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2. LTB curves with Lipschitz turn are par-regular

Author

Quentrec, Etienne Le, Mazo, Loïc, Baudrier, Étienne, and Tajine, Mohamed

Subjects
Computer Science - Computational Geometry, Computer Science - Computer Vision and Pattern Recognition, Computer Science - Discrete Mathematics
Abstract

Preserving the topology during a digitization process is a requirement of first importance. To this end, it is classical in Digital Geometry to assume the shape borders to be par-regular. Par-regularity was proved to be equivalent to having positive reach or to belong to the class C 1,1 of curves with Lipschitz derivative. Recently, we proposed to use a larger class that encompasses polygons with obtuse angles, the locally turn-bounded curves. The aim of this technical report is to define the class of par-regular curves inside the class of locally turn-bounded curves using only the notion of turn, that is of integral curvature. To be more precise, in a previous article, we have already proved that par-regular curves are locally turn-bounded. Incidentally this proof lead us to show that the turn of par-regular curves is a Lipschitz function of their length. We call the class of curves verifying this latter property the curves with Lipschitz turn. In this technical report, we prove the converse assertion : locally turn-bounded curves with Lipschitz turn are par-regular. The equivalence is stated in Theorem 3.1 and the converse assertion is proved in Lemma 3.2. In section 1, we recall the definition of par-regularity and equivalently of sets with positive reach. In section 2, we present the notions of curves locally turn-bounded and of curves with Lipschitz turn. Throughout this latter section, some of intermediate steps (Lemmas 2.3 and 2.11) are proved just after the introduction of their related notions. The last section (section 3) is dedicated to the proof of the equivalence of the notions.

Published
2021

3. Locally Turn-Bounded Curves Are Quasi-Regular

Author

Le Quentrec, Étienne, Mazo, Loïc, Baudrier, Étienne, Tajine, Mohamed, Goos, Gerhard, Founding Editor, Hartmanis, Juris, Founding Editor, Bertino, Elisa, Editorial Board Member, Gao, Wen, Editorial Board Member, Steffen, Bernhard, Editorial Board Member, Woeginger, Gerhard, Editorial Board Member, Yung, Moti, Editorial Board Member, Lindblad, Joakim, editor, Malmberg, Filip, editor, and Sladoje, Nataša, editor

Published
2021
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4. Multilabel, Multiscale Topological Transformation for Cerebral MRI Segmentation Post-processing

Author

Tor-Díez, Carlos, Faisan, Sylvain, Mazo, Loïc, Bednarek, Nathalie, Meunier, Hélène, Bloch, Isabelle, Passat, Nicolas, Rousseau, François, Hutchison, David, Editorial Board Member, Kanade, Takeo, Editorial Board Member, Kittler, Josef, Editorial Board Member, Kleinberg, Jon M., Editorial Board Member, Mattern, Friedemann, Editorial Board Member, Mitchell, John C., Editorial Board Member, Naor, Moni, Editorial Board Member, Pandu Rangan, C., Editorial Board Member, Steffen, Bernhard, Editorial Board Member, Terzopoulos, Demetri, Editorial Board Member, Tygar, Doug, Editorial Board Member, Goos, Gerhard, Founding Editor, Hartmanis, Juris, Founding Editor, Burgeth, Bernhard, editor, Kleefeld, Andreas, editor, Naegel, Benoît, editor, Passat, Nicolas, editor, and Perret, Benjamin, editor

Published
2019
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5. Local Turn-Boundedness: A Curvature Control for a Good Digitization

Author

Le Quentrec, Étienne, Mazo, Loïc, Baudrier, Étienne, Tajine, Mohamed, Hutchison, David, Series Editor, Kanade, Takeo, Series Editor, Kittler, Josef, Series Editor, Kleinberg, Jon M., Series Editor, Mattern, Friedemann, Series Editor, Mitchell, John C., Series Editor, Naor, Moni, Series Editor, Pandu Rangan, C., Series Editor, Steffen, Bernhard, Series Editor, Terzopoulos, Demetri, Series Editor, Tygar, Doug, Series Editor, Couprie, Michel, editor, Cousty, Jean, editor, Kenmochi, Yukiko, editor, and Mustafa, Nabil, editor

Published
2019
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7. Study on the Digitization Dual Combinatorics and Convex Case

