Publications of Gilles Aubert

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• 15 Articles
• 23 Conference articles
• 15 Technical and Research Reports

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15 Articles

1 - On the Illumination Invariance of the Level Lines under Directed Light: Application to Change Detection. P. Weiss and A. Fournier and L. Blanc-Féraud and G. Aubert. SIAM Journal on Imaging Sciences, 4(1): pages 448-471, March 2011. Keywords : Level Lines, topographic map, illumination invariance, Change detection, contrast equalization, remote sensing. @ARTICLE{SIIMS_2011, author = {Weiss, P. and Fournier, A. and Blanc-Féraud, L. and Aubert, G.}, title = {On the Illumination Invariance of the Level Lines under Directed Light: Application to Change Detection}, year = {2011}, month = {March}, journal = {SIAM Journal on Imaging Sciences}, volume = {4}, number = {1}, pages = {448-471}, url = {http://www.math.univ-toulouse.fr/~weiss/Publis/SIIMS_2011_Weiss.pdf}, pdf = {http://www.math.univ-toulouse.fr/~weiss/Publis/SIIMS_2011_Weiss.pdf}, keyword = {Level Lines, topographic map, illumination invariance, Change detection, contrast equalization, remote sensing} } Abstract : We analyze the illumination invariance of the level lines of an image. We show that if the scene surface has Lambertian reflectance and the light is directed, then a necessary and sufficient condition for the level lines to be illumination invariant is that the three-dimensional scene be developable and that its albedo satisfy some geometrical constraints. We then show that the level lines are “almost” invariant for piecewise developable surfaces. Such surfaces fit most of the urban structures. This allows us to devise a fast and simple algorithm that detects changes between pairs of remotely sensed images of urban areas, independently of the lighting conditions. We show the effectiveness of the algorithm both on synthetic OpenGL scenes and real QuickBird images. The synthetic results illustrate the theory developed in this paper. The two real QuickBird images show that the proposed change detection algorithm is discriminant. For easy scenes it achieves a rate of 85% detected changes for 10% false positives, while it reaches a rate of 75% detected changes for 25% false positives on demanding scenes.

2 - A formal Gamma-convergence approach for the detection of points in 2-D biological images. D. Graziani and G. Aubert and L. Blanc-Féraud. SIAM Journal on Imaging Sciences, 3(3): pages 578-594, September 2010. Keywords : points detection, curvature-depending functionals, divergence-measure fields. @ARTICLE{2, author = {Graziani, D. and Aubert, G. and Blanc-Féraud, L.}, title = {A formal Gamma-convergence approach for the detection of points in 2-D biological images}, year = {2010}, month = {September}, journal = {SIAM Journal on Imaging Sciences}, volume = {3}, number = {3}, pages = {578-594}, url = {http://hal.inria.fr/inria-00503152/}, keyword = {points detection, curvature-depending functionals, divergence-measure fields} } Abstract : We propose a new variational model to locate points in 2-dimensional biological images. To this purpose we introduce a suitable functional whose minimizers are given by the points we want to detect. In order to provide numerical experiments we replace this energy with a sequence of a more treatable functionals by means of the notion of Gamma-convergence.

3 - Variational approximation for detecting point-like target problems. D. Graziani and G. Aubert. COCV: Esaim Control Optimization and Calculus of Variations DOI: 10.1051/cocv/2010029, August 2010. Keywords : points detection, Biological images, divergence-measure fields. @ARTICLE{COCV2010, author = {Graziani, D. and Aubert, G.}, title = {Variational approximation for detecting point-like target problems}, year = {2010}, month = {August}, journal = {COCV: Esaim Control Optimization and Calculus of Variations DOI: 10.1051/cocv/2010029}, url = {http://dx.doi.org/10.1051/cocv/2010029}, keyword = {points detection, Biological images, divergence-measure fields} } Abstract : The aim of this paper is to provide a rigorous variational formulation for the detection of points in 2-d biological images. To this purpose we introduce a new functional whose minimizers give the points we want to detect. Then we define an approximating sequence of functionals for which we prove the Γ-convergence to the initial one.

4 - Efficient schemes for total variation minimization under constraints in image processing. P. Weiss and L. Blanc-Féraud and G. Aubert. SIAM journal on Scientific Computing, 31(3): pages 2047-2080, 2009. Keywords : Total variation, l1 norm, nesterov scheme, Rudin Osher Fatemi, fast optimization, real time. Copyright : Copyright Siam Society for Industrial and Applied @ARTICLE{SIAM_JSC_PWEISS, author = {Weiss, P. and Blanc-Féraud, L. and Aubert, G.}, title = {Efficient schemes for total variation minimization under constraints in image processing}, year = {2009}, journal = {SIAM journal on Scientific Computing}, volume = {31}, number = {3}, pages = {2047-2080}, url = {http://www.math.univ-toulouse.fr/~weiss/Publis/SIAM_JSC09_PWEISS.pdf}, pdf = {http://www.math.univ-toulouse.fr/~weiss/Publis/SIAM_JSC09_PWEISS.pdf}, keyword = {Total variation, l1 norm, nesterov scheme, Rudin Osher Fatemi, fast optimization, real time} }

5 - An approximation of the Mumford-Shah energy by a family of dicrete edge-preserving functionals. G. Aubert and L. Blanc-Féraud and R. March. Nonlinear Analysis, 64: pages 1908-1930, 2006. Keywords : Gamma Convergence, Finite Element, Segmentation. @ARTICLE{laure-na05, author = {Aubert, G. and Blanc-Féraud, L. and March, R.}, title = {An approximation of the Mumford-Shah energy by a family of dicrete edge-preserving functionals}, year = {2006}, journal = {Nonlinear Analysis}, volume = {64}, pages = {1908-1930}, pdf = {ftp://ftp-sop.inria.fr/ariana/Articles/2006_laure-na05.pdf}, keyword = {Gamma Convergence, Finite Element, Segmentation} } Abstract : We show the Gamma-convergence of a family of discrete functionals to the Mumford and Shah image segmentation functional. The functionals of the family are constructed by modifying the elliptic approximating functionals proposed by Ambrosio and Tortorelli. The quadratic term of the energy related to the edges of the segmentation is replaced by a nonconvex functional.

