François Malgouyres

SEMINAIRE DU 2 JUIN 2016 – 15H @ LTCI – C49

Optimisation de transformée rapide structurée en arbre convolutionnel

Je commencerai par une introduction à la réduction de dimension en traitement d’images et en résolution de problèmes inverses. Cette présentation débute par l’approximation non-linéaire puis l’échantillonnage compressé. Je poursuivrai en présentant les principes de l’apprentissage de dictionnaire et nos travaux récents sur l’optimisation de transformée rapide.

L’ORATEUR

François Malgouyres est Professeur des Universités à l’Institut de Mathématiques de Toulouse.

Rémy Abergel

SEMINAIRE DU 2 juin 2016 – 16H @ LTCI – C49

The Shannon total variation

In image processing problems, the minimization of total variation (TV) based energies requires discretization schemes, such as the commonly used finite differences approach. Unfortunately, such schemes generally lead to images which are difficult to interpolate at sub-pixel scales, which can be extremely problematic for subsequent processing. In this talk, we study a Fourier-based estimate called Shannon total variation (STV), which behaves much better in terms of sub-pixel accuracy. We will first explain how the STV regularization can be efficiently handled with modern dual algorithms, and show that replacing the classical discrete TV model by this STV variant does not raise any theoretical or numerical difficulties. We will then consider many classical TV-based restoration models, such as image denoising (Rudin, Osher and Fatemi), image deblurring and spectrum extrapolation (Guichard-Malgouyres), where the improved behavior of the Shannon total variation yields images that are easy to interpolate. Lastly, we will propose a new STV-regularized optimization problem (involving a data-fidelity term formulated in the frequency domain), which can be used to remove aliasing from an image, or given an image which is difficult to interpolate, can produce a visually similar image which can be easily interpolated. The experimental results that we provide show the interesting perspectives opened by this model in applications where the correct sampling of the data (according to the Shannon sampling theory) is carefully taken into account.

L’ORATEUR

Rémy Abergel est doctorant au MAP5, Université Paris Descartes.

Gabriel Peyré

SéMINAIRE DU 7 AVRIL 2016 – 15H @ LTCI – C49

Transport optimal numérique et applications

Le transport optimal s’est maintenant imposé comme un outil fondamental pour analyser différents problèmes théoriques à l’interface entre le calcul des variations, les équations aux dérivées partielles et les probabilités. Il a cependant fallu plus de temps pour cette notion devienne également incontournable dans les applications numériques. Ceci est en grande partie du au cout du calcul liés aux problèmes d’optimisation correspondants. On commence cependant à voir émerger de nombreuses applications dans des champs aussi divers que l’astrophysique, la vision par ordinateur, l’infographie, le traitement d’image, les statistiques et l’apprentissage. Dans cet exposé, je ferai un tour d’horizon d’une classe de méthodes rapides pour l’approximation de fonctionnelles de transport optimal. Ces méthodes utilisent une régularisation entropique des problèmes variationnels considérés pour approcher la solution à l’aide d’algorithmes rapides et parallelisable. Ils permettent pour la première fois de calculer des barycentres au sens de la distance de transport d’un grand nombre de densités définis sur des gilles de millions de points. Ceci offre de nouvelles perspectives pour l’application du transport optimal en apprentissage (classification d’histogrammes d’images ou de mots) et en infographie (transfert de couleurs, de textures et de formes). Ces algorithmes permettent aussi de calculer des flots gradients pour la métrique de transport, permettant par exemple d’approximer avec un algorithme rapide le modèle de mouvement de foules proposé par Maury, Roudneff Chupin et Santambrogio. Il s’agit d’un travail en collaboration avec J.D. Benamou, G. Carlier, M. Cuturi et L. Nenna.

L’Orateur

Gabriel Peyré est Directeur de Recherche CNRS au CEREMADE, Université Paris-Dauphine. Il fait aussi partie de l’équipe-projet MOKAPLAN.

Blanche Buet

SEMINAIRE DU 7 avril 2016 – 16H @ LTCI – C49

Approximation de surfaces par des varifolds discrets

Il existe de très nombreuses façons de représenter et discrétiser une courbe ou une surface, en raison notamment des applications envisagées et des modes d’acquisitions des données (nuages de points, approximations volumiques, triangulations…). Le but de cet exposé sera de présenter un cadre commun pour l’approximation des surfaces, dans l’esprit de la théorie géométrique de la mesure, à travers la notion de varifold discret.
Ce cadre nous a notamment permis de dégager une notion de courbure moyenne discrète (à une échelle donnée) unifiée dont on présentera les propriétés de convergence et qu’on illustrera numériquement sur des nuages de points.

L’Orateur

Blanche Buet est maîtresse de conférences dans le groupe d’Analyse Harmonique du Laboratoire de Mathématiques d’Orsay (Université Paris Sud, France).