Author

Mazo, Loïc, Baudrier, Étienne, Hutchison, David, Series editor, Kanade, Takeo, Series editor, Kittler, Josef, Series editor, Kleinberg, Jon M., Series editor, Mattern, Friedemann, Series editor, Mitchell, John C., Series editor, Naor, Moni, Series editor, Pandu Rangan, C., Series editor, Steffen, Bernhard, Series editor, Terzopoulos, Demetri, Series editor, Tygar, Doug, Series editor, Weikum, Gerhard, Series editor, Kropatsch, Walter G., editor, Artner, Nicole M., editor, and Janusch, Ines, editor

Published
2017
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10. nD Quasi-Affine Transformations

Author

Jacob-Da Col, Marie-Andrée, Mazo, Loïc, Hutchison, David, Series editor, Kanade, Takeo, Series editor, Kittler, Josef, Series editor, Kleinberg, Jon M., Series editor, Mattern, Friedemann, Series editor, Mitchell, John C., Series editor, Naor, Moni, Series editor, Pandu Rangan, C., Series editor, Steffen, Bernhard, Series editor, Terzopoulos, Demetri, Series editor, Tygar, Doug, Series editor, Weikum, Gerhard, Series editor, Normand, Nicolas, editor, Guédon, Jeanpierre, editor, and Autrusseau, Florent, editor

Published
2016
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11. Curve Digitization Variability

Author

Baudrier, Étienne, Mazo, Loïc, Hutchison, David, Series editor, Kanade, Takeo, Series editor, Kittler, Josef, Series editor, Kleinberg, Jon M., Series editor, Mattern, Friedemann, Series editor, Mitchell, John C., Series editor, Naor, Moni, Series editor, Pandu Rangan, C., Series editor, Steffen, Bernhard, Series editor, Terzopoulos, Demetri, Series editor, Tygar, Doug, Series editor, Weikum, Gerhard, Series editor, Normand, Nicolas, editor, Guédon, Jeanpierre, editor, and Autrusseau, Florent, editor

Published
2016
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13. About Multigrid Convergence of Some Length Estimators

Author

Mazo, Loïc, Baudrier, Étienne, Hutchison, David, Series editor, Kanade, Takeo, Series editor, Kittler, Josef, Series editor, Kleinberg, Jon M., Series editor, Kobsa, Alfred, Series editor, Mattern, Friedemann, Series editor, Mitchell, John C., Series editor, Naor, Moni, Series editor, Nierstrasz, Oscar, Series editor, Pandu Rangan, C., Series editor, Steffen, Bernhard, Series editor, Terzopoulos, Demetri, Series editor, Tygar, Doug, Series editor, Weikum, Gerhard, Series editor, Barcucci, Elena, editor, Frosini, Andrea, editor, and Rinaldi, Simone, editor

Published
2014
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15. A Framework for Label Images

Author

Mazo, Loïc, Hutchison, David, editor, Kanade, Takeo, editor, Kittler, Josef, editor, Kleinberg, Jon M., editor, Mattern, Friedemann, editor, Mitchell, John C., editor, Naor, Moni, editor, Nierstrasz, Oscar, editor, Pandu Rangan, C., editor, Steffen, Bernhard, editor, Sudan, Madhu, editor, Terzopoulos, Demetri, editor, Tygar, Doug, editor, Vardi, Moshe Y., editor, Weikum, Gerhard, editor, Ferri, Massimo, editor, Frosini, Patrizio, editor, Landi, Claudia, editor, Cerri, Andrea, editor, and Di Fabio, Barbara, editor

Published
2012
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16. A Unified Topological Framework for Digital Imaging

Author

Mazo, Loïc, Passat, Nicolas, Couprie, Michel, Ronse, Christian, Hutchison, David, Series editor, Kanade, Takeo, Series editor, Kittler, Josef, Series editor, Kleinberg, Jon M., Series editor, Mattern, Friedemann, Series editor, Mitchell, John C., Series editor, Naor, Moni, Series editor, Nierstrasz, Oscar, Series editor, Pandu Rangan, C., Series editor, Steffen, Bernhard, Series editor, Sudan, Madhu, Series editor, Terzopoulos, Demetri, Series editor, Tygar, Doug, Series editor, Vardi, Moshe Y., Series editor, Weikum, Gerhard, Series editor, Debled-Rennesson, Isabelle, editor, Domenjoud, Eric, editor, Kerautret, Bertrand, editor, and Even, Philippe, editor

Published
2011
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17. Topology-Preserving Thinning in 2-D Pseudomanifolds