6 - 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. @ARTICLE{laure-ijcv05, author = {Aubert, G. and Aujol, J.F. and Blanc-Féraud, L.}, title = {Detecting codimension-two objects in an image with Ginzburg-Landau models}, year = {2005}, month = {November}, journal = {International Journal of Computer Vision}, volume = {65}, number = {1-2}, pages = {29-42}, pdf = {ftp://ftp-sop.inria.fr/ariana/Articles/GL_IJCV_5.pdf}, keyword = {Ginzburg-Landau model, Point Detection, Segmentation, PDE, Biological images, SAR Images} } 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.

7 - 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. @ARTICLE{AujolAubert, author = {Aubert, G. and Aujol, J.F.}, title = {Modeling very Oscillating Signals. Application to Image Processing}, year = {2005}, month = {March}, journal = {Applied Mathematics and Optimization}, volume = {51}, number = {2}, pages = {163--182}, pdf = {http://www.springerlink.com/media/n9a4b9ftvj6jvlc2ux4g/contributions/5/t/5/4/5t543cht7hn2xchw.pdf}, keyword = {} }

8 - Optimal Partitions, Regularized Solutions, and Application to Image Classification. G. Aubert and J.F. Aujol. Applicable Analysis, 84(1): pages 15--35, January 2005. @ARTICLE{AujolAubertclassif, author = {Aubert, G. and Aujol, J.F.}, title = {Optimal Partitions, Regularized Solutions, and Application to Image Classification}, year = {2005}, month = {January}, journal = {Applicable Analysis}, volume = {84}, number = {1}, pages = {15--35}, keyword = {} }

9 - 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. @ARTICLE{BLA05, author = {Aujol, J.F. and Aubert, G. and Blanc-Féraud, L. and Chambolle, A.}, title = {Image Decomposition into a Bounded Variation Component and an Oscillating Component}, year = {2005}, month = {January}, journal = {Journal of Mathematical Imaging and Vision}, volume = {22}, number = {1}, pages = {71--88}, pdf = {http://springerlink.metapress.com/media/6n99d5dtvj6juld5tw5w/contributions/h/2/0/3/h20366rj1r34567m.pdf}, keyword = {} }

10 - Gamma-convergence of discrete functionals with nonconvex perturbation for image classification. G. Aubert and L. Blanc-Féraud and R. March. SIAM Journal on Numerical Analysis, 12(3): pages 1128--1145, 2004. @ARTICLE{BLA04, author = {Aubert, G. and Blanc-Féraud, L. and March, R.}, title = {Gamma-convergence of discrete functionals with nonconvex perturbation for image classification}, year = {2004}, journal = {SIAM Journal on Numerical Analysis}, volume = {12}, number = {3}, pages = {1128--1145}, keyword = {} }

11 - 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. @ARTICLE{aujolGL, author = {Aujol, J.F. and Aubert, G. and Blanc-Féraud, L.}, title = {Wavelet-based Level Set Evolution for Classification of Textured Images}, year = {2003}, journal = {IEEE Trans. Image Processing}, volume = {12}, number = {12}, pdf = {http://ieeexplore.ieee.org/iel5/83/28122/01257399.pdf?tp=&arnumber=1257399&isnumber=28122}, keyword = {} }

12 - A variational model for image classification and restoration. C. Samson and L. Blanc-Féraud and G. Aubert and J. Zerubia. IEEE Trans. Pattern Analysis ans Machine Intelligence, 22(5): pages 460-472, May 2000. @ARTICLE{cs00, author = {Samson, C. and Blanc-Féraud, L. and Aubert, G. and Zerubia, J.}, title = {A variational model for image classification and restoration}, year = {2000}, month = {May}, journal = {IEEE Trans. Pattern Analysis ans Machine Intelligence}, volume = {22}, number = {5}, pages = {460-472}, keyword = {} }

13 - A Level Set Model for Image Classification. C. Samson and L. Blanc-Féraud and G. Aubert and J. Zerubia. International Journal of Computer Vision, 40(3): pages 187-198, 2000. @ARTICLE{cs00b, author = {Samson, C. and Blanc-Féraud, L. and Aubert, G. and Zerubia, J.}, title = {A Level Set Model for Image Classification}, year = {2000}, journal = {International Journal of Computer Vision}, volume = {40}, number = {3}, pages = {187-198}, keyword = {} }

14 - Some remarks on the equivalence between 2D and 3D classical snakes and geodesic active contours. L. Blanc-Féraud and G. Aubert. International Journal of Computer Vision, 34(1): pages 19-28, September 1999. @ARTICLE{lbf99a, author = {Blanc-Féraud, L. and Aubert, G.}, title = {Some remarks on the equivalence between 2D and 3D classical snakes and geodesic active contours}, year = {1999}, month = {September}, journal = {International Journal of Computer Vision}, volume = {34}, number = {1}, pages = {19-28}, keyword = {} }