Charles Kervrann

SEMINAIRE DU 3 DÉCEMBRE 2015 – 15H @ LTCI

PEWA: Patch-based Exponentially Weighted Aggregation for image denoising

Patch-based methods have been widely used for noise reduction in recent years. In this talk, we present a general statistical aggregation method which combines image patches denoised with several commonly-used algorithms. We show that weakly denoised versions of the input image obtained with standard methods, can serve to compute an efficient patch-based aggregated estimator. In our approach, we evaluate the Stein’s Unbiased Risk Estimator (SURE) of each denoised candidate image patch and use this information to compute the exponential weighted aggregation (EWA) estimator. The aggregation method is flexible enough to combine any standard denoising algorithm and has an interpretation with Gibbs distribution. The denoising algorithm (PEWA) is based on a MCMC sampling and is able to produce results that are comparable to the current state of-the-art. The methodology is applied to 2D, 2D+time and 3D images corrupted by white Gaussian noise and can be adapted to remove mixed Poisson-Gaussian noise in fluorescence microscopy and live cell imaging.

L’ORATEUR

Charles Kervrann est Directeur de Recherche à l’INRIA où il dirige l’équipe SERPICO.

Référence Bibliographique

C. Kervrann. « PEWA: Patch-based Exponentially Weighted Aggregation for image denoising. » Proc. Neural Information Processing Systems (NIPS’14), Montreal, Canada, Décembre 2014. [preprint][article]

Emmanuel Soubies

SEMINAIRE DU 3 décembre 2015 – 16H @ LTCI

The Continuous Exact \(\ell_0\) (CEL0) penalty: An alternative to \(\ell_0\)-norm.

Many signal/image processing applications are concerned with sparse estimation/recovery such as compressed sensing, source separation, variable selection, image separation among many others. Usually, such a sparsity prior is modeled using the « \(\ell_0\)-pseudo norm » counting the nonzero entries of a given vector. This leads to nonconvex optimization problems which are well-known to be NP-hard.
After an introduction on existing solutions to find a good approximate solution of this problem, such that \(\ell_1\) relaxation or greedy algorithms, we will focus on nonconvex continuous penalties approximating the \(\ell_0\)-norm for the \(\ell_0\) regularized least squares problem. Within this framework, we will present the Continuous Exact \(\ell_0\) penalty (CEL0), an approximation of the \(\ell_0\) norm leading to a tight continuous relaxation of the \(\ell_2 – \ell_0\) criteria and equal to its convex-hull when the linear operator, in the quadratic term, is orthogonal. Moreover, for any linear operator, global minimizers of \(\ell_2\)-CEL0 contain those of the \(\ell_0\) penalized least-squares functional. We will also show that from each local minimizer of this relaxed functional, one can easily extract a local minimizer for \(\ell_2 – \ell_0\) while the reciprocal is false and some local minimizers of the initial functional are eliminated with \(\ell_2\)-CEL0. Hence, the CEL0 functional provides a good continuous alternative to the \(\ell_2 – \ell_0\) criteria since it is continuous and convex with respect to each variable. Finally, recent nonsmooth nonconvex algorithms are used to address this relaxed problem within a macro algorithm ensuring the convergence to a point which is both a critical point of \(\ell_2\)-CEL0 and a (local) minimizer of the initial \(\ell_2 – \ell_0\) problem.

L’ORATEUR

Emmanuel Soubies est en thèse dans l’équipe Morphème à l’INRIA. Il est encadré par Laure Blanc-Féraud et Sébastien Schaub.

Référence Bibliographique

Emmanuel Soubies, Laure Blanc-Féraud, Gilles Aubert. « A Continuous Exact l0 penalty (CEL0) for least squares regularized problem. » SIAM J. on Imaging Science (SIIMS), SIAM, 2015, 8 (3), pp. 1607-1639 (33 p.). <doi-10.1137/151003714> <hal-01102492v2>

Jalal Fadili

SEMINAIRE DU 1 OCTOBRE 2015 – 15H @ LTCI – SALLE C49

Finite identification and local linear convergence of proximal splitting algorithms.

Convex nonsmooth optimization has become ubiquitous in most quantitative disciplines of science. Proximal splitting algorithms are very popular to solve structured convex optimization problems. Within these algorithms, the Forward-Backward and its variants (e.g. inertial FB, FISTA, Tseng’s FBF), Douglas-Rachford and ADMM are widely used. The goal of this work is to investigate the local convergence behavior of these schemes when the involved functions are partly smooth relative to the associated active manifolds. In particular, we show that (i) all the aforementioned splitting algorithms correctly identify the active manifolds in a finite number of iterations (finite activity identification), and (ii) then enter a local linear convergence regime, which we characterize precisely in terms of the structure of the involved active manifolds. For problems involving quadratic and polyhedral functions, we show how to get finite termination of Forward-Backward-type splitting. These results may have numerous applications including in signal/image processing, sparse recovery and machine learning. Indeed, the obtained results explain the typical behaviour that has been observed numerically for many problems in these fields such as the Lasso, the group Lasso, the fused Lasso and the nuclear norm regularization to name only a few.