Author

Passat, Nicolas, Couprie, Michel, Mazo, Loïc, Bertrand, Gilles, Hutchison, David, Series editor, Kanade, Takeo, Series editor, Kittler, Josef, Series editor, Kleinberg, Jon M., Series editor, Mattern, Friedemann, Series editor, Mitchell, John C., Series editor, Naor, Moni, Series editor, Nierstrasz, Oscar, Series editor, Pandu Rangan, C., Series editor, Steffen, Bernhard, Series editor, Sudan, Madhu, Series editor, Terzopoulos, Demetri, Series editor, Tygar, Doug, Series editor, Vardi, Moshe Y., Series editor, Weikum, Gerhard, Series editor, Brlek, Srečko, editor, Reutenauer, Christophe, editor, and Provençal, Xavier, editor

Published
2009
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22. LTB curves with Lipschitz turn are par-regular

Author

Le Quentrec, Etienne, Mazo, Loïc, Baudrier, Étienne, Tajine, Mohamed, Aix Marseille Université (AMU), Université de Strasbourg (UNISTRA), and Laboratoire ICube, université de Strasbourg

Subjects
[INFO.INFO-CV]Computer Science [cs]/Computer Vision and Pattern Recognition [cs.CV], [INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM]
Abstract

Preserving the topology during a digitization process is a requirement of first importance. To this end, it is classical in Digital Geometry to assume the shape borders to be par-regular. Par-regularity was proved to be equivalent to having positive reach or to belong to the class C 1,1 of curves with Lipschitz derivative. Recently, we proposed to use a larger class that encompasses polygons with obtuse angles, the locally turn-bounded curves. The aim of this technical report is to define the class of par-regular curves inside the class of locally turn-bounded curves using only the notion of turn, that is of integral curvature. To be more precise, in a previous article, we have already proved that par-regular curves are locally turn-bounded. Incidentally this proof lead us to show that the turn of par-regular curves is a Lipschitz function of their length. We call the class of curves verifying this latter property the curves with Lipschitz turn. In this technical report, we prove the converse assertion : locally turn-bounded curves with Lipschitz turn are par-regular. The equivalence is stated in Theorem 3.1 and the converse assertion is proved in Lemma 3.2. In section 1, we recall the definition of par-regularity and equivalently of sets with positive reach. In section 2, we present the notions of curves locally turn-bounded and of curves with Lipschitz turn. Throughout this latter section, some of intermediate steps (Lemmas 2.3 and 2.11) are proved just after the introduction of their related notions. The last section (section 3) is dedicated to the proof of the equivalence of the notions.

Published
2021

23. Monotonic sampling of a continuous closed curve from its Gauss digitization. Application to length estimation

Author

Le Quentrec, Étienne, Mazo, Loïc, Baudrier, Étienne, Tajine, Mohamed, Laboratoire des sciences de l'ingénieur, de l'informatique et de l'imagerie (ICube), Institut National des Sciences Appliquées - Strasbourg (INSA Strasbourg), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)-École Nationale du Génie de l'Eau et de l'Environnement de Strasbourg (ENGEES)-Réseau nanophotonique et optique, Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)-Université de Haute-Alsace (UHA) Mulhouse - Colmar (Université de Haute-Alsace (UHA))-Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)-Université de Haute-Alsace (UHA) Mulhouse - Colmar (Université de Haute-Alsace (UHA))-Matériaux et nanosciences d'Alsace, Centre National de la Recherche Scientifique (CNRS)-Institut National de la Santé et de la Recherche Médicale (INSERM)-Université de Haute-Alsace (UHA) Mulhouse - Colmar (Université de Haute-Alsace (UHA))-Université de Strasbourg (UNISTRA)-Institut National de la Santé et de la Recherche Médicale (INSERM)-Université de Strasbourg (UNISTRA), École Nationale du Génie de l'Eau et de l'Environnement de Strasbourg (ENGEES)-Université de Strasbourg (UNISTRA)-Institut National des Sciences Appliquées - Strasbourg (INSA Strasbourg), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Institut National de Recherche en Informatique et en Automatique (Inria)-Les Hôpitaux Universitaires de Strasbourg (HUS)-Centre National de la Recherche Scientifique (CNRS)-Matériaux et Nanosciences Grand-Est (MNGE), Université de Strasbourg (UNISTRA)-Université de Haute-Alsace (UHA) Mulhouse - Colmar (Université de Haute-Alsace (UHA))-Institut National de la Santé et de la Recherche Médicale (INSERM)-Institut de Chimie du CNRS (INC)-Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)-Université de Haute-Alsace (UHA) Mulhouse - Colmar (Université de Haute-Alsace (UHA))-Institut National de la Santé et de la Recherche Médicale (INSERM)-Institut de Chimie du CNRS (INC)-Centre National de la Recherche Scientifique (CNRS)-Réseau nanophotonique et optique, Université de Strasbourg (UNISTRA)-Université de Haute-Alsace (UHA) Mulhouse - Colmar (Université de Haute-Alsace (UHA))-Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS), Mazo, Loïc, Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)-Université de Haute-Alsace (UHA) Mulhouse - Colmar (Université de Haute-Alsace (UHA))-Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)-Université de Haute-Alsace (UHA) Mulhouse - Colmar (Université de Haute-Alsace (UHA))-Matériaux et nanosciences d'Alsace (FMNGE), and Institut de Chimie du CNRS (INC)-Université de Strasbourg (UNISTRA)-Université de Haute-Alsace (UHA) Mulhouse - Colmar (Université de Haute-Alsace (UHA))-Institut National de la Santé et de la Recherche Médicale (INSERM)-Centre National de la Recherche Scientifique (CNRS)-Institut de Chimie du CNRS (INC)-Université de Strasbourg (UNISTRA)-Institut National de la Santé et de la Recherche Médicale (INSERM)-Centre National de la Recherche Scientifique (CNRS)

