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

P. Bazin and D. L. Pham, Topology-Preserving Tissue Classification of Magnetic Resonance Brain Images, IEEE Transactions on Medical Imaging, vol.26, issue.4, pp.487-496, 2007.
DOI : 10.1109/TMI.2007.893283

B. Zitová and J. Flusser, Image registration methods: a survey, Image and Vision Computing, vol.21, issue.11, pp.977-1000, 2003.
DOI : 10.1016/S0262-8856(03)00137-9

A. Yilmaz, O. Javed, and M. Shah, Object tracking, ACM Computing Surveys, vol.38, issue.4, pp.1-45, 2006.
DOI : 10.1145/1177352.1177355

D. L. Pham, P. Bazin, and J. L. Prince, Digital Topology in Brain Imaging, IEEE Signal Processing Magazine, vol.27, issue.4, pp.51-59, 2010.
DOI : 10.1109/MSP.2010.936729

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

C. A. Rothwell, J. L. Mundy, W. Hoffman, and V. Nguyen, Driving vision by topology, Proceedings of International Symposium on Computer Vision, ISCV, pp.395-400, 1995.
DOI : 10.1109/ISCV.1995.477034

URL : https://hal.archives-ouvertes.fr/inria-00074231

A. Rosenfeld and J. L. Pfaltz, Sequential operations in digital picture processing, J ACM, vol.13, issue.4, pp.471-494, 1966.

O. Duda, P. E. Hart, and J. H. Munson, Graphical data processing research study and experimental investigation, 1967.

P. Bazin, L. M. Ellingsen, and D. L. Pham, Digital Homeomorphisms in Deformable Registration, IPMI, ser. LNCS, pp.211-222, 2007.
DOI : 10.1007/978-3-540-73273-0_18

S. Faisan, N. Passat, V. Noblet, R. Chabrier, and C. Meyer, Topology Preserving Warping of 3-D Binary Images According to Continuous One-to-One Mappings, IEEE Transactions on Image Processing, vol.20, issue.8, pp.2135-2145, 2011.
DOI : 10.1109/TIP.2011.2158338

V. Noblet, C. Heinrich, F. Heitz, and J. Armspach, 3-D deformable image registration: a topology preservation scheme based on hierarchical deformation models and interval analysis optimization, IEEE Transactions on Image Processing, vol.14, issue.5, pp.553-566, 2005.
DOI : 10.1109/TIP.2005.846026

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.127.9352

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

E. Andres, The Quasi-Shear rotation, DGCI, ser. LNCS, pp.307-314, 1996.
DOI : 10.1007/3-540-62005-2_26

M. S. Richman, Understanding discrete rotations, 1997 IEEE International Conference on Acoustics, Speech, and Signal Processing, pp.2057-2060, 1997.
DOI : 10.1109/ICASSP.1997.599351

URL : http://cam.cornell.edu/richman/ic97paper.ps.Z

B. Nouvel and E. Rémila, Incremental and Transitive Discrete Rotations, IWCIA, ser. LNCS, pp.199-213, 2006.
DOI : 10.1007/11774938_16

URL : https://hal.archives-ouvertes.fr/hal-00016037

Y. Thibault, Y. Kenmochi, and A. Sugimoto, Computing upper and lower bounds of rotation angles from digital images, Pattern Recognition, vol.42, issue.8, pp.1708-1717, 2009.
DOI : 10.1016/j.patcog.2008.12.027

URL : https://hal.archives-ouvertes.fr/hal-00622416

B. Nouvel and E. Rémila, Configurations induced by discrete rotations: periodicity and quasi-periodicity properties, Discrete Applied Mathematics, vol.147, issue.2-3, pp.3-325, 2005.
DOI : 10.1016/j.dam.2004.09.018

P. Ngo, Y. Kenmochi, N. Passat, and H. Talbot, Sufficient conditions for topological invariance of 2D digital images under rigid transformations, DGCI, ser. LNCS, pp.155-168, 2013.

