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Publications of Jean-François Aujol
Result of the query in the list of publications :
6 Articles |
1 - Detecting codimension-two objects in an image with Ginzburg-Landau models.
G. Aubert and J.F. Aujol and L. Blanc-Féraud. International Journal of Computer Vision, 65(1-2): pages 29-42, November 2005.
Keywords : Ginzburg-Landau model, Point Detection, Segmentation, PDE, Biological images, SAR Images.
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Abstract :
In this paper, we propose a new mathematical model for detecting in an image singularities of codimension greater than or equal to two. This means we want to detect points in a 2-D image or points and curves in a 3-D image. We drew one's inspiration from
Ginzburg-Landau (G-L) models which have proved their efficiency for modeling many phenomena in physics. We introduce the model, state its
mathematical properties and give some experimental results demonstrating its capability in image processing. |
|
2 - Dual Norms and Image Decomposition Models.
J.F. Aujol and A. Chambolle. International Journal of Computer Vision, 63(1): pages 85-104, June 2005.
Keywords : Image decomposition.
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3 - Modeling very Oscillating Signals. Application to Image Processing.
G. Aubert and J.F. Aujol. Applied Mathematics and Optimization, 51(2): pages 163--182, March 2005.
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4 - Optimal Partitions, Regularized Solutions, and Application to Image Classification.
G. Aubert and J.F. Aujol. Applicable Analysis, 84(1): pages 15--35, January 2005.
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5 - Image Decomposition into a Bounded Variation Component and an Oscillating Component.
J.F. Aujol and G. Aubert and L. Blanc-Féraud and A. Chambolle. Journal of Mathematical Imaging and Vision, 22(1): pages 71--88, January 2005.
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6 - Wavelet-based Level Set Evolution for Classification of Textured Images.
J.F. Aujol and G. Aubert and L. Blanc-Féraud. IEEE Trans. Image Processing, 12(12), 2003.
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|
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PhD Thesis and Habilitation |
1 - Contribution à l'Analyse de Textures en Traitement d'Images par Méthodes Variationnelles et Equations aux Dérivées Partielles.
J.F. Aujol. PhD Thesis, Universite de Nice Sophia Antipolis, June 2004.
Keywords : Image decomposition, Classification, Restoration, Fonctional analysis, Bounded Variation Space, Sobolev space.
@PHDTHESIS{JFAujol,
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Résumé :
Cette thèse est un travail en mathématiques appliquées. Elle aborde quelques problèmes en analyse d'images et utilise des outils mathématiques spécifiques.
L'objectif des deux premières parties de cette thèse est de proposer un modèle pour décomposer une image f'en trois composantes : f=u+v+w. Notre approche repose sur l'utilisation d'espaces mathématiques adaptés à chaque composante: l'espace BV des fonctions à variations bornées pour u, un espace G'proche du dual de BV pour les textures, et un espace de Besov d'exposant négatif E'pour le bruit. Nous effectuons l'étude mathématique complète des différents modèles que nous proposons. Nous illustrons notre approche par de nombreux exemples.Dans la troisième et dernière partie de cette thèse, nous nous intéressons spécifiquement à la composante texturée. Nous proposons un algorithme de classification supervisée pour les images texturées. |
Abstract :
This Ph.D. thesis is a work in applied mathematics. It deals with image processing problems, and uses specific mathematical tools.
The aim of the two first parts is to propose a model for decomposing an image f'into three components : f=u+v+w. Our approach relies on the use of mathematical spaces adapted to each component : the space BV of functions with bounded variations for u, a space G'close to the dual space of BV for v, and a negative Besov space E'for w. We carry out the complete mathematical analysis of the different models we propose. We illustrate our approach with many numerical examples. In the third and last part, we only deal with the texture component of an image. We propose a supervised classification algorithm for textured images. |
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4 Conference articles |
1 - Simultaneous structure and texture compact representation.
J.F. Aujol and B. Matei. In Proc. Advanced Concepts for Intelligent Vision Systems, Brussels, Belgium, September 2004.
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2 - Décomposition D'images: Application Aux Images RSO.