Subjects
integral curvature, perimeter discrete estimation, [MATH.MATH-GM]Mathematics [math]/General Mathematics [math.GM], [INFO.INFO-TI] Computer Science [cs]/Image Processing [eess.IV], digitization, [INFO.INFO-TI]Computer Science [cs]/Image Processing [eess.IV], Jordan curve, [MATH.MATH-GM] Mathematics [math]/General Mathematics [math.GM], turn
Abstract

Our study is about perimeter estimation of continuous shapes based on their digitization (i.e. their pixelated representation). To study the estimation error, we are led to propose the notion of monotonically sampling back-digitization, a mapping between the boundary of a digitized Jordan curve and the Jordan curve itself. This mapping ensures that the digital curve and the continuous curve are traveled together in a cyclic order manner. Then, we use back-digitization to prove the multigrid convergence of perimeter estimators that are based on polygons inscribed in the digitization. Furthermore, convergence speed is given for this class of estimators. The convergence is guaranteed provided the polygon edge lengths tend toward 0 and their edge sizes relative to the grid step tends towards infinity (thus the sampling of the digital boundary has to be both tight and sparse). All the proofs are established in the framework of locally turn bounded curves (LTB) presented in the previous article Le Quentrec et al., Local turn-boundedness: A curvature control for a good digitization, DGCI 2019. If, moreover, the continuous curves also have a Lipschitz turn, an explicit error bound is calculated. Besides, an equivalence between LTB-curves with a Lipschitz turn and curves of class C 1,1 is subsequently established.

Published
2020

33. Density based graph denoising for manifold learning

Author

Michels, Yves, Baudrier, Etienne, Mazo, Loïc, Tajine, Mohamed, Laboratoire des sciences de l'ingénieur, de l'informatique et de l'imagerie (ICube), Institut National des Sciences Appliquées - Strasbourg (INSA Strasbourg), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)-École Nationale du Génie de l'Eau et de l'Environnement de Strasbourg (ENGEES)-Réseau nanophotonique et optique, Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)-Université de Haute-Alsace (UHA) Mulhouse - Colmar (Université de Haute-Alsace (UHA))-Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)-Université de Haute-Alsace (UHA) Mulhouse - Colmar (Université de Haute-Alsace (UHA))-Matériaux et nanosciences d'Alsace (FMNGE), Institut de Chimie du CNRS (INC)-Université de Strasbourg (UNISTRA)-Université de Haute-Alsace (UHA) Mulhouse - Colmar (Université de Haute-Alsace (UHA))-Institut National de la Santé et de la Recherche Médicale (INSERM)-Centre National de la Recherche Scientifique (CNRS)-Institut de Chimie du CNRS (INC)-Université de Strasbourg (UNISTRA)-Institut National de la Santé et de la Recherche Médicale (INSERM)-Centre National de la Recherche Scientifique (CNRS), Baudrier, Etienne, École Nationale du Génie de l'Eau et de l'Environnement de Strasbourg (ENGEES)-Université de Strasbourg (UNISTRA)-Institut National des Sciences Appliquées - Strasbourg (INSA Strasbourg), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Institut National de Recherche en Informatique et en Automatique (Inria)-Les Hôpitaux Universitaires de Strasbourg (HUS)-Centre National de la Recherche Scientifique (CNRS)-Matériaux et Nanosciences Grand-Est (MNGE), Université de Strasbourg (UNISTRA)-Université de Haute-Alsace (UHA) Mulhouse - Colmar (Université de Haute-Alsace (UHA))-Institut National de la Santé et de la Recherche Médicale (INSERM)-Institut de Chimie du CNRS (INC)-Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)-Université de Haute-Alsace (UHA) Mulhouse - Colmar (Université de Haute-Alsace (UHA))-Institut National de la Santé et de la Recherche Médicale (INSERM)-Institut de Chimie du CNRS (INC)-Centre National de la Recherche Scientifique (CNRS)-Réseau nanophotonique et optique, and Université de Strasbourg (UNISTRA)-Université de Haute-Alsace (UHA) Mulhouse - Colmar (Université de Haute-Alsace (UHA))-Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)

