N. Boutry, L. Najman, and T. Géraud, About the equivalence between AWCness and DWCness
URL : https://hal.archives-ouvertes.fr/hal-01375621

O. N. Monge-;-lrde-laboratoire-de-recherche-et-de-développement-de-l-'epita, T. Boutry, L. Géraud, and . Najman, url: https://hal-upec-upem.archives-ouvertes.fr/hal-01375621. communications [1] Nicolas Boutry, Thierry Géraud, and Laurent Najman Une généralisation du bien-composé composé`composéà la dimension n How to make n-D images wellcomposed without interpolation, IEEE International Conference on Image Processing, pp.2149-2153, 2014.

N. Boutry, T. Géraud, and L. Najman, How to Make nD Functions Digitally Well-Composed in a Self-dual Way, International Symposium on Mathematical Morphology and Its Applications to Signal and Image Processing, pp.561-572, 2015.
DOI : 10.1007/978-3-319-18720-4_47

URL : https://hal.archives-ouvertes.fr/hal-01168723

N. Boutry, T. Géraud, and L. Najman, On making n-D images wellcomposed by a self-dual local interpolation, International Conference on Discrete Geometry for Computer Imagery, pp.320-331, 2014.
DOI : 10.1007/978-3-319-09955-2_27

URL : https://hal.archives-ouvertes.fr/hal-01071624/document

T. Géraud, Y. Xu, E. Carlinet, N. Géraud, and L. Najman, Introducing the Dahu pseudo-distance (submitted) . In: International Symposium on Mathematical Morphology and Its Applications to Signal and Image Processing Digitally well-composed sets and functions on the n-D cubical grid (in preparation), Journal of Mathematical Imaging and Vision, 2017.

W. James and . Alexander, A proof and extension of the Jordan-Brouwer separation theorem, In: Transactions of the American Mathematical Society, vol.234, pp.333-349, 1922.

W. James and . Alexander, The combinatorial theory of complexes, Annals of Mathematics, pp.292-320, 1930.

P. Alexandroff and H. Hopf, Topologie I: Erster Band. Grundbegriffe der Mengentheoretischen , Topologie ,Topologie der Komplexe· Topologische Invarianzsätze und Anschliessende Begriffsbildungen· Verschlingungen im n-Dimensionalen Euklidischen Raum Stetige Abbildungen von Polyedern, 2013.
DOI : 10.1007/978-3-662-02021-0

O. Alexandrov and F. Santosa, A topology-preserving level set method for shape optimization, Journal of Computational Physics, vol.204, issue.1, pp.121-130, 2005.
DOI : 10.1016/j.jcp.2004.10.005

URL : http://arxiv.org/abs/math/0405142

P. Sergue¨?evitchsergue¨?evitch and A. , Diskrete Räume, Matematicheskij Sbornik 2, pp.44-501, 1937.

C. Arcelli, Pattern thinning by contour tracing, Computer Graphics and Image Processing, vol.17, issue.2, pp.130-144, 1981.
DOI : 10.1016/0146-664X(81)90021-6

E. Artzy, G. Frieder, T. Gabor, and . Herman, The theory, design, implementation and evaluation of a three-dimensional surface detection algorithm, Computer Graphics and Image Processing, vol.151, pp.1-24, 1981.

J. Aubin-andhéì-ene-frankowska, Set-valued analysis, 2009.

C. Ballester, V. Caselles, and P. Monasse, The tree of shapes of an image, ESAIM: Control, Optimisation and Calculus of Variations, vol.9, pp.1-18, 2003.
DOI : 10.1051/cocv:2002069

P. Bazin, L. M. Ellingsen, L. Dzung, and . Pham, Digital Homeomorphisms in Deformable Registration, International Conference on Information Processing in Medical Imaging, pp.211-222, 2007.
DOI : 10.1007/978-3-540-73273-0_18

O. Gordon, . Berg, H. William, R. Julian, F. Mines et al., The constructive Jordan curve theorem, In: Rocky Mountain Journal of Mathematics, vol.5, issue.2, pp.225-236, 1975.

G. Bertrand, A Boolean characterization of three-dimensional simple points, Pattern Recognition Letters, vol.17, issue.2, pp.115-124, 1996.
DOI : 10.1016/0167-8655(95)00100-X

URL : https://hal.archives-ouvertes.fr/hal-00621994

G. Bertrand, New Notions for Discrete Topology, pp.218-228, 1999.
DOI : 10.1007/3-540-49126-0_17

URL : https://hal.archives-ouvertes.fr/hal-00621992

G. Bertrand, Simple points, topological numbers and geodesic neighborhoods in cubic grids, Pattern Recognition Letters, vol.15, issue.10, pp.1003-1011, 1994.
DOI : 10.1016/0167-8655(94)90032-9

URL : https://hal.archives-ouvertes.fr/hal-00621999

G. Bertrand, J. Everat, and M. Couprie, Image segmentation through operators based on topology, Journal of Electronic Imaging, vol.6, issue.4, pp.395-405, 1997.
DOI : 10.1117/12.276856