J.F. Aujol and G. Aubert and L. Blanc-Féraud and A. Chambolle. In Proc. GRETSI Symposium on Signal and Image Processing, Paris, France, September 2003.
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3 - Wavelet-Based Level Set Evolution for Classification of Textured Images.
J.F. Aujol and G. Aubert and L. Blanc-Féraud. In Proc. IEEE International Conference on Image Processing (ICIP), Barcelona, Spain, September 2003.
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4 - Decomposing an Image: Application to SAR Images.
J.F. Aujol and G. Aubert and L. Blanc-Féraud and A. Chambolle. In Proc. Scale-Space, Vol. 2695, series Lecture No, June 2003.
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7 Technical and Research Reports |
1 - Detecting Codimension-two Objects in an Image with Ginzburg-Landau Models.
G. Aubert and J.F. Aujol and L. Blanc-Féraud. Research Report 5254, INRIA, France, July 2004.
Keywords : Ginzburg-Landau model, Biological images, Segmentation, Partial differential equation.
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Résumé :
Dans cet article, nous proposons a nouveau modèle mathématique pour détecter dans une image les singularités de codimension supérieure ou égale à deux. Cela signifie que nous voulons détecter des points dans des images 2-D, ou des points et des courbes dans des images 3-D. Nous nous inspirons des modèles de Ginzburg-Landau (GL). Ces derniers se sont révélés efficace pour modéliser de nombreux phénomènes physiques. Nous introduisons le modèle, nous énonçons ses propriétés mathématiques, et nous donnons des résultats expérimentaux illustrant les performances du modèle. |
Abstract :
In this paper, we propose a new mathematical model for detecting in an image singularities of codimension greater than or equal to two. This means we want to detect points in a 2-D image or points and curves in a 3-D image. We drew one's inspiration from Ginzburg-Landau (G-L) models which have proved their efficiency for modeling many phenomena in physics. We introduce the model, state its mathematical properties and give some experimental results demonstrating its capability. |
|
2 - Dual Norms and Image Decomposition Models.
J.F. Aujol and A. Chambolle. Research Report 5130, INRIA, France, March 2004.
Keywords : Total variation, Bounded Variation Space, Image decomposition.
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Résumé :
Inspiré par [16], de nombreux modèles de décomposition d'images en une composante géométrique et une composante texturée ont été proposés en traitement d'images. Dans de telles approches, les normes d'espaces de Sobolev d'exposant négatif ont paru intéressantes pour modéliser les éléments oscillants. Dans ce papier, nous comparons les propriétés de différentes normes qui sont duales de normes de Sobolev ou de Besov. Nous proposons ensuite un modèle de décomposition qui sépare une image en deux composantes, une première contenant les structures de l'image, une seconde les textures de l'image, et une troisième le bruit. Notre modèle de décomposition repose sur l'utilisation de trois semi-normes différentes: la variation totale pour la composante géométrique, une norme de Sobolev négative pour la texture, et une norme de Besov négative pour le bruit. Nous illustrons notre étude par des exemples numériques. |
Abstract :
Following [16], decomposition models into a geometrical component and a textured component have recently been proposed in image processing. In such approaches, negative Sobolev norms have seemed to be useful to modelize oscillating patterns. In this paper, we compare the properties of various norms that are dual of Sobolev or Besov norms. We then propose a decomposition model which splits an image into three components: a first one containing the structure of the image, a second one the texture of the image, and a third one the noise. Our decomposition model relies on the use of three different semi-norms: the total variation for the geometrical componant, a negative Sobolev norm for the texture, and a negative Besov norm for the noise. We illustrate our study with numerical examples. |
|
3 - Structure and Texture Compression.
J.F. Aujol and B. Matei. Research Report 5076, INRIA, France, January 2004.
Keywords : Bounded Variation Space, Image decomposition, Texture, Structure.