15 - Variational approach for edge preserving regularization using coupled PDE's. S. Teboul and L. Blanc-Féraud and G. Aubert and M. Barlaud. IEEE Trans. Image Processing, 7(3): pages 387-397, March 1998. @ARTICLE{lbf98, author = {Teboul, S. and Blanc-Féraud, L. and Aubert, G. and Barlaud, M.}, title = {Variational approach for edge preserving regularization using coupled PDE's}, year = {1998}, month = {March}, journal = {IEEE Trans. Image Processing}, volume = {7}, number = {3}, pages = {387-397}, keyword = {} }

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23 Conference articles

1 - A new variational method for preserving point-like and curve-like singularities in 2d images. D. Graziani and L. Blanc-Féraud and G. Aubert. In Proc. IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), Prague, Czech Republic, May 2011. Keywords : Convex optimization, nesterov scheme, laplacian operator. @INPROCEEDINGS{ICASSP_Graziani11, author = {Graziani, D. and Blanc-Féraud, L. and Aubert, G.}, title = {A new variational method for preserving point-like and curve-like singularities in 2d images}, year = {2011}, month = {May}, booktitle = {Proc. IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)}, address = {Prague, Czech Republic}, url = {http://hal.inria.fr/inria-00592603/fr/}, keyword = {Convex optimization, nesterov scheme, laplacian operator} } Abstract : We propose a new variational method to restore point-like and curve-like singularities in 2-D images. As points and open curves are fine structures, they are difficult to restore by means of first order derivative operators computed in the noisy image. In this paper we propose to use the Laplacian operator of the observed intensity, since it becomes singular at points and curves. Then we propose to restore these singularities by introducing suitable regularization involving the l-1-norm of the Laplacian operator. Results are shown on synthetic an real data.

2 - Detection and tracking of threats in aerial infrared images by a minimal path approach. G. Aubert and A. Baudour and L. Blanc-Féraud and L. Guillot and Y. Le Guilloux. In Proc. IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), Dallas, Texas, USA, March 2010. @INPROCEEDINGS{ICASSP10, author = {Aubert, G. and Baudour, A. and Blanc-Féraud, L. and Guillot, L. and Le Guilloux, Y.}, title = {Detection and tracking of threats in aerial infrared images by a minimal path approach}, year = {2010}, month = {March}, booktitle = {Proc. IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)}, address = {Dallas, Texas, USA}, keyword = {} }

3 - A new variational method to detect points in biological images. D. Graziani and L. Blanc-Féraud and G. Aubert. In ISBI'09, Org. IEEE International Symposium on Biomedical Imaging, Boston, USA, June 2009. Keywords : Biological images, points detection, Gamma-convergence. @INPROCEEDINGS{GRAZIANI_ISBI2009, author = {Graziani, D. and Blanc-Féraud, L. and Aubert, G.}, title = {A new variational method to detect points in biological images}, year = {2009}, month = {June}, booktitle = {ISBI'09}, organization = {IEEE International Symposium on Biomedical Imaging}, address = {Boston, USA}, url = {http://dx.doi.org/10.1109/ISBI.2009.5193301}, keyword = {Biological images, points detection, Gamma-convergence} } Abstract : We propose a new variational method to isolate points in biological images. As points are fine structures they are difficult to detect by derivative operators computed in the noisy image. In this paper we propose to compute a vector field from the observed intensity so that its divergence explodes at points. As the image could contains spots but also noise and curves where the divergence also blows up, we propose to capture spots by introducing suitable energy whose minimizers are given by the points we want to detect. In order to provide numerical experiments we approximate this energy by means of a sequence of more treatable functionals by a Gamma-convergence approach. Results are shown on synthetic and biological images.

4 - A contrast equalization procedure for change detection algorithms: applications to remotely sensed images of urban areas. A. Fournier and P. Weiss and L. Blanc-Féraud and G. Aubert. In International Conference on Pattern Recognition (ICPR), Tampa, USA, December 2008. Keywords : Change detection, Level Lines, remote sensing. Copyright : ©2008 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE. @INPROCEEDINGS{l_lines_icpr08, author = {Fournier, A. and Weiss, P. and Blanc-Féraud, L. and Aubert, G.}, title = {A contrast equalization procedure for change detection algorithms: applications to remotely sensed images of urban areas}, year = {2008}, month = {December}, booktitle = {International Conference on Pattern Recognition (ICPR)}, address = {Tampa, USA}, url = {http://www.math.univ-toulouse.fr/~weiss/Publis/Conferences/icpr2008.pdf}, pdf = {http://www.math.univ-toulouse.fr/~weiss/Publis/Conferences/icpr2008.pdf}, keyword = {Change detection, Level Lines, remote sensing} }

5 - Sur la complexite et la rapidite d’algorithmes pour la minimisation de la variation totale sous contraintes. P. Weiss and L. Blanc-Féraud and G. Aubert. In Proc. Symposium on Signal and Image Processing (GRETSI), Troyes, France, September 2007. Keywords : l1 norm minimization, compression noise denoising, optimal algorithm, convex analysis, Total variation, nesterov scheme. @INPROCEEDINGS{Pierre Weiss, author = {Weiss, P. and Blanc-Féraud, L. and Aubert, G.}, title = {Sur la complexite et la rapidite d’algorithmes pour la minimisation de la variation totale sous contraintes}, year = {2007}, month = {September}, booktitle = {Proc. Symposium on Signal and Image Processing (GRETSI)}, address = {Troyes, France}, url = {http://www.math.univ-toulouse.fr/~weiss/Publis/Conferences/Gretsi_WeissBlancFeraudAubert_2010.PDF}, pdf = {ftp://ftp-sop.inria.fr/ariana/Articles/2007_Pierre Weiss.pdf}, keyword = {l1 norm minimization, compression noise denoising, optimal algorithm, convex analysis, Total variation, nesterov scheme} }