Subjects
neighborhood graph !, [INFO.INFO-TI] Computer Science [cs]/Image Processing [eess.IV], [INFO.INFO-LG]Computer Science [cs]/Machine Learning [cs.LG], Shortcut detection, structure learning, manifold learning, [INFO.INFO-TI]Computer Science [cs]/Image Processing [eess.IV], [INFO.INFO-LG] Computer Science [cs]/Machine Learning [cs.LG], unsupervised learning
Abstract

Processing high dimension data often makes use of a dimension reduction step. Indeed, high dimension data generally rely on a low dimension underlying structure. When the data are noisy, dimension reduction may fail because of shortcuts appearing on the graph catching the underlying structure. Our paper presents a method to suppress shortcuts in the underlying structure graph. The method is based on a skeleton graph that approximates the data and that is built using a data probability density estimation. This approximating graph is then used to select the edges of the underlying structure graph used in the dimension reduction. The proposed algorithm is tested on the capacity to suppress shortcuts and to conserve the underlying structure geodesic distance. Our method outperforms the state-of-the-art methods in the experiments on six 3D synthetic dataset and one tomographic dataset with different noise levels.

Published
2019

38. Homotopic deformations in n-ary digital images

Author

Mazo, Loïc and Mazo, Loïc

Subjects
Simple point, Image de labels, [INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM], Ensemble partiellement ordonné, Homotopie, Poset, [INFO.INFO-TI] Computer Science [cs]/Image Processing [eess.IV], Point simple, Digital topology, Topologie digitale, Label image, Homotopy, Treillis
Abstract

In a digital image, when performing processes such as registration, deformation or thinning, the preservation of the topological properties of the objects contained in the image (e.g., connected components, tunnels, cavities, etc.) is an important requirement. For 50 years, several tools enabling the analysis and the modification under topological constraints of binary images have been proposed and used. Nevertheless, in many fields an image is generally composed of several objects, and it is often important to understand or maintain their topological properties all together, that is the topology of each and the topology of the scene. In such images, the objects are characterised by specific labels on which there generally exists no meaningful order relation (unlike grey-level images for instance). In this thesis, we focus on homotopic deformations that preserve all the topological relations of such partitions of the space. After describing our theoretical framework for binary images, and its compatibility with usual approaches in digital imaging, we propose two models to be used with n-ary images. In these models, the regions of interest compose a sub-lattice of the power set of the labeled regions. Thereby, we can define some elementary modifications that preserve individual and "collective" topologies., De nombreux domaines applicatifs utilisent des techniques de traitement d'images basées sur l'analyse de la topologie des images discrètes, en particulier pour des opérations devant préserver cette topologie. Si beaucoup de travaux théoriques et méthodologiques ont été menés dans le cadre des images binaires, les questions relatives à la modélisation et à la gestion simultanées des propriétés topologiques de plusieurs éléments sémantiques dans une même image discrète reste à l'heure actuelle un problème peu exploré. Dans cette thèse, nous avons porté notre attention sur la définition de déformations homotopiques compatibles avec la présence de plusieurs éléments non hiérarchisés dont les relations spatiales peuvent être significatives. Après avoir décrit le cadre théorique retenu pour les images binaires, et après avoir montré sa compatibilité avec les approches les plus fréquentes en imagerie, nous proposons des modélisations des images n-aires appuyées sur ce cadre théorique et dont les objets d'intérêts forment un sous-treillis de l'ensemble des parties de la partition initiale en régions de sémantiques distinctes. Ainsi, nous sommes en mesure de décrire quelques transformations élémentaires des images n-aires respectueuses non seulement des topologies individuelles des différents objets mais aussi des topologies "collectives" qui traduisent les inter-relations des objets.