G. Bertrand, J. Everat, and M. Couprie, Topological approach to image segmentation, SPIE's 1996 International Symposium on Optical Science, Engineering , and Instrumentation. International Society for Optics and Photonics, pp.65-76, 1996.
DOI : 10.1117/12.251813

URL : https://hal.archives-ouvertes.fr/hal-00622005

G. Bertrand and G. Malandain, A new characterization of three-dimensional simple points, Pattern Recognition Letters, vol.15, issue.2, pp.169-175, 1994.
DOI : 10.1016/0167-8655(94)90046-9

URL : https://hal.archives-ouvertes.fr/inria-00615050

S. Beucher and C. Lantuéjoul, Use of watersheds in contour detection, International Workshop on Image Processing: Real-time Edge and Motion Detection, 1979.

S. Beucher and F. Meyer, The morphological approach to segmentation: The watershed transformation, In: Optical Engineering Marcel Dekker Incorporated, vol.34, pp.433-433, 1992.

H. Bieri and W. Nef, Algorithms for the Euler characteristic and related additive functionals of digital objects " . In: Computer vision, graphics, and image processing 28, pp.166-175, 1984.

E. Bishop, S. Douglas, and . Bridges, Constructive analysis, 2012.
DOI : 10.1007/978-3-642-61667-9

D. Ethan and . Bloch, A first course in geometric topology and differential geometry, 1997.

I. Bloch, H. Heijmans, and C. Ronse, Mathematical Morphology, pp.857-944, 2007.
DOI : 10.1007/978-1-4020-5587-4_14

URL : https://hal.archives-ouvertes.fr/hal-00556180

N. Boutry, T. Géraud, and L. Najman, How to Make nD Functions Digitally Well-Composed in a Self-dual Way, International Symposium on Mathematical Morphology and Its Applications to Signal and Image Processing, pp.561-572, 2015.
DOI : 10.1007/978-3-319-18720-4_47

URL : https://hal.archives-ouvertes.fr/hal-01168723

N. Boutry, T. Géraud, and L. Najman, How to make n-D images wellcomposed without interpolation, IEEE International Conference on Image Processing. IEEE. 2015, pp.2149-2153
DOI : 10.1109/icip.2015.7351181

URL : https://hal.archives-ouvertes.fr/hal-01134166/document

N. Boutry, T. Géraud, and L. Najman, On making n-D images wellcomposed by a self-dual local interpolation, Discrete Geometry for Computer Imagery, pp.320-331, 2014.
DOI : 10.1007/978-3-319-09955-2_27

URL : https://hal.archives-ouvertes.fr/hal-01071624/document

J. Braquelaire and L. Brun, Image Segmentation with Topological Maps and Inter-pixel Representation, Journal of Visual Communication and Image Representation, vol.9, issue.1, pp.62-79, 1998.
DOI : 10.1006/jvci.1998.0374

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.12.8542

M. Busta, L. Neumann, and J. Matas, FASText: Efficient unconstrained scene text detector, IEEE International Conference on Computer Vision. IEEE. 2015, pp.1206-1214

V. Caselles and P. Monasse, Geometric description of images as topographic maps, ser, Lecture Notes in Mathematics, 1984.
DOI : 10.1007/978-3-642-04611-7

V. Caselles and P. Monasse, Grain Filters, Journal of Mathematical Imaging and Vision, vol.173, pp.249-270, 2002.
DOI : 10.1007/978-3-642-04611-7_3

T. Christopher and C. , Numerical methods for partial differential equations involving discontinuities, 2003.

L. Chen, Algorithms for Computing Topological Invariants in 2D and 3D Digital Spaces, 2013.
DOI : 10.1109/icpr.2008.4761192

URL : http://arxiv.org/abs/0804.1982

L. Chen, Genus computing for 3D digital objects: Algorithm and implementation, 2009.

A. Coates, B. Carpenter, C. Case, S. Satheesh, B. Suresh et al., Text Detection and Character Recognition in Scene Images with Unsupervised Feature Learning, 2011 International Conference on Document Analysis and Recognition, pp.440-445, 2011.
DOI : 10.1109/ICDAR.2011.95

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.220.8072

Y. Cointepas, Modélisation homotopique et segmentation tridimensionnelles du cortex cérébraì a partir d'IRM pour la résolution desprobì emes directs et inverses en EEG et en MEG, 1999.