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Résumé :
Dans ce papier, nous nous intéressons au problème de la compression d'image. Les ondelettes se sont révélées être un outil particulièremment efficace . Récemment, de nombreux algorithmes ont été proposés pour amméliorer la compression par ondelettes en essayant de prendre en compte les strucutres présentes dans l'image. De telles méthodes se révèlents très efficaces pour les images géométriques. Nous construisons un algorithme de compression d'images qui prend en compte la géométrie de l'image tout en étant capable d'être performant sur des images contenant à la fois des structures et des textures. Pour cela, nous utilisons un algorithme de décomposition d'image récemment introduit dans . Cet algorithme permet de séparer une image en deux composantes, une première composante contenant l'information géométrique de l'image, et une deuxième contenant les éléments oscillants de l'image. L'idée de notre méthode de compression est la suivante. Nous commen ons par décomposer l'image à compresser en sa partie géométrique et sa partie oscillante. Nous effectuons ensuite la compression de la partie géométrique à l'aide de l'algorithme introduit dans , ce dernier étant particulièrement bien adapté pour la compression des structures d'une image. Pour la partie oscillante de l'image, nous utilisons l'algorithme classique de compression par ondelettes biorthogonales. sur les zones régulières d'une image). l'image. Notre nouvel algorithme de compression s'avère plus performant que la méthode classique par ondelettes biorthogonales. meilleurs à la fois en PSNR, et aussi visuellement (les bords sont plus précis et les textures sont mieux conservées). |
Abstract :
In this paper, we tackle the problem of image compression. During the last past years, many algorithms have been proposed to take advantage of the geometry of the image. We intend here to propose a new compression algorithm which would take into account the structures in the image, and which would be powerful even when the original image has some textured areas. To this end, we first split our image into two components, a first one containing the structures of the image, and a second one the oscillating patterns. We then perform the compression of each component separately. Our final compressed image is the sum of these two compressed components. This new compression algorithm outperforms the standard biorthogonal wavelets compession. |
|
4 - Modeling very Oscillating Signals : Application to Image Processing.
G. Aubert and J.F. Aujol. Research Report 4878, INRIA, France, July 2003.
Keywords : Bounded Variation Space, Sobolev space, Image decomposition, Optimization, Partial differential equation.
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Résumé :
Cet article complète le travail présenté dans cite{Aujol[3]} dans lequel nous avions développé l'analyse numérique d'un modéle variationnel, initialement introduit par L. Rudin, S. Osher and E. Fatemi cite{Rudin[1]}, et revisité depuis par Y. Meyer cite{Meyer[1]}, pour supprimer le bruit et isoler les textures dans une image. Dans un tel modèle, on décompose l'image f en deux composantes (u+v), u et v minimisant une énergie. La première composante u appartient à BV et contient l'information géométrique de l'image, alors que la seconde v appartient à un espace G qui contient les signaux à fortes oscillations, i.e. le bruit et les textures. Dans cite{Meyer[1]}, Y. Meyer effectue son étude dans ^2 entier, et son approche repose principalement sur des outils d'analyse harmonique. Nous nous pla ons dans le cas d'un ouvert borné de ^2, ce qui constitue le cadre adapté au traitement d'images, et notre approche repose sur des arguments d'analyse fonctionnelle. Nous définissons l'espace G dans ce cadre puis donnons quelques unes de ses propriétés. Nous étudions ensuite la fonctionnelle permettant de calculer les composantes u et v. |
Abstract :
This article is a companion paper of a previous work cite{Aujol[3]} where we have developed the numerical analysis of a variational model first introduced by L. Rudin, S. Osher and E. Fatemi cite{Rudin[1]} and revisited by Y. Meyer cite{Meyer[1]} for removing the noise and capturing textures in an image. The basic idea in this model is to decompose f into two components (u+v) and then to search for (u,v) as a minimizer of an energy functional. The first component u belongs to BV and contains geometrical informations while the second one v is sought in a space G which contains signals with large oscillations, i.e. noise and textures. In Y. Meyer carried out his study in the whole ^2 and his approach is rather built on harmonic analysis tools. We place ourselves in the case of a bounded set of ^2 which is the proper setting for image processing and our approach is based upon functional analysis arguments. We define in this context the space G, give some of its properties and then study in this continuous setting the energy functional which allows us to recover the components u and v. model signals with strong oscillations. For instance, in an image, this space models noises and textures. case of a bounded open set of ^2 which is the proper setting for image processing. We give a definition of G adapted to our case, and we show that it still has good properties to model signals with strong oscillations. In cite{Meyer[1]}, the author had also paved the way to a new model to decompose an image into two components: one in BV (the space of bounded variations) which contains the geometrical information, and one in G which consists in the noises ad the textures. An algorithm to perform this decomposition has been proposed in cite{Meyer[1]}. We show here its relevance in a continuous setting. |
|
5 - Image Decomposition : Application to Textured Images and SAR Images.