6 - Detection and Completion of Filaments: A Vector Field and PDE Approach. A. Baudour and G. Aubert and L. Blanc-Féraud. In SSVM 2007, LNCS 4485 proceedings, 2007. @INPROCEEDINGS{ssvm2007, author = {Baudour, A. and Aubert, G. and Blanc-Féraud, L.}, title = {Detection and Completion of Filaments: A Vector Field and PDE Approach}, year = {2007}, booktitle = { SSVM 2007, LNCS 4485 proceedings}, pdf = {ftp://ftp-sop.inria.fr/ariana/Articles/2007_ssvm2007.pdf}, keyword = {} }

7 - Détection et Complétion de Filaments: une approche variationelle et vectorielle. A. Baudour and G. Aubert and L. Blanc-Féraud. In Colloque Gretsi Troyes, 2007, 2007. @INPROCEEDINGS{ Gretsi 2007, author = {Baudour, A. and Aubert, G. and Blanc-Féraud, L.}, title = {Détection et Complétion de Filaments: une approche variationelle et vectorielle}, year = {2007}, booktitle = {Colloque Gretsi Troyes, 2007}, pdf = {ftp://ftp-sop.inria.fr/ariana/Articles/2007_ Gretsi 2007.pdf}, keyword = {} }

8 - Image Disocclusion Using a Probabilistic Gradient Orientation. E. Villéger and G. Aubert and L. Blanc-Féraud. In Proc. International Conference on Pattern Recognition (ICPR), Cambridge, United Kingdom, August 2004. @INPROCEEDINGS{Villeger04, author = {Villéger, E. and Aubert, G. and Blanc-Féraud, L.}, title = {Image Disocclusion Using a Probabilistic Gradient Orientation}, year = {2004}, month = {August}, booktitle = {Proc. International Conference on Pattern Recognition (ICPR)}, address = {Cambridge, United Kingdom}, keyword = {} }

9 - A $l^1$-unified variational framework for image restoration. J. Bect and L. Blanc-Féraud and G. Aubert and A. Chambolle. In Proc. European Conference on Computer Vision (ECCV), Vol. LNCS 3024, pages 1--13, Ed. T. Pajdla and J. Matas, Publ. Springer, Prague, Czech Republic, May 2004. @INPROCEEDINGS{eccv04, author = {Bect, J. and Blanc-Féraud, L. and Aubert, G. and Chambolle, A.}, title = {A $l^1$-unified variational framework for image restoration}, year = {2004}, month = {May}, booktitle = {Proc. European Conference on Computer Vision (ECCV)}, volume = {LNCS 3024}, pages = {1--13}, editor = {T. Pajdla and J. Matas}, publisher = {Springer}, address = {Prague, Czech Republic}, keyword = {} }

10 - 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. @INPROCEEDINGS{jf_gretsi, author = {Aujol, J.F. and Aubert, G. and Blanc-Féraud, L. and Chambolle, A.}, title = {Décomposition D'images: Application Aux Images RSO}, year = {2003}, month = {September}, booktitle = {Proc. GRETSI Symposium on Signal and Image Processing}, address = {Paris, France}, keyword = {} }

11 - 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. @INPROCEEDINGS{jf_icip, author = {Aujol, J.F. and Aubert, G. and Blanc-Féraud, L.}, title = {Wavelet-Based Level Set Evolution for Classification of Textured Images}, year = {2003}, month = {September}, booktitle = {Proc. IEEE International Conference on Image Processing (ICIP)}, address = {Barcelona, Spain}, keyword = {} }

12 - Filtering interferometric phase images by anisotropic diffusion. C. Lacombe and G. Aubert and L. Blanc-Féraud and P. Kornprobst. In Proc. IEEE International Conference on Image Processing (ICIP), Barcelona, September 2003. @INPROCEEDINGS{ClacombGALBFPK, author = {Lacombe, C. and Aubert, G. and Blanc-Féraud, L. and Kornprobst, P.}, title = {Filtering interferometric phase images by anisotropic diffusion}, year = {2003}, month = {September}, booktitle = {Proc. IEEE International Conference on Image Processing (ICIP)}, address = {Barcelona}, keyword = {} }

13 - 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. @INPROCEEDINGS{jf_scalespace, author = {Aujol, J.F. and Aubert, G. and Blanc-Féraud, L. and Chambolle, A.}, title = {Decomposing an Image: Application to SAR Images}, year = {2003}, month = {June}, booktitle = {Proc. Scale-Space}, volume = {2695}, series = {Lecture No}, keyword = {} }

14 - Diffusion anisotrope et filtrage interférométrique. C. Lacombe and G. Aubert and L. Blanc-Féraud and P. Kornprobst. In Congrès National d'Analyse Numérique, Montpellier, June 2003. @INPROCEEDINGS{Lacombe-Canum03, author = {Lacombe, C. and Aubert, G. and Blanc-Féraud, L. and Kornprobst, P.}, title = {Diffusion anisotrope et filtrage interférométrique}, year = {2003}, month = {June}, booktitle = {Congrès National d'Analyse Numérique}, address = {Montpellier}, keyword = {} }