Published
2011

42. A framework for digital label images

Author

Mazo, Loïc, Laboratoire des sciences de l'ingénieur, de l'informatique et de l'imagerie (ICube), Institut National des Sciences Appliquées - Strasbourg (INSA Strasbourg), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)-École Nationale du Génie de l'Eau et de l'Environnement de Strasbourg (ENGEES)-Réseau nanophotonique et optique, Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)-Université de Haute-Alsace (UHA) Mulhouse - Colmar (Université de Haute-Alsace (UHA))-Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)-Université de Haute-Alsace (UHA) Mulhouse - Colmar (Université de Haute-Alsace (UHA))-Matériaux et nanosciences d'Alsace (FMNGE), Institut de Chimie du CNRS (INC)-Université de Strasbourg (UNISTRA)-Université de Haute-Alsace (UHA) Mulhouse - Colmar (Université de Haute-Alsace (UHA))-Institut National de la Santé et de la Recherche Médicale (INSERM)-Centre National de la Recherche Scientifique (CNRS)-Institut de Chimie du CNRS (INC)-Université de Strasbourg (UNISTRA)-Institut National de la Santé et de la Recherche Médicale (INSERM)-Centre National de la Recherche Scientifique (CNRS), École Nationale du Génie de l'Eau et de l'Environnement de Strasbourg (ENGEES)-Université de Strasbourg (UNISTRA)-Institut National des Sciences Appliquées - Strasbourg (INSA Strasbourg), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Institut National de Recherche en Informatique et en Automatique (Inria)-Les Hôpitaux Universitaires de Strasbourg (HUS)-Centre National de la Recherche Scientifique (CNRS)-Matériaux et Nanosciences Grand-Est (MNGE), Université de Strasbourg (UNISTRA)-Université de Haute-Alsace (UHA) Mulhouse - Colmar (Université de Haute-Alsace (UHA))-Institut National de la Santé et de la Recherche Médicale (INSERM)-Institut de Chimie du CNRS (INC)-Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)-Université de Haute-Alsace (UHA) Mulhouse - Colmar (Université de Haute-Alsace (UHA))-Institut National de la Santé et de la Recherche Médicale (INSERM)-Institut de Chimie du CNRS (INC)-Centre National de la Recherche Scientifique (CNRS)-Réseau nanophotonique et optique, and Université de Strasbourg (UNISTRA)-Université de Haute-Alsace (UHA) Mulhouse - Colmar (Université de Haute-Alsace (UHA))-Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)

Subjects
[INFO.INFO-TI]Computer Science [cs]/Image Processing [eess.IV], [INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM]
Abstract

International audience; Digital label images - partitions of a discrete space - need a specific topological model to take into account not only the topologies of the regions but also the topology of the partition. In this article, we propose a topological framework for label images in which all the regions of the initial partition and of any coarser partition of the space can be explicitly represented accordingly to any classical adjacency relation. Moreover, we define a notion of simple point which enables atomic changes in the partition without breaking any topology. Finally, we discuss about the implementation of the framework.

Published
2012

43. Minimal simple sets: A new concept for topology-preserving transformations

Author

Passat, Nicolas, Couprie, Michel, Mazo, Loïc, Bertrand, Gilles, Laboratoire des Sciences de l'Image, de l'Informatique et de la Télédétection (LSIIT), Centre National de la Recherche Scientifique (CNRS), Laboratoire d'Informatique Gaspard-Monge (LIGM), Centre National de la Recherche Scientifique (CNRS)-Fédération de Recherche Bézout-ESIEE Paris-École des Ponts ParisTech (ENPC)-Université Paris-Est Marne-la-Vallée (UPEM), and Passat, Nicolas

Subjects
[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM], [INFO.INFO-CV] Computer Science [cs]/Computer Vision and Pattern Recognition [cs.CV], topology preservation, [MATH.MATH-GT]Mathematics [math]/Geometric Topology [math.GT], binary image reduction, n-D cubical complexes, [INFO.INFO-CV]Computer Science [cs]/Computer Vision and Pattern Recognition [cs.CV], [INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM], [MATH.MATH-GT] Mathematics [math]/Geometric Topology [math.GT]
Abstract

International audience; Preserving topological properties of binary objects during thinning procedures is an important issue in the field of image analysis. In this context, we present the new notion of simple set which extends the well-known notion of simple point. Similarly to simple points, simple sets have the property that the homotopy type of the object in which they lie is not changed when such sets are removed. This paper defines and justifies the concept of simple sets and of a sub-family of such sets called minimal simple sets. It also briefly describes some first results and work in progress on this subject.