Y. Cointepas, I. Bloch, and L. Garnero, A cellular model for multi-objects multi-dimensional homotopic deformations, Pattern Recognition, vol.34, issue.9, pp.1785-1798, 2001.
DOI : 10.1016/S0031-3203(00)00106-0

J. Cousty and G. Bertrand, Uniqueness of the Perfect Fusion Grid on Z d, Journal of Mathematical Imaging and Vision, vol.343, pp.291-306, 2009.
URL : https://hal.archives-ouvertes.fr/hal-00511818

J. Cousty, M. Couprie, L. Najman, and G. Bertrand, Grayscale Watersheds on Perfect Fusion Graphs, pp.60-73, 2006.
DOI : 10.1007/11774938_6

URL : https://hal.archives-ouvertes.fr/hal-00622038

S. Crozet and T. Géraud, A first parallel algorithm to compute the morphological tree of shapes of nD images, 2014 IEEE International Conference on Image Processing (ICIP), pp.2933-2937
DOI : 10.1109/ICIP.2014.7025593

X. Daragon, Surfaces discrètes etfrontì eres d'objets dans les ordres, 2005.

X. Daragon, M. Couprie, and G. Bertrand, Discrete Surfaces and Frontier Orders, Journal of Mathematical Imaging and Vision, vol.147, issue.2???3, pp.379-399, 2005.
DOI : 10.1117/12.364107

URL : https://hal.archives-ouvertes.fr/hal-00622397

X. Daragon, M. Couprie, and G. Bertrand, Marching chains algorithm for Alexandroff-Khalimsky spaces, International Symposium on Optical Science and Technology . International Society for Optics and Photonics, pp.51-62, 2002.
DOI : 10.1117/12.453595

URL : https://hal.archives-ouvertes.fr/hal-00622041

M. Pierrot-deseilligny, G. Stamon, Y. Ching, and . Suen, Veinerization: a new shape description for flexible skeletonization, IEEE Transactions on Pattern Analysis and Machine Intelligence, vol.20, issue.5, pp.505-521, 1998.
DOI : 10.1109/34.682180

K. Tamal, S. Dey, and . Guha, Computing homology groups of simplicial complexes in R 3, Journal of the ACM, vol.452, pp.266-287, 1998.

O. Richard, . Duda, H. John, and . Munson, Graphical-data-processing research study and experimental investigation, 1967.

R. Charles and . Dyer, Computing the Euler number of an image from its quadtree " . In: Computer graphics and image processing 13, pp.270-276, 1980.

H. Edelsbrunner, Geometry and topology for mesh generation, 2001.
DOI : 10.1115/1.1445302

H. Edelsbrunner and J. Harer, Computational topology: An introduction, 2010.
DOI : 10.1090/mbk/069

B. Epshtein, E. Ofek, and Y. Wexler, Detecting text in natural scenes with stroke width transform, 2010 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, pp.2963-2970, 2010.
DOI : 10.1109/CVPR.2010.5540041

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.188.187

V. Alexander, R. Evako, . Kopperman, V. Yurii, and . Mukhin, Dimensional properties of graphs and digital spaces, Journal of Mathematical Imaging and Vision, vol.6, pp.2-3, 1996.

J. Fabrizio, M. Robert-seidowsky, S. Dubuisson, and S. Calarasanu, TextCatcher: a method to detect curved and challenging text in natural scenes, International Journal on Document Analysis and Recognition (IJDAR), vol.36, issue.5, pp.99-117, 2016.
DOI : 10.1109/ICDM.2011.66

N. Sylvain-faisan, V. Passat, R. Noblet, C. Chabrier, and . Meyer, Topology preserving warping of 3D binary images according to continuous one-to-one mappings, IEEE Transactions on Image Processing, vol.208, pp.2135-2145, 2011.

E. Stepanovitch and F. , Course of Crystallography, Saint-Petersburg (in Russian), 1901.

C. Fiorio, A topologically consistent representation for image analysis: The frontiers topological graph " . In: Discrete Geometry for Computer Imagery, pp.151-162, 1996.
URL : https://hal.archives-ouvertes.fr/lirmm-01168321

C. Fiorio, Approche interpixel en analyse d'images, une topologie et des algorithmes de segmentation, 1995.
URL : https://hal.archives-ouvertes.fr/tel-01168523

A. Flores, ¨ Uber n-dimensionale Komplexe, die im R 2k +1 absolut selbstverschlungen sind, Ergebnisse eines Mathematischen Kolloquiums, vol.34, pp.4-6, 1933.

T. Géraud, E. Carlinet, and S. Crozet, Self-duality and Digital Topology: Links Between the Morphological Tree of Shapes and Well-Composed Gray-Level Images, International Symposium on Mathematical Morphology and Its Applications to Signal and Image Processing, pp.573-584, 2015.
DOI : 10.1007/978-3-319-18720-4_48

T. Géraud, E. Carlinet, S. Crozet, and L. Najman, A quasilinear algorithm to compute the tree of shapes of n-D images, International Symposium on Mathematical Morphology and Its Applications to Signal and Image Processing, pp.98-110, 2013.

T. Géraud, J. Mangin, I. Bloch, and H. Ma??trema??tre, Segmenting internal structures in 3D MR images of the brain by Markovian relaxation on a watershed based adjacency graph, Proceedings., International Conference on Image Processing
DOI : 10.1109/ICIP.1995.537693

R. González-díaz, M. Jiménez, and B. Medrano, 3D well-composed polyhedral complexes, Discrete Applied Mathematics, vol.183, pp.59-77, 2015.
DOI : 10.1016/j.dam.2014.08.036

R. González-díaz, M. Jiménez, and B. Medrano, Cohomology ring of 3D cubical complexes, In: IWCIA Special Track on Applications, pp.139-150, 2009.