15 - Filtrage adaptatif des interférogrammes par diffusion anisotrope. C. Lacombe and G. Aubert and L. Blanc-Féraud and P. Kornprobst. In Proc. Journées des jeunes chercheurs en vision par ordinateur, Gerardmer, May 2003. @INPROCEEDINGS{CLGALBFPK, author = {Lacombe, C. and Aubert, G. and Blanc-Féraud, L. and Kornprobst, P.}, title = {Filtrage adaptatif des interférogrammes par diffusion anisotrope}, year = {2003}, month = {May}, booktitle = {Proc. Journées des jeunes chercheurs en vision par ordinateur}, address = {Gerardmer}, keyword = {} }

16 - A variational approach to one dimensional phase unwrapping. C. Lacombe and P. Kornprobst and G. Aubert and L. Blanc-Féraud. In Proc. International Conference on Pattern Recognition (ICPR), Québec, Canada, August 2002. @INPROCEEDINGS{lacombekronp, author = {Lacombe, C. and Kornprobst, P. and Aubert, G. and Blanc-Féraud, L.}, title = {A variational approach to one dimensional phase unwrapping}, year = {2002}, month = {August}, booktitle = {Proc. International Conference on Pattern Recognition (ICPR)}, address = {Québec, Canada}, keyword = {} }

17 - Two variational models for multispectral image classification. C. Samson and L. Blanc-Féraud and G. Aubert and J. Zerubia. In Proc. Energy Minimization Methods in Computer Vision and Pattern Recognition (EMMCVPR), Sophia Antipolis, France, September 2001. @INPROCEEDINGS{lbf01a, author = {Samson, C. and Blanc-Féraud, L. and Aubert, G. and Zerubia, J.}, title = {Two variational models for multispectral image classification}, year = {2001}, month = {September}, booktitle = {Proc. Energy Minimization Methods in Computer Vision and Pattern Recognition (EMMCVPR)}, address = {Sophia Antipolis, France}, keyword = {} }

18 - Multiphase evolution and variational image classification. L. Blanc-Féraud and C. Samson and G. Aubert and J. Zerubia. In Congress SIMAI, Ischia, Italie, June 2000. @INPROCEEDINGS{lbf00a, author = {Blanc-Féraud, L. and Samson, C. and Aubert, G. and Zerubia, J.}, title = {Multiphase evolution and variational image classification}, year = {2000}, month = {June}, booktitle = {Congress SIMAI}, address = {Ischia, Italie}, keyword = {} }

19 - Une approche variationnelle pour la classification d'images par régions actives. C. Samson and L. Blanc-Féraud and G. Aubert and J. Zerubia. In Proc. Reconnaissance des formes et Intelligence Artificielle, Paris, France, February 2000. @INPROCEEDINGS{cs00a, author = {Samson, C. and Blanc-Féraud, L. and Aubert, G. and Zerubia, J.}, title = {Une approche variationnelle pour la classification d'images par régions actives}, year = {2000}, month = {February}, booktitle = {Proc. Reconnaissance des formes et Intelligence Artificielle}, address = {Paris, France}, keyword = {} }

20 - A Level Set Model for Image Classification. C. Samson and L. Blanc-Féraud and G. Aubert and J. Zerubia. In Proc. Scale-Space, Corfu, Grèce, September 1999. @INPROCEEDINGS{cs99d, author = {Samson, C. and Blanc-Féraud, L. and Aubert, G. and Zerubia, J.}, title = {A Level Set Model for Image Classification}, year = {1999}, month = {September}, booktitle = {Proc. Scale-Space}, address = {Corfu, Grèce}, keyword = {} }

21 - Simultaneous Image Classification and Restoration Using a Variational Approach. C. Samson and L. Blanc-Féraud and G. Aubert and J. Zerubia. In Proc. IEEE Computer Vision and Pattern Recognition (CVPR), Fort Collins, Colorado, USA, June 1999. @INPROCEEDINGS{cs99c, author = {Samson, C. and Blanc-Féraud, L. and Aubert, G. and Zerubia, J.}, title = {Simultaneous Image Classification and Restoration Using a Variational Approach}, year = {1999}, month = {June}, booktitle = {Proc. IEEE Computer Vision and Pattern Recognition (CVPR)}, address = {Fort Collins, Colorado, USA}, keyword = {} }

22 - Classification et Restauration d'Images par Approche Variationnelle. C. Samson and L. Blanc-Féraud and G. Aubert and J. Zerubia. In Proc. Journées des jeunes chercheurs en vision par ordinateur, Aussois, France, 1999. @INPROCEEDINGS{cs99b, author = {Samson, C. and Blanc-Féraud, L. and Aubert, G. and Zerubia, J.}, title = {Classification et Restauration d'Images par Approche Variationnelle}, year = {1999}, booktitle = {Proc. Journées des jeunes chercheurs en vision par ordinateur}, address = {Aussois, France}, keyword = {} }

23 - Active contour models for segmentation and reconstruction on medical images. S. Teboul and L. Blanc-Féraud and G. Aubert and M. Barlaud. In Asilomar Conference, USA, November 1998. @INPROCEEDINGS{lbf98g, author = {Teboul, S. and Blanc-Féraud, L. and Aubert, G. and Barlaud, M.}, title = {Active contour models for segmentation and reconstruction on medical images}, year = {1998}, month = {November}, booktitle = {Asilomar Conference}, address = {USA}, keyword = {} }