Published
2008

44. Des grumeaux dans la topologie : une introduction à la notion d'ensemble simple

Author

Passat, Nicolas, Couprie, Michel, Bertrand, Gilles, Mazo, Loïc, Laboratoire des Sciences de l'Image, de l'Informatique et de la Télédétection (LSIIT), Centre National de la Recherche Scientifique (CNRS), Laboratoire d'Informatique Gaspard-Monge (LIGM), Centre National de la Recherche Scientifique (CNRS)-Fédération de Recherche Bézout-ESIEE Paris-École des Ponts ParisTech (ENPC)-Université Paris-Est Marne-la-Vallée (UPEM), and Passat, Nicolas

Subjects
[INFO.INFO-CV] Computer Science [cs]/Computer Vision and Pattern Recognition [cs.CV], [MATH.MATH-GT]Mathematics [math]/Geometric Topology [math.GT], [INFO.INFO-CV]Computer Science [cs]/Computer Vision and Pattern Recognition [cs.CV], ComputingMilieux_MISCELLANEOUS, [MATH.MATH-GT] Mathematics [math]/Geometric Topology [math.GT]
Abstract

National audience

Published
2007

49. Some representations of real numbers using integer sequences

Author

Loïc Mazo, Marie-Andrée Da Col-Jacob, Laurent Fuchs, Nicolas Magaud, Gaëlle Skapin, Laboratoire des sciences de l'ingénieur, de l'informatique et de l'imagerie (ICube), Institut National des Sciences Appliquées - Strasbourg (INSA Strasbourg), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)-École Nationale du Génie de l'Eau et de l'Environnement de Strasbourg (ENGEES)-Réseau nanophotonique et optique, Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)-Université de Haute-Alsace (UHA) Mulhouse - Colmar (Université de Haute-Alsace (UHA))-Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)-Université de Haute-Alsace (UHA) Mulhouse - Colmar (Université de Haute-Alsace (UHA))-Matériaux et nanosciences d'Alsace (FMNGE), Institut de Chimie du CNRS (INC)-Université de Strasbourg (UNISTRA)-Université de Haute-Alsace (UHA) Mulhouse - Colmar (Université de Haute-Alsace (UHA))-Institut National de la Santé et de la Recherche Médicale (INSERM)-Centre National de la Recherche Scientifique (CNRS)-Institut de Chimie du CNRS (INC)-Université de Strasbourg (UNISTRA)-Institut National de la Santé et de la Recherche Médicale (INSERM)-Centre National de la Recherche Scientifique (CNRS), XLIM (XLIM), Université de Limoges (UNILIM)-Centre National de la Recherche Scientifique (CNRS), Mazo, Loïc, École Nationale du Génie de l'Eau et de l'Environnement de Strasbourg (ENGEES)-Université de Strasbourg (UNISTRA)-Institut National des Sciences Appliquées - Strasbourg (INSA Strasbourg), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Institut National de Recherche en Informatique et en Automatique (Inria)-Les Hôpitaux Universitaires de Strasbourg (HUS)-Centre National de la Recherche Scientifique (CNRS)-Matériaux et Nanosciences Grand-Est (MNGE), Université de Strasbourg (UNISTRA)-Université de Haute-Alsace (UHA) Mulhouse - Colmar (Université de Haute-Alsace (UHA))-Institut National de la Santé et de la Recherche Médicale (INSERM)-Institut de Chimie du CNRS (INC)-Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)-Université de Haute-Alsace (UHA) Mulhouse - Colmar (Université de Haute-Alsace (UHA))-Institut National de la Santé et de la Recherche Médicale (INSERM)-Institut de Chimie du CNRS (INC)-Centre National de la Recherche Scientifique (CNRS)-Réseau nanophotonique et optique, Université de Strasbourg (UNISTRA)-Université de Haute-Alsace (UHA) Mulhouse - Colmar (Université de Haute-Alsace (UHA))-Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS), and univOAK, Archive ouverte

Subjects
Mathematics (miscellaneous), [INFO.INFO-TI] Computer Science [cs]/Image Processing [eess.IV], [INFO.INFO-TI]Computer Science [cs]/Image Processing [eess.IV], Stream of integers, [INFO.INFO-AO]Computer Science [cs]/Computer Arithmetic, Constructive mathematics, [INFO.INFO-AO] Computer Science [cs]/Computer Arithmetic, Real arithmetic, Arithmetization, Computer Science Applications
Abstract

The paper describes three models of the real field based on subsets of the integer sequences. The three models are compared to the Harthong–Reeb line. Two of the new models, contrary to the Harthong–Reeb line, provide accurate integer “views” on real numbers at a sequence of growing scales $B^n$ ( $B\ge2$ ).