R. González-díaz, M. Jiménez, and B. Medrano, Cubical cohomology ring of 3D photographs, International Journal of Imaging Systems and Technology, vol.54, issue.1, pp.76-85, 2011.
DOI : 10.2307/1969485

R. González-díaz, M. Jiménez, and B. Medrano, Encoding Specific 3D Polyhedral Complexes Using 3D Binary Images, pp.268-281, 2016.
DOI : 10.1007/978-3-319-32360-2_21

R. González-díaz, M. Jiménez, and B. Medrano, Well-Composed Cell Complexes, pp.153-162, 2011.
DOI : 10.1007/978-3-642-19867-0_13

R. González-díaz, J. Lamar, and R. Umble, Cup Products on Polyhedral Approximations of 3D Digital Images, International Workshop on Combinatorial Image Analysis, pp.107-119, 2011.
DOI : 10.2307/1969485

R. González-díaz and P. Real, On the cohomology of 3D digital images, Discrete Applied Mathematics, vol.147, issue.2-3, pp.245-263, 2005.
DOI : 10.1016/j.dam.2004.09.014

R. González-díaz and P. Real, Towards Digital Cohomology, pp.92-101, 2003.
DOI : 10.1007/978-3-540-39966-7_8

B. Stephen and . Gray, Local properties of binary images in two dimensions, IEEE Transactions on Computers, vol.1005, pp.551-561, 1971.

J. Marvin and . Greenberg, Lectures on algebraic topology, 1967.

A. Gross and L. J. Latecki, Digitizations Preserving Topological and Differential Geometric Properties, Computer Vision and Image Understanding, vol.62, issue.3, pp.370-381, 1995.
DOI : 10.1006/cviu.1995.1061

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.51.8473

X. Han, C. Xu, and J. L. Prince, A topology preserving deformable model using level sets, IEEE Conference on Computer Vision and Pattern Recognition, pp.765-770, 2001.

X. Han, C. Xu, and J. L. Prince, A topology preserving level set method for geometric deformable models, IEEE Transactions on Pattern Analysis and Machine Intelligence, vol.256, pp.755-768, 2003.

H. Heijmans, Morphological image operators, Advances in Electronics and Electron Physics Supplement, 1994.

H. Heijmans, Theoretical aspects of gray-level morphology, IEEE Transactions on Pattern Analysis and Machine Intelligence, vol.13, issue.6, pp.568-582, 1991.
DOI : 10.1109/34.87343

T. Gabor and . Herman, Discrete multidimensional Jordan surfaces, Graphical Models and Image Processing, vol.546, pp.507-515, 1992.

F. John and . Hudson, Piecewise linear topology, 1969.

Y. Lê-duy-huynh, T. Xu, and . Géraud, A Morphological Hierarchical Representation with Application to Text Segmentation in Natural Images, 2016.

L. Janos and A. Rosenfeld, Digital connectedness: An algebraic approach, Pattern Recognition Letters, vol.1, issue.3, pp.135-139, 1983.
DOI : 10.1016/0167-8655(83)90052-1

C. Jordan, Cours d'Analyse de l'Ecole Polytechnique, pp.587-594

T. Ju, F. Losasso, S. Schaefer, and J. Warren, Dual contouring of Hermite data, In: ACM Transactions on Graphics, vol.21, issue.3, pp.339-346, 2002.
DOI : 10.1145/566654.566586

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.19.3605

L. John and . Kelley, General Topology. The University Series in Higher Mathematics, 1955.

E. Khalimsky, Applications of connected ordered topological spaces in topology, 1970.

E. Khalimsky, Ordered Topological Spaces, 1977.

E. Khalimsky, R. Kopperman, R. Paul, and . Meyer, Computer graphics and connected topologies on finite ordered sets, Topology and its Applications, vol.36, issue.1, pp.1-17, 1990.
DOI : 10.1016/0166-8641(90)90031-V

URL : http://doi.org/10.1016/0166-8641(90)90031-v

M. Robert, V. Kirby, . Pascucci, T. Cláudio, . Silva et al., Topology verification for isosurface extraction, IEEE Transactions on Visualization and Computer Graphics, vol.6, pp.952-965, 2012.

T. Yung, K. , and A. Rosenfeld, Digital Topology, Computer Vision, Graphics, and Image Processing, pp.357-393, 1989.
DOI : 10.1016/B978-044450355-8/50106-9

T. Yung, K. , and A. Rosenfeld, If we use 4-or 8-connectedness for both the objects and the background, the Euler characteristics is not locally computable, In: Pattern Recognition Letters, vol.11, issue.4, pp.231-232, 1990.