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15 Technical and Research Reports

1 - A formal Gamma-convergence approach for the detection of points in 2-D images. D. Graziani and L. Blanc-Féraud and G. Aubert. Research Report 7038, INRIA, May 2009. Note : to appear Siam Journal of Imaging Science Keywords : points detection, curvature-depending functionals, divergence-measure fields, Gamma-convergence, biological 2-D images. @TECHREPORT{GRAZIANI_GAMMA_POINTS, author = {Graziani, D. and Blanc-Féraud, L. and Aubert, G.}, title = {A formal Gamma-convergence approach for the detection of points in 2-D images}, year = {2009}, month = {May}, institution = {INRIA}, type = {Research Report}, number = {7038}, note = {to appear Siam Journal of Imaging Science}, url = {http://hal.inria.fr/index.php?halsid=vocj692ukdgrhpvo85in6ob5m5&view_this_doc=inria-00418526&version=1}, keyword = {points detection, curvature-depending functionals, divergence-measure fields, Gamma-convergence, biological 2-D images} }

2 - On the illumination invariance of the level lines under directed light. Application to change detection. P. Weiss and A. Fournier and L. Blanc-Féraud and G. Aubert. Research Report 6612, INRIA, 2008. Keywords : Level Lines, illumination invariance, topographic map, Change detection, remote sensing, Urban areas. Copyright : @TECHREPORT{RR-6612, author = {Weiss, P. and Fournier, A. and Blanc-Féraud, L. and Aubert, G.}, title = {On the illumination invariance of the level lines under directed light. Application to change detection}, year = {2008}, institution = {INRIA}, type = {Research Report}, number = {6612}, url = {http://hal.archives-ouvertes.fr/index.php?halsid=atooirf8jo0ggefoedhebmhnr0&view_this_doc=inria-00310383&version=1}, pdf = {http://hal.inria.fr/docs/00/31/03/83/PDF/RR-6612.pdf}, keyword = {Level Lines, illumination invariance, topographic map, Change detection, remote sensing, Urban areas} } Abstract : We analyze the illumination invariance of the level lines of an image. We show that if the scene surface has Lambertian reflectance and the light is directed, then a necessary condition for the level lines to be illumination invariant is that the 3D scene be developable and that its albedo satisfies some geometrical constraints. We then show that the level lines are ``almost'' invariant for piecewise developable surfaces. Such surfaces fit most of the urban structures. In a second part, this allows us to devise a very fast algorithm that detects changes between pairs of remotely sensed images of urban areas, independently of the lighting conditions. We show the effectiveness of the algorithm both on synthetic OpenGL scenes and real Quickbird images. We compare the efficiency of the proposed algorithm with other classical approaches and show that it is superior both in practice and in theory.

3 - Efficient schemes for total variation minimization under constraints in image processing. P. Weiss and L. Blanc-Féraud and G. Aubert. Research Report 6260, INRIA, July 2007. Keywords : l1 norm, total variation minimization, duality lp norms, gradient and subgradient descent, nesterov scheme, texture + geometry decomposition. @TECHREPORT{RR-6260, author = {Weiss, P. and Blanc-Féraud, L. and Aubert, G.}, title = {Efficient schemes for total variation minimization under constraints in image processing}, year = {2007}, month = {July}, institution = {INRIA}, type = {Research Report}, number = {6260}, url = {http://hal.inria.fr/inria-00166096/fr/}, pdf = {http://hal.inria.fr/docs/00/26/16/35/PDF/RR-6260.pdf}, ps = {http://hal.inria.fr/docs/00/26/16/35/PS/RR-6260.ps}, keyword = {l1 norm, total variation minimization, duality lp norms, gradient and subgradient descent, nesterov scheme, texture + geometry decomposition} } Résumé : Ce papier présente de nouveaux algorithmes pour minimiser la variation totale, et plus généralement des normes l^1, sous des contraintes convexes. Ces algorithmes proviennent d'une avancée récente en optimisation convexe proposée par Yurii Nesterov. Suivant la régularité de l'attache aux données, nous résolvons soit un problème primal, soit un problème dual. Premièrement, nous montrons que les schémas standard de premier ordre permettent d'obtenir des solutions de précision epsilon en O(frac1epsilon^2) itérations au pire des cas. Pour une contrainte convexe quelconque, nous proposons un schéma qui permet d'obtenir une solution de précision epsilon en O(frac1epsilon) itérations. Pour une contrainte fortement convexe, nous résolvons un problème dual avec un schéma qui demande O(frac1sqrtepsilon) itérations pour obtenir une solution de précision epsilon. Suivant la contrainte, nous gagnons donc un à deux ordres dans la rapidité de convergence par rapport à des approches standard. Finalement, nous faisons quelques expériences numériques qui confirment les résultats théoriques sur de nombreux problèmes. Abstract : This paper presents new algorithms to minimize total variation and more generally l^1-norms under a general convex constraint. The algorithms are based on a recent advance in convex optimization proposed by Yurii Nesterov citeNESTEROV. Depending on the regularity of the data fidelity term, we solve either a primal problem, either a dual problem. First we show that standard first order schemes allow to get solutions of precision epsilon in O(frac1epsilon^2) iterations at worst. For a general convex constraint, we propose a scheme that allows to obtain a solution of precision epsilon in O(frac1epsilon) iterations. For a strongly convex constraint, we solve a dual problem with a scheme that requires O(frac1sqrtepsilon) iterations to get a solution of precision epsilon. Thus, depending on the regularity of the data term, we gain from one to two orders of magnitude in the convergence rates with respect to standard schemes. Finally we perform some numerical experiments which confirm the theoretical results on various problems.