J.F. Aujol and G. Aubert and L. Blanc-Féraud and A. Chambolle. Research Report 4704, INRIA, France, January 2003.
Keywords : Total variation, Bounded Variation Space, Texture, Classification, Restoration, Synthetic Aperture Radar (SAR).
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Résumé :
Dans ce rapport, nous présentons un nouvel algorithme pour décomposer une imagef en u+v, u étant à variation bornée, et v contenant les textures et le bruit de l'image originale. Nous introduisons une fonctionnelle adaptée à ce problème. Le minimum de cette fonctionnelle correspond à la décomposition cherchée de l'image. Le calcul de ce minimum se fait par minimisation successive par rapport à chacune des variables, chaque minimisati- on étant réalisée à l'aide d'un algorithme de projection. Nous faisons l'étude théorique de notre modèle, et nous présentons des résultats numériques. D'une part, nous montrons comment la composante v peut être utilisée pour faire de la classification d'images texturées, et d'autre part nous montrons comment la composante u peut être utilisée en restauration d'images SAR. |
Abstract :
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. |
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6 - Supervised Classification for Textured Images.
J.F. Aujol and G. Aubert and L. Blanc-Féraud. Research Report 4640, Inria, France, November 2002.
Keywords : Texture, Classification, Wavelets, Partial differential equation, Level sets.
@TECHREPORT{4640,
|
| author |
= |
{Aujol, J.F. and Aubert, G. and Blanc-Féraud, L.}, |
| title |
= |
{Supervised Classification for Textured Images}, |
| year |
= |
{2002}, |
| month |
= |
{November}, |
| institution |
= |
{Inria}, |
| type |
= |
{Research Report}, |
| number |
= |
{4640}, |
| address |
= |
{France}, |
| url |
= |
{https://hal.inria.fr/inria-00071945}, |
| pdf |
= |
{https://hal.inria.fr/file/index/docid/71945/filename/RR-4640.pdf}, |
| ps |
= |
{https://hal.inria.fr/docs/00/07/19/45/PS/RR-4640.ps}, |
| keyword |
= |
{Texture, Classification, Wavelets, Partial differential equation, Level sets} |
| } |
Résumé :
Dans ce rapport, nous présentons un modèle de classification supervisée basé sur une approche variationnelle. Ce modèle s'applique spécifiquement aux images texturées. Nous souhaitons obtenir une partition optimale de l'image constituée de textures séparées par des interfaces régulières. Pour cela, nous représentons les régions définies par les classes ainsi que leurs interfaces par des fonctions d'ensemble de niveaux. Nous définissons une fonctionnelle sur ces ensembles de niveaux dont le minimum est une partition optimale. Cette fonctionnelle comporte en particulier un terme d'attache aux données spécifique aux textures. Nous utilisons une transformée en paquets d'ondelettes pour analyser les textures, ces dernières étant caractérisées par la distribution de leur énergie dans chaque sous-bande de la décompositon. Les équations aux dérivées partielles (EDP) relatives à la minimisation de la fonctionnelle sont couplées et plongées dans un schéma dynamique. En fixant un ensemble de niveaux initial, les différents termes des EDP guident l'évolution des interfaces (ensemble de niveau zéro) vers les frontières de la partion optimale, par le biais de forces externes (régularité de l'interface) et internes (attache aux données et contraintes partition). Nous avons effectué des tests sur des images synthétiques et sur des images réelles. |
Abstract :
In this report, we present a supervised classification model based on a variational approach. This model is specifically devoted to textured images. We want to get an optimal partition of an image which is composed of textures separated by regular interfaces. To reach this goal, we represent the regions defined by the classes as well as their interfaces by level set functions. We define a functional on these level sets whose minimizers define an optimal partition. In particular, this functional owns a data term specific to textures. We use a packet wavelet transform to analyze the textures, these ones being characterized by their energy distribution in each sub-band of the decomposition. The partial differential equations (PDE) related to the minimization of the functional are embeded in a dynamical scheme. Given an initial interface set (zero level set), the different terms of the PDE's govern the motion of interfaces such that, at convergence, we get an optimal partition as defined above. Each interface is guided by external forces (regularity of the interface), and internal ones (data term and partition constraints). We have conducted several experiments on both synthetic and real images. |
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7 - Signed Distance Functions and Viscosity Solutions of Discontinuous Hamilton-Jacobi Equations.