R. Kopperman, The Khalimsky Line as a Foundation for Digital Topology, Shape in Picture, pp.3-20, 1994.
DOI : 10.1007/978-3-662-03039-4_2

R. Kopperman, R. Paul, . Meyer, G. Richard, and . Wilson, A Jordan surface theorem for three-dimensional digital spaces, Discrete & Computational Geometry, vol.86, issue.2, pp.155-161, 1991.
DOI : 10.2307/2321290

K. Ullrich and . Othe, Generische Programmierung f ¨ ur die Bildverarbeitung, BoD?Books on Demand, 2000.

A. Vladimir and . Kovalevsky, Finite topology as applied to image analysis, Computer Vision, Graphics, and Image Processing 46, pp.141-161, 1989.

C. Kuratowski, Sur leprobì eme des courbes gauches en topologie, Fundamenta Mathematicae, vol.151, pp.271-283, 1930.

J. Lachaud and A. Montanvert, Continuous Analogs of Digital Boundaries: A Topological Approach to Iso-Surfaces, Graphical Models, vol.62, issue.3, pp.129-164, 2000.
DOI : 10.1006/gmod.2000.0522

J. Longin and . Latecki, 3D well-composed pictures, Graphical Models and Image Processing, vol.593, pp.164-172, 1997.

J. Longin and . Latecki, Discrete representation of spatial objects in computer vision, 1998.

J. Longin and . Latecki, Multicolor well-composed pictures, Photonics for Industrial Applications. International Society for Optics and Photonics, pp.63-70, 1995.

J. Longin and . Latecki, Well-composed sets, Advances in Electronics and Electron Physics, pp.95-163, 2000.

C. Longin-jan-latecki, A. Conrad, and . Gross, Preserving topology by a digitization process, Journal of Mathematical Imaging and Vision, vol.8, issue.2, pp.131-159, 1998.
DOI : 10.1023/A:1008273227913

U. Longin-jan-latecki, A. Eckhardt, and . Rosenfeld, Well-composed sets, Computer Vision and Image Understanding, vol.611, pp.70-83, 1995.

C. L. , G. Luminita, and A. Vese, Self-repelling snakes for topology-preserving segmentation models, IEEE Transactions on Image Processing, vol.175, pp.767-779, 2008.

C. Lee, T. Poston, and A. Rosenfeld, Winding and Euler numbers for 2D and 3D digital images, CVGIP: Graphical Models and Image Processing, vol.53, issue.6, pp.522-537, 1991.
DOI : 10.1016/1049-9652(91)90003-3

C. Lee and A. Rosenfeld, Computing the Euler number of a 3D image, Center for Automation Research Technical Report CAR-TR-205, 1986.

J. Lee, Introduction to topological manifolds, 2010.
DOI : 10.1007/978-1-4419-7940-7

J. Lee, . Pyoung-hean-lee, A. L. Seong-whan-lee, C. Yuille, and . Koch, AdaBoost for Text Detection in Natural Scene, 2011 International Conference on Document Analysis and Recognition, pp.429-434, 2011.
DOI : 10.1109/ICDAR.2011.93

J. Luis-diaz-de, L. , and J. H. Sossa-azuela, On the computation of the Euler number of a binary object, In: Pattern Recognition, vol.293, pp.471-476, 1996.

R. Levillain, T. Géraud, and L. Najman, Why and how to design a generic and efficient image processing framework: The case of the Milena library, IEEE International Conference on Image Processing, pp.1941-1944, 2010.
URL : https://hal.archives-ouvertes.fr/hal-00622480

R. Levillain, T. Géraud, and L. Najman, Writing Reusable Digital Topology Algorithms in a Generic Image Processing Framework, Applications of Discrete Geometry and Mathematical Morphology, pp.140-153, 2012.
DOI : 10.1007/978-3-642-32313-3_10

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.258.4924

W. Bernard and R. Lickorish, Simplicial moves on complexes and manifolds, Geometry and Topology Monographs, vol.2, pp.299-320, 1999.

L. Elon and . Lima, The Jordan-Brouwer separation theorem for smooth hypersurfaces, The American Mathematical Monthly, vol.951, pp.39-42, 1988.

E. William, . Lorensen, E. Harvey, and . Cline, Marching cubes: A high resolution 3D surface construction algorithm, In: Special Interest Group on Computer GRAPHics and Interactive Techniques, vol.21, issue.4, pp.163-169, 1987.

H. Wolfram, . Lunscher, P. Michael, and . Beddoes, Fast binary-image boundary extraction, Computer Vision, Graphics, and Image Processing, pp.229-257, 1987.