4 - Some applications of L infinite norms in image processing. P. Weiss and G. Aubert and L. Blanc-Féraud. Research Report 6115, INRIA, September 2006. Keywords : projected subgradient descent, convergence rate, Total variation, compression bounded noise, meyer G norm, fast l1 minimization. @TECHREPORT{Some applications of L infinite constraints, author = {Weiss, P. and Aubert, G. and Blanc-Féraud, L.}, title = {Some applications of L infinite norms in image processing}, year = {2006}, month = {September}, institution = {INRIA}, type = {Research Report}, number = {6115}, url = {http://www.math.univ-toulouse.fr/~weiss/Publis/RR-6115.pdf}, pdf = {ftp://ftp-sop.inria.fr/ariana/Articles/2006_Some applications of L infinite constraints.pdf}, keyword = {projected subgradient descent, convergence rate, Total variation, compression bounded noise, meyer G norm, fast l1 minimization} }

5 - 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. @TECHREPORT{5254, author = {Aubert, G. and Aujol, J.F. and Blanc-Féraud, L.}, title = {Detecting Codimension-two Objects in an Image with Ginzburg-Landau Models}, year = {2004}, month = {July}, institution = {INRIA}, type = {Research Report}, number = {5254}, address = {France}, url = {http://www.inria.fr/rrrt/rr-5254.html}, pdf = {ftp://ftp.inria.fr/INRIA/publication/publi-pdf/RR/RR-5254.pdf}, ps = {ftp://ftp.inria.fr/INRIA/publication/publi-ps-gz/RR/RR-5254.ps.gz}, keyword = {Ginzburg-Landau model, Biological images, Segmentation, Partial differential equation} } 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.

6 - 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. @TECHREPORT{4878, author = {Aubert, G. and Aujol, J.F.}, title = {Modeling very Oscillating Signals : Application to Image Processing}, year = {2003}, month = {July}, institution = {INRIA}, type = {Research Report}, number = {4878}, address = {France}, url = {http://www.inria.fr/rrrt/rr-4878.html}, pdf = {ftp://ftp.inria.fr/INRIA/publication/publi-pdf/RR/RR-4878.pdf}, ps = {ftp://ftp.inria.fr/INRIA/publication/publi-ps-gz/RR/RR-4878.ps.gz}, keyword = {Bounded Variation Space, Sobolev space, Image decomposition, Optimization, Partial differential equation} } 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.

7 - 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). @TECHREPORT{4704, author = {Aujol, J.F. and Aubert, G. and Blanc-Féraud, L. and Chambolle, A.}, title = {Image Decomposition : Application to Textured Images and SAR Images}, year = {2003}, month = {January}, institution = {INRIA}, type = {Research Report}, number = {4704}, address = {France}, url = {http://www.inria.fr/rrrt/rr-4704.html}, pdf = {ftp://ftp.inria.fr/INRIA/publication/publi-pdf/RR/RR-4704.pdf}, ps = {ftp://ftp.inria.fr/INRIA/publication/publi-ps-gz/RR/RR-4704.ps.gz}, keyword = {Total variation, Bounded Variation Space, Texture, Classification, Restoration, Synthetic Aperture Radar (SAR)} } 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.

8 - 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 = {http://www.inria.fr/rrrt/rr-4640.html}, pdf = {ftp://ftp.inria.fr/INRIA/publication/publi-pdf/RR/RR-4640.pdf}, ps = {ftp://ftp.inria.fr/INRIA/publication/publi-ps-gz/RR/RR-4640.ps.gz}, 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.

9 - Gamma-Convergence of Discrete Functionals with non Convex Perturbation for Image Classification. G. Aubert and L. Blanc-Féraud and R. March. Research Report 4560, Inria, France, September 2002. Keywords : Generalised Gaussians, Classification, Regularization. @TECHREPORT{4560, author = {Aubert, G. and Blanc-Féraud, L. and March, R.}, title = {Gamma-Convergence of Discrete Functionals with non Convex Perturbation for Image Classification}, year = {2002}, month = {September}, institution = {Inria}, type = {Research Report}, number = {4560}, address = {France}, url = {http://www.inria.fr/rrrt/rr-4560.html}, pdf = {ftp://ftp.inria.fr/INRIA/publication/publi-pdf/RR/RR-4560.pdf}, ps = {ftp://ftp.inria.fr/INRIA/publication/publi-ps-gz/RR/RR-4560.ps.gz}, keyword = {Generalised Gaussians, Classification, Regularization} } Résumé : Ce rapport contient la justification mathématique du modèle variationnel proposé en traitement d'image pour la classification supervisée. A partir des travaux effectués en mécanique des fluides pour les transitions de phase, nous avons développé un modèle de classification par minimisation d'une suite de fonctionnelles. Le résultat est une image de classes formée de régions homogènes séparées par des contours réguliers. Ce modèle diffère de ceux utilisés en mécanique des fluides car la perturbation utilisée n'est pas quadratique mais correspond à une fonction de régularisation d'image préservant les contours. La gamma-convergence de cette nouvelle suite de fonctionnelles est prouvée. Abstract : The purpose of this report is to show the theoretical soundness of a variation- al method proposed in image processing for supervised classification. Based on works developed for phase transitions in fluid mechanics, the classification is obtained by minimizing a sequence of functionals. The method provides an image composed of homogeneous regions with regular boundaries, a region being defined as a set of pixels belonging to the same class. In this paper, we show the gamma-convergence of the sequence of functionals which differ from the ones proposed in fluid mechanics in the sense that the perturbation term is not quadratic but has a finite asymptote at infinity, corresponding to an edge preserving regularization term in image processing.