J.F. Aujol and G. Aubert. Research Report 4507, Inria, France, July 2002.
Keywords : Partial differential equation, Signed distance function, Hamilton-Jacobi equation, Skeleton.
@TECHREPORT{4507,
|
| author |
= |
{Aujol, J.F. and Aubert, G.}, |
| title |
= |
{Signed Distance Functions and Viscosity Solutions of Discontinuous Hamilton-Jacobi Equations}, |
| year |
= |
{2002}, |
| month |
= |
{July}, |
| institution |
= |
{Inria}, |
| type |
= |
{Research Report}, |
| number |
= |
{4507}, |
| address |
= |
{France}, |
| url |
= |
{https://hal.inria.fr/inria-00072081}, |
| pdf |
= |
{https://hal.inria.fr/file/index/docid/72081/filename/RR-4507.pdf}, |
| ps |
= |
{https://hal.inria.fr/docs/00/07/20/81/PS/RR-4507.ps}, |
| keyword |
= |
{Partial differential equation, Signed distance function, Hamilton-Jacobi equation, Skeleton} |
| } |
Résumé :
Dans ce travail, nous commençons par revoir quelques propriétés de la fonction distance signée. En particulier, nous examinons le squelette d'une courbe de ^2, et nous obtenons une description complète de sa fermeture. Nous donnons aussi une condition suffisante pour que l'adhérence du squelette soit de mesure de Lebesgue nulle. Nous menons alors une étude complète de l'EDP: du/dt +sign(u_0(x))(|Du|-1)=0 , cette dernière étant reliée étroitement à la fonction distance signée. Les articles spécialisés ne fournissent pas de résultats mathématiques pour ce genre d'EDP. En effet, nous sommes confrontés à un Hamiltonien discontinu. Nous nous intéressons ensuite à une classe d'EDP plus générale: du/dt +sign(u_0(x))H(Du)=0 , où H est un opérateur convexe. En se plaçant dans le cadre d'hypothèses techniques raisonnables, nous obtenons le même genre de résultats que précédemment. A notre connaissance, il s'agit de résultats nouveaux pour des opérateurs hamiltoniens discontinus. |
Abstract :
In this paper, we first review some properties of the signed distance function. In particular, we examine the skeleton of a curve in ^2 and get a complete description of its closure. We also give a sufficient condition for the closure of the skeleton to be of zero Lebesgue's measure. We then make a complete study of the PDE: du/dt +sign(u_0(x))(|Du|-1)=0 , which is closely related to the signed distance function. The existing literature provides no mathematical results for such PDEs. Indeed, we face the difficulty of considering a discontinuous Hamiltonian operator with respect to the space variable. We state an existence and uniqueness theorem, giving in particular an explicit Hopf-Lax formula for the solution as well as its asymptotic behaviour. This generalizes classical results for continous Hamitonian. We then get interested in a more general class of PDEs: du/dt +sign(u_0(x))H(D- u)=0, with H convex Under some technical but reasonable assumptions, we obtain the same kind of results. As far as we know, they are new for discontinuous Hamiltonians. |
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