J. Mangin, O. Coulon, and V. Frouin, Robust brain segmentation using histogram scale-space analysis and mathematical morphology, International Conference on Medical Image Computing and Computer-Assisted Intervention, pp.1230-1241, 1998.
DOI : 10.1109/34.19041

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.16.8898

J. Marchadier, S. Didierarqù, and . Michelin, Thinning grayscale wellcomposed images, Pattern Recognition Letters, vol.255, pp.581-590, 2004.
DOI : 10.1016/j.patrec.2003.12.005

D. Martin, C. Fowlkes, D. Tal, and J. Malik, A database of human segmented natural images and its application to evaluating segmentation algorithms and measuring ecological statistics, Proceedings Eighth IEEE International Conference on Computer Vision. ICCV 2001, pp.416-423, 2001.
DOI : 10.1109/ICCV.2001.937655

L. Mazo, Déformations homotopiques dans les images digitales n-aires, 2011.

L. Mazo, N. Passat, M. Couprie, and C. Ronse, Digital imaging: A unified topological framework, Journal of Mathematical Imaging and Vision, vol.441, pp.19-37, 2012.
DOI : 10.1007/s10851-011-0308-9

URL : https://hal.archives-ouvertes.fr/hal-00728915

E. Meinhardt-llopis, Morphological and statistical techniques for the analysis of 3D images, 2011.

F. Meyer, Skeletons and perceptual graphs, Signal Processing 16, pp.335-363, 1989.
DOI : 10.1016/0165-1684(89)90030-3

F. Meyer and P. Maragos, Morphological Scale-Space Representation with Levelings, International Conference on Scale-Space Theories in Computer Vision, pp.187-198, 1999.
DOI : 10.1007/3-540-48236-9_17

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.104.4948

P. Monasse and F. Guichard, Fast computation of a contrast-invariant image representation, IEEE Transactions on Image Processing, vol.9, issue.5, pp.860-872, 2000.
DOI : 10.1109/83.841532

P. John, T. Mylopoulos, and . Pavlidis, On the topological properties of quantized spaces, I. The notion of dimension, Journal of the ACM, vol.18, issue.2, pp.239-246, 1971.

P. John, T. Mylopoulos, and . Pavlidis, On the topological properties of quantized spaces, II. Connectivity and order of connectivity, Journal of the ACM, vol.18, issue.2, pp.247-254, 1971.

L. Najman and M. Couprie, Watershed Algorithms and Contrast Preservation, pp.62-71, 2003.
DOI : 10.1007/978-3-540-39966-7_5

URL : https://hal.archives-ouvertes.fr/hal-00622112

L. Najman and T. Géraud, Discrete Set-Valued Continuity and Interpolation, Mathematical Morphology and Its Applications to Signal and Image Processing, pp.37-48, 2013.
DOI : 10.1007/978-3-642-38294-9_4

URL : https://hal.archives-ouvertes.fr/hal-00798574

L. Najman and M. Schmitt, Watershed of a continuous function, Signal Processing, vol.38, issue.1, pp.99-112, 1994.
DOI : 10.1016/0165-1684(94)90059-0

URL : https://hal.archives-ouvertes.fr/hal-00622129

L. Najman and H. Talbot, Mathematical Morphology, 2013.
DOI : 10.1002/9781118600788

URL : https://hal.archives-ouvertes.fr/hal-00622499

M. Nakahara, Geometry, topology and physics, 2003.
DOI : 10.1887/0750306068

M. Herman and A. Newman, Elements of the topology of plane sets of points, 1939.

P. Ngo, Y. Kenmochi, N. Passat, and H. Talbot, Combinatorial structure of rigid transformations in 2D digital images, Computer Vision and Image Understanding, vol.117, issue.4, pp.393-408, 2013.
DOI : 10.1016/j.cviu.2012.08.014

URL : https://hal.archives-ouvertes.fr/hal-00643734

P. Ngo, Y. Kenmochi, N. Passat, and H. Talbot, Sufficient Conditions for Topological Invariance of 2D Images under Rigid Transformations, Discrete Geometry for Computer Imagery, pp.155-168, 2013.
DOI : 10.1007/978-3-642-37067-0_14

URL : https://hal.archives-ouvertes.fr/hal-00827195

P. Ngo, N. Passat, Y. Kenmochi, and H. Talbot, Topology-Preserving Rigid Transformation of 2D Digital Images, IEEE Transactions on Image Processing, vol.23, issue.2, pp.885-897, 2014.
DOI : 10.1109/TIP.2013.2295751

URL : https://hal.archives-ouvertes.fr/hal-00795054

P. Ngo, N. Passat, Y. Kenmochi, and H. Talbot, Well-composed images and rigid transformations, 2013 IEEE International Conference on Image Processing, pp.3035-3039, 2013.
DOI : 10.1109/ICIP.2013.6738625

URL : https://hal.archives-ouvertes.fr/hal-00779262

S. Papert and M. Minsky, Perceptrons: An Introduction to Computational Geometry, 1969.

T. Pavlidis, Algorithms for Graphics and Image Processing, 2012.
DOI : 10.1007/978-3-642-93208-3

L. Dzung, P. Pham, J. L. Bazin, and . Prince, Digital topology in brain imaging, Signal Processing Magazine, pp.51-59, 2010.

B. Jos, A. Roerdink, and . Meijster, The watershed transform: Definitions, algorithms and parallelization strategies, Fundamenta Informaticae, vol.411, issue.2, pp.187-228, 2000.