10 - Mathematical Statement to one Dimensional Phase Unwrapping : a Variational Approach. C. Lacombe and G. Aubert and L. Blanc-Féraud. Research Report 4521, Inria, France, July 2002. Keywords : Sobolev space, Bounded Variation Space, Synthetic Aperture Radar (SAR), Interferometry, Phase unwrapping. @TECHREPORT{4521, author = {Lacombe, C. and Aubert, G. and Blanc-Féraud, L.}, title = {Mathematical Statement to one Dimensional Phase Unwrapping : a Variational Approach}, year = {2002}, month = {July}, institution = {Inria}, type = {Research Report}, number = {4521}, address = {France}, url = {http://www.inria.fr/rrrt/rr-4521.html}, pdf = {ftp://ftp.inria.fr/INRIA/publication/publi-pdf/RR/RR-4521.pdf}, ps = {ftp://ftp.inria.fr/INRIA/publication/publi-ps-gz/RR/RR-4521.ps.gz}, keyword = {Sobolev space, Bounded Variation Space, Synthetic Aperture Radar (SAR), Interferometry, Phase unwrapping} } Résumé : Beaucoup d'alogorithmes de déroulement de phase ont été développés et formulés dans le domaine discret durant ces dix dernières années. Nous proposons ici, une formulation variationnelle pour résoudre le problème. Cette étude dans le domaine continu va nous permettre d'imposer quelques contraintes sur la régularité de la solution et de les implémenter efficacement. Cette méthode est présentée dans le cas unidimensionnel, et servira de base pour nos développement futurs pour le cas réel en 2D. Abstract : Over the past ten years, many phase unwrapping algorithms have been developed and formulated in a discrete setting. Here we propose a variational formulatio- n to solve the problem. This continuous framework will allow us to impose some constraints on the smoothness of the solution and to implement them efficiently. This method is presented in the one dimensional case, and will serve as a basis for future developments in the real 2D case.

11 - 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 = {http://www.inria.fr/rrrt/rr-4507.html}, pdf = {ftp://ftp.inria.fr/INRIA/publication/publi-pdf/RR/RR-4507.pdf}, ps = {ftp://ftp.inria.fr/INRIA/publication/publi-ps-gz/RR/RR-4507.ps.gz}, 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.

12 - Classification d'Images Multibandes par Modèles Variationnels. C. Samson and L. Blanc-Féraud and G. Aubert and J. Zerubia. Research Report 4010, Inria, September 2000. Keywords : Variational methods, Classification, Active contour, Level sets, Gamma Convergence. @TECHREPORT{cs99e, author = {Samson, C. and Blanc-Féraud, L. and Aubert, G. and Zerubia, J.}, title = {Classification d'Images Multibandes par Modèles Variationnels}, year = {2000}, month = {September}, institution = {Inria}, type = {Research Report}, number = {4010}, url = {http://www.inria.fr/rrrt/rr-4010.html}, pdf = {ftp://ftp.inria.fr/INRIA/publication/publi-pdf/RR/RR-4010.pdf}, ps = {ftp://ftp.inria.fr/INRIA/publication/publi-psgz/RR/RR-4010.ps.gz}, keyword = {Variational methods, Classification, Active contour, Level sets, Gamma Convergence} } Résumé : Dans ce rapport, nous proposons deux modèles variationnels pour la classificat- ion d'images multibandes. Le premier modèle présenté repose sur la minimisation d'une famille de critères dont la suite de solutions converge vers une partition des données composée de classes homogènes séparées par des contours réguliers. Parallèlement à cette approche, nous avons développé un second modèle de classification mettant en jeu un ensemble de régions et contours actifs. Nous utilisons une approche par ensembles de niveaux pour définir le critère à minimiser. Le critère proposé contient des termes reliés à l'information sur les régions ainsi qu'à l'information sur les contours. L'imagerie multispectrale permet de prendre en compte, et de combiner, l'information des différentes bandes spectrales renvoyée par un capteur satellitaire ou aérien. L'extension au cas multispectral intervient à des niveaux différents pour les deux modèles proposés dans ce rapport. Nous traitons une application réelle sur une scène SPOT en mode XS pour laquelle nous disposons d'une vérité terrain. Nous comparons les deux modèles variationnels que nous proposons à d'autres approches dont un modèle stochastique hiérarchique, récemment développé à l'IRISA au sein du projet VISTA. Abstract : Herein, we propose two variational models for multiband image classification. \The first model we propose herein is based on the minimization of a criterion family whose set of solutions is converging to a partition of the data set composed of homogeneous regions with regularized boundaries. The second model we propose is based on a set of active regions and contours. We use a level set formulation to define the criterion we want to minimize. Each class and its associated set of regions and boundaries is defined thanks to a level set function. The extension of these two models to the multispectral case is presented in this report. The extension of the dynamic model is quite straightforward whereas the one of the first model is more tricky. We have conducted experiments on SPOT XS data whose ground truth is given. We compare the results we obtain with other approaches, in particular we compare the proposed models to a stochastic hierarchical model recently developed within the VISTA group from IRISA.

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