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

T. Y. Kong and A. Rosenfeld, Digital topology: Introduction and survey, Computer Vision, Graphics, and Image Processing, vol.48, issue.3, pp.357-393, 1989.
DOI : 10.1016/0734-189X(89)90147-3

L. Mazo, N. Passat, M. Couprie, and C. Ronse, Digital Imaging: A Unified Topological Framework, Journal of Mathematical Imaging and Vision, vol.32, issue.9, pp.19-37, 2012.
DOI : 10.1007/s10851-011-0308-9

URL : https://hal.archives-ouvertes.fr/hal-00622529

E. Khalimsky, Topological structures in computer science, Journal of Applied Mathematics and Simulation, vol.1, issue.1, pp.25-40, 1987.
DOI : 10.1155/S1048953388000036

V. A. Kovalevsky, Finite topology as applied to image analysis, Computer Vision, Graphics, and Image Processing, vol.46, issue.2, pp.141-161, 1989.
DOI : 10.1016/0734-189X(89)90165-5

L. Mazo, N. Passat, M. Couprie, and C. Ronse, Paths, Homotopy and Reduction in Digital Images, Acta Applicandae Mathematicae, vol.55, issue.1, pp.167-193, 2011.
DOI : 10.1007/s10440-010-9591-5

URL : https://hal.archives-ouvertes.fr/hal-00622495

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

L. J. Latecki, U. Eckhardt, and A. Rosenfeld, Well-Composed Sets, Computer Vision and Image Understanding, vol.61, issue.1, pp.70-83, 1995.
DOI : 10.1006/cviu.1995.1006

T. Y. Kong, A digital fundamental group, Computers & Graphics, vol.13, issue.2, pp.159-166, 1989.
DOI : 10.1016/0097-8493(89)90058-7

C. Ronse, A topological characterization of thinning, Theoretical Computer Science, vol.43, issue.1, pp.31-41, 2007.
DOI : 10.1016/0304-3975(86)90164-7

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

G. Bertrand, On P-simple points, CR Acad Sci I-Math, vol.I, issue.321, pp.1077-1084, 1995.
URL : https://hal.archives-ouvertes.fr/hal-00621996

N. Passat and L. Mazo, An introduction to simple sets, Pattern Recognition Letters, vol.30, issue.15, pp.1366-1377, 2009.
DOI : 10.1016/j.patrec.2009.07.008

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

G. Bertrand, M. Couprie, and N. Passat, A note on 3-D simple points and simple-equivalence, Information Processing Letters, vol.109, issue.13, pp.700-704, 2009.
DOI : 10.1016/j.ipl.2009.03.002

URL : https://hal.archives-ouvertes.fr/hal-00622395

A. Rosenfeld, T. 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

J. Marchadier, D. Arqù, and S. Michelin, Thinning grayscale well-composed images, Pattern Recognition Letters, vol.25, issue.5, pp.581-590, 2004.
DOI : 10.1016/j.patrec.2003.12.005

P. Salembier, A. Oliveras, and L. Garrido, Antiextensive connected operators for image and sequence processing, IEEE Transactions on Image Processing, vol.7, issue.4, pp.555-570, 1998.
DOI : 10.1109/83.663500

P. Salembier and M. H. Wilkinson, Connected operators, IEEE Signal Processing Magazine, vol.26, issue.6, pp.136-157, 2009.
DOI : 10.1109/MSP.2009.934154

L. J. Latecki, Multicolor well-composed pictures, Pattern Recognition Letters, vol.16, issue.4, pp.425-431, 1995.
DOI : 10.1016/0167-8655(94)00104-B

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

G. Damiand, A. Dupas, and J. Lachaud, Fully deformable 3D digital partition model with topological control, Pattern Recognition Letters, vol.32, issue.9, pp.1374-1383, 2011.
DOI : 10.1016/j.patrec.2010.09.005

URL : https://hal.archives-ouvertes.fr/hal-00591065

L. Mazo, A Framework for Label Images, CTIC, ser. LNCS, pp.1-12, 2012.
DOI : 10.1007/978-3-642-30238-1_1

URL : https://hal.archives-ouvertes.fr/hal-00962186

E. R. Davies and A. P. Plummer, Thinning algorithms: A critique and a new methodology, Pattern Recognition, vol.14, issue.1-6, pp.53-63, 1981.
DOI : 10.1016/0031-3203(81)90045-5

J. Mangin, V. Frouin, I. Bloch, J. Régis, and J. López-krahe, From 3D magnetic resonance images to structural representations of the cortex topography using topology preserving deformations, Journal of Mathematical Imaging and Vision, vol.44, issue.6, pp.297-318, 1995.
DOI : 10.1007/BF01250286

X. Han, C. Xu, and J. L. Prince, A topology preserving level set method for geometric deformable models, IEEE T Pattern Anal, vol.25, issue.6, pp.755-768, 2003.