C. Ronse, Flat Morphological Operators on Arbitrary Power Lattices, Geometry , Morphology, and Computational Imaging, pp.1-21, 2003.
DOI : 10.1007/3-540-36586-9_1

A. Rosenfeld, Adjacency in digital pictures, Information and Control, vol.26, issue.1, pp.24-33, 1974.
DOI : 10.1016/S0019-9958(74)90696-2

A. Rosenfeld, Arcs and Curves in Digital Pictures, Journal of the ACM, vol.20, issue.1, pp.81-87, 1973.
DOI : 10.1145/321738.321745

A. Rosenfeld, Connectivity in Digital Pictures, Journal of the ACM, vol.17, issue.1, pp.146-160, 1970.
DOI : 10.1145/321556.321570

A. Rosenfeld, Digital Topology, The American Mathematical Monthly, vol.86, issue.8, pp.621-630, 1979.
DOI : 10.2307/2321290

A. Rosenfeld, Fuzzy digital topology, Information and Control, vol.40, issue.1, pp.76-87, 1979.
DOI : 10.1016/S0019-9958(79)90353-X

A. Rosenfeld, On connectivity properties of grayscale pictures, Pattern Recognition, vol.16, issue.1, pp.47-50, 1983.
DOI : 10.1016/0031-3203(83)90007-9

A. Rosenfeld, Picture Languages-Formal Model of Picture Recognition, 1979.

A. Rosenfeld, Y. Kong, and A. Nakamura, Topology-Preserving Deformations of Two-Valued Digital Pictures, Graphical Models and Image Processing, vol.60, issue.1, pp.24-34, 1998.
DOI : 10.1006/gmip.1997.0459

A. Rosenfeld, Y. Kong, and A. Y. Wu, Digital surfaces, Graphical Models and Image Processing 53, pp.305-312, 1991.
DOI : 10.1016/1049-9652(91)90034-H

A. Rosenfeld, L. John, and . Pfaltz, Sequential operations in digital picture processing, Journal of the ACM, vol.134, pp.471-494, 1966.

K. Punam, R. Saha, G. Strand, and . Borgefors, Digital topology and geometry in medical imaging: A survey, IEEE Transactions on Medical Imaging, vol.34, issue.9, pp.1940-1964, 2015.

P. Salembier and J. Serra, Flat zones filtering, connected operators, and filters by reconstruction, IEEE Transactions on Image Processing, vol.4, issue.8, pp.1153-1160, 1995.
DOI : 10.1109/83.403422

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.17.9460

F. Ségonne, Active contours under topology control ? Genus preserving level sets, International Journal of Computer Vision, vol.792, pp.107-117, 2008.

J. Serra and R. Kiran, Digitization of Partitions and Tessellations, pp.323-334, 2016.
DOI : 10.1007/978-3-319-32360-2_25

URL : https://hal.archives-ouvertes.fr/hal-01306414

R. Shekhar, E. Fayyad, and R. Yagel, Octree-based decimation of marching cubes surfaces, Proceedings of Seventh Annual IEEE Visualization '96, pp.335-342, 1996.
DOI : 10.1109/VISUAL.1996.568127

M. Siqueira, L. J. Latecki, and J. Gallier, Making 3D binary digital images well-composed, Electronic Imaging 2005. International Society for Optics and Photonics, pp.150-163, 2005.
DOI : 10.1117/12.596447

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.109.4285

M. Siqueira, L. J. Latecki, N. Tustison, J. Gallier, and J. Gee, Topological Repairing of 3D Digital Images, Journal of Mathematical Imaging and Vision, vol.17, issue.3, pp.249-274, 2008.
DOI : 10.1007/978-94-015-9002-0

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.110.8479

L. Snidaro and G. L. Foresti, Real-time thresholding with Euler numbers, Pattern Recognition Letters, vol.24, issue.9-10, pp.1533-1544, 2003.
DOI : 10.1016/S0167-8655(02)00392-6

P. Soille, Morphological Image Analysis: Principles and Applications, 2013.

P. Soille and M. Pesaresi, Advances in mathematical morphology applied to geoscience and remote sensing, IEEE Transactions on Geoscience and Remote Sensing, vol.40, issue.9, pp.2042-2055, 2002.
DOI : 10.1109/TGRS.2002.804618

J. Humberto-sossa-azuela, R. Santiago-montero, M. Pérez-cisneros, and E. Rubio-espino, Computing the Euler number of a binary image based on a vertex codification, Journal of Applied Research and Technology, vol.113, pp.360-370, 2013.

