Personnes associées (1)

Personnes associées

Chargement

Unités associées (1)

Unités associées

Chargement

Concepts associés (16)

Concepts associés

Chargement

Cours associés (6)

Cours associés

Chargement

Séances de cours associées (7)

Séances de cours associées

Chargement

Publications associées (11)

On the optimal sampling design for the Graph Total Variation decoder: recovering piecewise-constant graph signals from a few vertex measurements

Rodrigo Cerqueira Gonzalez Pena

Compressed Sensing teaches us that measurements can be traded for offline computation if the signal being sensed has a simple enough representation. Proper decoders can exactly recover the high-dimensional signal of interest from a lower-dimensional vector of that signal's observations. In graph domains -- like social, similarity, or interaction networks -- the relevant signals often have to do with the network's cluster structure. Partitioning a graph into different communities induces a piecewise-constant signal, an object that can be decoded via Graph Total Variation (G-TV) minimization even if it is not fully observed. In fact, assume that such a signal can only be accessed by querying vertices at random. Then, we could sensibly ask: what are the sampling probabilities that minimize the number of queries required for a successful G-TV recovery? This thesis is an attempt to answer this question through the study of the success conditions in G-TV minimization programs. I show that the recovery error in these programs undergoes a phase transition in terms of the number of measurements, with a threshold that explicitly depends on the vertex-sampling probabilities. It suffices to minimize this threshold to obtain an optimal sampling design. Yet, sampling optimally in practice has problems of its own. While numerical experiments reveal that it is important to focus on the places of the graph where the signal varies, implementing the optimal design without actually knowing the signal-to-be-sampled remains an open issue.

EPFL2020

Chargement

When a strict local martingale is projected onto a subfiltration to which it is not adapted, the local martingale property may be lost, and the finite variation part of the projection may have singular paths. This phenomenon has consequences for arbitrage theory in mathematical finance. In this paper it is shown that the loss of the local martingale property is related to a measure extension problem for the associated Follmer measure. When a solution exists, the finite variation part of the projection can be interpreted as the compensator, under the extended measure, of the explosion time of the original local martingale. In a topological setting, this leads to intuitive conditions under which its paths are singular. The measure extension problem is then solved in a Brownian framework, allowing an explicit treatment of several interesting examples.

Inst Mathematical Statistics2014

Chargement

Fast high-resolution brain metabolite mapping on a clinical 3T MRI by accelerated H-1-FID-MRSI and low-rank constrained reconstruction

Sébastien Courvoisier, Matthieu Guerquin-Kern, Jeffrey Adam Kasten, Michel Kocher, François Lazeyras, Dimitri Nestor Alice Van De Ville

Purpose: Epitomizing the advantages of ultra short echo time and no chemical shift displacement error, high-resolution-free induction decay magnetic resonance spectroscopic imaging (FID-MRSI) sequences have proven to be highly effective in providing unbiased characterizations of metabolite distributions. However, its merits are often overshadowed in high-resolution settings by reduced signal-to-noise ratios resulting from the smaller voxel volumes procured by extensive phase encoding and the related acquisition times. Methods: To address these limitations, we here propose an acquisition and reconstruction scheme that offers both implicit dataset denoising and acquisition acceleration. Specifically, a slice selective high-resolution FID-MRSI sequence was implemented. Spectroscopic datasets were processed to remove fat contamination, and then reconstructed using a total generalized variation (TGV) regularized low-rank model. We further measured reconstruction performance for random under-sampled data to assess feasibility of a compressed-sensing SENSE acceleration scheme. Performance of the lipid suppression was assessed using an ad hoc phantom, while that of the low-rank TGV reconstruction model was benchmarked using simulated MRSI data. To assess real-world performance, 2D FID-MRSI acquisitions of the brain in healthy volunteers were reconstructed using the proposed framework. Results: Results from the phantom and simulated data demonstrate that skull lipid contamination is effectively removed and that data reconstruction quality is improved with the low-rank TGV model. Also, we demonstrated that the presented acquisition and reconstruction methods are compatible with a compressed-sensing SENSE acceleration scheme. Conclusions: An original reconstruction pipeline for 2D H-1-FID-MRSI datasets was presented that places high-resolution metabolite mapping on 3T MR scanners within clinically feasible limits.

WILEY2019

Chargement

Afficher plus

Fast high-resolution brain metabolite mapping on a clinical 3T MRI by accelerated H-1-FID-MRSI and low-rank constrained reconstruction

Sébastien Courvoisier, Matthieu Guerquin-Kern, Jeffrey Adam Kasten, Michel Kocher, François Lazeyras, Dimitri Nestor Alice Van De Ville

WILEY2019