M. Couprie and G. Bertrand, New Characterizations of Simple Points in 2D, 3D, and 4D Discrete Spaces, IEEE Transactions on Pattern Analysis and Machine Intelligence, vol.31, issue.4, pp.637-648, 2009.
DOI : 10.1109/TPAMI.2008.117

URL : https://hal.archives-ouvertes.fr/hal-00622393

M. Couprie, F. N. Bezerra, and G. Bertrand, Topological operators for grayscale image processing, Journal of Electronic Imaging, vol.10, issue.4, pp.1003-1015, 2001.
DOI : 10.1117/1.1408316

URL : https://hal.archives-ouvertes.fr/hal-00622474

M. Siqueira, L. J. Latecki, N. J. Tustison, J. H. Gallier, and J. C. Gee, Topological Repairing of 3D Digital Images, Journal of Mathematical Imaging and Vision, vol.17, issue.3, pp.249-274, 2008.
DOI : 10.1007/s10851-007-0054-1

J. J. Staal, M. D. Abramoff, M. Niemeijer, M. A. Viergever, and B. Van-ginneken, Ridge-Based Vessel Segmentation in Color Images of the Retina, IEEE Transactions on Medical Imaging, vol.23, issue.4, pp.501-509, 2004.
DOI : 10.1109/TMI.2004.825627

C. Kurtz, N. Passat, P. Gançarski, and A. Puissant, Extraction of complex patterns from multiresolution remote sensing images: A hierarchical top-down methodology, Pattern Recognition, vol.45, issue.2, pp.685-706, 2012.
DOI : 10.1016/j.patcog.2011.07.017

J. Inglada and J. Michel, Qualitative Spatial Reasoning for High-Resolution Remote Sensing Image Analysis, IEEE Transactions on Geoscience and Remote Sensing, vol.47, issue.2, pp.599-612, 2009.
DOI : 10.1109/TGRS.2008.2003435

URL : https://hal.archives-ouvertes.fr/hal-00594525

P. Soille, Morphological image compositing, IEEE Transactions on Pattern Analysis and Machine Intelligence, vol.28, issue.5, pp.673-683, 2006.
DOI : 10.1109/TPAMI.2006.99

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.468.4436

A. Gross and L. 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

L. J. Latecki, C. Conrad, and A. 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

L. J. Latecki, 3D Well-Composed Pictures, Graphical Models and Image Processing, vol.59, issue.3, pp.164-172, 1997.
DOI : 10.1006/gmip.1997.0422

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.304.9284

C. Ronse, Set-theoretical algebraic approaches to connectivity in continuous or digital spaces, Journal of Mathematical Imaging and Vision, vol.8, issue.1, pp.41-58, 1998.
DOI : 10.1023/A:1008210216583

J. Serra, Connectivity on Complete Lattices, J Math Imaging Vis, vol.9, issue.3, pp.231-251, 1998.
DOI : 10.1007/978-1-4613-0469-2_11

U. Braga-neto and J. K. Goutsias, A theoretical tour of connectivity in image processing and analysis, Journal of Mathematical Imaging and Vision, vol.19, issue.1, pp.5-31, 2003.
DOI : 10.1023/A:1024476403183

Y. Kenmochi-received-her, B. Eng, M. Eng, and D. Eng, degrees in information and computer sciences from Chiba University, Japan, in 1993, 1995, and 1998, respectively. She joined Japan Advanced Institute of Science and Technology as a research associate in 1998, she has been a CNRS research associate at Laboratoire d'Informatique Gaspard-Monge Est, and a member of the research group A3SI. Her research interests include discrete geometry for computer imagery, 2003.