Published
2021

50. Local Turn-Boundedness, a curvature control for continuous curves with application to digitization

Author

Étienne Baudrier, Mohamed Tajine, Étienne Le Quentrec, Loïc Mazo, Laboratoire des sciences de l'ingénieur, de l'informatique et de l'imagerie (ICube), Institut National des Sciences Appliquées - Strasbourg (INSA Strasbourg), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)-École Nationale du Génie de l'Eau et de l'Environnement de Strasbourg (ENGEES)-Réseau nanophotonique et optique, Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)-Université de Haute-Alsace (UHA) Mulhouse - Colmar (Université de Haute-Alsace (UHA))-Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)-Université de Haute-Alsace (UHA) Mulhouse - Colmar (Université de Haute-Alsace (UHA))-Matériaux et nanosciences d'Alsace (FMNGE), Institut de Chimie du CNRS (INC)-Université de Strasbourg (UNISTRA)-Université de Haute-Alsace (UHA) Mulhouse - Colmar (Université de Haute-Alsace (UHA))-Institut National de la Santé et de la Recherche Médicale (INSERM)-Centre National de la Recherche Scientifique (CNRS)-Institut de Chimie du CNRS (INC)-Université de Strasbourg (UNISTRA)-Institut National de la Santé et de la Recherche Médicale (INSERM)-Centre National de la Recherche Scientifique (CNRS), École Nationale du Génie de l'Eau et de l'Environnement de Strasbourg (ENGEES)-Université de Strasbourg (UNISTRA)-Institut National des Sciences Appliquées - Strasbourg (INSA Strasbourg), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Institut National de Recherche en Informatique et en Automatique (Inria)-Les Hôpitaux Universitaires de Strasbourg (HUS)-Centre National de la Recherche Scientifique (CNRS)-Matériaux et Nanosciences Grand-Est (MNGE), Université de Strasbourg (UNISTRA)-Université de Haute-Alsace (UHA) Mulhouse - Colmar (Université de Haute-Alsace (UHA))-Institut National de la Santé et de la Recherche Médicale (INSERM)-Institut de Chimie du CNRS (INC)-Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)-Université de Haute-Alsace (UHA) Mulhouse - Colmar (Université de Haute-Alsace (UHA))-Institut National de la Santé et de la Recherche Médicale (INSERM)-Institut de Chimie du CNRS (INC)-Centre National de la Recherche Scientifique (CNRS)-Réseau nanophotonique et optique, Université de Strasbourg (UNISTRA)-Université de Haute-Alsace (UHA) Mulhouse - Colmar (Université de Haute-Alsace (UHA))-Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS), univOAK, Archive ouverte, and Mazo, Loïc

Subjects
Statistics and Probability, Property (programming), 02 engineering and technology, [INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM], Curvature, [MATH.MATH-GM]Mathematics [math]/General Mathematics [math.GM], Turn (geometry), 0202 electrical engineering, electronic engineering, information engineering, Total curvature, Control (linguistics), Digitization, Mathematics, Smoothness, Applied Mathematics, Mathematical analysis, [MATH.MATH-GM] Mathematics [math]/General Mathematics [math.GM], Function (mathematics), Condensed Matter Physics, Informatique [cs]/Mathématique discrète [cs.DM], [INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM], [INFO.INFO-TI] Computer Science [cs]/Image Processing [eess.IV], Modeling and Simulation, [INFO.INFO-TI]Computer Science [cs]/Image Processing [eess.IV], 020201 artificial intelligence & image processing, Geometry and Topology, Computer Vision and Pattern Recognition
Abstract

International audience; This article focuses on the classical problem of the control of information loss during the digitization step. The properties proposed in the literature rely on smoothness hypotheses that are not satisfied by the curves including angular points. The notion of turn introduced by Milnor in the article On the Total Curvature of Knots generalizes the notion of integral curvature to continuous curves. Thanks to the turn, we are able to define the local turn-boundedness. This promising property of curves does not require smoothness hypotheses and shares several properties with the par(r)-regularity, in particular well-composed digitizations. Besides, the local turn-boundedness enables to constrain spatially the continuous curve as a function of its digitization.

Published
2020

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