L’Orateur

Jalal Fadili est Professeur de traitement d’images et du signal au ENSICAEN  et membre junior de l’Institut Universitaire de France.

Alasdair Newson

SEMINAIRE DU 1 octobre 2015 – 16H @ LTCI – Salle C49

Low-Rank Video Segmentation for Background Estimation, and Multi-Temporal Foreground Detection in Videos

Background and foreground separation in videos is a ubiquitous task in many computer vision problems such as object recognition to tracking, and consequently much work has been done on this subject.
Recently, Candès et al. showed that such an estimation task could be addressed as a convex optimisation problem. The framework which they proposed is called Robust Principle Component Analysis (RPCA), and separates a matrix into the sum of a low-rank matrix (the background) and a sparse matrix (the foreground). The low-rank nature of the first matrix means that global lighting changes are handled. In this talk, we adapt and improve upon this framework in two ways.

Firstly, we propose an algorithm to achieve a « local » version of RPCA. Indeed, a considerable drawback of the standard RPCA is its poor ability to handle local lighting changes. We propose to model the background as piece-wise low-rank, each « piece » corresponding to spatio-temporally localised lighting conditions. The main challenge in this task is to identify the coherent regions, which we do with a region-merging approach based on spectral (graph) clustering. We show that this local RPCA allows for greater robustness to both local lighting changes and foreground elements which may remain static for a certain time.

Secondly, we present an online version of RPCA for the task of detecting foreground at multiple timescales. The goal of this analysis is to identify objects which remain in a certain position for a (user-defined) timescale. With visual examples, we show that this algorithm can be used for applications such as detecting the fluidity of traffic or detecting immobile people who may require assistance.

THE SPEAKER

Aladair Newson is currently a postdoctoral researcher at Université Paris-Descartes.

Laurent Condat

SEMINAIRE DU 7 mai 2015 – 15H @ LTCI – Amphi Saphir

Primal-dual forward-backward splitting for large-scale convex optimization

A wide array of estimation and restoration problems, in particular inverse imaging problems, can be formulated as large-scale convex optimization problems in Hilbert spaces: the goal is to minimize a sum of convex functions, possibly composed with linear operators. The forward-backward splitting technique, when applied in primal-dual product spaces, is a powerful umbrella that encompasses the classical forward-backward, Douglas-Rachford, and Chambolle-Pock algorithms. A useful extension with variable metric is discussed. Some applications in imaging are shown.

Slides  [pdf]

The Speaker

Laurent Condat is a permanent CNRS researcher at GIPSA-lab (Grenoble, France).

Related material

  • L. Condat, “A Generic Proximal Algorithm for Convex Optimization – Application to Total Variation Minimization,” IEEE Signal Proc. Letters, vol. 21, no. 8, pp. 1054-1057, Aug. 2014.  PDF.  Matlab files:  optimization.zip
  • L. Condat, “A primal-dual splitting method for convex optimization involving Lipschitzian, proximable and linear composite terms,” J. Optimization Theory and Applications, vol. 158, no. 2, pp. 460-479, 2013.  PDF
  • L. Condat, “A direct algorithm for 1D total variation denoising,” IEEE Signal Proc. Letters, vol. 20, no. 11, pp. 1054-1057, Nov. 2013.  PDF.  C file:  condat_fast_tv.c

 

Arthur Leclaire

SEMINAIRE DU 7 mai 2015 – 16H @ LTCI – AMPHI SAPHIR

A Texton for Fast and Flexible Gaussian Texture Synthesis

Gaussian textures can be easily simulated by convolving an image sample with a conveniently normalized white noise. However, this procedure is not very flexible (it does not allow for non-uniform grids in particular), and becomes computationally heavy for very large domains. A Gaussian texture can also be approximated by a high-intensity discrete spot noise (DSN), obtained by summing randomly-shifted copies of a kernel along the points of a Poisson process.

In this talk, we propose an algorithm that summarizes a texture sample into a synthesis-oriented texton, that is, a small image for which the DSN simulation is more efficient than the classical convolution algorithm. Using this synthesis-oriented texture summary, Gaussian textures can be generated on-demand in a faster, simpler, and more flexible way.

Slides [pdf]

Related material

Project Page (Synthesis-Oriented Texton)

The Speaker

Arthur Leclaire is finishing his PhD Thesis and working as a young teacher and researcher at Université Paris-Descartes and ENS Cachan.