P. Stelldinger, Image Digitization and its Influence on Shape Properties in Finite Dimensions, 2008.

P. Stelldinger, K. Ullrich, and . Othe, Towards a general sampling theory for shape preservation, Image and Vision Computing, vol.23, issue.2, pp.237-248, 2005.
DOI : 10.1016/j.imavis.2004.06.003

P. Stelldinger and L. J. Latecki, 3D object digitization: Majority interpolation and marching cubes, IEEE International Conference on Pattern Recognition, pp.1173-1176, 2006.
DOI : 10.1109/icpr.2006.29

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.161.4646

P. Stelldinger, L. J. Latecki, and M. Siqueira, Topological Equivalence between a 3D Object and the Reconstruction of Its Digital Image, IEEE Transactions on Pattern Analysis and Machine Intelligence, vol.29, issue.1, pp.126-140, 2007.
DOI : 10.1109/TPAMI.2007.250604

P. Stelldinger and R. Strand, Topology Preserving Digitization with FCC and BCC Grids, International Workshop on Combinatorial Image Analysis, pp.226-240, 2006.
DOI : 10.1007/11774938_18

L. Neff and S. , Two discrete forms of the Jordan curve theorem, The American Mathematical Monthly 95, pp.332-336, 1988.

G. Sundaramoorthi and A. Yezzi, Global Regularizing Flows With Topology Preservation for Active Contours and Polygons, IEEE Transactions on Image Processing, vol.16, issue.3, pp.803-812, 2007.
DOI : 10.1109/TIP.2007.891071

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.122.1889

M. Tajine and C. Ronse, Topological Properties of Hausdorff Discretizations, Mathematical Morphology and its Applications to Image and Signal Processing, pp.41-50, 2002.
DOI : 10.1007/0-306-47025-X_6

J. Toriwaki and H. Yoshida, Fundamentals of three-dimensional digital image processing, 2009.
DOI : 10.1007/978-1-84800-172-5

J. Nicholas, . Tustison, B. Brian, M. Avants, . Siqueira et al., Topological well-composedness and glamorous glue: A digital gluing algorithm for topologically constrained front propagation, IEEE Transactions onImage Processing, vol.206, pp.1756-1761, 2011.

H. Tverberg, A Proof of the Jordan Curve Theorem, Bulletin of the London Mathematical Society, vol.12, issue.1, pp.34-38, 1980.
DOI : 10.1112/blms/12.1.34

R. Egbert and . Van-kampen, Komplexe in Euklidischen Räumen, Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, vol.9, issue.1, pp.72-78, 1933.

J. Lucas, . Van-vliet, T. Ian, . Young, L. Guus et al., An edge detection model based on non-linear Laplace filtering, In: International workshop on Pattern Recognition and Artificial Intelligence, 1988.

K. Voss, Images, objects, and surfaces in Z n, International Journal of Pattern Recognition and Artificial Intelligence, vol.5, pp.5-797, 1991.

T. Wang, J. David, A. Wu, . Coates, Y. Andrew et al., End-to-end text recognition with convolutional neural networks, International Conference on Pattern Recognition. IEEE. 2012, pp.3304-3308

Y. Wang and P. Bhattacharya, Digital Connectivity and Extended Well-Composed Sets for Gray Images, Computer Vision and Image Understanding, vol.68, issue.3, pp.330-345, 1997.
DOI : 10.1006/cviu.1997.0551

R. Paul and . Wilson, Euler formulas and geometric modeling, IEEE Computer Graphics and Applications, vol.85, pp.24-36, 1985.

Y. Xu, T. Géraud, and L. Najman, Context-based energy estimator: Application to object segmentation on the tree of shapes, 2012 19th IEEE International Conference on Image Processing, pp.1577-1580
DOI : 10.1109/ICIP.2012.6467175

URL : https://hal.archives-ouvertes.fr/hal-00762289

Y. Xu, T. Géraud, and L. Najman, Morphological filtering in shape spaces: Applications using tree-based image representations, International Conference on Pattern Recognition. IEEE. 2012, pp.485-488
DOI : 10.1109/tpami.2015.2441070

URL : https://hal.archives-ouvertes.fr/hal-00714847

Y. Xu, T. Géraud, and L. Najman, Two Applications of Shape-Based Morphology: Blood Vessels Segmentation and a Generalization of Constrained Connectivity, International Symposium on Mathematical Morphology and Its Applications to Signal and Image Processing, pp.390-401, 2013.
DOI : 10.1007/978-3-642-38294-9_33

URL : https://hal.archives-ouvertes.fr/hal-00798625

S. Hyun, S. Yang, and . Sengupta, Intelligent shape recognition for complex industrial tasks, IEEE Control Systems Magazine, vol.83, pp.23-30, 1988.

Q. Ye and D. Doermann, Text Detection and Recognition in Imagery: A Survey, IEEE Transactions on Pattern Analysis and Machine Intelligence, vol.37, issue.7, pp.1480-1500, 2015.
DOI : 10.1109/TPAMI.2014.2366765

S. Yokoi, J. Toriwaki, and T. Fukumura, An Analysis of Topological Properties of Digitized Binary Pictures Using Local Features, Computer Graphics and Image Processing, vol.4, issue.1, pp.63-73, 1975.
DOI : 10.1016/0146-664X(75)90022-2

Y. Zhu, C. Yao, and X. Bai, Scene text detection and recognition: recent advances and future trends, Frontiers of Computer Science, vol.88, issue.2, pp.19-36, 2016.
DOI : 10.1007/s11263-009-0275-4